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fff3436c-7de5-4b4d-bd43-c91ff64e6c44	StampyAI/alignment-research-dataset/arxiv	Arxiv	Rational Consensus 1 Introduction --------------- Consensus is a fundamental problem in distributed computing; it plays a key role in state machine replication, transaction commitment, and many other tasks where agreement among processes is required. It is well known that consensus cannot be deterministically achieved in asynchronous systems [[9](#bib.bib9)], but can be achieved in synchronous systems even if we allow Byzantine failures (see, e.g., [[16](#bib.bib16)]). The assumption in all these solutions is that the reason that processes do not follow the protocol is that they have been taken over by some adversary. There has been a great deal of interest recently in viewing at least some of the processes as being under the control of rational agents, who try to influence outcomes in a way that promotes their self interest. Halpern and Teague \citeyearHT04 were perhaps the first to do this. Their focus was on secret sharing and multiparty computation. Following \shortciteADH13,AGLS14,BCZ12,GKTZ12, we are interested in applying these ideas to standard problems in game theory. And like \shortciteADGH06,Alvisi05,BCZ12, we are interested in what happens when there is a mix of rational and faulty agents. For the purposes of this paper, we restrict to crash failures. As we shall see, a number of subtle issues arise even in this relatively simple setting. We focus on the fair consensus problem, where fairness means that the input of every agent is selected with equal probability. Fairness seems critical in applications where we do not want agents to be able to influence an outcome unduly. For instance, when agents must decide whether to commit or abort a transaction, it is useful to ensure that the outcome reflects the preferences of the agents, so that if a majority of agents prefers a particular outcome, it is selected with higher probability. Abraham, Dolev and Halpern \citeyearADH13 present a protocol for fair leader election that even tolerates coalitions of rational agents. That is, in equilibrium, a leader is elected, and each agent is elected with equal probability. Fair leader election can be used to solve fair consensus (for example, once a leader is elected, the consensus value can be taken to be the leader’s value). However, the protocol of [[2](#bib.bib2)] assumes that there are no faulty agents. Groce et al. \citeyearGKTZ12 directly provide protocols for consensus with rational agents, but again, they do not consider faulty agents and do not require fairness. Afek et al. \citeyearAGLS14 and Bei, Chen, and Zhang \citeyearBCZ12 provide protocols for consensus with crash failures and rational agents. However, Afek et al.’s protocol works only under strong assumptions about agents’ preferences, such as an agent having a strict preference for outcomes where it learns the input of other agents, while Bei, Chen, and Zhang require that their protocol be robust to deviations (that is, it achieves agreement even if rational agents deviate), a requirement that we view as unreasonably strong (see Section [3](#S3 "3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")). Neither of these protocols satisfy the fairness requirement. Moreover, the protocol proposed by Afek et al. is not even an equilibrium if some agent knows the input of other agents. As we show, this is not an accident. To explain our result, we need to briefly recall the standard notion of ex post Nash equilibrium. In a setting where we have an adversary, a protocol is an ex post equilibrium if no agent has any incentive to deviate no matter what the adversary does. Formally, “no matter what the adversary does” is captured by saying that even if we fix the adversary’s choice (so that the agents essentially know what the adversary does), agents have no incentive to deviate. Abraham, Dolev, and Halpern \citeyearADH13 provide protocols for leader election (and hence consensus) that achieve ex post Nash equilibrium if there are no failures. Here, we show that even in synchronous systems, there is no consensus protocol that is an ex post Nash equilibrium if there can be even one crash failure. In the case of crash failures, the adversary can be viewed as choosing two things: the failure pattern—which agents fail, when they fail, and which other agents they send a message to in the round that they fail, and the initial configuration—what the initial preference of each of the agents is. Roughly speaking, the reason that we cannot obtain an ex post Nash equilibrium is that if the failure pattern and initial configuration have a specific form, a rational agent i𝑖iitalic\_i can take advantage of knowing this to increase the probability of obtaining consensus on its preferred value. There might seem to be an inconsistency here. It is well known that we can achieve consensus in synchronous systems with crash failures, so it seems that we shouldn’t have any difficulty dealing with one possibly faulty agent and one rational agent who does not follow the protocol. After all, we can view a rational agent who deviates from the protocol as a faulty agent. But there is no contradiction. When the agent deviates from the purported equilibrium, consensus is still reached, just on a different value. That is, a rational agent may want to deviate so as to bias the decision, although a consensus is still reached. To get around our impossibility result, rather than trying to achieve ex post Nash equilibrium, we assume that there is some distribution π𝜋\piitalic\_π on contexts: pairs (F,v→)𝐹→𝑣(F,\vec{v})( italic\_F , over→ start\_ARG italic\_v end\_ARG ) consisting of a failure pattern F𝐹Fitalic\_F and an initial configuration v→→𝑣\vec{v}over→ start\_ARG italic\_v end\_ARG. We show that under appropriate assumptions about π𝜋\piitalic\_π, if agents care only about consensus—specifically, if (a) an agent’s utility depends only on the consensus value achieved (and not, for example, on the number of messages the agent sends) and (b) agents strictly prefer reaching consensus to not reaching consensus—then there is a Nash equilibrium that tolerates up to f𝑓fitalic\_f failures, as long as f+1<n𝑓1𝑛f+1<nitalic\_f + 1 < italic\_n, where n𝑛nitalic\_n is the total number of agents. Specifically, we make two assumptions about π𝜋\piitalic\_π, namely, we assume that π𝜋\piitalic\_π supports reachability and is uniform. Roughly speaking, we say that π𝜋\piitalic\_π supports reachability if it attributes small probability to particular failure patterns that prevent information from one agent reaching an agent that has not crashed by the end of the protocol; we say that π𝜋\piitalic\_π is uniform if it attributes equal probability to equivalent failures of different agents. We believe that these assumptions apply in many practical systems; we discuss this further in Section [4](#S4 "4 Discussion ‣ Rational Consensus"). Our Nash equilibrium strategy relies on “threats”; the threat that there will be no consensus if an agent deviates (and is caught). There might be some concern that these are empty threats, which will never be carried out. The notion of sequential equilibrium [[15](#bib.bib15)] is intended to deal with empty threats. Roughly speaking, a strategy is a sequential equilibrium if all agents are best responding to what the others are doing even off the equilibrium path. We generalize sequential equilibrium to our setting, where there might be failures, and show that the strategy that gives a Nash equilibrium can be slightly extended to give a sequential equilibrium that tolerates up to f𝑓fitalic\_f failures. The rest of this paper is organized as follows. In Section [2](#S2 "2 Model ‣ Rational Consensus") we discuss the model that we are using. Our main technical results on Nash equilibrium and sequential equilibrium are given in Section [3](#S3 "3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). We conclude with some discussion of the assumptions in Section [4](#S4 "4 Discussion ‣ Rational Consensus"). 2 Model -------- We consider a synchronous message-passing system with n𝑛nitalic\_n agents and reliable communication channels between each pair of agents. Time is divided into synchronous rounds. Each round is divided into a send phase, where agents send messages to other agents, a receive phase, where agents receive messages sent by other agents in the send phase of that round, and an update phase, where agents update the value of variables based on what they have sent and received. We denote by N𝑁Nitalic\_N the set of agents and assume that they have commonly-known identifiers in {0,…,n−1}0…𝑛1\{0,\ldots,n-1\}{ 0 , … , italic\_n - 1 }. Round m𝑚mitalic\_m takes place between time m𝑚mitalic\_m and time m+1𝑚1m+1italic\_m + 1. We now formalize the notion of run. We take a round-m𝑚mitalic\_m history for agent i𝑖iitalic\_i to be a sequence of form (v,t1,…,tm−1)𝑣subscript𝑡1…subscript𝑡𝑚1(v,t\_{1},\ldots,t\_{m-1})( italic\_v , italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_t start\_POSTSUBSCRIPT italic\_m - 1 end\_POSTSUBSCRIPT ), where v𝑣vitalic\_v is agent i𝑖iitalic\_i’s initial preference and tjsubscript𝑡𝑗t\_{j}italic\_t start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT has the form (sj,rj,dj)subscript𝑠𝑗subscript𝑟𝑗subscript𝑑𝑗(s\_{j},r\_{j},d\_{j})( italic\_s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT , italic\_d start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ), where sjsubscript𝑠𝑗s\_{j}italic\_s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT is the set of messages that i𝑖iitalic\_i sent in round j𝑗jitalic\_j tagged by who they were sent to, rjsubscript𝑟𝑗r\_{j}italic\_r start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT is the set of messages that i𝑖iitalic\_i received in round j𝑗jitalic\_j tagged by who they were sent by, and dj∈{λ}∪Vsubscript𝑑𝑗𝜆𝑉d\_{j}\in\{\lambda\}\cup Vitalic\_d start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ∈ { italic\_λ } ∪ italic\_V is i𝑖iitalic\_i’s decision (where λ𝜆\lambdaitalic\_λ denotes that no decision has been made yet and V𝑉Vitalic\_V is the set of decision values). A global (round-m𝑚mitalic\_m) history has the form (h1,…,hn)subscriptℎ1…subscriptℎ𝑛(h\_{1},\ldots,h\_{n})( italic\_h start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_h start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ) where hisubscriptℎ𝑖h\_{i}italic\_h start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is a round-m𝑚mitalic\_m history, if j𝑗jitalic\_j receives a message 𝐦𝐦{\bf m}bold\_m from i𝑖iitalic\_i in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT of hjsubscriptℎ𝑗h\_{j}italic\_h start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT, then i𝑖iitalic\_i sends 𝐦𝐦{\bf m}bold\_m to j𝑗jitalic\_j in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT in history hisubscriptℎ𝑖h\_{i}italic\_h start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. A run r𝑟ritalic\_r is a function from time (which ranges over the natural numbers) to global histories such that (a) r(m)𝑟𝑚r(m)italic\_r ( italic\_m ) is a global round-m𝑚mitalic\_m history and (b) if m<m′𝑚superscript𝑚′m<m^{\prime}italic\_m < italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, then for each agent i𝑖iitalic\_i, i𝑖iitalic\_i’s history in r(m)𝑟𝑚r(m)italic\_r ( italic\_m ) is a prefix of i𝑖iitalic\_i’s history in r(m′)𝑟superscript𝑚′r(m^{\prime})italic\_r ( italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ). Agents are either correct or faulty in a run. An agent fails only by crashing. If it crashes in round m𝑚mitalic\_m of run r𝑟ritalic\_r, then it may send a message to some subset of agents in round m𝑚mitalic\_m, but from then on, it sends no further messages. We assume that all messages sent are received in the round in which they are sent. Thus, we take a failure f of agent i𝑖iitalic\_i to be a tuple (i,m,A)𝑖𝑚𝐴(i,m,A)( italic\_i , italic\_m , italic\_A ), where m𝑚mitalic\_m is a round number (intuitively, the round at which i𝑖iitalic\_i crashes) and A𝐴Aitalic\_A is a set of agents (intuitively, the set of agents j𝑗jitalic\_j to whom i𝑖iitalic\_i can send a message before it fails). We assume that if m>1𝑚1m>1italic\_m > 1, then A𝐴Aitalic\_A is non-empty, so that i𝑖iitalic\_i sends a message to at least one agent in round m𝑚mitalic\_m if i𝑖iitalic\_i fails in round m𝑚mitalic\_m. (Intuitively, if m>1𝑚1m>1italic\_m > 1, we are identifying the failure pattern where i𝑖iitalic\_i crashes in round m𝑚mitalic\_m and sends no message with the failure pattern where i𝑖iitalic\_i crashes in round m−1𝑚1m-1italic\_m - 1 and sends messages to all the agents.) A failure pattern F𝐹Fitalic\_F is a set of failures of distinct agents i𝑖iitalic\_i. A run r𝑟ritalic\_r has context (F,v→)𝐹normal-→𝑣(F,\vec{v})( italic\_F , over→ start\_ARG italic\_v end\_ARG ) if (a) v→→𝑣\vec{v}over→ start\_ARG italic\_v end\_ARG describes the initial preferences of the agents in r𝑟ritalic\_r, (b) if (i,m,A)∈F𝑖𝑚𝐴𝐹(i,m,A)\in F( italic\_i , italic\_m , italic\_A ) ∈ italic\_F, then i𝑖iitalic\_i sends all messages according to its protocol in each round m′<msuperscript𝑚′𝑚m^{\prime}<mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m, sends no messages in each round m′>msuperscript𝑚′𝑚m^{\prime}>mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT > italic\_m, and sends messages according to its protocol only to the agents in A𝐴Aitalic\_A in round m𝑚mitalic\_m, and (c) all messages sent in r𝑟ritalic\_r are received in the round that they are sent. Let ℛ(F,v→)ℛ𝐹→𝑣\mathcal{R}(F,\vec{v})caligraphic\_R ( italic\_F , over→ start\_ARG italic\_v end\_ARG ) consist of all runs r𝑟ritalic\_r that have context (F,v→)𝐹→𝑣(F,\vec{v})( italic\_F , over→ start\_ARG italic\_v end\_ARG ). Let ℛ(F)ℛ𝐹\mathcal{R}(F)caligraphic\_R ( italic\_F ) consist of all runs that have F𝐹Fitalic\_F as the set of failures. In the consensus problem, we assume that each agent i𝑖iitalic\_i has an initial preference visubscript𝑣𝑖v\_{i}italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT in some set V𝑉Vitalic\_V. For ease of exposition, we take V={0,1}𝑉01V=\{0,1\}italic\_V = { 0 , 1 }. (Our results can easily be extended to deal with larger sets of possible values.) A protocol achieves consensus if it satisfies the following properties [[9](#bib.bib9)]: * • Agreement: No two correct agents decide different values. * • Termination: Every correct agent eventually decides. * • Integrity: All agents decide at most once. * • Validity: If an agent decides v𝑣vitalic\_v, then v𝑣vitalic\_v was the initial preference of some agent. We are interested in one other property: fairness. Note that, once we fix a context, a protocol for the agents generates a probability on runs, and hence on outcomes, in the obvious way. Fairness just says that each agent has probability at least 1/n1𝑛1/n1 / italic\_n of having its value be the consensus value, no matter what the context. More precisely, we have the following condition: * • Fairness: For each context (F,v→)𝐹→𝑣(F,\vec{v})( italic\_F , over→ start\_ARG italic\_v end\_ARG ), if c𝑐citalic\_c of the nonfaulty agents in F𝐹Fitalic\_F have initial preference v𝑣vitalic\_v, then the probability of v𝑣vitalic\_v being the consensus decision conditional on ℛ(F,v→)ℛ𝐹→𝑣\mathcal{R}(F,\vec{v})caligraphic\_R ( italic\_F , over→ start\_ARG italic\_v end\_ARG ) is at least c/n𝑐𝑛c/nitalic\_c / italic\_n. It is straightforward to view a consensus problem as a game once we associate a utility function uisubscript𝑢𝑖u\_{i}italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT with each agent i𝑖iitalic\_i, where uisubscript𝑢𝑖u\_{i}italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT maps each outcome to a utility for i𝑖iitalic\_i. Technically, it is an extensive-form Bayesian game. In a Bayesian game, agents have types, which encode private information. In consensus, an agent’s type is its initial preference. A strategy for agent i𝑖iitalic\_i in this game is just a protocol: a function from information sets to actions. As usual, we view an extensive-form game as being defined by a game tree, with the nodes where an agent i𝑖iitalic\_i moves into information sets where, intuitively, two nodes are in the same information set of agent i𝑖iitalic\_i if i𝑖iitalic\_i has the same information at both. In our setting, the nodes in a game tree correspond to global histories, and agent i𝑖iitalic\_i’s information set at a global history is determined by i𝑖iitalic\_i’s history in that global history; that is, we can take i𝑖iitalic\_i’s information set at a global history hℎhitalic\_h to consist of all global histories where i𝑖iitalic\_i’s history is the same as it is at hℎhitalic\_h. Thus, we identify an information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for agent i𝑖iitalic\_i with a history hisubscriptℎ𝑖h\_{i}italic\_h start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for agent i𝑖iitalic\_i. If Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is the information set associated with history hisubscriptℎ𝑖h\_{i}italic\_h start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, we denote by ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) the set of runs r𝑟ritalic\_r where i𝑖iitalic\_i has history hisubscriptℎ𝑖h\_{i}italic\_h start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT in r(m)𝑟𝑚r(m)italic\_r ( italic\_m ). In game theory, a strategy for agent i𝑖iitalic\_i is a function that associates with each information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for agent i𝑖iitalic\_i a distribution over the actions that i𝑖iitalic\_i can take at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. In distributed computing, a protocol for agent i𝑖iitalic\_i is a function that associates with each history hisubscriptℎ𝑖h\_{i}italic\_h start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for agent i𝑖iitalic\_i a distribution over the actions that i𝑖iitalic\_i can take at hisubscriptℎ𝑖h\_{i}italic\_h start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Since we are identifying histories for agent i𝑖iitalic\_i with information sets, it is clear that a protocol for agent i𝑖iitalic\_i can be identified with a strategy for agent i𝑖iitalic\_i. In consensus, the actions involve sending messages and deciding on values. We assume that there is a special value ⊥bottom\bot⊥ that an agent can decide on. By deciding on ⊥bottom\bot⊥, an agent guarantees that there is no consensus. If we assume that an agent prefers to reach consensus on some value to not reaching consensus at all, in the language of Ben Porath \citeyearBp03, this means that each agent has a punishment strategy. We next want to define an appropriate solution concept for our setting. The standard approach is to say that an equilibrium is a strategy profile (i.e., a tuple of strategies, one for each agent) where no agent can do better by deviating. “Doing better” is typically taken to mean “gets a higher expected utility”. However, if we do not have a probability on contexts, we cannot compute an agent’s expected utility. We thus consider two families of solution concepts. In the first, we take “doing better” to mean that, for each fixed context, no agent can do better by deviating. Once we fix the context, the strategy profile generates a probability distribution on runs, and we can compute the expected utility. In the second approach we assume a distribution on contexts. A strategy profile σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG is an ϵitalic-ϵ\epsilonitalic\_ϵ–f𝑓fitalic\_f-Nash equilibrium if, for each fixed context (F,v→)𝐹→𝑣(F,\vec{v})( italic\_F , over→ start\_ARG italic\_v end\_ARG ) where there are at most f𝑓fitalic\_f faulty agents in F𝐹Fitalic\_F, and all agents i𝑖iitalic\_i, there is no strategy σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT for agent i𝑖iitalic\_i such that i𝑖iitalic\_i can improve its expected utility by more than ϵitalic-ϵ\epsilonitalic\_ϵ. Formally, if ui(τ→∣ℛ(F,v→))subscript𝑢𝑖conditional→𝜏ℛ𝐹→𝑣u\_{i}(\vec{\tau}\mid\mathcal{R}(F,\vec{v}))italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_τ end\_ARG ∣ caligraphic\_R ( italic\_F , over→ start\_ARG italic\_v end\_ARG ) ) denotes i𝑖iitalic\_i’s expected utility if strategy profile τ→→𝜏\vec{\tau}over→ start\_ARG italic\_τ end\_ARG is played, conditional on the run being in ℛ(F,v→)ℛ𝐹→𝑣\mathcal{R}(F,\vec{v})caligraphic\_R ( italic\_F , over→ start\_ARG italic\_v end\_ARG ), we require that for all strategies σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT for i𝑖iitalic\_i, ui((σi′,σ→−i)∣ℛ(F,v→))≤ui(σ→∣ℛ(F,v→))+ϵsubscript𝑢𝑖conditionalsuperscriptsubscript𝜎𝑖′subscript→𝜎𝑖ℛ𝐹→𝑣subscript𝑢𝑖conditional→𝜎ℛ𝐹→𝑣italic-ϵu\_{i}((\sigma\_{i}^{\prime},\vec{\sigma}\_{-i})\mid\mathcal{R}(F,\vec{v}))\leq u\_{i}(\vec{\sigma}\mid\mathcal{R}(F,\vec{v}))+\epsilonitalic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) ∣ caligraphic\_R ( italic\_F , over→ start\_ARG italic\_v end\_ARG ) ) ≤ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_σ end\_ARG ∣ caligraphic\_R ( italic\_F , over→ start\_ARG italic\_v end\_ARG ) ) + italic\_ϵ. An f𝑓fitalic\_f-Nash equilibrium is a 00–f𝑓fitalic\_f-Nash equilibrium. The notion of f𝑓fitalic\_f-Nash equilibrium extends the notion of ex post Nash equilibrium by allowing up to f𝑓fitalic\_f faulty agents; a 00-Nash equilibrium is an ex post Nash equilibrium.111This definition is in the spirit of the notion of (k,t)𝑘𝑡(k,t)( italic\_k , italic\_t )-robustness as defined by Abraham et al. [[1](#bib.bib1)], where coalitions of size k𝑘kitalic\_k are allowed in addition to t𝑡titalic\_t “faulty” agents, but here we restrict the behavior of the faulty agents to crash failures rather than allowing the faulty agents to follow an arbitrary protocol, and take k=1𝑘1k=1italic\_k = 1. We also allow the deviating agents to fail (this assumption has no impact on our results). Given a distribution π𝜋\piitalic\_π on contexts and a strategy profile σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG, π𝜋\piitalic\_π and σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG determine a probability on runs denoted πσ→subscript𝜋→𝜎\pi\_{\vec{\sigma}}italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG end\_POSTSUBSCRIPT in the obvious way. We say that σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG is an ϵitalic-ϵ\epsilonitalic\_ϵ–π𝜋\piitalic\_π-Nash equilibrium if, for all agents i𝑖iitalic\_i and all strategies σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT for i𝑖iitalic\_i, we have ui(σi′,σ→−i)≤ui(σ→)+ϵsubscript𝑢𝑖superscriptsubscript𝜎𝑖′subscript→𝜎𝑖subscript𝑢𝑖→𝜎italic-ϵu\_{i}(\sigma\_{i}^{\prime},\vec{\sigma}\_{-i})\leq u\_{i}(\vec{\sigma})+\epsilonitalic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) ≤ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_σ end\_ARG ) + italic\_ϵ, where now the expectation is taken with respect to the probability πσ→subscript𝜋→𝜎\pi\_{\vec{\sigma}}italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG end\_POSTSUBSCRIPT. A π𝜋\piitalic\_π-Nash equilibrium is a 0–π𝜋\piitalic\_π-Nash equilibrium. If π𝜋\piitalic\_π puts probability 1 on there being no failures, then we get the standard notion of (ϵitalic-ϵ\epsilonitalic\_ϵ-) Nash equilibrium. 3 Possibility and Impossibility Results for Consensus ------------------------------------------------------ In this section, we consider the consensus problem from a game-theoretic viewpoint. We focus on the case where agents care only about consensus, since this type of utility function seems to capture many situations of interest. For the rest of this section, let β0isubscript𝛽0𝑖\beta\_{0i}italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT be i𝑖iitalic\_i’s utility if its initial preference is decided, let β1isubscript𝛽1𝑖\beta\_{1i}italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT be i𝑖iitalic\_i’s utility if there is consensus, but not on i𝑖iitalic\_i’s initial preference, and let β2isubscript𝛽2𝑖\beta\_{2i}italic\_β start\_POSTSUBSCRIPT 2 italic\_i end\_POSTSUBSCRIPT be i𝑖iitalic\_i’s utility if there is no consensus. The assumption that agents care only about consensus means that, for all i𝑖iitalic\_i, β0i>β1i>β2isubscript𝛽0𝑖subscript𝛽1𝑖subscript𝛽2𝑖\beta\_{0i}>\beta\_{1i}>\beta\_{2i}italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT > italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT > italic\_β start\_POSTSUBSCRIPT 2 italic\_i end\_POSTSUBSCRIPT. Note that although we assume that agents prefer consensus to no consensus, unlike Bei, Chen, and Zhang. \citeyearBCZ12, we do not require that our algorithms guarantee consensus when rational agents deviate. Our algorithm does guarantee that there will be consensus if there are no deviations. On the other hand, we allow for the possibility that a deviation by a rational agent will result in there being no consensus. For example, suppose that a rational agent pretends to fail in a setting where there is a bound f𝑓fitalic\_f on the number of crash failures. That means that if f𝑓fitalic\_f other agents actually do crash, then some agent will detect that f+1𝑓1f+1italic\_f + 1 agents seem to have crashed. Our algorithm requires that if an agent detects such an inconsistency, then it aborts. If the probability that f𝑓fitalic\_f agents actually crash is low, in our framework, a rational agent may decide that it is worth the risk of pretending to crash if the potential gain is sufficiently large. Bei, Chen, and Zhang would not permit this, since they require consensus even if rational agents deviate from the algorithm. This requirement thus severely limits the possible deviations. ### 3.1 An Impossibility Result We start by showing that there is no fair consensus protocol that is an f𝑓fitalic\_f-Nash equilibrium. ###### Theorem 1. If σ→normal-→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG solves fair consensus, agents care only about consensus, and f≥1𝑓1f\geq 1italic\_f ≥ 1, then σ→normal-→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG is not an f𝑓fitalic\_f-Nash equilibrium ###### Proof. Consider the initial configuration v→→𝑣\vec{v}over→ start\_ARG italic\_v end\_ARG where all agents but i𝑖iitalic\_i have initial preference 0 and i𝑖iitalic\_i has initial preference 1. If F1superscript𝐹1F^{1}italic\_F start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT is the failure pattern where no agent fails, by Fairness, the agents must decide 1 with positive probability in context (v→,F1)→𝑣superscript𝐹1(\vec{v},F^{1})( over→ start\_ARG italic\_v end\_ARG , italic\_F start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT ). It follows that there must be a failure pattern F2superscript𝐹2F^{2}italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT where only agent i𝑖iitalic\_i fails but the agents decide 1 with positive probability in context (v→,F2)→𝑣superscript𝐹2(\vec{v},F^{2})( over→ start\_ARG italic\_v end\_ARG , italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ). (In F2superscript𝐹2F^{2}italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT, i𝑖iitalic\_i fails only after a decision has been made in F1superscript𝐹1F^{1}italic\_F start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT.) If F0superscript𝐹0F^{0}italic\_F start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT is the failure pattern where only i𝑖iitalic\_i fails, and i𝑖iitalic\_i fails immediately, before sending any messages, then it is clear that no agents can distinguish this context from one where all agents have initial preference 0, so all agents must decide 0, by the Validity requirement. Put a partial order ≤\leq≤ on failure patterns where only i𝑖iitalic\_i crashes by taking F≤F′𝐹superscript𝐹′F\leq F^{\prime}italic\_F ≤ italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT if either i𝑖iitalic\_i crashes in an earlier round in F𝐹Fitalic\_F than in F′superscript𝐹′F^{\prime}italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, or i𝑖iitalic\_i crashes in the same round m𝑚mitalic\_m in both F𝐹Fitalic\_F and F′superscript𝐹′F^{\prime}italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, but the set of agents to whom i𝑖iitalic\_i sends a message in F𝐹Fitalic\_F is a subset of the set of agents to whom i𝑖iitalic\_i sends a message in F′superscript𝐹′F^{\prime}italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. Clearly F0<F2superscript𝐹0superscript𝐹2F^{0}<F^{2}italic\_F start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT < italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT. Thus, there exists a minimal failure pattern F\superscript𝐹F^{\}italic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT such that F0<F\≤F2superscript𝐹0superscript𝐹superscript𝐹2F^{0}<F^{\}\leq F^{2}italic\_F start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT < italic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ≤ italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT, only i𝑖iitalic\_i fails in F\superscript𝐹F^{\}italic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, the consensus is on 1 with positive probability in context (v→,F\)→𝑣superscript𝐹(\vec{v},F^{\})( over→ start\_ARG italic\_v end\_ARG , italic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ), the consensus is 0 with probability 1 in all contexts (F,v→)𝐹→𝑣(F,\vec{v})( italic\_F , over→ start\_ARG italic\_v end\_ARG ) where only agent i𝑖iitalic\_i fails in F𝐹Fitalic\_F and F<F\𝐹superscript𝐹F<F^{\}italic\_F < italic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT. We can assume without loss of generality that i𝑖iitalic\_i sends a message to some agent j𝑗jitalic\_j in the round m𝑚mitalic\_m in which i𝑖iitalic\_i fails. To see this, note that if i𝑖iitalic\_i crashes in the first round then i𝑖iitalic\_i must send a message to some agent (otherwise F\=F0superscript𝐹superscript𝐹0F^{\}=F^{0}italic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT = italic\_F start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT and the decision is 0 with probability 1). And if i𝑖iitalic\_i crashes in round m>1𝑚1m>1italic\_m > 1, we have assumed that i𝑖iitalic\_i sends at least one message before crashing (recall that we identify an agent crashing at round m>1𝑚1m>1italic\_m > 1 and sending no messages with the agent crashing at round m−1𝑚1m-1italic\_m - 1 and sending to all agents). Now suppose that an agent j𝑗jitalic\_j that receives a message from i𝑖iitalic\_i in round m𝑚mitalic\_m pretends not to receive that message. This makes the situation indistinguishable from the context (F,v→)𝐹→𝑣(F,\vec{v})( italic\_F , over→ start\_ARG italic\_v end\_ARG ) where F𝐹Fitalic\_F is just like F\superscript𝐹F^{\}italic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT except that i𝑖iitalic\_i does not send a message to j𝑗jitalic\_j in round m𝑚mitalic\_m. Since F0≤F<F\superscript𝐹0𝐹superscript𝐹F^{0}\leq F<F^{\}italic\_F start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT ≤ italic\_F < italic\_F start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, the decision must be 0 with probability 1 in context (v→,F)→𝑣𝐹(\vec{v},F)( over→ start\_ARG italic\_v end\_ARG , italic\_F ). Since j𝑗jitalic\_j has initial preference 00 in v→→𝑣\vec{v}over→ start\_ARG italic\_v end\_ARG, j𝑗jitalic\_j can increase its expected utility by this pretense, so σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG is not an f𝑓fitalic\_f-Nash equilibrium. ∎ ### 3.2 Obtaining a π𝜋\piitalic\_π-Nash equilibrium We now prove a positive result. If we are willing to assume that there is a distribution π𝜋\piitalic\_π on contexts with some reasonable properties, then we can get a fair π𝜋\piitalic\_π-Nash equilibrium. But, as we show below, there are some subtle problems in doing this. Before discussing these problems, it is useful to recall some results from social choice theory. Consider a setting with n𝑛nitalic\_n agents where each has a preference order (i.e., a total order) over some set O𝑂Oitalic\_O of outcomes. A social-choice function is a (possibly randomized) function that maps a profile of preference orders to an outcome. For example, we can consider agents trying to elect a leader, where each agent has a preference order over the candidates; the social-choice function chooses a leader as a function of the expressed preferences. A social-choice function is incentive compatible if no agent can do better by lying about its preferences. The well-known Gibbard-Satterthwaite theorem [[10](#bib.bib10), [18](#bib.bib18)] says that if there are at least three possible outcomes, then the only incentive-compatible deterministic social-choice function f𝑓fitalic\_f is a dictatorship; i.e., the function f𝑓fitalic\_f just chooses a player i𝑖iitalic\_i and takes the outcome to be i𝑖iitalic\_i’s most-preferred candidate, ignoring all other agents’ preferences. Gibbard \citeyearGibbard77 extends this result to show that if there are at least three outcomes, then the only randomized incentive-compatible social-choice function is a random dictatorship, which essentially amounts to choosing some player i𝑖iitalic\_i according to some probability distribution and then choosing i𝑖iitalic\_i’s value. Bei, Chen, and Zehang \citeyearBCZ12 point out that a strategy profile that solves consensus can be viewed as a social-choice function: agents have preferences over three outcomes, 0, 1, and ⊥bottom\bot⊥, and the consensus value (or ⊥bottom\bot⊥, if there is no consensus) can be viewed as the outcome chosen by the function. A strategy profile that is a Nash equilibrium is clearly incentive-compatible; no agent has an incentive to lie about its preferences. Thus, it follows from Gibbard’s \citeyearGibbard77 result that a solution to rational consensus must be a randomized dictatorship. And, indeed, our protocols can be viewed as implementing a randomized dictatorship: one agent is chosen at random, and its value becomes the consensus value. However, implementing such a randomized dictatorship in our setting is nontrivial because of the possibility of failures.222We remark that Theorem 1 of Bei, Chen, and Zhang \citeyearBCZ12 claims that, given a fixed failure pattern, a strategy profile for consensus that is a Nash equilibrium must implement a dictatorship, rather than randomized dictatorship. While this is true if we restrict to deterministic strategies, neither we nor Bei, Chen, and Zhang do so. We have not checked carefully whether results of Bei, Chen, and Zhang that depend on their Theorem 1 continue to hold once we allow for randomized dictatorships. #### 3.2.1 A naive protocol We start with a protocol that, while not solving the problem, has many of the essential features of our solution, and also helps to point out the subtleties. Consider the following slight variant of one of the early protocols for consensus [[8](#bib.bib8)]: In round 1, each agent i𝑖iitalic\_i broadcasts a tuple (i,vi,xi0,…,xif)𝑖subscript𝑣𝑖subscript𝑥𝑖0…subscript𝑥𝑖𝑓(i,v\_{i},x\_{i0},\ldots,x\_{if})( italic\_i , italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_x start\_POSTSUBSCRIPT italic\_i 0 end\_POSTSUBSCRIPT , … , italic\_x start\_POSTSUBSCRIPT italic\_i italic\_f end\_POSTSUBSCRIPT ), where visubscript𝑣𝑖v\_{i}italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is i𝑖iitalic\_i’s initial preference, and xitsubscript𝑥𝑖𝑡x\_{it}italic\_x start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT is a random element in {0,…,n−t}0…𝑛𝑡\{0,\ldots,n-t\}{ 0 , … , italic\_n - italic\_t }. For round 2,…,f+12…𝑓12,\ldots,f+12 , … , italic\_f + 1, each agent i𝑖iitalic\_i broadcasts all the tuples (j,vj,x→j)𝑗subscript𝑣𝑗subscript→𝑥𝑗(j,v\_{j},\vec{x}\_{j})( italic\_j , italic\_v start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT , over→ start\_ARG italic\_x end\_ARG start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) that i𝑖iitalic\_i received and did not already forward in earlier rounds. At the end of round f+1𝑓1f+1italic\_f + 1, each agent checks for consistency; specifically, it checks that it has received tuples from at least n−f𝑛𝑓n-fitalic\_n - italic\_f agents and that it has not received distinct tuples claimed to have been sent by some agent j𝑗jitalic\_j. If i𝑖iitalic\_i detects an inconsistency, then i𝑖iitalic\_i decides ⊥bottom\bot⊥. Otherwise, suppose that i𝑖iitalic\_i received tuples from n−t𝑛𝑡n-titalic\_n - italic\_t agents. Then i𝑖iitalic\_i computes the sum mod n−t𝑛𝑡n-titalic\_n - italic\_t of the values xjtsubscript𝑥𝑗𝑡x\_{jt}italic\_x start\_POSTSUBSCRIPT italic\_j italic\_t end\_POSTSUBSCRIPT for each agent j𝑗jitalic\_j from which it received a tuple. If the sum is S𝑆Sitalic\_S, then i𝑖iitalic\_i decides on the value of the agent with the (S+1)𝑆1(S+1)( italic\_S + 1 )st highest id among the n−t𝑛𝑡n-titalic\_n - italic\_t agents from which it received tuples. (Here is where we are implementing the random dictatorship.) Note that the random value xjtsubscript𝑥𝑗𝑡x\_{jt}italic\_x start\_POSTSUBSCRIPT italic\_j italic\_t end\_POSTSUBSCRIPT is used by i𝑖iitalic\_i in computing the consensus value if exactly t𝑡titalic\_t faulty agents are discovered; the remaining random values sent by agent j𝑗jitalic\_j in the first round are discarded. It is straightforward to check that if all nonfaulty agents follow this protocol, then they will all agree on the set of tuples received (see the proof of Theorem [2](#Thmtheorem2 "Theorem 2. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") for an argument similar in spirit), and so will choose the same decision value, and each agent whose value is considered has an equal chance of having their value determine the outcome. But this will not be in general a π𝜋\piitalic\_π-Nash equilibrium if π𝜋\piitalic\_π allows up to f𝑓fitalic\_f failures, that is, π𝜋\piitalic\_π puts probability 0 on all failure patterns that have more than f𝑓fitalic\_f failures and f≥2𝑓2f\geq 2italic\_f ≥ 2. Consider a distribution π𝜋\piitalic\_π that puts positive probability on all contexts with at most f𝑓fitalic\_f failures, and an initial configuration where agent 1 prefers 1, but all other agents prefer 0. Agent 1 follows the protocol in the first round, and receives a message from all the other agents. We claim that agent 1 may have an incentive to pretend to fail (without sending any messages) at this point. Agent 1 can gain by doing this if one of the other agents, say agent 2, crashed in the first round and sent a message only to agent 1. In this case, if 1 pretends to crash, no other agent will learn 2’s initial preference, so 1’s initial preference will have a somewhat higher probability (at least 1n−1−1n1𝑛11𝑛\frac{1}{n-1}-\frac{1}{n}divide start\_ARG 1 end\_ARG start\_ARG italic\_n - 1 end\_ARG - divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG) of becoming the consensus decision. Of course, there is a risk in pretending to crash: if f𝑓fitalic\_f agents really do crash, then an inconsistency will be detected, and the decision will be ⊥bottom\bot⊥. Let α<fsubscript𝛼absent𝑓\alpha\_{<f}italic\_α start\_POSTSUBSCRIPT < italic\_f end\_POSTSUBSCRIPT be the probability of there being fewer than f𝑓fitalic\_f failures and at least one agent crashing in the first round who does not send to any agent other than 1 (this is the probability that 1 gains some utility by its action); let α=fsubscript𝛼absent𝑓\alpha\_{=f}italic\_α start\_POSTSUBSCRIPT = italic\_f end\_POSTSUBSCRIPT be the probability of there being f𝑓fitalic\_f crashes other than 1 (this is an upper bound on the probability that 1 loses utility by its action). Then 1’s expected gain by deviating is at least \| \| \| \| \| --- \| --- \| --- \| \| \| (β0i−β1i)(1n−1−1n)α<f−(β0i−β2i)α=f.subscript𝛽0𝑖subscript𝛽1𝑖1𝑛11𝑛subscript𝛼absent𝑓subscript𝛽0𝑖subscript𝛽2𝑖subscript𝛼absent𝑓(\beta\_{0i}-\beta\_{1i})\left(\frac{1}{n-1}-\frac{1}{n}\right)\alpha\_{<f}-(\beta\_{0i}-\beta\_{2i})\alpha\_{=f}.( italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT - italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT ) ( divide start\_ARG 1 end\_ARG start\_ARG italic\_n - 1 end\_ARG - divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ) italic\_α start\_POSTSUBSCRIPT < italic\_f end\_POSTSUBSCRIPT - ( italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT - italic\_β start\_POSTSUBSCRIPT 2 italic\_i end\_POSTSUBSCRIPT ) italic\_α start\_POSTSUBSCRIPT = italic\_f end\_POSTSUBSCRIPT . \| \| This is a small quantity. However, if f𝑓fitalic\_f is reasonably large and failures are unlikely, we would expect α=fsubscript𝛼absent𝑓\alpha\_{=f}italic\_α start\_POSTSUBSCRIPT = italic\_f end\_POSTSUBSCRIPT to be much smaller than α<fsubscript𝛼absent𝑓\alpha\_{<f}italic\_α start\_POSTSUBSCRIPT < italic\_f end\_POSTSUBSCRIPT, so as the number f𝑓fitalic\_f of failures that the protocol is designed to handle increases, deviating becomes more and more likely to produce a (small) gain. #### 3.2.2 A π𝜋\piitalic\_π-Nash equilibrium There are three problems with the preceding protocol. The first is that, even if 1 pretends to fail, 1’s value will be considered a potential consensus value, since everyone received the value before 1 failed. This means that there is little downside in pretending to fail. Roughly speaking, we deal with this problem by taking into consideration only the values of nonfaulty agents when deciding on a consensus value. The second problem is that since agents learn the random values (xi0,…,xif)subscript𝑥𝑖0…subscript𝑥𝑖𝑓(x\_{i0},\ldots,x\_{if})( italic\_x start\_POSTSUBSCRIPT italic\_i 0 end\_POSTSUBSCRIPT , … , italic\_x start\_POSTSUBSCRIPT italic\_i italic\_f end\_POSTSUBSCRIPT ) that will be used in determining the consensus value in round 1, they may be able to guess with high probability the value that will be decided on at a point when they can still influence the outcome. To address this problem, agents do not send these random values in the first round; instead, they use secret sharing [[19](#bib.bib19)], so as to allow the nonfaulty agents to reconstruct these random values when they need to decide on the consensus value. This prevents agents from being able to guess with high probability what the decision will be too early. The third problem is that in some cases agents can safely lie about the messages sent by other agents (e.g., i𝑖iitalic\_i can pretend that another agent did not crash). We could solve this by assuming that messages can be signed using unforgeable signatures. We do not need this or any other cryptographic assumption. Instead, we use some randomization to ensure that if an agent lies about a message that was sent, it will be caught with high probability. Thus, in our algorithm, an agent i𝑖iitalic\_i generates random numbers for two reasons. The first is that it generates f+1𝑓1f+1italic\_f + 1 random numbers (xi0,…,xif)subscript𝑥𝑖0…subscript𝑥𝑖𝑓(x\_{i0},\ldots,x\_{if})( italic\_x start\_POSTSUBSCRIPT italic\_i 0 end\_POSTSUBSCRIPT , … , italic\_x start\_POSTSUBSCRIPT italic\_i italic\_f end\_POSTSUBSCRIPT ), where xitsubscript𝑥𝑖𝑡x\_{it}italic\_x start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT is used in choosing the consensus value if there are exactly t𝑡titalic\_t faulty agents discovered, and then, as we suggested above, shares them using secret sharing, so that the numbers can be reconstructed at the appropriate time (see below). The second is that it generates n−1𝑛1n-1italic\_n - 1 additional random numbers, denoted zijm[i]superscriptsubscript𝑧𝑖𝑗𝑚delimited-[]𝑖z\_{ij}^{m}[i]italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_i ], one for each agent j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i, in each round m𝑚mitalic\_m, and sends them to j𝑗jitalic\_j in round m𝑚mitalic\_m. Then if agent j𝑗jitalic\_j claims that it got a message in round m𝑚mitalic\_m from i𝑖iitalic\_i, it will have to also provide zijm[i]superscriptsubscript𝑧𝑖𝑗𝑚delimited-[]𝑖z\_{ij}^{m}[i]italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_i ] as proof. In more detail, we proceed as follows. Initially, each agent i𝑖iitalic\_i generates a random tuple (xi0,…,xif)subscript𝑥𝑖0…subscript𝑥𝑖𝑓(x\_{i0},\ldots,x\_{if})( italic\_x start\_POSTSUBSCRIPT italic\_i 0 end\_POSTSUBSCRIPT , … , italic\_x start\_POSTSUBSCRIPT italic\_i italic\_f end\_POSTSUBSCRIPT ), where xitsubscript𝑥𝑖𝑡x\_{it}italic\_x start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT is in {0,…,n−t}0…𝑛𝑡\{0,\ldots,n-t\}{ 0 , … , italic\_n - italic\_t }. It then computes f+1𝑓1f+1italic\_f + 1 random polynomials qi0,…,qifsubscript𝑞𝑖0…subscript𝑞𝑖𝑓q\_{i0},\ldots,q\_{if}italic\_q start\_POSTSUBSCRIPT italic\_i 0 end\_POSTSUBSCRIPT , … , italic\_q start\_POSTSUBSCRIPT italic\_i italic\_f end\_POSTSUBSCRIPT, each of degree 1111, such that qit(0)=xitsubscript𝑞𝑖𝑡0subscript𝑥𝑖𝑡q\_{it}(0)=x\_{it}italic\_q start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT ( 0 ) = italic\_x start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT. It then sends (qi0(j),…,qif(j))subscript𝑞𝑖0𝑗…subscript𝑞𝑖𝑓𝑗(q\_{i0}(j),\ldots,q\_{if}(j))( italic\_q start\_POSTSUBSCRIPT italic\_i 0 end\_POSTSUBSCRIPT ( italic\_j ) , … , italic\_q start\_POSTSUBSCRIPT italic\_i italic\_f end\_POSTSUBSCRIPT ( italic\_j ) ) to agent j𝑗jitalic\_j. The upshot of this is that no agent will be able to compute xitsubscript𝑥𝑖𝑡x\_{it}italic\_x start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT given this information (since one point on a degree-1 polynomial qitsubscript𝑞𝑖𝑡q\_{it}italic\_q start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT gives no information regarding qit(0)subscript𝑞𝑖𝑡0q\_{it}(0)italic\_q start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT ( 0 )). In addition, in round 1, each agent i𝑖iitalic\_i sends visubscript𝑣𝑖v\_{i}italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT to each agent j𝑗jitalic\_j, just as in the naive algorithm; it also generates the random number zij1[i]superscriptsubscript𝑧𝑖𝑗1delimited-[]𝑖z\_{ij}^{1}[i]italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_i ] and a special random number z𝑧zitalic\_z, and sends each agent the vector zij1=(zij1[1],…,zij1[n])superscriptsubscript𝑧𝑖𝑗1superscriptsubscript𝑧𝑖𝑗1delimited-[]1…superscriptsubscript𝑧𝑖𝑗1delimited-[]𝑛z\_{ij}^{1}=(z\_{ij}^{1}[1],\ldots,z\_{ij}^{1}[n])italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT = ( italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ 1 ] , … , italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_n ] ), where zij1[j′]=0superscriptsubscript𝑧𝑖𝑗1delimited-[]superscript𝑗′0z\_{ij}^{1}[j^{\prime}]=0italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ] = 0 for j′≠i,jsuperscript𝑗′𝑖𝑗j^{\prime}\neq i,jitalic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_i , italic\_j and zij1[j]=zsuperscriptsubscript𝑧𝑖𝑗1delimited-[]𝑗𝑧z\_{ij}^{1}[j]=zitalic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_j ] = italic\_z. (As we said, these random numbers form a “signature”; their role will become clearer in the proof.) Finally, in round 1, agent i𝑖iitalic\_i sends a status report SRi1𝑆subscriptsuperscript𝑅1𝑖SR^{1}\_{i}italic\_S italic\_R start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT; we discuss this in more detail below. In the receive phase of round 1, agent i𝑖iitalic\_i adds all the values received from other agents to the set STi𝑆subscript𝑇𝑖ST\_{i}italic\_S italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. In round m𝑚mitalic\_m with 2≤m≤f2𝑚𝑓2\leq m\leq f2 ≤ italic\_m ≤ italic\_f, i𝑖iitalic\_i again sends a status report SRim𝑆superscriptsubscript𝑅𝑖𝑚SR\_{i}^{m}italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT and a vector zijmsuperscriptsubscript𝑧𝑖𝑗𝑚z\_{ij}^{m}italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT. For each agent j𝑗jitalic\_j, SRim[j]𝑆superscriptsubscript𝑅𝑖𝑚delimited-[]𝑗SR\_{i}^{m}[j]italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_j ] is a tuple of the form (m,x)𝑚𝑥(m,x)( italic\_m , italic\_x ), where m𝑚mitalic\_m is the first round that i𝑖iitalic\_i knows that j𝑗jitalic\_j crashed (m=∞𝑚m=\inftyitalic\_m = ∞ if i𝑖iitalic\_i believes that j𝑗jitalic\_j has not yet crashed), and x𝑥xitalic\_x is either the vector zjim−1superscriptsubscript𝑧𝑗𝑖𝑚1z\_{ji}^{m-1}italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 1 end\_POSTSUPERSCRIPT of random values sent by j𝑗jitalic\_j in m−1𝑚1m-1italic\_m - 1 (if i𝑖iitalic\_i believes that j𝑗jitalic\_j has not yet crashed) or an agent that told i𝑖iitalic\_i that j𝑗jitalic\_j crashed in round m𝑚mitalic\_m. The tuple zijmsuperscriptsubscript𝑧𝑖𝑗𝑚z\_{ij}^{m}italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT is computed by setting zijm[l]superscriptsubscript𝑧𝑖𝑗𝑚delimited-[]𝑙z\_{ij}^{m}[l]italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_l ] for l≠i,j𝑙𝑖𝑗l\neq i,jitalic\_l ≠ italic\_i , italic\_j to be zlim−1[l]superscriptsubscript𝑧𝑙𝑖𝑚1delimited-[]𝑙z\_{li}^{m-1}[l]italic\_z start\_POSTSUBSCRIPT italic\_l italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 1 end\_POSTSUPERSCRIPT [ italic\_l ], the random number sent by l𝑙litalic\_l in the previous round (this will be used to prove that i𝑖iitalic\_i really got a message from l𝑙litalic\_l in the previous round—it is our replacement for unforgeable signatures); again, zijm[i]superscriptsubscript𝑧𝑖𝑗𝑚delimited-[]𝑖z\_{ij}^{m}[i]italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_i ] is a random value generated by i𝑖iitalic\_i. In round f+1𝑓1f+1italic\_f + 1, i𝑖iitalic\_i also sends j𝑗jitalic\_j the secret shares ylitsuperscriptsubscript𝑦𝑙𝑖𝑡y\_{li}^{t}italic\_y start\_POSTSUBSCRIPT italic\_l italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT it received in round 1 from each agent l𝑙litalic\_l (i.e., the value qlt(i)superscriptsubscript𝑞𝑙𝑡𝑖q\_{l}^{t}(i)italic\_q start\_POSTSUBSCRIPT italic\_l end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_i ) that it received from l𝑙litalic\_l, assuming that l𝑙litalic\_l did not lie). This enables j𝑗jitalic\_j to compute the polynomials qitsubscript𝑞𝑖𝑡q\_{it}italic\_q start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT, and hence the secret qit(0)=xitsubscript𝑞𝑖𝑡0subscript𝑥𝑖𝑡q\_{it}(0)=x\_{it}italic\_q start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT ( 0 ) = italic\_x start\_POSTSUBSCRIPT italic\_i italic\_t end\_POSTSUBSCRIPT for 0≤t≤f0𝑡𝑓0\leq t\leq f0 ≤ italic\_t ≤ italic\_f. If i𝑖iitalic\_i detects an inconsistency in round m≤f+1𝑚𝑓1m\leq f+1italic\_m ≤ italic\_f + 1, then i𝑖iitalic\_i decides ⊥bottom\bot⊥, where i𝑖iitalic\_i detects an inconsistency in round m𝑚mitalic\_m if the messages received by i𝑖iitalic\_i are inconsistent with all agents following the protocol except that up to f𝑓fitalic\_f agents may crash. This can happen if 1. 1. j𝑗jitalic\_j sends incorrectly formatted messages; 2. 2. m=2𝑚2m=2italic\_m = 2 and agents j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and j′′≠isuperscript𝑗′′𝑖j^{\prime\prime}\neq iitalic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT ≠ italic\_i disagree about the random values zjj′1[j′]superscriptsubscript𝑧𝑗superscript𝑗′1delimited-[]superscript𝑗′z\_{jj^{\prime}}^{1}[j^{\prime}]italic\_z start\_POSTSUBSCRIPT italic\_j italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ] and zjj′′1[j′′]superscriptsubscript𝑧𝑗superscript𝑗′′1delimited-[]superscript𝑗′′z\_{jj^{\prime\prime}}^{1}[j^{\prime\prime}]italic\_z start\_POSTSUBSCRIPT italic\_j italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT ] sent by j𝑗jitalic\_j in round 1111; 3. 3. m>2𝑚2m>2italic\_m > 2 and some agent j′≠jsuperscript𝑗′𝑗j^{\prime}\neq jitalic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_j reports that j𝑗jitalic\_j sent a value zjj′m−1[i]superscriptsubscript𝑧𝑗superscript𝑗′𝑚1delimited-[]𝑖z\_{jj^{\prime}}^{m-1}[i]italic\_z start\_POSTSUBSCRIPT italic\_j italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 1 end\_POSTSUPERSCRIPT [ italic\_i ] in round m−1𝑚1m-1italic\_m - 1 different from the value zijm−2[i]superscriptsubscript𝑧𝑖𝑗𝑚2delimited-[]𝑖z\_{ij}^{m-2}[i]italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 2 end\_POSTSUPERSCRIPT [ italic\_i ] sent by i𝑖iitalic\_i to j𝑗jitalic\_j in round m−2𝑚2m-2italic\_m - 2; 4. 4. m=f+1𝑚𝑓1m=f+1italic\_m = italic\_f + 1 and it is not possible to interpolate a polynomial qjtsuperscriptsubscript𝑞𝑗𝑡q\_{j}^{t}italic\_q start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT through the shares yjitsuperscriptsubscript𝑦𝑗𝑖𝑡y\_{ji}^{t}italic\_y start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT received by i𝑖iitalic\_i from j𝑗jitalic\_j in round 1 and the values yjltsuperscriptsubscript𝑦𝑗𝑙𝑡y\_{jl}^{t}italic\_y start\_POSTSUBSCRIPT italic\_j italic\_l end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT received from l≠j𝑙𝑗l\neq jitalic\_l ≠ italic\_j in round f+1𝑓1f+1italic\_f + 1. 5. 5. some agent j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT sends i𝑖iitalic\_i a status report in round m𝑚mitalic\_m that says that j𝑗jitalic\_j crashed in some round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and either i𝑖iitalic\_i receives a message from j𝑗jitalic\_j in round m′′>m′superscript𝑚′′superscript𝑚′m^{\prime\prime}>m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT > italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT or some agent j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT sends i𝑖iitalic\_i a status report saying that it received a message from j𝑗jitalic\_j in a round m′′>m′superscript𝑚′′superscript𝑚′m^{\prime\prime}>m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT > italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT; 6. 6. for some agents j𝑗jitalic\_j, j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, and j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT, j𝑗jitalic\_j sends i𝑖iitalic\_i a status report in round m𝑚mitalic\_m that says that j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and that j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT reported this, but j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT sends i𝑖iitalic\_i a status report in round m𝑚mitalic\_m that says that j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT did not crash before round m′′>m′superscript𝑚′′superscript𝑚′m^{\prime\prime}>m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT > italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT; 7. 7. for some agents j𝑗jitalic\_j, j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, and j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT, j𝑗jitalic\_j sends i𝑖iitalic\_i a status report in round m𝑚mitalic\_m that says that j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT did not crash by round m−1𝑚1m-1italic\_m - 1 and j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT crashed in some round m′<msuperscript𝑚′𝑚m^{\prime}<mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m, while j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT sends i𝑖iitalic\_i a status report in round m−1𝑚1m-1italic\_m - 1 saying that j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT crashed in round m′′<m′superscript𝑚′′superscript𝑚′m^{\prime\prime}<m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT < italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT (so either j𝑗jitalic\_j ignored the report about j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT sent by j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT or j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT lied to j𝑗jitalic\_j); 8. 8. more than f𝑓fitalic\_f crashes are detected by i𝑖iitalic\_i by round m𝑚mitalic\_m (i.e., f𝑓fitalic\_f or more agents have not sent messages to i𝑖iitalic\_i or were reported to crash in some round up to and including m𝑚mitalic\_m). If agent i𝑖iitalic\_i does not detect an inconsistency at some round m≤f+1𝑚𝑓1m\leq f+1italic\_m ≤ italic\_f + 1, i𝑖iitalic\_i proceeds as follows in round f+1𝑓1f+1italic\_f + 1. For each round 1≤m≤f+11𝑚𝑓11\leq m\leq f+11 ≤ italic\_m ≤ italic\_f + 1 in a run r𝑟ritalic\_r, agent i𝑖iitalic\_i computes NCm(r)𝑁subscript𝐶𝑚𝑟NC\_{m}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT ( italic\_r ), the set of agents that it believes did not crash up to and including round m𝑚mitalic\_m. Take NC0(r)=N𝑁subscript𝐶0𝑟𝑁NC\_{0}(r)=Nitalic\_N italic\_C start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT ( italic\_r ) = italic\_N (the set of all agents). Say that round m𝑚mitalic\_m in run r𝑟ritalic\_r seems clean if NCm−1(r)=NCm(r)𝑁subscript𝐶𝑚1𝑟𝑁subscript𝐶𝑚𝑟NC\_{m-1}(r)=NC\_{m}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m - 1 end\_POSTSUBSCRIPT ( italic\_r ) = italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT ( italic\_r ). As we show (Theorem [2](#Thmtheorem2 "Theorem 2. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")), if no inconsistency is detected in run r𝑟ritalic\_r, then there must be a round in r𝑟ritalic\_r that seems clean. Moreover, we show that if m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is the first round in r𝑟ritalic\_r that seems clean to a nonfaulty agent i𝑖iitalic\_i, then all the nonfaulty agents agree that m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is the first round that seems clean in r𝑟ritalic\_r, and they agree on the initial preference of all agents in NCm\(r)𝑁subscript𝐶superscript𝑚𝑟NC\_{m^{\}}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_r ), and the random numbers sent by these agents in round 1111 messages in run r𝑟ritalic\_r. The agents then use these random numbers to choose an agent j𝑗jitalic\_j among the agents in NCm\(r)𝑁subscript𝐶superscript𝑚𝑟NC\_{m^{\}}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_r ) and take vjsubscript𝑣𝑗v\_{j}italic\_v start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT to be the consensus value. The pseudocode for the strategy (protocol) σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT that implements this idea is given in Figure [1](#alg1 "Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). Lines [1](#alg1.l1 "1 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")–[14](#alg1.l14 "14 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") initialize the values of ST𝑆𝑇STitalic\_S italic\_T and SR1[j]𝑆superscript𝑅1delimited-[]𝑗SR^{1}[j]italic\_S italic\_R start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_j ], as well as the random numbers required in round 1111; that is, i𝑖iitalic\_i generates xi[t]subscript𝑥𝑖delimited-[]𝑡x\_{i}[t]italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT [ italic\_t ] and the corresponding polynomial qitsuperscriptsubscript𝑞𝑖𝑡q\_{i}^{t}italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT used for secret sharing for 0≤t≤f0𝑡𝑓0\leq t\leq f0 ≤ italic\_t ≤ italic\_f, and random vectors (zij1[1],…,zij1[n])superscriptsubscript𝑧𝑖𝑗1delimited-[]1…superscriptsubscript𝑧𝑖𝑗1delimited-[]𝑛(z\_{ij}^{1}[1],\ldots,z\_{ij}^{1}[n])( italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ 1 ] , … , italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_n ] ) for j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i, where zij1[l]∈{0,…,n−1}superscriptsubscript𝑧𝑖𝑗1delimited-[]𝑙0…𝑛1z\_{ij}^{1}[l]\in\{0,\ldots,n-1\}italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_l ] ∈ { 0 , … , italic\_n - 1 }. In phase 1111 (the “sending” phase) of round m𝑚mitalic\_m, i𝑖iitalic\_i sends SRim𝑆superscriptsubscript𝑅𝑖𝑚SR\_{i}^{m}italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT and zijmsuperscriptsubscript𝑧𝑖𝑗𝑚z\_{ij}^{m}italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT. If m=1𝑚1m=1italic\_m = 1, then i𝑖iitalic\_i also sends visubscript𝑣𝑖v\_{i}italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT and (yij0,…,yijf)superscriptsubscript𝑦𝑖𝑗0…superscriptsubscript𝑦𝑖𝑗𝑓(y\_{ij}^{0},\ldots,y\_{ij}^{f})( italic\_y start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT , … , italic\_y start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_f end\_POSTSUPERSCRIPT ) to j𝑗jitalic\_j, where yijt=qit(j)superscriptsubscript𝑦𝑖𝑗𝑡superscriptsubscript𝑞𝑖𝑡𝑗y\_{ij}^{t}=q\_{i}^{t}(j)italic\_y start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT = italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_j ); that is, yijtsuperscriptsubscript𝑦𝑖𝑗𝑡y\_{ij}^{t}italic\_y start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT is j𝑗jitalic\_j’s share of the secret xitsuperscriptsubscript𝑥𝑖𝑡x\_{i}^{t}italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT. Finally, if m=f+1𝑚𝑓1m=f+1italic\_m = italic\_f + 1, instead of sending zijmsuperscriptsubscript𝑧𝑖𝑗𝑚z\_{ij}^{m}italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT to j𝑗jitalic\_j, i𝑖iitalic\_i sends all the shares ylitsuperscriptsubscript𝑦𝑙𝑖𝑡y\_{li}^{t}italic\_y start\_POSTSUBSCRIPT italic\_l italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT it has received from other agents, so that all agents can compute the secret (lines [17](#alg1.l17 "17 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")-[22](#alg1.l22 "22 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")). In phase 2 (the “receive” phase) of round m𝑚mitalic\_m, i𝑖iitalic\_i processes all the messages received and keeps track of all agents who have crashed (lines [24](#alg1.l24 "24 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")-[40](#alg1.l40 "40 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")). If i𝑖iitalic\_i receives a round m𝑚mitalic\_m message from j𝑗jitalic\_j, then i𝑖iitalic\_i adds (j,vj)𝑗subscript𝑣𝑗(j,v\_{j})( italic\_j , italic\_v start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) to STi𝑆subscript𝑇𝑖ST\_{i}italic\_S italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT if m=1𝑚1m=1italic\_m = 1, includes in SRim[j]𝑆superscriptsubscript𝑅𝑖𝑚delimited-[]𝑗SR\_{i}^{m}[j]italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_j ] the vector z𝑧zitalic\_z sent by j𝑗jitalic\_j to i𝑖iitalic\_i, and updates the status report SRim[l]𝑆superscriptsubscript𝑅𝑖𝑚delimited-[]𝑙SR\_{i}^{m}[l]italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_l ] of each agent l𝑙litalic\_l. Specifically, if j𝑗jitalic\_j reports that j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT crashed in a round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and i𝑖iitalic\_i earlier considered it possible that j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT was still nonfaulty at round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, then i𝑖iitalic\_i includes in SRim[l]𝑆superscriptsubscript𝑅𝑖𝑚delimited-[]𝑙SR\_{i}^{m}[l]italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_l ] the fact that j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT crashed and that j𝑗jitalic\_j is an agent that reported this fact (lines [31](#alg1.l31 "31 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")-[35](#alg1.l35 "35 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")); if i𝑖iitalic\_i does not receive a round m𝑚mitalic\_m message from j𝑗jitalic\_j and i𝑖iitalic\_i believed that j𝑗jitalic\_j did not crash before, then i𝑖iitalic\_i marks j𝑗jitalic\_j as crashed (line [37](#alg1.l37 "37 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")). In phase 3 (the “update” phase) of round m≤f𝑚𝑓m\leq fitalic\_m ≤ italic\_f, i𝑖iitalic\_i generates the random value zijm+1[i]subscriptsuperscript𝑧𝑚1𝑖𝑗delimited-[]𝑖z^{m+1}\_{ij}[i]italic\_z start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT [ italic\_i ] for the next round. If i𝑖iitalic\_i detects an inconsistency, then i𝑖iitalic\_i decides ⊥bottom\bot⊥ (line [44](#alg1.l44 "44 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")); if no inconsistency is detected by the end of round f+1𝑓1f+1italic\_f + 1, then i𝑖iitalic\_i decides on a value (lines [49](#alg1.l49 "49 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")-[59](#alg1.l59 "59 ‣ Algorithm 1 ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) by computing the set NCm′𝑁subscript𝐶superscript𝑚′NC\_{m^{\prime}}italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT for every round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, determining the earliest round m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT that seems clean (NCm\=NCm\−1𝑁subscript𝐶superscript𝑚𝑁subscript𝐶superscript𝑚1NC\_{m^{\}}=NC\_{m^{\}-1}italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT = italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT - 1 end\_POSTSUBSCRIPT), computing a random number S∈{0,…,n−t−1}𝑆0…𝑛𝑡1S\in\{0,\ldots,n-t-1\}italic\_S ∈ { 0 , … , italic\_n - italic\_t - 1 }, where t𝑡titalic\_t is the number of crashes that occurred before m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, by summing the random numbers xj[t]subscript𝑥𝑗delimited-[]𝑡x\_{j}[t]italic\_x start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT [ italic\_t ] of j∈NCm\𝑗𝑁subscript𝐶superscript𝑚j\in NC\_{m^{\}}italic\_j ∈ italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT (computed by interpolating the polynomials), and deciding on the value of the agent in NCm\𝑁subscript𝐶superscript𝑚NC\_{m^{\}}italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT with the (S+1)𝑆1(S+1)( italic\_S + 1 )st highest id. Algorithm 1 σi𝑐𝑜𝑛𝑠(vi)superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠subscript𝑣𝑖\sigma\_{i}^{\mathit{cons}}(v\_{i})italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ( italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ): i𝑖iitalic\_i’s consensus protocol with initial value visubscript𝑣𝑖v\_{i}italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT 1:decided ←←\leftarrow← 𝑓𝑎𝑙𝑠𝑒𝑓𝑎𝑙𝑠𝑒\mathit{false}italic\_false 2:STi𝑆subscript𝑇𝑖ST\_{i}italic\_S italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ←←\leftarrow← {(i,vi)}𝑖subscript𝑣𝑖\{(i,v\_{i})\}{ ( italic\_i , italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) } 3:z𝑧zitalic\_z ←←\leftarrow← random in {0…n−1}0…𝑛1\{0\,\ldots\,n-1\}{ 0 … italic\_n - 1 } 4:for all j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i do 5: SRi1[j]𝑆superscriptsubscript𝑅𝑖1delimited-[]𝑗SR\_{i}^{1}[j]italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_j ] ←←\leftarrow← (∞,⊥)bottom(\infty,\bot)( ∞ , ⊥ )▷▷\triangleright▷ All agents are initially active 6: for all l≠j,i𝑙𝑗𝑖l\neq j,iitalic\_l ≠ italic\_j , italic\_i do 7: zij1[l]superscriptsubscript𝑧𝑖𝑗1delimited-[]𝑙z\_{ij}^{1}[l]italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_l ] ←←\leftarrow← 00 8: zij1[i]subscriptsuperscript𝑧1𝑖𝑗delimited-[]𝑖z^{1}\_{ij}[i]italic\_z start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT [ italic\_i ] ←←\leftarrow← random in {0…n−1}0…𝑛1\{0\,\ldots\,n-1\}{ 0 … italic\_n - 1 }▷▷\triangleright▷ Random number to be used by j𝑗jitalic\_j in round 2222 9: zij1[j]subscriptsuperscript𝑧1𝑖𝑗delimited-[]𝑗z^{1}\_{ij}[j]italic\_z start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT [ italic\_j ] ←←\leftarrow← z𝑧zitalic\_z▷▷\triangleright▷ Proves that i𝑖iitalic\_i sends round 1 message to j𝑗jitalic\_j 10:for all 0≤t≤f0𝑡𝑓0\leq t\leq f0 ≤ italic\_t ≤ italic\_f do 11: xi[t]subscript𝑥𝑖delimited-[]𝑡x\_{i}[t]italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT [ italic\_t ] ←←\leftarrow← random in {0,…,n−t−1}0…𝑛𝑡1\{0,\ldots,n-t-1\}{ 0 , … , italic\_n - italic\_t - 1 }▷▷\triangleright▷ A random number for each possible value of t𝑡titalic\_t 12: qitsuperscriptsubscript𝑞𝑖𝑡q\_{i}^{t}italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ←←\leftarrow← random polynomial of degree 1111 with qit(0)=xi[t]superscriptsubscript𝑞𝑖𝑡0subscript𝑥𝑖delimited-[]𝑡q\_{i}^{t}(0)=x\_{i}[t]italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( 0 ) = italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT [ italic\_t ] 13: for all j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i do 14: yij[t]subscript𝑦𝑖𝑗delimited-[]𝑡y\_{ij}[t]italic\_y start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT [ italic\_t ] ←←\leftarrow← qit(j)superscriptsubscript𝑞𝑖𝑡𝑗q\_{i}^{t}(j)italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_j ) 15: 16:for all round 1≤m≤f+11𝑚𝑓11\leq m\leq f+11 ≤ italic\_m ≤ italic\_f + 1 such that ¬decideddecided\neg\mbox{decided}¬ decided do 17: Phase 1: send phase 18: for all j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i do 19: if m=1𝑚1m=1italic\_m = 1 then Send ⟨vi,SRim,(yij0,…,yijf),zijm⟩subscript𝑣𝑖𝑆superscriptsubscript𝑅𝑖𝑚superscriptsubscript𝑦𝑖𝑗0…superscriptsubscript𝑦𝑖𝑗𝑓subscriptsuperscript𝑧𝑚𝑖𝑗\langle v\_{i},SR\_{i}^{m},(y\_{ij}^{0},\ldots,y\_{ij}^{f}),z^{m}\_{ij}\rangle⟨ italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT , ( italic\_y start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT , … , italic\_y start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_f end\_POSTSUPERSCRIPT ) , italic\_z start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT ⟩ to j𝑗jitalic\_j 20: if 2≤m≤f2𝑚𝑓2\leq m\leq f2 ≤ italic\_m ≤ italic\_f then Send ⟨SRim,zijm⟩𝑆superscriptsubscript𝑅𝑖𝑚subscriptsuperscript𝑧𝑚𝑖𝑗\langle SR\_{i}^{m},z^{m}\_{ij}\rangle⟨ italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT , italic\_z start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT ⟩ to j𝑗jitalic\_j 21: if m=f+1𝑚𝑓1m=f+1italic\_m = italic\_f + 1 then Send ⟨SRim,(yli0,…,ylif)l≠j⟩𝑆superscriptsubscript𝑅𝑖𝑚subscriptsubscriptsuperscript𝑦0𝑙𝑖…subscriptsuperscript𝑦𝑓𝑙𝑖𝑙𝑗\langle SR\_{i}^{m},(y^{0}\_{li},\ldots,y^{f}\_{li})\_{l\neq j}\rangle⟨ italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT , ( italic\_y start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l italic\_i end\_POSTSUBSCRIPT , … , italic\_y start\_POSTSUPERSCRIPT italic\_f end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l italic\_i end\_POSTSUBSCRIPT ) start\_POSTSUBSCRIPT italic\_l ≠ italic\_j end\_POSTSUBSCRIPT ⟩ to j𝑗jitalic\_j 22: EndPhase 23: 24: Phase 2: receive phase 25: SRim+1𝑆superscriptsubscript𝑅𝑖𝑚1SR\_{i}^{m+1}italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT ←←\leftarrow← SRim𝑆superscriptsubscript𝑅𝑖𝑚SR\_{i}^{m}italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT 26: for all j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i do 27: if receive valid message from j𝑗jitalic\_j then 28: if m=1𝑚1m=1italic\_m = 1 then STi𝑆subscript𝑇𝑖ST\_{i}italic\_S italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ←←\leftarrow← STi∪{(j,vj)}𝑆subscript𝑇𝑖𝑗subscript𝑣𝑗ST\_{i}\cup\{(j,v\_{j})\}italic\_S italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ∪ { ( italic\_j , italic\_v start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) }▷▷\triangleright▷ STi𝑆subscript𝑇𝑖ST\_{i}italic\_S italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT contains all the values that i𝑖iitalic\_i has seen 29: SRim+1[j]←(∞,zjim)←𝑆superscriptsubscript𝑅𝑖𝑚1delimited-[]𝑗subscriptsuperscript𝑧𝑚𝑗𝑖SR\_{i}^{m+1}[j]\leftarrow(\infty,z^{m}\_{ji})italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT [ italic\_j ] ← ( ∞ , italic\_z start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT )▷▷\triangleright▷ Note that j𝑗jitalic\_j is still active 30: for all l≠i,j𝑙𝑖𝑗l\neq i,jitalic\_l ≠ italic\_i , italic\_j do 31: if SRjm[l]=(m′,j′)𝑆superscriptsubscript𝑅𝑗𝑚delimited-[]𝑙superscript𝑚′superscript𝑗′SR\_{j}^{m}[l]=(m^{\prime},j^{\prime})italic\_S italic\_R start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_l ] = ( italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) and SRim[l]=(m′′,j′′)𝑆superscriptsubscript𝑅𝑖𝑚delimited-[]𝑙superscript𝑚′′superscript𝑗′′SR\_{i}^{m}[l]=(m^{\prime\prime},j^{\prime\prime})italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_l ] = ( italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT , italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT ) and m′<m′′superscript𝑚′superscript𝑚′′m^{\prime}<m^{\prime\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT then 32: SRim+1[l]←(m′,j)←𝑆superscriptsubscript𝑅𝑖𝑚1delimited-[]𝑙superscript𝑚′𝑗SR\_{i}^{m+1}[l]\leftarrow(m^{\prime},j)italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT [ italic\_l ] ← ( italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_j ) ▷▷\triangleright▷ l𝑙litalic\_l crashed earlier than previously thought 33: zilm+1[j]←⊥←superscriptsubscript𝑧𝑖𝑙𝑚1delimited-[]𝑗bottomz\_{il}^{m+1}[j]\leftarrow\botitalic\_z start\_POSTSUBSCRIPT italic\_i italic\_l end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT [ italic\_j ] ← ⊥ 34: else if SRjm[l]=(∞,zjm)𝑆superscriptsubscript𝑅𝑗𝑚delimited-[]𝑙superscriptsubscript𝑧𝑗𝑚SR\_{j}^{m}[l]=(\infty,z\_{j}^{m})italic\_S italic\_R start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_l ] = ( ∞ , italic\_z start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT ) then 35: zilm+1[j]=zjim[j]superscriptsubscript𝑧𝑖𝑙𝑚1delimited-[]𝑗superscriptsubscript𝑧𝑗𝑖𝑚delimited-[]𝑗z\_{il}^{m+1}[j]=z\_{ji}^{m}[j]italic\_z start\_POSTSUBSCRIPT italic\_i italic\_l end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT [ italic\_j ] = italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_j ] 36: else if SRim+1[j]=(∞,z′)𝑆superscriptsubscript𝑅𝑖𝑚1delimited-[]𝑗superscript𝑧′SR\_{i}^{m+1}[j]=(\infty,z^{\prime})italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT [ italic\_j ] = ( ∞ , italic\_z start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) for some z′superscript𝑧′z^{\prime}italic\_z start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT then 37: SRim+1[j]←(m,i)←𝑆superscriptsubscript𝑅𝑖𝑚1delimited-[]𝑗𝑚𝑖SR\_{i}^{m+1}[j]\leftarrow(m,i)italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT [ italic\_j ] ← ( italic\_m , italic\_i )▷▷\triangleright▷ i𝑖iitalic\_i detects a crash of j𝑗jitalic\_j 38: for all l≠i𝑙𝑖l\neq iitalic\_l ≠ italic\_i do 39: zilm+1[j]←⊥←superscriptsubscript𝑧𝑖𝑙𝑚1delimited-[]𝑗bottomz\_{il}^{m+1}[j]\leftarrow\botitalic\_z start\_POSTSUBSCRIPT italic\_i italic\_l end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT [ italic\_j ] ← ⊥ 40: EndPhase 41: 42: Phase 3: update phase 43: if an inconsistency is detected then 44: Decide(⊥bottom\bot⊥)▷▷\triangleright▷ Punishment 45: decided ←𝑡𝑟𝑢𝑒←absent𝑡𝑟𝑢𝑒\leftarrow\mathit{true}← italic\_true 46: else if m≤f𝑚𝑓m\leq fitalic\_m ≤ italic\_f then 47: for all j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i do 48: zijm+1[i]subscriptsuperscript𝑧𝑚1𝑖𝑗delimited-[]𝑖z^{m+1}\_{ij}[i]italic\_z start\_POSTSUPERSCRIPT italic\_m + 1 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT [ italic\_i ] ←←\leftarrow← random in {0,…,n−1}0…𝑛1\{0,\ldots,n-1\}{ 0 , … , italic\_n - 1 } 49: else if decided = 𝑓𝑎𝑙𝑠𝑒𝑓𝑎𝑙𝑠𝑒\mathit{false}italic\_false then 50: NC0=N𝑁subscript𝐶0𝑁NC\_{0}=Nitalic\_N italic\_C start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT = italic\_N 51: for all 1≤m′≤f+11superscript𝑚′𝑓11\leq m^{\prime}\leq f+11 ≤ italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≤ italic\_f + 1 do 52: NCm′𝑁subscript𝐶superscript𝑚′NC\_{m^{\prime}}italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ←←\leftarrow← {j∈N−{i}∣∀m′′≤m′,l(SRif+2[j]≠(m′′,l))∪{i}\{j\in N-\{i\}\mid\forall m^{\prime\prime}\leq m^{\prime},l(SR\_{i}^{f+2}[j]\neq(m^{\prime\prime},l))\cup\{i\}{ italic\_j ∈ italic\_N - { italic\_i } ∣ ∀ italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT ≤ italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_l ( italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_f + 2 end\_POSTSUPERSCRIPT [ italic\_j ] ≠ ( italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT , italic\_l ) ) ∪ { italic\_i }▷▷\triangleright▷ Agents that did not crash up to round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT 53: m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ←←\leftarrow← first round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT such that NCm′=NCm′−1𝑁subscript𝐶superscript𝑚′𝑁subscript𝐶superscript𝑚′1NC\_{m^{\prime}}=NC\_{m^{\prime}-1}italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT = italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT - 1 end\_POSTSUBSCRIPT▷▷\triangleright▷ First round that seems clean 54: t𝑡titalic\_t ←←\leftarrow← n−\|NCm\\|𝑛𝑁subscript𝐶superscript𝑚n-\|NC\_{m^{\}}\|italic\_n - \| italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT \|▷▷\triangleright▷ Number of crashes prior to m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT 55: for all j∈NCm\𝑗𝑁subscript𝐶superscript𝑚j\in NC\_{m^{\}}italic\_j ∈ italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT do 56: qjtsuperscriptsubscript𝑞𝑗𝑡q\_{j}^{t}italic\_q start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ←←\leftarrow← unique polynomial interpolating the values yjltsuperscriptsubscript𝑦𝑗𝑙𝑡y\_{jl}^{t}italic\_y start\_POSTSUBSCRIPT italic\_j italic\_l end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT received▷▷\triangleright▷ otherwise, an inconsistency was detected 57: xj[t]←qjt(0)←subscript𝑥𝑗delimited-[]𝑡superscriptsubscript𝑞𝑗𝑡0x\_{j}[t]\leftarrow q\_{j}^{t}(0)italic\_x start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT [ italic\_t ] ← italic\_q start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( 0 ) 58: S←∑j∈NCm\xj[t] mod (n−t)←𝑆subscript𝑗𝑁subscript𝐶superscript𝑚subscript𝑥𝑗delimited-[]𝑡 mod 𝑛𝑡S\leftarrow\sum\_{j\in NC\_{m^{\}}}x\_{j}[t]\mbox{ mod }(n-t)italic\_S ← ∑ start\_POSTSUBSCRIPT italic\_j ∈ italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT italic\_x start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT [ italic\_t ] mod ( italic\_n - italic\_t )▷▷\triangleright▷ Calculate a random number in 0,…,n−t−10…𝑛𝑡10,\ldots,n-t-10 , … , italic\_n - italic\_t - 1 59: Decide(vjsubscript𝑣𝑗v\_{j}italic\_v start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT), where j𝑗jitalic\_j is the (S+1)𝑆1(S+1)( italic\_S + 1 )st highest id in NCm\𝑁subscript𝐶superscript𝑚NC\_{m^{\}}italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT 60: EndPhase We now prove that σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT gives a π𝜋\piitalic\_π-Nash equilibrium, under reasonable assumptions about π𝜋\piitalic\_π. We first prove that the protocol satisfies all the properties of fair consensus without making any assumptions about π𝜋\piitalic\_π. ###### Theorem 2. σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT solves fair consensus if at most f𝑓fitalic\_f agents crash, f+1<n𝑓1𝑛f+1<nitalic\_f + 1 < italic\_n, and all the remaining agents follow the protocol. ###### Proof. Consider a run r𝑟ritalic\_r where all agents follow σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT and at most f𝑓fitalic\_f agents crash. It is easy to see that no inconsistency is detected in r𝑟ritalic\_r. Since an agent crashes in at most one round and there are at most f𝑓fitalic\_f faulty agents, there must exist a round 1≤m≤f+11𝑚𝑓11\leq m\leq f+11 ≤ italic\_m ≤ italic\_f + 1 when no agent crashes. Let m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT be the first such round. We prove that for all nonfaulty agents i𝑖iitalic\_i and j𝑗jitalic\_j, NCmi(r)=NCmj(r)𝑁superscriptsubscript𝐶𝑚𝑖𝑟𝑁superscriptsubscript𝐶𝑚𝑗𝑟NC\_{m}^{i}(r)=NC\_{m}^{j}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_i end\_POSTSUPERSCRIPT ( italic\_r ) = italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_j end\_POSTSUPERSCRIPT ( italic\_r ) for all m≤m\𝑚superscript𝑚m\leq m^{\}italic\_m ≤ italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT (where NCmi(r)𝑁superscriptsubscript𝐶𝑚𝑖𝑟NC\_{m}^{i}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_i end\_POSTSUPERSCRIPT ( italic\_r ) denotes i𝑖iitalic\_i’s version of NCm(r)𝑁subscript𝐶𝑚𝑟NC\_{m}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT ( italic\_r ) in run r𝑟ritalic\_r, and similarly for j𝑗jitalic\_j). To see this, fix two nonfaulty agents i𝑖iitalic\_i and j𝑗jitalic\_j. Agent i𝑖iitalic\_i adds agent l𝑙litalic\_l to NCmi(r)𝑁superscriptsubscript𝐶𝑚𝑖𝑟NC\_{m}^{i}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_i end\_POSTSUPERSCRIPT ( italic\_r ) iff i𝑖iitalic\_i receives a message from l𝑙litalic\_l in every round m′≤msuperscript𝑚′𝑚m^{\prime}\leq mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≤ italic\_m of run r𝑟ritalic\_r, and i𝑖iitalic\_i receives no status report indicating that l𝑙litalic\_l crashed in some round m′≤msuperscript𝑚′𝑚m^{\prime}\leq mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≤ italic\_m. If m<m\𝑚superscript𝑚m<m^{\}italic\_m < italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, then it must be the case that j𝑗jitalic\_j also received a message from l𝑙litalic\_l in every round m′<msuperscript𝑚′𝑚m^{\prime}<mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m of r𝑟ritalic\_r and neither received nor sent a status report indicating that l𝑙litalic\_l crashed in a round m′≤msuperscript𝑚′𝑚m^{\prime}\leq mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≤ italic\_m; otherwise j𝑗jitalic\_j would have learned about this crash by round m𝑚mitalic\_m and would have told i𝑖iitalic\_i by round f+1𝑓1f+1italic\_f + 1 that l𝑙litalic\_l was faulty (since j𝑗jitalic\_j is nonfaulty). Thus, l∈NCmj(r)𝑙𝑁superscriptsubscript𝐶𝑚𝑗𝑟l\in NC\_{m}^{j}(r)italic\_l ∈ italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_j end\_POSTSUPERSCRIPT ( italic\_r ). If m=m\𝑚superscript𝑚m=m^{\}italic\_m = italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, then l𝑙litalic\_l sends a round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT message to all agents for all m′<m\superscript𝑚′superscript𝑚m^{\prime}<m^{\}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT; and since no agents fail in round m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, by assumption, we again have l∈NCmj(r)𝑙𝑁superscriptsubscript𝐶𝑚𝑗𝑟l\in NC\_{m}^{j}(r)italic\_l ∈ italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_j end\_POSTSUPERSCRIPT ( italic\_r ). Thus, NCmi(r)⊆NCmj(r)𝑁superscriptsubscript𝐶𝑚𝑖𝑟𝑁superscriptsubscript𝐶𝑚𝑗𝑟NC\_{m}^{i}(r)\subseteq NC\_{m}^{j}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_i end\_POSTSUPERSCRIPT ( italic\_r ) ⊆ italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_j end\_POSTSUPERSCRIPT ( italic\_r ); similar arguments give the opposite inclusion. Note that since no agent crashes in round m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, it is easy to see that we must have NCm\i(r)=NCm\−1i(r)𝑁subscriptsuperscript𝐶𝑖superscript𝑚𝑟𝑁subscriptsuperscript𝐶𝑖superscript𝑚1𝑟NC^{i}\_{m^{\}}(r)=NC^{i}\_{m^{\}-1}(r)italic\_N italic\_C start\_POSTSUPERSCRIPT italic\_i end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_r ) = italic\_N italic\_C start\_POSTSUPERSCRIPT italic\_i end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT - 1 end\_POSTSUBSCRIPT ( italic\_r ) for all nonfaulty agents i𝑖iitalic\_i, so round m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT seems clean. With these observations, we can now prove that σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT satisfies each requirement of Fair Consensus in r𝑟ritalic\_r. Validity: Since no inconsistency is detected, every agent i𝑖iitalic\_i decides a value different from ⊥bottom\bot⊥ in r𝑟ritalic\_r. Agent i𝑖iitalic\_i always finds some round m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT that seems clean, computes a nonempty set NCm\(r)𝑁subscript𝐶superscript𝑚𝑟NC\_{m^{\}}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_r ), which includes at least i𝑖iitalic\_i, and knows the random numbers sent by these agents in round m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT. Since STi𝑆subscript𝑇𝑖ST\_{i}italic\_S italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT contains only initial preferences, i𝑖iitalic\_i decides the initial preference of some agent in NCm\(r)𝑁subscript𝐶superscript𝑚𝑟NC\_{m^{\}}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_r ). Termination and Integrity: Every agent either crashes before deciding or decides exactly once at the end of round f+1𝑓1f+1italic\_f + 1. Agreement: We have shown that all nonfaulty agents i𝑖iitalic\_i and j𝑗jitalic\_j agree on NCm(r)𝑁subscript𝐶𝑚𝑟NC\_{m}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT ( italic\_r ) for all m≤m\𝑚superscript𝑚m\leq m^{\}italic\_m ≤ italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT. We thus omit the superscripts i𝑖iitalic\_i and j𝑗jitalic\_j on NCm(r)𝑁subscript𝐶𝑚𝑟NC\_{m}(r)italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT ( italic\_r ) from here on in. Given this, they agree on whether each round m≤m\𝑚superscript𝑚m\leq m^{\}italic\_m ≤ italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT seems clean and thus agree that some m¯≤m\¯𝑚superscript𝑚\overline{m}\leq m^{\}over¯ start\_ARG italic\_m end\_ARG ≤ italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is the first round that seems clean in r𝑟ritalic\_r. Moreover, i𝑖iitalic\_i and j𝑗jitalic\_j receive identical round 1111 messages from the agents in NCm¯(r)𝑁subscript𝐶¯𝑚𝑟NC\_{\overline{m}}(r)italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT ( italic\_r ). It follows that i𝑖iitalic\_i adds a tuple (l,vl)𝑙subscript𝑣𝑙(l,v\_{l})( italic\_l , italic\_v start\_POSTSUBSCRIPT italic\_l end\_POSTSUBSCRIPT ) to STi𝑆subscript𝑇𝑖ST\_{i}italic\_S italic\_T start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for l∈NCm¯(r)𝑙𝑁subscript𝐶¯𝑚𝑟l\in NC\_{\overline{m}}(r)italic\_l ∈ italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT ( italic\_r ) iff j𝑗jitalic\_j adds that tuple to STj𝑆subscript𝑇𝑗ST\_{j}italic\_S italic\_T start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT. Suppose that \|NCm¯(r)\|=n−t𝑁subscript𝐶¯𝑚𝑟𝑛𝑡\|NC\_{\overline{m}}(r)\|=n-t\| italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT ( italic\_r ) \| = italic\_n - italic\_t. Since NCm¯(r)𝑁subscript𝐶¯𝑚𝑟NC\_{\overline{m}}(r)italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT ( italic\_r ) must include all the nonfaulty agents, we must have t≤f𝑡𝑓t\leq fitalic\_t ≤ italic\_f. Clearly, if l∈NCm¯(r)𝑙𝑁subscript𝐶¯𝑚𝑟l\in NC\_{\overline{m}}(r)italic\_l ∈ italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT ( italic\_r ), then i𝑖iitalic\_i and j𝑗jitalic\_j must receive the values ylitsuperscriptsubscript𝑦𝑙𝑖𝑡y\_{li}^{t}italic\_y start\_POSTSUBSCRIPT italic\_l italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT and yljtsuperscriptsubscript𝑦𝑙𝑗𝑡y\_{lj}^{t}italic\_y start\_POSTSUBSCRIPT italic\_l italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT in round 1111 messages sent by l𝑙litalic\_l. Agents i𝑖iitalic\_i and j𝑗jitalic\_j also receive yll′tsuperscriptsubscript𝑦𝑙superscript𝑙′𝑡y\_{ll^{\prime}}^{t}italic\_y start\_POSTSUBSCRIPT italic\_l italic\_l start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT from each nonfaulty agent l′superscript𝑙′l^{\prime}italic\_l start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. Since there are at least n−f≥2𝑛𝑓2n-f\geq 2italic\_n - italic\_f ≥ 2 nonfaulty agents, and l𝑙litalic\_l follows σl𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑙𝑐𝑜𝑛𝑠\sigma\_{l}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_l end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT, i𝑖iitalic\_i and j𝑗jitalic\_j will be able to interpolate the polynomial qltsuperscriptsubscript𝑞𝑙𝑡q\_{l}^{t}italic\_q start\_POSTSUBSCRIPT italic\_l end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT, and compute xl[t]=qlt(0)subscript𝑥𝑙delimited-[]𝑡superscriptsubscript𝑞𝑙𝑡0x\_{l}[t]=q\_{l}^{t}(0)italic\_x start\_POSTSUBSCRIPT italic\_l end\_POSTSUBSCRIPT [ italic\_t ] = italic\_q start\_POSTSUBSCRIPT italic\_l end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( 0 ). Consequently, i𝑖iitalic\_i and j𝑗jitalic\_j agree on the information relevant to the consensus decision, so must decide on the same value. Fairness: The probability of the initial preference of each agent in NCm¯(r)𝑁subscript𝐶¯𝑚𝑟NC\_{\overline{m}}(r)italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT ( italic\_r ) being decided is 1/\|NCm¯(r)\|1𝑁subscript𝐶¯𝑚𝑟1/\|NC\_{\overline{m}}(r)\|1 / \| italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT ( italic\_r ) \|. Since \|NCm¯(r)\|≤n𝑁subscript𝐶¯𝑚𝑟𝑛\|NC\_{\overline{m}}(r)\|\leq n\| italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT ( italic\_r ) \| ≤ italic\_n, if c𝑐citalic\_c nonfaulty agents in NCm¯𝑁subscript𝐶¯𝑚NC\_{\overline{m}}italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT initially have preference v𝑣vitalic\_v, then the probability of v𝑣vitalic\_v being decided is at least c/\|NCm¯(r)\|≥c/n𝑐𝑁subscript𝐶¯𝑚𝑟𝑐𝑛c/\|NC\_{\overline{m}(r)}\|\geq c/nitalic\_c / \| italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG ( italic\_r ) end\_POSTSUBSCRIPT \| ≥ italic\_c / italic\_n. Since NCm¯(r)𝑁subscript𝐶¯𝑚𝑟NC\_{\overline{m}}(r)italic\_N italic\_C start\_POSTSUBSCRIPT over¯ start\_ARG italic\_m end\_ARG end\_POSTSUBSCRIPT ( italic\_r ) contains all the nonfaulty agents, Fairness holds. ∎ It remains to show that σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT is a π𝜋\piitalic\_π-Nash equilibrium. We show that σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT is a π𝜋\piitalic\_π-Nash equilibrium under appropriate assumptions about π𝜋\piitalic\_π. Specifically, we assume that π𝜋\piitalic\_π supports reachability and is uniform, notions that we now define. The reachability assumption has three parts. The first two parts consider how likely it is that some information that an agent j𝑗jitalic\_j has will reach an agent that will decide on a value; the third part is quite similar, and considers how likely it is that a nonfaulty agent becomes aware that an agent j𝑗jitalic\_j failed in round m𝑚mitalic\_m. Of course, the answer to these questions depends in part on whether agents are supposed to send messages in every round (as is the case with σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT). In the formal definition, we implicitly assume that this is the case. (So, effectively, the reachability assumption is appropriate only for protocols where agents send messages in every round.) Given agents i𝑖iitalic\_i and j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i, a round-m𝑚mitalic\_m information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for i𝑖iitalic\_i, a failure pattern F𝐹Fitalic\_F compatible with Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, in that ℛ(F)∩ℛ(Ii)≠∅ℛ𝐹ℛsubscript𝐼𝑖\mathcal{R}(F)\cap\mathcal{R}(I\_{i})\neq\emptysetcaligraphic\_R ( italic\_F ) ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ≠ ∅, and m′≥msuperscript𝑚′𝑚m^{\prime}\geq mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≥ italic\_m, say that a nonfaulty agent l≠i𝑙𝑖l\neq iitalic\_l ≠ italic\_i is reachable from j𝑗jitalic\_j without i𝑖iitalic\_i between rounds m′superscript𝑚normal-′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and f+1𝑓1f+1italic\_f + 1 given F𝐹Fitalic\_F if there is a sequence jm′,…,jf+1subscript𝑗superscript𝑚′…subscript𝑗𝑓1j\_{m^{\prime}},\ldots,j\_{f+1}italic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT , … , italic\_j start\_POSTSUBSCRIPT italic\_f + 1 end\_POSTSUBSCRIPT of agents different from i𝑖iitalic\_i such that j=jm′𝑗subscript𝑗superscript𝑚′j=j\_{m^{\prime}}italic\_j = italic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT, for m′′={m′,…,f}superscript𝑚′′superscript𝑚′…𝑓m^{\prime\prime}=\{m^{\prime},\ldots,f\}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT = { italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , … , italic\_f }, jm′′subscript𝑗superscript𝑚′′j\_{m^{\prime\prime}}italic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT has not failed prior to round m′′superscript𝑚′′m^{\prime\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT according to F𝐹Fitalic\_F, and either does not fail in round m′′superscript𝑚′′m^{\prime\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT or, if m′′<f+1superscript𝑚′′𝑓1m^{\prime\prime}<f+1italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT < italic\_f + 1, jm′′subscript𝑗superscript𝑚′′j\_{m^{\prime\prime}}italic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT fails in round m′′superscript𝑚′′m^{\prime\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT but sends a message to jm′′+1subscript𝑗superscript𝑚′′1j\_{m^{\prime\prime}+1}italic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT + 1 end\_POSTSUBSCRIPT before failing (i.e., if (jm′′,m′′,A)∈Fsubscript𝑗superscript𝑚′′superscript𝑚′′𝐴𝐹(j\_{m^{\prime\prime}},m^{\prime\prime},A)\in F( italic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT , italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT , italic\_A ) ∈ italic\_F, then jm′′+1∈Asubscript𝑗superscript𝑚′′1𝐴j\_{m^{\prime\prime}+1}\in Aitalic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT + 1 end\_POSTSUBSCRIPT ∈ italic\_A), and l=jf+1𝑙subscript𝑗𝑓1l=j\_{f+1}italic\_l = italic\_j start\_POSTSUBSCRIPT italic\_f + 1 end\_POSTSUBSCRIPT. Note that if j𝑗jitalic\_j is nonfaulty according to F𝐹Fitalic\_F, then a nonfaulty agent is certainly reachable from j𝑗jitalic\_j without i𝑖iitalic\_i between rounds m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and f+1𝑓1f+1italic\_f + 1; just take jm′=⋯=jf+1=jsubscript𝑗superscript𝑚′⋯subscript𝑗𝑓1𝑗j\_{m^{\prime}}=\cdots=j\_{f+1}=jitalic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT = ⋯ = italic\_j start\_POSTSUBSCRIPT italic\_f + 1 end\_POSTSUBSCRIPT = italic\_j. But even if j𝑗jitalic\_j fails in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT according to F𝐹Fitalic\_F, as long j𝑗jitalic\_j can send a message to a nonfaulty agent other than i𝑖iitalic\_i, or there is an appropriate chain of agents, then a nonfaulty agent is reachable from j𝑗jitalic\_j without i𝑖iitalic\_i by round f+1𝑓1f+1italic\_f + 1. The probability of there being a failure pattern for which a nonfaulty agent is reachable from j𝑗jitalic\_j without i𝑖iitalic\_i depends in part on how many agents are known to have failed in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT; the more agents are known not to have failed, the more likely we would expect a nonfaulty agent to be reachable from j𝑗jitalic\_j without i𝑖iitalic\_i. We also want this condition to hold even conditional on a set of failure patterns, provided that the set of failure patterns does not favor particular agents failing. To make this precise, we need a few more definitions. Say that an agent j𝑗jitalic\_j is known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT if j𝑗jitalic\_j is faulty in all runs in ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ); thus, j𝑗jitalic\_j is known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT if j𝑗jitalic\_j did not send a message to i𝑖iitalic\_i at round m−1𝑚1m-1italic\_m - 1 according to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Say that a set ℱℱ\mathcal{F}caligraphic\_F of failure patterns satisfies the permutation assumption with respect to a set F𝐹Fitalic\_F of failures and an information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT if, for all permutations g𝑔gitalic\_g of the agents that keep fixed the agents that fail in F𝐹Fitalic\_F or are known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, if F′∈ℱsuperscript𝐹′ℱF^{\prime}\in\mathcal{F}italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ∈ caligraphic\_F, then so is g(F′)𝑔superscript𝐹′g(F^{\prime})italic\_g ( italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ), where g(F′)𝑔superscript𝐹′g(F^{\prime})italic\_g ( italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) is the failure pattern that results by replacing each triple (j,m′′,A)∈F′𝑗superscript𝑚′′𝐴superscript𝐹′(j,m^{\prime\prime},A)\in F^{\prime}( italic\_j , italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT , italic\_A ) ∈ italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT by (g(j),m′′,g(A))𝑔𝑗superscript𝑚′′𝑔𝐴(g(j),m^{\prime\prime},g(A))( italic\_g ( italic\_j ) , italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT , italic\_g ( italic\_A ) ). ℱℱ\mathcal{F}caligraphic\_F satisfies the permutation assumption with respect to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT if ℱℱ\mathcal{F}caligraphic\_F satisfies it with respect to the empty set of failures and Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Let ℛ(ℱ)=∪F∈ℱℛ(F)ℛℱsubscript𝐹ℱℛ𝐹\mathcal{R}(\mathcal{F})=\cup\_{F\in\mathcal{F}}\mathcal{R}(F)caligraphic\_R ( caligraphic\_F ) = ∪ start\_POSTSUBSCRIPT italic\_F ∈ caligraphic\_F end\_POSTSUBSCRIPT caligraphic\_R ( italic\_F ). We say that π𝜋\piitalic\_π supports reachability if for all agents i𝑖iitalic\_i, all time-m𝑚mitalic\_m information sets Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT such that M𝑀Mitalic\_M agents are not known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, failure pattern F𝐹Fitalic\_F, and all sets ℱℱ\mathcal{F}caligraphic\_F of failure patterns that satisfy the permutation assumption with respect to F𝐹Fitalic\_F and Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, we have that 1. 1. if j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i is not known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT and is not in F𝐹Fitalic\_F, then \| \| \| \| \| --- \| --- \| --- \| \| \| π(no nonfaulty agent l≠i is reachable from j without ibetween rounds m and f+1∣ℛ(Ii)∩ℛ(ℱ)∩ℛ(F))≤12M;\begin{array}[]{l}\pi(\mbox{no nonfaulty agent $l\neq i$ is reachable from $j$ without $i$}\\ \ \ \ \mbox{between rounds $m$ and $f+1$}\mid\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(F))\leq\frac{1}{2M};\end{array}start\_ARRAY start\_ROW start\_CELL italic\_π ( no nonfaulty agent italic\_l ≠ italic\_i is reachable from italic\_j without italic\_i end\_CELL end\_ROW start\_ROW start\_CELL between rounds italic\_m and italic\_f + 1 ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_F ) ) ≤ divide start\_ARG 1 end\_ARG start\_ARG 2 italic\_M end\_ARG ; end\_CELL end\_ROW end\_ARRAY \| \| 2. 2. if j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i is not known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT and is not in F𝐹Fitalic\_F, then \| \| \| \| \| --- \| --- \| --- \| \| \| π(no nonfaulty agent l≠i is reachable from j without ibetween rounds m−1 and f+1∣ℛ(Ii)∩ℛ(ℱ)∩ℛ(F))≤12M;\begin{array}[]{ll}\pi(\mbox{no nonfaulty agent $l\neq i$ is reachable from $j$ without $i$}\\ \ \ \ \mbox{between rounds $m-1$ and $f+1$}\mid\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(F))\leq\frac{1}{2M};\end{array}start\_ARRAY start\_ROW start\_CELL italic\_π ( no nonfaulty agent italic\_l ≠ italic\_i is reachable from italic\_j without italic\_i end\_CELL start\_CELL end\_CELL end\_ROW start\_ROW start\_CELL between rounds italic\_m - 1 and italic\_f + 1 ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_F ) ) ≤ divide start\_ARG 1 end\_ARG start\_ARG 2 italic\_M end\_ARG ; end\_CELL start\_CELL end\_CELL end\_ROW end\_ARRAY \| \| 3. 3. if a message from some agent j𝑗jitalic\_j not in F𝐹Fitalic\_F was received up to and including round m−2𝑚2m-2italic\_m - 2 but not in round m−1𝑚1m-1italic\_m - 1, then \| \| \| \| \| --- \| --- \| --- \| \| \| π(no nonfaulty agent l≠i is reachable from an agent j′≠i that did not receive a messagefrom j in round m−1 without i between rounds m and f+1∣ℛ(Ii)∩ℛ(ℱ)∩ℛ(F))≤12M.\begin{array}[]{ll}\pi(\mbox{no nonfaulty agent $l\neq i$}\mbox{ is reachable from an agent $j^{\prime}\neq i$ that did not receive a message}\\ \ \ \ \mbox{from $j$ in round $m-1$ without $i$ between rounds $m$ and $f+1$}\mid\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(F))\leq\frac{1}{2M}.\end{array}start\_ARRAY start\_ROW start\_CELL italic\_π ( no nonfaulty agent l≠i is reachable from an agent j′≠i that did not receive a message end\_CELL start\_CELL end\_CELL end\_ROW start\_ROW start\_CELL from italic\_j in round italic\_m - 1 without italic\_i between rounds italic\_m and italic\_f + 1 ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_F ) ) ≤ divide start\_ARG 1 end\_ARG start\_ARG 2 italic\_M end\_ARG . end\_CELL start\_CELL end\_CELL end\_ROW end\_ARRAY \| \| The first two requirements essentially say that if i𝑖iitalic\_i hears from j𝑗jitalic\_j in round m−1𝑚1m-1italic\_m - 1, then it is likely that other agents will hear from j𝑗jitalic\_j as well in a way that affects the decision, even if i𝑖iitalic\_i does not forward j𝑗jitalic\_j’s information. That is, it is unlikely that j𝑗jitalic\_j will fail right away, and do so in a way that prevents its information from having an effect. Similarly, the third requirement says that if i𝑖iitalic\_i does not hear from j𝑗jitalic\_j in round m−1𝑚1m-1italic\_m - 1 (as reflected in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT), then it is likely that other agents will hear that j𝑗jitalic\_j crashed at or before round m−1𝑚1m-1italic\_m - 1 even if i𝑖iitalic\_i does not report this fact. We next define the notion of uniformity. Given two failure patterns F1superscript𝐹1F^{1}italic\_F start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT and F2superscript𝐹2F^{2}italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT, we say that F1superscript𝐹1F^{1}italic\_F start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT and F2superscript𝐹2F^{2}italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT are equivalent if there is a permutation g𝑔gitalic\_g of the agents such that F2=g(F1)superscript𝐹2𝑔superscript𝐹1F^{2}=g(F^{1})italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT = italic\_g ( italic\_F start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT ). We say that π𝜋\piitalic\_π is uniform if, for all equivalent failure patterns F1superscript𝐹1F^{1}italic\_F start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT and F2superscript𝐹2F^{2}italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT and vectors v→→𝑣\vec{v}over→ start\_ARG italic\_v end\_ARG of initial preferences, we have π(F1,v→)=π(F2,v→)𝜋superscript𝐹1→𝑣𝜋superscript𝐹2→𝑣\pi(F^{1},\vec{v})=\pi(F^{2},\vec{v})italic\_π ( italic\_F start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_v end\_ARG ) = italic\_π ( italic\_F start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_v end\_ARG ). Intuitively, if π𝜋\piitalic\_π is uniform, then the probability of each failure pattern depends only on the number of messages omitted by each agent in each round; it does not depend on the identity of faulty agents. The following lemma will prove useful in the argument, and shows where the uniformity assumption comes into play. Roughly speaking, the lemma says that if the agents run σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT, then each agent i𝑖iitalic\_i’s expected value of its initial preference being the consensus value is just its current knowledge about the fraction of nonfaulty agents that have its initial preference. The lemma’s claim is somewhat stronger, because it allows for expectations conditional on certain sets of agents failing. Before stating the lemma, we need some definitions. Let ℛ(D≥m)ℛsubscript𝐷absent𝑚\mathcal{R}(D\_{\geq m})caligraphic\_R ( italic\_D start\_POSTSUBSCRIPT ≥ italic\_m end\_POSTSUBSCRIPT ) consist of all runs where a decision is made and the first round that seems clean is m′≥msuperscript𝑚′𝑚m^{\prime}\geq mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≥ italic\_m. A set ℱℱ\mathcal{F}caligraphic\_F of failure patterns, a failure pattern F𝐹Fitalic\_F, a round-m𝑚mitalic\_m information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for i𝑖iitalic\_i, and m′≥msuperscript𝑚′𝑚m^{\prime}\geq mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≥ italic\_m are compatible if (a) all the failures in F𝐹Fitalic\_F happen before round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, (b) m′≤f+1superscript𝑚′𝑓1m^{\prime}\leq f+1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≤ italic\_f + 1, and (c) ℱℱ\mathcal{F}caligraphic\_F satisfies the permutation assumption with respect to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT and F𝐹Fitalic\_F. Given an agent i𝑖iitalic\_i and a run r𝑟ritalic\_r where consensus is reached, let nc(r)𝑛𝑐𝑟nc(r)italic\_n italic\_c ( italic\_r ) be the number of agents who apparently have not crashed in the first round of r𝑟ritalic\_r that seems clean (i.e., if m𝑚mitalic\_m is the first clean round in r𝑟ritalic\_r, then nc(r)=\|NCm(r)\|𝑛𝑐𝑟𝑁subscript𝐶𝑚𝑟nc(r)=\|NC\_{m}(r)\|italic\_n italic\_c ( italic\_r ) = \| italic\_N italic\_C start\_POSTSUBSCRIPT italic\_m end\_POSTSUBSCRIPT ( italic\_r ) \|), and let ac(r)𝑎𝑐𝑟ac(r)italic\_a italic\_c ( italic\_r ) be the number of these agents in r𝑟ritalic\_r that have initial preference 1111. Given an information set Ii∈ℐisubscript𝐼𝑖subscriptℐ𝑖I\_{i}\in\mathcal{I}\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ∈ caligraphic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT and a failure pattern F𝐹Fitalic\_F, let AFsubscript𝐴𝐹A\_{F}italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT be the set of agents who are faulty in F𝐹Fitalic\_F; let A𝐴Aitalic\_A consist of the agents known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT; let n(Ii,F)=n−\|A∪AF\|𝑛subscript𝐼𝑖𝐹𝑛𝐴subscript𝐴𝐹n(I\_{i},F)=n-\|A\cup A\_{F}\|italic\_n ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) = italic\_n - \| italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT \|; and let a(Ii,F)𝑎subscript𝐼𝑖𝐹a(I\_{i},F)italic\_a ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) be the agents not in A∪AF𝐴subscript𝐴𝐹A\cup A\_{F}italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT that have initial preference 1. Note that nc𝑛𝑐ncitalic\_n italic\_c and ac𝑎𝑐acitalic\_a italic\_c are random variables on runs (i.e., functions from runs to numbers); technically, a(Ii,F)𝑎subscript𝐼𝑖𝐹a(I\_{i},F)italic\_a ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) and n(Ii,F)𝑛subscript𝐼𝑖𝐹n(I\_{i},F)italic\_n ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) are also random variables on runs, but n(Ii,F)𝑛subscript𝐼𝑖𝐹n(I\_{i},F)italic\_n ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) is constant on runs in ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ), while a(Ii,F)𝑎subscript𝐼𝑖𝐹a(I\_{i},F)italic\_a ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) is constant on runs in ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) if m≥2𝑚2m\geq 2italic\_m ≥ 2, since then Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT contains the initial values of nonfaulty agents. ###### Lemma 1. If i𝑖iitalic\_i is an agent who is nonfaulty at the beginning of round m≤f+1𝑚𝑓1m\leq f+1italic\_m ≤ italic\_f + 1 and has information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT (so that Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is a round-m𝑚mitalic\_m information set), F𝐹Fitalic\_F is a failure pattern, m′≥msuperscript𝑚normal-′𝑚m^{\prime}\geq mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≥ italic\_m, ℱℱ\mathcal{F}caligraphic\_F is a set of failure patterns such that ℱℱ\mathcal{F}caligraphic\_F, F𝐹Fitalic\_F, Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, and m′superscript𝑚normal-′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT are compatible, π𝜋\piitalic\_π is a distribution that supports reachability and is uniform, and πσ→𝑐𝑜𝑛𝑠(ℛ(Ii)∩ℛ(F)∩ℛ(ℱ)∩ℛ(D≥m′))>0subscript𝜋superscriptnormal-→𝜎𝑐𝑜𝑛𝑠ℛsubscript𝐼𝑖ℛ𝐹ℛℱℛsubscript𝐷absentsuperscript𝑚normal-′0\pi\_{{\vec{\sigma}}^{\mathit{cons}}}(\mathcal{R}(I\_{i})\cap\mathcal{R}(F)\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(D\_{\geq m^{\prime}}))>0italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_D start\_POSTSUBSCRIPT ≥ italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ) ) > 0, then \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| E[ac/nc∣ℛ(Ii)∩ℛ(F)∩ℛ(ℱ)∩ℛ(D≥m′)]=E[a(Ii,F)/n(Ii,F)∣ℛ(Ii)],𝐸delimited-[]conditional𝑎𝑐𝑛𝑐ℛsubscript𝐼𝑖ℛ𝐹ℛℱℛsubscript𝐷absentsuperscript𝑚′𝐸delimited-[]conditional𝑎subscript𝐼𝑖𝐹𝑛subscript𝐼𝑖𝐹ℛsubscript𝐼𝑖E[ac/nc\mid\mathcal{R}(I\_{i})\cap\mathcal{R}(F)\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(D\_{\geq m^{\prime}})]=E[a(I\_{i},F)/n(I\_{i},F)\mid\mathcal{R}(I\_{i})],italic\_E [ italic\_a italic\_c / italic\_n italic\_c ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_D start\_POSTSUBSCRIPT ≥ italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ) ] = italic\_E [ italic\_a ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) / italic\_n ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ] , \| \| (1) \| where the expectation is taken with respect to πσ→𝑐𝑜𝑛𝑠subscript𝜋superscriptnormal-→𝜎𝑐𝑜𝑛𝑠\pi\_{\vec{\sigma}^{\mathit{cons}}}italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT. ###### Proof. Let f′=\|A∪AF\|=n−n(Ii,F)superscript𝑓′𝐴subscript𝐴𝐹𝑛𝑛subscript𝐼𝑖𝐹f^{\prime}=\|A\cup A\_{F}\|=n-n(I\_{i},F)italic\_f start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = \| italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT \| = italic\_n - italic\_n ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ). For all f′′superscript𝑓′′f^{\prime\prime}italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT with f′≤f′′≤fsuperscript𝑓′superscript𝑓′′𝑓f^{\prime}\leq f^{\prime\prime}\leq fitalic\_f start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≤ italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT ≤ italic\_f, let ℛf′′subscriptℛsuperscript𝑓′′\mathcal{R}\_{f^{\prime\prime}}caligraphic\_R start\_POSTSUBSCRIPT italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT consists of all runs r𝑟ritalic\_r where agents are using σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT such that exactly f′′superscript𝑓′′f^{\prime\prime}italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT agents are viewed as faulty in the first round that seems clean. We claim that, for all f′′superscript𝑓′′f^{\prime\prime}italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT, we have \| \| \| \| \| --- \| --- \| --- \| \| \| E[ac/nc∣ℛf′′∩ℛ(Ii)∩ℛ(F)∩ℛ(ℱ)∩ℛ(D≥m′)]=E[a(Ii,F)/n(Ii,F)∣ℛ(Ii)).E[ac/nc\mid\mathcal{R}\_{f^{\prime\prime}}\cap\mathcal{R}(I\_{i})\cap\mathcal{R}(F)\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(D\_{\geq m^{\prime}})]=E[a(I\_{i},F)/n(I\_{i},F)\mid\mathcal{R}(I\_{i})).italic\_E [ italic\_a italic\_c / italic\_n italic\_c ∣ caligraphic\_R start\_POSTSUBSCRIPT italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_D start\_POSTSUBSCRIPT ≥ italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ) ] = italic\_E [ italic\_a ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) / italic\_n ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ) . \| \| Clearly, ([1](#S3.E1 "1 ‣ Lemma 1. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) follows immediately from this claim. We can calculate the relevant expectations using algebra, but there is an easier way to see that the claim holds. First suppose that m′>1superscript𝑚′1m^{\prime}>1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT > 1 (so that a(Ii,F)𝑎subscript𝐼𝑖𝐹a(I\_{i},F)italic\_a ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) and n(Ii,F)𝑛subscript𝐼𝑖𝐹n(I\_{i},F)italic\_n ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) are constants on ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT )). If the first clean round occurs at or after m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, then it is easy to see that all the agents in A∪AF𝐴subscript𝐴𝐹A\cup A\_{F}italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT will be viewed as faulty in that round (by all nonfaulty agents), since all these agents fail before round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. Note that the set of agents viewed as faulty in the first clean round of run r𝑟ritalic\_r is completely determined by the failure pattern in r𝑟ritalic\_r. Moreover, it easily follows from the uniformity assumption, the fact that σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT treats agents uniformly, and the fact that ℱℱ\mathcal{F}caligraphic\_F satisfies the permutation assumption that each set B𝐵Bitalic\_B of cardinality f′′superscript𝑓′′f^{\prime\prime}italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT that includes A∪AF𝐴subscript𝐴𝐹A\cup A\_{F}italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT is equally likely to be the set of agents viewed as faulty in the first clean round of a run in ℛf′′∩ℛ(Ii)∩ℛ(F)∩ℛ(ℱ)∩ℛ(D≥m)subscriptℛsuperscript𝑓′′ℛsubscript𝐼𝑖ℛ𝐹ℛℱℛsubscript𝐷absent𝑚\mathcal{R}\_{f^{\prime\prime}}\cap\mathcal{R}(I\_{i})\cap\mathcal{R}(F)\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(D\_{\geq m})caligraphic\_R start\_POSTSUBSCRIPT italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_D start\_POSTSUBSCRIPT ≥ italic\_m end\_POSTSUBSCRIPT ). Consider the following experiment: choose a set B𝐵Bitalic\_B of f′′superscript𝑓′′f^{\prime\prime}italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT agents containing A∪AF𝐴subscript𝐴𝐹A\cup A\_{F}italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT uniformly at random, and then choose one more agent j∉B𝑗𝐵j\notin Bitalic\_j ∉ italic\_B at random. Assign a pair (B,j)𝐵𝑗(B,j)( italic\_B , italic\_j ) value 1 if the agent j𝑗jitalic\_j chosen has initial preference 1 in all runs of Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT; otherwise, assign it value 0. It is easy to see that the expected value of a pair is precisely E[ac/nc∣ℛf′′∩ℛ(Ii)∩ℛ(F)∩ℛ(ℱ)∩ℛ(D≥m)]𝐸delimited-[]conditional𝑎𝑐𝑛𝑐subscriptℛsuperscript𝑓′′ℛsubscript𝐼𝑖ℛ𝐹ℛℱℛsubscript𝐷absent𝑚E[ac/nc\mid\mathcal{R}\_{f^{\prime\prime}}\cap\mathcal{R}(I\_{i})\cap\mathcal{R}(F)\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(D\_{\geq m})]italic\_E [ italic\_a italic\_c / italic\_n italic\_c ∣ caligraphic\_R start\_POSTSUBSCRIPT italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_D start\_POSTSUBSCRIPT ≥ italic\_m end\_POSTSUBSCRIPT ) ]. The f′′superscript𝑓′′f^{\prime\prime}italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT agents in B𝐵Bitalic\_B constitute the set of faulty agents. The fact that B𝐵Bitalic\_B is chosen uniformly at random (among sets of cardinality f′′superscript𝑓′′f^{\prime\prime}italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT containing A∪AF𝐴subscript𝐴𝐹A\cup A\_{F}italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT) corresponds to the assumption that all choices of B𝐵Bitalic\_B are equally likely. The last agent chosen determines the consensus value; as long as there is at least one nonfaulty agent, the procedure used in runs of σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT guarantees that all choices of j𝑗jitalic\_j are equally likely. Now switch the order that the choices are made: we first choose a nonfaulty agent not in A∪AF𝐴subscript𝐴𝐹A\cup A\_{F}italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT uniformly at random and then choose f′′−\|A∪AF\|superscript𝑓′′𝐴subscript𝐴𝐹f^{\prime\prime}-\|A\cup A\_{F}\|italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT - \| italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT \| other agents not in A∪AF𝐴subscript𝐴𝐹A\cup A\_{F}italic\_A ∪ italic\_A start\_POSTSUBSCRIPT italic\_F end\_POSTSUBSCRIPT who will fail uniformly at random. It is clear that there is a one-to-one correspondence between the choices in the first experiment and the second experiment: in corresponding choices, the same set of f′′−f′superscript𝑓′′superscript𝑓′f^{\prime\prime}-f^{\prime}italic\_f start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT - italic\_f start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT agents fail and the same other agent is chosen to determine the consensus value. Moreover, corresponding choices are equally likely. With the second experiment, it is immediate that the expected value is a(Ii,F)/n(Ii,F)𝑎subscript𝐼𝑖𝐹𝑛subscript𝐼𝑖𝐹a(I\_{i},F)/n(I\_{i},F)italic\_a ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ) / italic\_n ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_F ). If m′≤1superscript𝑚′1m^{\prime}\leq 1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≤ 1, then the argument is the same, except that the value of (B,j)𝐵𝑗(B,j)( italic\_B , italic\_j ) is chosen according to the distribution of initial preferences of agents j∉B𝑗𝐵j\notin Bitalic\_j ∉ italic\_B in runs where the faulty agents are exactly the ones in B𝐵Bitalic\_B. This concludes the proof. ∎ Theorem [3](#Thmtheorem3 "Theorem 3. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") shows that σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT is a π𝜋\piitalic\_π-Nash equilibrium, as long as f+1<n𝑓1𝑛f+1<nitalic\_f + 1 < italic\_n and π𝜋\piitalic\_π supports reachability and is uniform. ###### Theorem 3. If f+1<n𝑓1𝑛f+1<nitalic\_f + 1 < italic\_n, π𝜋\piitalic\_π is a distribution that supports reachability, is uniform, and allows up to f𝑓fitalic\_f failures, and agents care only about consensus, then σ→𝑐𝑜𝑛𝑠superscriptnormal-→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT is a π𝜋\piitalic\_π-Nash equilibrium. ###### Proof. Fix an agent i𝑖iitalic\_i and a strategy σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. We must show that we have \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ui(σ→𝑐𝑜𝑛𝑠)≥ui(σi,σ→−i𝑐𝑜𝑛𝑠).subscript𝑢𝑖superscript→𝜎𝑐𝑜𝑛𝑠subscript𝑢𝑖subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖u\_{i}(\vec{\sigma}^{\mathit{cons}})\geq u\_{i}(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i}).italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ≥ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) . \| \| (2) \| Suppose, by way of contradiction, that ([2](#S3.E2 "2 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) does not hold. Then i𝑖iitalic\_i must deviate from σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT at some round m𝑚mitalic\_m. Consider all the ways that i𝑖iitalic\_i can deviate in round m𝑚mitalic\_m that can affect the outcome (we discuss what it means to affect the outcome shortly): 1. 1. i𝑖iitalic\_i pretends to crash; it does not send messages to some subset of agents in round m𝑚mitalic\_m (and then does not not send messages from then on). 2. 2. m=1𝑚1m=1italic\_m = 1 and i𝑖iitalic\_i sends (i,1−vi)𝑖1subscript𝑣𝑖(i,1-v\_{i})( italic\_i , 1 - italic\_v start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) to some agent j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i (i.e., i𝑖iitalic\_i lies about its initial preference to at least one agent). 3. 3. i𝑖iitalic\_i sends an incorrectly formatted message to j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i (i.e., i𝑖iitalic\_i sends a message that is different in format from that required by σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT). 4. 4. m=1𝑚1m=1italic\_m = 1 and i𝑖iitalic\_i sends values yijtsuperscriptsubscript𝑦𝑖𝑗𝑡y\_{ij}^{t}italic\_y start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT to an agent j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i such that there is no polynomial qitsuperscriptsubscript𝑞𝑖𝑡q\_{i}^{t}italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT of degree 1111 that interpolates them all or does not choose the polynomials qitsuperscriptsubscript𝑞𝑖𝑡q\_{i}^{t}italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT at random. 5. 5. i𝑖iitalic\_i does not choose some zijmsuperscriptsubscript𝑧𝑖𝑗𝑚z\_{ij}^{m}italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT appropriately (as specified by σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT). 6. 6. m<f+1𝑚𝑓1m<f+1italic\_m < italic\_f + 1 and i𝑖iitalic\_i decides on a value in {0,1}01\{0,1\}{ 0 , 1 } in round m𝑚mitalic\_m or m=f+1𝑚𝑓1m=f+1italic\_m = italic\_f + 1 and i𝑖iitalic\_i decides on an incorrect value on the equilibrium path. 7. 7. m=f+1𝑚𝑓1m=f+1italic\_m = italic\_f + 1 and i𝑖iitalic\_i sends a value yjitsuperscriptsubscript𝑦𝑗𝑖𝑡y\_{ji}^{t}italic\_y start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT to j′≠isuperscript𝑗′𝑖j^{\prime}\neq iitalic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_i different from the value yjitsuperscriptsubscript𝑦𝑗𝑖𝑡y\_{ji}^{t}italic\_y start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT that i𝑖iitalic\_i received from j𝑗jitalic\_j in round 1. 8. 8. i𝑖iitalic\_i does not send a round m′<msuperscript𝑚′𝑚m^{\prime}<mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m message to some agent j𝑗jitalic\_j that i𝑖iitalic\_i does not know at round m𝑚mitalic\_m to have been faulty in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, and sends a round m𝑚mitalic\_m message to j′≠isuperscript𝑗′𝑖j^{\prime}\neq iitalic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_i. 9. 9. i𝑖iitalic\_i lies about j𝑗jitalic\_j’s status to j′≠isuperscript𝑗′𝑖j^{\prime}\neq iitalic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_i; that is, i𝑖iitalic\_i sends j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT a status report SR¯imsuperscriptsubscript¯𝑆𝑅𝑖𝑚\overline{SR}\_{i}^{m}over¯ start\_ARG italic\_S italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT such that SR¯im[j]≠SRim[j]superscriptsubscript¯𝑆𝑅𝑖𝑚delimited-[]𝑗𝑆superscriptsubscript𝑅𝑖𝑚delimited-[]𝑗\overline{SR}\_{i}^{m}[j]\neq SR\_{i}^{m}[j]over¯ start\_ARG italic\_S italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_j ] ≠ italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_j ]. Note that in a deviation of type 8, we did not consider the case where i𝑖iitalic\_i deviates by not sending a message to j𝑗jitalic\_j in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and then sending a message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT if i𝑖iitalic\_i knows that j𝑗jitalic\_j failed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. In this case, i𝑖iitalic\_i’s deviation is undetectable, and will not affect the outcome. Clearly if i𝑖iitalic\_i performs only such undetectable deviations, then σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is equivalent to σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT, so we do not need to worry about these deviations. We consider these deviations one by one, and show that none of them makes i𝑖iitalic\_i better off. More precisely, we show that if σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT involves only deviations 1–d𝑑ditalic\_d on the list above for appropriate choices of d𝑑ditalic\_d, then ([2](#S3.E2 "2 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds. But even this “brute force” argument requires some care, using a somewhat delicate induction on the number of deviations that i𝑖iitalic\_i is better off not deviating. We now prove ([2](#S3.E2 "2 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")). We start with the first type of deviation; that is, suppose that σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT involves only i𝑖iitalic\_i pretending to crash and that if Ii\superscriptsubscript𝐼𝑖I\_{i}^{\}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT is a time-m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT information set for i𝑖iitalic\_i, ℱℱ\mathcal{F}caligraphic\_F is a set of failure patterns that satisfies the permutation assumption relative to Ii\superscriptsubscript𝐼𝑖I\_{i}^{\}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, πσ→𝑐𝑜𝑛𝑠(ℛ(Ii\)∩ℛ(ℱ))>0subscript𝜋superscript→𝜎𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖ℛℱ0\pi\_{\vec{\sigma}^{\mathit{cons}}}(\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F}))>0italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) > 0, and either there are no deviations in runs in ℛ(Ii\)ℛsuperscriptsubscript𝐼𝑖\mathcal{R}(I\_{i}^{\})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) or the first deviation in a run in ℛ(Ii\)ℛsuperscriptsubscript𝐼𝑖\mathcal{R}(I\_{i}^{\})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) occurs at or after information set Ii\superscriptsubscript𝐼𝑖I\_{i}^{\}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, then \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ui(σ→𝑐𝑜𝑛𝑠∣ℛ(Ii\)∩ℛ(ℱ))≥ui((σi,σ→−i𝑐𝑜𝑛𝑠)∣ℛ(Ii\)∩ℛ(ℱ)).subscript𝑢𝑖conditionalsuperscript→𝜎𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖ℛℱsubscript𝑢𝑖conditionalsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖ℛℱu\_{i}(\vec{\sigma}^{\mathit{cons}}\mid\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F}))\geq u\_{i}((\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})\mid\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F})).italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) ≥ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) . \| \| (3) \| ([2](#S3.E2 "2 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) clearly follows from ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) by taking Ii\superscriptsubscript𝐼𝑖I\_{i}^{\}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT to be the initial information set and letting ℱℱ\mathcal{F}caligraphic\_F be the set of all failure patterns compatible with Ii\superscriptsubscript𝐼𝑖I\_{i}^{\}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT. Given a strategy profile σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG, let ℛ(σ→)ℛ→𝜎\mathcal{R}(\vec{\sigma})caligraphic\_R ( over→ start\_ARG italic\_σ end\_ARG ) denote the possible runs of σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG. If there are no runs in ℛ(σi,σ→−i𝑐𝑜𝑛𝑠)∩ℛ(Ii\)ℛsubscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖ℛsubscriptsuperscript𝐼𝑖\mathcal{R}(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})\cap\mathcal{R}(I^{\}\_{i})caligraphic\_R ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_I start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) in which i𝑖iitalic\_i pretends to fail, then conditional on ℛ(Ii\)ℛsubscriptsuperscript𝐼𝑖\mathcal{R}(I^{\}\_{i})caligraphic\_R ( italic\_I start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ), σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT and σioscsuperscriptsubscript𝜎𝑖𝑜𝑠𝑐\sigma\_{i}^{osc}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_o italic\_s italic\_c end\_POSTSUPERSCRIPT agree, so ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds. If there are runs in ℛ(σi,σ→−i𝑐𝑜𝑛𝑠)∩ℛ(Ii\)ℛsubscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖ℛsuperscriptsubscript𝐼𝑖\mathcal{R}(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})\cap\mathcal{R}(I\_{i}^{\})caligraphic\_R ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) in which i𝑖iitalic\_i pretends to fail, then we proceed by induction on the number of information sets Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT at or after Ii\subscriptsuperscript𝐼𝑖I^{\}\_{i}italic\_I start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT at which i𝑖iitalic\_i first pretends to crash such that π(σi,σ→−i𝑐𝑜𝑛𝑠)(ℛ(Ii\)∩ℛ(ℱ)∩ℛ(Ii))>0subscript𝜋subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖ℛsuperscriptsubscript𝐼𝑖ℛℱℛsubscript𝐼𝑖0\pi\_{(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})}(\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(I\_{i}))>0italic\_π start\_POSTSUBSCRIPT ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) end\_POSTSUBSCRIPT ( caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ) > 0. Suppose that i𝑖iitalic\_i first pretends to crash at some information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT that comes at or after Ii\subscriptsuperscript𝐼𝑖I^{\}\_{i}italic\_I start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT and π(σi,σ→−i𝑐𝑜𝑛𝑠)(ℛ(Ii\)∩ℛ(ℱ)∩ℛ(Ii))>0subscript𝜋subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖ℛsuperscriptsubscript𝐼𝑖ℛℱℛsubscript𝐼𝑖0\pi\_{(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})}(\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F})\cap\mathcal{R}(I\_{i}))>0italic\_π start\_POSTSUBSCRIPT ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) end\_POSTSUBSCRIPT ( caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ) > 0. Thus, there are no runs in ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) in which i𝑖iitalic\_i pretends to crash prior to information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Let σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT be identical to σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT except that i𝑖iitalic\_i does not pretend to fail at or after Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. By ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")), \| \| \| \| \| --- \| --- \| --- \| \| \| ui(σ→𝑐𝑜𝑛𝑠∣ℛ(Ii\)∩ℛ(ℱ))≥ui((σi′,σ→−i𝑐𝑜𝑛𝑠)∣ℛ(Ii\)∩ℛ(ℱ)).subscript𝑢𝑖conditionalsuperscript→𝜎𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖ℛℱsubscript𝑢𝑖conditionalsuperscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖ℛℱu\_{i}(\vec{\sigma}^{\mathit{cons}}\mid\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F}))\geq u\_{i}((\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})\mid\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F})).italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) ≥ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) . \| \| We now show that \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ui((σi′,σ→−i𝑐𝑜𝑛𝑠)∣ℛ(Ii\)∩ℛ(ℱ))≥ui((σi,σ→−i𝑐𝑜𝑛𝑠)∣ℛ(Ii\)∩ℛ(ℱ)).subscript𝑢𝑖conditionalsuperscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖ℛℱsubscript𝑢𝑖conditionalsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖ℛℱmissing-subexpression\begin{array}[]{ll}u\_{i}((\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})\mid\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F}))\geq u\_{i}((\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})\mid\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F})).\end{array}start\_ARRAY start\_ROW start\_CELL italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) ≥ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) . end\_CELL start\_CELL end\_CELL end\_ROW end\_ARRAY \| \| (4) \| ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) follows immediately. To prove ([4](#S3.E4 "4 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")), since ℛ(Ii\)ℛsuperscriptsubscript𝐼𝑖\mathcal{R}(I\_{i}^{\})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) is the union of all the time-m𝑚mitalic\_m information sets for i𝑖iitalic\_i that follow Ii\superscriptsubscript𝐼𝑖I\_{i}^{\}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, it suffices to prove that for all time-m𝑚mitalic\_m information sets Ii′superscriptsubscript𝐼𝑖′I\_{i}^{\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT for i𝑖iitalic\_i that follow Ii\superscriptsubscript𝐼𝑖I\_{i}^{\}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT, we have \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ui((σi′,σ→−i𝑐𝑜𝑛𝑠)∣ℛ(Ii′)∩ℛ(ℱ))≥ui((σi,σ→−i𝑐𝑜𝑛𝑠)∣ℛ(Ii′)∩ℛ(ℱ))subscript𝑢𝑖conditionalsuperscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖′ℛℱsubscript𝑢𝑖conditionalsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖′ℛℱu\_{i}((\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})\mid\mathcal{R}(I\_{i}^{\prime})\cap\mathcal{R}(\mathcal{F}))\geq u\_{i}((\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})\mid\mathcal{R}(I\_{i}^{\prime})\cap\mathcal{R}(\mathcal{F}))italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) ≥ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) \| \| (5) \| (provided, of course, that πσ→𝑐𝑜𝑛𝑠(ℛ(Ii′)∩ℛ(ℱ))>0subscript𝜋superscript→𝜎𝑐𝑜𝑛𝑠ℛsuperscriptsubscript𝐼𝑖′ℛℱ0\pi\_{\vec{\sigma}^{\mathit{cons}}}(\mathcal{R}(I\_{i}^{\prime})\cap\mathcal{R}(\mathcal{F}))>0italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ) ) > 0; in the future, we take it for granted that the relevant results apply only if we are conditioning on a set with positive measure). ([4](#S3.E4 "4 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) clearly follows from ([5](#S3.E5 "5 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")), since the time-m𝑚mitalic\_m information sets for i𝑖iitalic\_i partition ℛ(Ii\)∩ℛ(ℱ)ℛsuperscriptsubscript𝐼𝑖ℛℱ\mathcal{R}(I\_{i}^{\})\cap\mathcal{R}(\mathcal{F})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F ). If Ii′≠Iisuperscriptsubscript𝐼𝑖′subscript𝐼𝑖I\_{i}^{\prime}\neq I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, then ([5](#S3.E5 "5 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds trivially, since in that case σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT agrees with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT at Ii′superscriptsubscript𝐼𝑖′I\_{i}^{\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and all subsequent information sets. Thus, it suffices to prove ([5](#S3.E5 "5 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) in the case that Ii′=Iisuperscriptsubscript𝐼𝑖′subscript𝐼𝑖I\_{i}^{\prime}=I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. We can assume without loss of generality that i𝑖iitalic\_i’s actions at and after Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT are deterministic. If i𝑖iitalic\_i is better off by pretending to fail at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT with some probability, then i𝑖iitalic\_i is better off by pretending to fail at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT with probability 1. Note that (a) whether or not there is a seemingly clean round, (b) which is the first seemingly clean round if there is one, and (c) which agents are considered nonfaulty at that round are completely determined by the failure pattern. Specifically, a particular failure pattern F′∈ℱsuperscript𝐹′ℱF^{\prime}\in\mathcal{F}italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ∈ caligraphic\_F determines the first seemingly clean round m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT. We partition the set ℱℱ\mathcal{F}caligraphic\_F into four sets, ℱ1,…,ℱ4subscriptℱ1…subscriptℱ4\mathcal{F}\_{1},\ldots,\mathcal{F}\_{4}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT, and show that conditional on ℛ(Ii)∩ℛ(Fj)ℛsubscript𝐼𝑖ℛsubscript𝐹𝑗\mathcal{R}(I\_{i})\cap\mathcal{R}(F\_{j})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ), agent i𝑖iitalic\_i does at least as well by using σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT as it does by using σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, for j=1,…,4𝑗1…4j=1,\ldots,4italic\_j = 1 , … , 4. ℱ1subscriptℱ1\mathcal{F}\_{1}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT deals with a trivial case; the remaining elements of the partition consider the first seemingly clean round of (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i}^{\prime},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) and (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ). ([5](#S3.E5 "5 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) in the case that Ii′=Iisuperscriptsubscript𝐼𝑖′subscript𝐼𝑖I\_{i}^{\prime}=I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT clearly follows from this. 1. (a) ℱ1subscriptℱ1\mathcal{F}\_{1}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT consists of the failure patterns in ℱℱ\mathcal{F}caligraphic\_F where with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) an inconsistency is detected (because f+1𝑓1f+1italic\_f + 1 agents seem to fail). Clearly, conditional on ℛ(Ii)∩ℛ(ℱ1)ℛsubscript𝐼𝑖ℛsubscriptℱ1\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{1})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i’s utility is at least as high with (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) as with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ). It may be that with some failure patterns in ℱ1subscriptℱ1\mathcal{F}\_{1}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, no inconsistency is detected if i𝑖iitalic\_i uses (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i}^{\prime},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ). But if the failure pattern is such that an inconsistency is detected with σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, then an inconsistency is certainly detected with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Thus, in all the remaining runs, we consider no inconsistency is detected with either σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT or σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. 2. (b) ℱ2subscriptℱ2\mathcal{F}\_{2}caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT consists of the failure patterns F′∈ℱ−ℱ1superscript𝐹′ℱsubscriptℱ1F^{\prime}\in\mathcal{F}-\mathcal{F}\_{1}italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ∈ caligraphic\_F - caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT such that in all runs r′superscript𝑟′r^{\prime}italic\_r start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT in ℛ((σi′,σ→−i𝑐𝑜𝑛𝑠))∩ℛ(Ii)∩ℛ(F′))\mathcal{R}((\sigma\_{i}^{\prime},\vec{\sigma}^{\mathit{cons}}\_{-i}))\cap\mathcal{R}(I\_{i})\cap\mathcal{R}(F^{\prime}))caligraphic\_R ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) ) ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) ), the first clean round occurs at some round m1<msubscript𝑚1𝑚m\_{1}<mitalic\_m start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT < italic\_m. It is easy to check that in a run r𝑟ritalic\_r of ℛ((σi,σ→−i𝑐𝑜𝑛𝑠))ℛsubscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖\mathcal{R}((\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i}))caligraphic\_R ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) ) corresponding to r′superscript𝑟′r^{\prime}italic\_r start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, the first clean round also occurs at m1subscript𝑚1m\_{1}italic\_m start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, so that all agents get the same utility at r𝑟ritalic\_r and r′superscript𝑟′r^{\prime}italic\_r start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. Thus, conditional on ℛ(Ii)∩ℛ(ℱ2)ℛsubscript𝐼𝑖ℛsubscriptℱ2\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{2})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i’s utility is the same with (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) and (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ). 3. (c) ℱ3subscriptℱ3\mathcal{F}\_{3}caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT consists of the failure patterns in ℱ−ℱ1ℱsubscriptℱ1\mathcal{F}-\mathcal{F}\_{1}caligraphic\_F - caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT that result in m𝑚mitalic\_m being the first seemingly clean round with both (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) and (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i}^{\prime},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ). This can happen in runs in ℛ(σi,σ→−i𝑐𝑜𝑛𝑠)ℛsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠\mathcal{R}(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})caligraphic\_R ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) only if the fact that i𝑖iitalic\_i started pretending to fail at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is not detected by any agent that does not crash (i.e., if no agent that decides is reachable from an agent that does not hear from i𝑖iitalic\_i in round m𝑚mitalic\_m). Conditional on ℛ(Ii)∩ℛ(ℱ3)ℛsubscript𝐼𝑖ℛsubscriptℱ3\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i’s utility is the same with (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) and (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ). 4. (d) ℱ4subscriptℱ4\mathcal{F}\_{4}caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT consists of the failure patterns in ℱ−ℱ1ℱsubscriptℱ1\mathcal{F}-\mathcal{F}\_{1}caligraphic\_F - caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT where the first seemingly clean round with (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i}^{\prime},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) comes at or after m𝑚mitalic\_m while with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ), the first clean round m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT comes strictly before m𝑚mitalic\_m or strictly after m𝑚mitalic\_m. Let M𝑀Mitalic\_M be the number of agents that are not known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, and let a𝑎aitalic\_a be the number of these that share i𝑖iitalic\_i’s initial preference. It is straightforward to check that ℱ4subscriptℱ4\mathcal{F}\_{4}caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT satisfies the permutation assumption with respect to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, so by Lemma [1](#Thmlemma1 "Lemma 1. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"), conditional on ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i’s expected utility with (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i}^{\prime},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) is aβ0iM+(M−a)β1iM𝑎subscript𝛽0𝑖𝑀𝑀𝑎subscript𝛽1𝑖𝑀\frac{a\beta\_{0i}}{M}+\frac{(M-a)\beta\_{1i}}{M}divide start\_ARG italic\_a italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_M end\_ARG + divide start\_ARG ( italic\_M - italic\_a ) italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_M end\_ARG. To compute i𝑖iitalic\_i’s expected utility with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ), we must first consider how we could have m\superscript𝑚m^{\}italic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT (the first seemingly clean round) occur before round m𝑚mitalic\_m. This can happen if (and only if) i𝑖iitalic\_i first learns in round m′′−1≥m−1superscript𝑚′′1𝑚1m^{\prime\prime}-1\geq m-1italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT - 1 ≥ italic\_m - 1 that some agent j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT crashed in round m′≤m−1superscript𝑚′𝑚1m^{\prime}\leq m-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≤ italic\_m - 1, no agent nonfaulty agent j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT (other than i𝑖iitalic\_i) will learn that j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT if i𝑖iitalic\_i pretends to crash, and, as a result, round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT will seem clean to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. This, in turn can happen if (and only if) either (i) m′=m−1superscript𝑚′𝑚1m^{\prime}=m-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_m - 1, and i𝑖iitalic\_i does not hear from j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT for the first time in round m−1𝑚1m-1italic\_m - 1, or (ii) m′<msuperscript𝑚′𝑚m^{\prime}<mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m, i𝑖iitalic\_i did not hear from j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT for the first time in round m′+1superscript𝑚′1m^{\prime}+1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + 1, and there is a chain j1,…,jm′′−m′ subscript𝑗1…subscript𝑗superscript𝑚′′superscript𝑚′j\_{1},\ldots,j\_{m^{\prime\prime}-m^{\prime}}italic\_j start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT of agents that “hides” the fact that j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT actually crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT from i𝑖iitalic\_i (and all other nonfaulty agents) until round m′′superscript𝑚′′m^{\prime\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT: j1subscript𝑗1j\_{1}italic\_j start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT does not hear from j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT; for h<m′′−mℎsuperscript𝑚′′𝑚h<m^{\prime\prime}-mitalic\_h < italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT - italic\_m, i𝑖iitalic\_i does not hear from jhsubscript𝑗ℎj\_{h}italic\_j start\_POSTSUBSCRIPT italic\_h end\_POSTSUBSCRIPT in round m′+hsuperscript𝑚′ℎm^{\prime}+hitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + italic\_h; but jh+1subscript𝑗ℎ1j\_{h+1}italic\_j start\_POSTSUBSCRIPT italic\_h + 1 end\_POSTSUBSCRIPT hears from jhsubscript𝑗ℎj\_{h}italic\_j start\_POSTSUBSCRIPT italic\_h end\_POSTSUBSCRIPT in round m′+1superscript𝑚′1m^{\prime}+1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + 1 (thus, j2subscript𝑗2j\_{2}italic\_j start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT hears that j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT from j1subscript𝑗1j\_{1}italic\_j start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT in round m′+1superscript𝑚′1m^{\prime}+1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + 1, j3subscript𝑗3j\_{3}italic\_j start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT hears about this from j2subscript𝑗2j\_{2}italic\_j start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT in round m′+2superscript𝑚′2m^{\prime}+2italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + 2, and so on), i𝑖iitalic\_i hears from jm′′−m′subscript𝑗superscript𝑚′′superscript𝑚′j\_{m^{\prime\prime}-m^{\prime}}italic\_j start\_POSTSUBSCRIPT italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT in round m′′superscript𝑚′′m^{\prime\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT (and so hears in round m′′superscript𝑚′′m^{\prime\prime}italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT that j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT); and there is no shorter chain like this from j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT to i𝑖iitalic\_i. Note that i𝑖iitalic\_i can tell by looking at its history at time m𝑚mitalic\_m whether it is possible that (i) or (ii) occurrred. Specifically, (i) can occur only if there is an agent j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT that i𝑖iitalic\_i does not hear from for the first time in round m𝑚mitalic\_m, and (ii) can occur only if there is a chain j1,…,jm−m′ subscript𝑗1…subscript𝑗𝑚superscript𝑚′j\_{1},\ldots,j\_{m-m^{\prime}}italic\_j start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_j start\_POSTSUBSCRIPT italic\_m - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT such that, for h′<m−m′superscriptℎ′𝑚superscript𝑚′h^{\prime}<m-m^{\prime}italic\_h start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, i𝑖iitalic\_i does not hear from jhsubscript𝑗ℎj\_{h}italic\_j start\_POSTSUBSCRIPT italic\_h end\_POSTSUBSCRIPT for the first time in round m′+hsuperscript𝑚′ℎm^{\prime}+hitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + italic\_h, either does not hear from jm−m′subscript𝑗𝑚superscript𝑚′j\_{m-m^{\prime}}italic\_j start\_POSTSUBSCRIPT italic\_m - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT in round m𝑚mitalic\_m or hears from jm−m′subscript𝑗𝑚superscript𝑚′j\_{m-m^{\prime}}italic\_j start\_POSTSUBSCRIPT italic\_m - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT that j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, and i𝑖iitalic\_i does not hear that j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT before round m𝑚mitalic\_m. Also note that in case (ii), i𝑖iitalic\_i’s history must be such that none of the rounds between m′+1superscript𝑚′1m^{\prime}+1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + 1 and m′′−1superscript𝑚′′1m^{\prime\prime}-1italic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT - 1 (inclusive) can seem clean to i𝑖iitalic\_i (or the other nonfaulty agents). Agent i𝑖iitalic\_i’s expected utility with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) conditional on ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ) depends on whether i𝑖iitalic\_i’s history (and hence Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT) is such that (i) or (ii) could have occurred. If (i) or (ii) could not have occurred, then we must have m\>msuperscript𝑚𝑚m^{\}>mitalic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT > italic\_m. To compute i𝑖iitalic\_i’s expected utility with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ), we can apply Lemma [1](#Thmlemma1 "Lemma 1. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"), but now we must include i𝑖iitalic\_i among the faulty agents (since in the first seemingly clean round in runs of ℛ(σi,σ→−i𝑐𝑜𝑛𝑠)∩ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})\cap\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i will be viewed as faulty by the nonfaulty agents). Let F𝐹Fitalic\_F be the failure pattern {(i,m,A)}𝑖𝑚𝐴\{(i,m,A)\}{ ( italic\_i , italic\_m , italic\_A ) }, where A𝐴Aitalic\_A is the set of agents to which i𝑖iitalic\_i sends a message in round m𝑚mitalic\_m according to σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Since m\>msuperscript𝑚𝑚m^{\}>mitalic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT > italic\_m, we have ℛ(Ii)∩ℛ(ℱ4)∩ℛ(F)=ℛ(Ii)∩ℛ(ℱ4)∩ℛ(F)∩ℛ(D≥m+1)ℛsubscript𝐼𝑖ℛsubscriptℱ4ℛ𝐹ℛsubscript𝐼𝑖ℛsubscriptℱ4ℛ𝐹ℛsubscript𝐷absent𝑚1\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})\cap\mathcal{R}(F)=\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})\cap\mathcal{R}(F)\cap\mathcal{R}(D\_{\geq m+1})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F ) = caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F ) ∩ caligraphic\_R ( italic\_D start\_POSTSUBSCRIPT ≥ italic\_m + 1 end\_POSTSUBSCRIPT ). Since Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, ℱ4subscriptℱ4\mathcal{F}\_{4}caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT, F𝐹Fitalic\_F, and m+1𝑚1m+1italic\_m + 1 are compatible, by Lemma [1](#Thmlemma1 "Lemma 1. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"), i𝑖iitalic\_i’s expected utility with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) conditional on ℛ(Ii)∩ℛ(F4)ℛsubscript𝐼𝑖ℛsubscript𝐹4\mathcal{R}(I\_{i})\cap\mathcal{R}(F\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ) is (a−1)β0iM−1+(M−a)β1iM−1𝑎1subscript𝛽0𝑖𝑀1𝑀𝑎subscript𝛽1𝑖𝑀1\frac{(a-1)\beta\_{0i}}{M-1}+\frac{(M-a)\beta\_{1i}}{M-1}divide start\_ARG ( italic\_a - 1 ) italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_M - 1 end\_ARG + divide start\_ARG ( italic\_M - italic\_a ) italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_M - 1 end\_ARG. Since β1i>β2isubscript𝛽1𝑖subscript𝛽2𝑖\beta\_{1i}>\beta\_{2i}italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT > italic\_β start\_POSTSUBSCRIPT 2 italic\_i end\_POSTSUBSCRIPT, conditional on ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i’s utility is higher with (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) than with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ). Now if Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is such that (i) or (ii) could happen, we use the reachability assumption to provide upper bounds on the probability that m\<msuperscript𝑚𝑚m^{\}<mitalic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT < italic\_m. Note that if (i) holds, m\<msuperscript𝑚𝑚m^{\}<mitalic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT < italic\_m only if no nonfaulty agent other than i𝑖iitalic\_i hears that j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. By part 3 of the reachability assumption, this happens with probabilty at most 1/2M12𝑀1/2M1 / 2 italic\_M. If (ii) holds, m\<msuperscript𝑚𝑚m^{\}<mitalic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT < italic\_m only if there is an appropriate chain. If m′′=msuperscript𝑚′′𝑚m^{\prime\prime}=mitalic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT = italic\_m, then agent jm−m′subscript𝑗𝑚superscript𝑚′j\_{m-m^{\prime}}italic\_j start\_POSTSUBSCRIPT italic\_m - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT in the chain is not known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, so by part 1 of the reachability assumption, the probability that no nonfaulty agent other than i𝑖iitalic\_i hears from jm−m′+1subscript𝑗𝑚superscript𝑚′1j\_{m-m^{\prime}+1}italic\_j start\_POSTSUBSCRIPT italic\_m - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + 1 end\_POSTSUBSCRIPT that j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, conditional on ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ) is again at most 1/2M12𝑀1/2M1 / 2 italic\_M. Similarly, if m′′>msuperscript𝑚′′𝑚m^{\prime\prime}>mitalic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT > italic\_m, then jm−m′+1subscript𝑗𝑚superscript𝑚′1j\_{m-m^{\prime}+1}italic\_j start\_POSTSUBSCRIPT italic\_m - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + 1 end\_POSTSUBSCRIPT is not known to be faulty in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, so by part 2 of the reachability assumption, the probability that no nonfaulty agent other than i𝑖iitalic\_i hears from jm−m′+1subscript𝑗𝑚superscript𝑚′1j\_{m-m^{\prime}+1}italic\_j start\_POSTSUBSCRIPT italic\_m - italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + 1 end\_POSTSUBSCRIPT that j\superscript𝑗j^{\}italic\_j start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, conditional on ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ) is again at most 1/2M12𝑀1/2M1 / 2 italic\_M. Thus, the probability that m\<msuperscript𝑚𝑚m^{\}<mitalic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT < italic\_m conditional on ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ) is at most 1/M1𝑀1/M1 / italic\_M, even if both (i) and (ii) can occur. In the runs of ℛ(σi,σ→−i𝑐𝑜𝑛𝑠)∩ℛ(Ii)∩ℛ(ℱ4)∩ℛ(F)ℛsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsubscript𝐼𝑖ℛsubscriptℱ4ℛ𝐹\mathcal{R}(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})\cap\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})\cap\mathcal{R}(F)caligraphic\_R ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F ) where the first seemingly clean round is m\<msuperscript𝑚𝑚m^{\}<mitalic\_m start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT < italic\_m, i𝑖iitalic\_i’s utility is at most β0isubscript𝛽0𝑖\beta\_{0i}italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT. If (i) or (ii) could happen and the first clean round in not before m𝑚mitalic\_m, then it must occur strictly after m𝑚mitalic\_m, as noted above. If it does occur after time m𝑚mitalic\_m, then by the argument above, i𝑖iitalic\_i’s expected utility is (a−1)β0iM−1+(M−a)β1iM−1𝑎1subscript𝛽0𝑖𝑀1𝑀𝑎subscript𝛽1𝑖𝑀1\frac{(a-1)\beta\_{0i}}{M-1}+\frac{(M-a)\beta\_{1i}}{M-1}divide start\_ARG ( italic\_a - 1 ) italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_M - 1 end\_ARG + divide start\_ARG ( italic\_M - italic\_a ) italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_M - 1 end\_ARG. Thus, if Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is such that (i) or (ii) could happen, then i𝑖iitalic\_i’s expected utility conditional on ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ) is at most \| \| \| \| \| --- \| --- \| --- \| \| \| (1M+M−1M⋅a−1M−1)β0i+M−1M⋅M−aM−1β1i=aMβ0i+M−aMβ1i.1𝑀⋅𝑀1𝑀𝑎1𝑀1subscript𝛽0𝑖⋅𝑀1𝑀𝑀𝑎𝑀1subscript𝛽1𝑖𝑎𝑀subscript𝛽0𝑖𝑀𝑎𝑀subscript𝛽1𝑖\left(\frac{1}{M}+\frac{M-1}{M}\cdot\frac{a-1}{M-1}\right)\beta\_{0i}+\frac{M-1}{M}\cdot\frac{M-a}{M-1}\beta\_{1i}=\frac{a}{M}\beta\_{0i}+\frac{M-a}{M}\beta\_{1i}.( divide start\_ARG 1 end\_ARG start\_ARG italic\_M end\_ARG + divide start\_ARG italic\_M - 1 end\_ARG start\_ARG italic\_M end\_ARG ⋅ divide start\_ARG italic\_a - 1 end\_ARG start\_ARG italic\_M - 1 end\_ARG ) italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT + divide start\_ARG italic\_M - 1 end\_ARG start\_ARG italic\_M end\_ARG ⋅ divide start\_ARG italic\_M - italic\_a end\_ARG start\_ARG italic\_M - 1 end\_ARG italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT = divide start\_ARG italic\_a end\_ARG start\_ARG italic\_M end\_ARG italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT + divide start\_ARG italic\_M - italic\_a end\_ARG start\_ARG italic\_M end\_ARG italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT . \| \| In either case, conditional on ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i’s utility is at least as high with (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) as with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ). The sets ℱ1,…,ℱ4subscriptℱ1…subscriptℱ4\mathcal{F}\_{1},\ldots,\mathcal{F}\_{4}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT form a partition of ℱℱ\mathcal{F}caligraphic\_F: they are clearly disjoint, and it is not possible for the first clean round with (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) to be strictly after m𝑚mitalic\_m while the first clean round with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) is m𝑚mitalic\_m. Thus, we have proved ([5](#S3.E5 "5 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) in the case that Ii′=Iisuperscriptsubscript𝐼𝑖′subscript𝐼𝑖I\_{i}^{\prime}=I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, as desired. This completes the argument for deviations of type 1. Now, consider a deviation of type 2. If σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is a strategy with deviations of only types 1 and 2, let σi′subscriptsuperscript𝜎′𝑖\sigma^{\prime}\_{i}italic\_σ start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT be the strategy identical to σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT except that i𝑖iitalic\_i does not lie about its initial value and behaves as if it had not deviated from σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT afterwards. There is a bijection between runs of (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) and runs of (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ), so that two corresponding runs r𝑟ritalic\_r and r′superscript𝑟′r^{\prime}italic\_r start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT are identical except that in run r𝑟ritalic\_r agent i𝑖iitalic\_i may lie about its initial value and in r′superscript𝑟′r^{\prime}italic\_r start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT agent i𝑖iitalic\_i does not. (So, among other things, the random choices made in r𝑟ritalic\_r and r′superscript𝑟′r^{\prime}italic\_r start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT are the same.) Again, the lie does not affect which round (if any) will be considered clean nor which agents will be viewed as nonfaulty in that round. If i𝑖iitalic\_i is not one of the agents considered nonfaulty in the clean round, or if i𝑖iitalic\_i is considered nonfaulty but i𝑖iitalic\_i is not the agent whose preference is chosen, then the outcome is the same in r𝑟ritalic\_r and r′superscript𝑟′r^{\prime}italic\_r start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. If i𝑖iitalic\_i is the agent whose value is chosen, then i𝑖iitalic\_i is worse off if it lies than if it doesn’t. Thus, i𝑖iitalic\_i does not gain if it lies about its initial value. Again, ([5](#S3.E5 "5 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds. Thus, ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds for deviations of types 1 and 2. Finally, we show that ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds if we allow deviations of types [3](#S3.I3.i3 "item 3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")–[9](#S3.I3.i9 "item 9 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). To deal with these, we proceed by induction on the number of deviations of types [3](#S3.I3.i3 "item 3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")–[9](#S3.I3.i9 "item 9 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") in σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, removing deviations starting from the earliest deviation. That is, we consider the information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT where the first deviation of type 3–9 occurs, so that the only deviations prior to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT are of type 1 or 2, and show that we can do better by removing the deviation at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Before getting into the details, we need to state carefully what counts as a deviation of type 1 or 2 prior to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. We try to “explain” as much as possible by i𝑖iitalic\_i pretending to fail, so as to delay the first deviation not of types 1 or 2 as late possible. Thus, if i𝑖iitalic\_i pretends to fail at information set Ii′superscriptsubscript𝐼𝑖′I\_{i}^{\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT (i.e., sends message according to σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT up to Ii′superscriptsubscript𝐼𝑖′I\_{i}^{\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, sends messages, again according to σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT, to some agents at Ii′superscriptsubscript𝐼𝑖′I\_{i}^{\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and does not send messages to some agents it does not know to be faulty), and then sends a message to some agent at some information set Ii′′superscriptsubscript𝐼𝑖′′I\_{i}^{\prime\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT after Ii′superscriptsubscript𝐼𝑖′I\_{i}^{\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, then we say that the first deviation not of types 1 and 2 occurs at Ii′′superscriptsubscript𝐼𝑖′′I\_{i}^{\prime\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT (it is a deviation of type [8](#S3.I3.i8 "item 8 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")). In the base case, σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT contains no deviations of type [3](#S3.I3.i3 "item 3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")–[9](#S3.I3.i9 "item 9 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"); we have already shown that ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds in this case. For the inductive step, let Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT be an information set at which σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT has a deviation of type [3](#S3.I3.i3 "item 3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")–[9](#S3.I3.i9 "item 9 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") and there are no deviations of type [3](#S3.I3.i3 "item 3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")–[9](#S3.I3.i9 "item 9 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") prior to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. We consider each deviation of type [3](#S3.I3.i3 "item 3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")–[9](#S3.I3.i9 "item 9 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") in turn. 1. [3](#S3.I3.i3 "item 3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). If i𝑖iitalic\_i sends an incorrectly formatted message to j𝑗jitalic\_j, then either j𝑗jitalic\_j receives this message and decides ⊥bottom\bot⊥ or j𝑗jitalic\_j crashes before sending any messages to an agent j′≠isuperscript𝑗′𝑖j^{\prime}\neq iitalic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_i (or before deciding, if m=f𝑚𝑓m=fitalic\_m = italic\_f). Let σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT be the strategy that is identical to σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT except i𝑖iitalic\_i sends a correctly formatted message to j𝑗jitalic\_j. In all cases, i𝑖iitalic\_i does at least as well if i𝑖iitalic\_i uses the strategy σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT as it does using σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Thus, ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) follows from the induction hypothesis. 2. [4](#S3.I3.i4 "item 4 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). If m=1𝑚1m=1italic\_m = 1 and i𝑖iitalic\_i sends values yijtsuperscriptsubscript𝑦𝑖𝑗𝑡y\_{ij}^{t}italic\_y start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT to an agent j𝑗jitalic\_j such that there is no polynomial qitsuperscriptsubscript𝑞𝑖𝑡q\_{i}^{t}italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT of degree 1111 that interpolates them then either an inconsistency is detected or i𝑖iitalic\_i would have done at least as well by choosing these values according to some polynomial. (Here and in the remainder of the proof, when we say “an inconsistency is detected”, we mean “an inconsistency is detected by a nonfaulty agent different from i𝑖iitalic\_i”.) If i𝑖iitalic\_i does not choose qitsuperscriptsubscript𝑞𝑖𝑡q\_{i}^{t}italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT at random, since f+1<n𝑓1𝑛f+1<nitalic\_f + 1 < italic\_n, there exists a nonfaulty agent j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i that sends values based on truly random polynomials. Thus, the agent whose preference determines the consensus value is chosen at random, even if qitsuperscriptsubscript𝑞𝑖𝑡q\_{i}^{t}italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT is not chosen at random. So choosing qitsuperscriptsubscript𝑞𝑖𝑡q\_{i}^{t}italic\_q start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT at random does not affect the expected outcome. Again, ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) follows from the induction hypothesis. 3. [5](#S3.I3.i5 "item 5 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). Suppose that i𝑖iitalic\_i does not choose zijmsuperscriptsubscript𝑧𝑖𝑗𝑚z\_{ij}^{m}italic\_z start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT according to protocol. From the perspective of an agent j′≠isuperscript𝑗′𝑖j^{\prime}\neq iitalic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_i following the protocol σj′𝑐𝑜𝑛𝑠superscriptsubscript𝜎superscript𝑗′𝑐𝑜𝑛𝑠\sigma\_{j^{\prime}}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT, it does not affect the outcome if these values are not chosen randomly. So, yet again, i𝑖iitalic\_i does just as well if i𝑖iitalic\_i chooses the numbers randomly, and ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds. 4. [6](#S3.I3.i6 "item 6 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). Clearly there is no benefit to i𝑖iitalic\_i deciding on a value other than ⊥bottom\bot⊥ early (it can decide the same value at round f+1𝑓1f+1italic\_f + 1) and no benefit in deciding an incorrect value (since this guarantees that there is no consensus). Thus, yet again, ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds. 5. [7](#S3.I3.i7 "item 7 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). Suppose that m=f+1𝑚𝑓1m=f+1italic\_m = italic\_f + 1 and i𝑖iitalic\_i lies about yjitsuperscriptsubscript𝑦𝑗𝑖𝑡y\_{ji}^{t}italic\_y start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT to some l≠i𝑙𝑖l\neq iitalic\_l ≠ italic\_i for j≠i𝑗𝑖j\neq iitalic\_j ≠ italic\_i. If it turns out that there are not n−t𝑛𝑡n-titalic\_n - italic\_t agents that seem to be nonfaulty in the first clean round, then the value of yjitsuperscriptsubscript𝑦𝑗𝑖𝑡y\_{ji}^{t}italic\_y start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT is irrelevant; it is not used in the calculation. If there are n−t𝑛𝑡n-titalic\_n - italic\_t seemingly nonfaulty agents in the clean round, then either an inconsistency is detected due to the lie (if yjitsuperscriptsubscript𝑦𝑗𝑖𝑡y\_{ji}^{t}italic\_y start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT is sent to some nonfaulty agent, who then cannot interpolate a polynomial through it and the other values received), in which case i𝑖iitalic\_i is clearly worse off, or the sum S𝑆Sitalic\_S computed will be a random element of {0,…,n−t−1}0…𝑛𝑡1\{0,\ldots,n-t-1\}{ 0 , … , italic\_n - italic\_t - 1 }, so the initial preference of each of the seeming nonfaulty agents is equally likely to be chosen whether or not i𝑖iitalic\_i lies. Thus, i𝑖iitalic\_i does not gain by lying about yjitsuperscriptsubscript𝑦𝑗𝑖𝑡y\_{ji}^{t}italic\_y start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT, so ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds. 6. [8](#S3.I3.i8 "item 8 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). Suppose that i𝑖iitalic\_i does not send a message in round m′<msuperscript𝑚′𝑚m^{\prime}<mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m to an agent j𝑗jitalic\_j that i𝑖iitalic\_i does not know (at round m𝑚mitalic\_m) to have been faulty at round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and then i𝑖iitalic\_i sends a message to j′≠isuperscript𝑗′𝑖j^{\prime}\neq iitalic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_i in round m𝑚mitalic\_m. If m′<m−1superscript𝑚′𝑚1m^{\prime}<m-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m - 1, since m𝑚mitalic\_m is the first round that a deviation of types [3](#S3.I3.i3 "item 3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")–[9](#S3.I3.i9 "item 9 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") occurs, and since i𝑖iitalic\_i does not know at any round m′′<msuperscript𝑚′′𝑚m^{\prime\prime}<mitalic\_m start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT < italic\_m that j𝑗jitalic\_j was faulty at round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT (since i𝑖iitalic\_i does not know it at round m𝑚mitalic\_m), i𝑖iitalic\_i does not send messages between rounds m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and m𝑚mitalic\_m. Thus, sending a round m𝑚mitalic\_m message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT either leads to an inconsistency being detected or does not affect the outcome (which can be the case if j𝑗jitalic\_j fails before deciding ⊥bottom\bot⊥). This means that i𝑖iitalic\_i does at least as well if i𝑖iitalic\_i does not send a message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT at round m𝑚mitalic\_m, so ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds. So we can assume without loss of generality that m′=m−1superscript𝑚′𝑚1m^{\prime}=m-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_m - 1, and that m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT is the first round that i𝑖iitalic\_i did not send a message to an agent j𝑗jitalic\_j. Similarly, we can assume that i𝑖iitalic\_i gets a message from j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT in round m−1𝑚1m-1italic\_m - 1; otherwise we can consider the strategy σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT where i𝑖iitalic\_i does send a message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT in round m−1𝑚1m-1italic\_m - 1, and otherwise agrees with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, and again the result follows from the induction hypothesis. The rest of the proof proceeds much in the spirit of the proof for deviations of type 1. We partition ℱℱ\mathcal{F}caligraphic\_F into subsets ℱ1,…,ℱ4 subscriptℱ1…subscriptℱ4\mathcal{F}\_{1},\ldots,\mathcal{F}\_{4}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT, and show that, for j=1,…,4𝑗 1…4j=1,\ldots,4italic\_j = 1 , … , 4, i𝑖iitalic\_i does at least as well with σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT as with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) conditional on ℛ(Ii)∩ℛ(Fj)ℛsubscript𝐼𝑖ℛsubscript𝐹𝑗\mathcal{R}(I\_{i})\cap\mathcal{R}(F\_{j})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( italic\_F start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ); ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) then follows. As in the case of type 1 failures, ℱ1subscriptℱ1\mathcal{F}\_{1}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT consists of the failure patterns in ℱℱ\mathcal{F}caligraphic\_F where, with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ), f+1𝑓1f+1italic\_f + 1 failures are detected. Clearly, conditional on ℛ(Ii)∩ℛ(ℱ1)ℛsubscript𝐼𝑖ℛsubscriptℱ1\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{1})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i’s utility is higher with (σ→𝑐𝑜𝑛𝑠)superscript→𝜎𝑐𝑜𝑛𝑠(\vec{\sigma}^{\mathit{cons}})( over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) than with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ). Let ℱ2subscriptℱ2\mathcal{F}\_{2}caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT be the set of failure patterns in ℱ−ℱ1ℱsubscriptℱ1\mathcal{F}-\mathcal{F}\_{1}caligraphic\_F - caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT such that in runs from ℛ(Ii)∩ℛ(ℱ2)ℛsubscript𝐼𝑖ℛsubscriptℱ2\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{2})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ), the agents that decide do not hear about i𝑖iitalic\_i’s round m𝑚mitalic\_m message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. Let σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT be identical to σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT except that at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT agent i𝑖iitalic\_i does not send a message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. It is not hard to check that ℱ2subscriptℱ2\mathcal{F}\_{2}caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT satisfies the permutation assumption with respect to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Clearly, i𝑖iitalic\_i gets the same utility with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT as with σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT conditional on ℛ(Ii)∩ℛ(ℱ2)ℛsubscript𝐼𝑖ℛsubscriptℱ2\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{2})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ). Since, with σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, i𝑖iitalic\_i has fewer deviations of types 3–9 than with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, by the induction hypothesis, ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds conditional on ℛ(Ii)∩ℛ(ℱ2)ℛsubscript𝐼𝑖ℛsubscriptℱ2\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{2})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ). Now let ℱ3subscriptℱ3\mathcal{F}\_{3}caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT consist of all failure patterns in ℱ−ℱ1ℱsubscriptℱ1\mathcal{F}-\mathcal{F}\_{1}caligraphic\_F - caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT such that, with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, the agents that decide hear both that i𝑖iitalic\_i sent a message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT in round m𝑚mitalic\_m and that i𝑖iitalic\_i did not send a message to some agents in round m−1𝑚1m-1italic\_m - 1. Thus, with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, an inconsistency will be detected, so i𝑖iitalic\_i does at least as well with σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT as with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT conditional on ℛ(Ii)∩ℛ(ℱ3)ℛsubscript𝐼𝑖ℛsubscriptℱ3\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ). ℱ3subscriptℱ3\mathcal{F}\_{3}caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT also satisfies the permutation assumption with respect to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, so ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds conditional on R(Ii)∩ℛ(ℱ3)𝑅subscript𝐼𝑖ℛsubscriptℱ3R(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3})italic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ) by the induction hypothesis. Finally, let ℱ4subscriptℱ4\mathcal{F}\_{4}caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT be the remaining failure patterns in ℱ−ℱ1ℱsubscriptℱ1\mathcal{F}-\mathcal{F}\_{1}caligraphic\_F - caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, the ones where agents that decide hear about the message sent by i𝑖iitalic\_i to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT but not about the omissions of i𝑖iitalic\_i in round m−1𝑚1m-1italic\_m - 1. Let Ii′superscriptsubscript𝐼𝑖′I\_{i}^{\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT be the round-(m−1)𝑚1(m-1)( italic\_m - 1 ) information set preceding Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, and let σi′′superscriptsubscript𝜎𝑖′′\sigma\_{i}^{\prime\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT be a strategy identical to σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, except that at at Ii′superscriptsubscript𝐼𝑖′I\_{i}^{\prime}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT i𝑖iitalic\_i does not deviate from σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT. Conditional on ℛ(Ii)∩ℛ(ℱ4)ℛsubscript𝐼𝑖ℛsubscriptℱ4\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i clearly gets the same utility with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) as with (σi′′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i}^{\prime\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ). It is not hard to show that ℱ4subscriptℱ4\mathcal{F}\_{4}caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT also satisfies the permutation assumption with respect to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. With σi′′superscriptsubscript𝜎𝑖′′\sigma\_{i}^{\prime\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT, i𝑖iitalic\_i does not deviate at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, so i𝑖iitalic\_i has fewer deviations of types 3–9 than with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Thus, by the induction hypothesis, ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds conditional on R(Ii)∩ℛ(ℱ4)𝑅subscript𝐼𝑖ℛsubscriptℱ4R(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{4})italic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPT ). This completes the argument for deviations of type 8. 7. [9](#S3.I3.i9 "item 9 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"). Suppose that i𝑖iitalic\_i lies about j𝑗jitalic\_j’s status to an agent j′≠isuperscript𝑗′𝑖j^{\prime}\neq iitalic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_i. That is, either (a) i𝑖iitalic\_i says that j𝑗jitalic\_j did not crash before round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT although i𝑖iitalic\_i knows that j𝑗jitalic\_j did crash in round m′−1superscript𝑚′1m^{\prime}-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT - 1; (b) i𝑖iitalic\_i says that j𝑗jitalic\_j crashed at or before round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT although i𝑖iitalic\_i received a message from j𝑗jitalic\_j in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and either m′=m−1superscript𝑚′𝑚1m^{\prime}=m-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_m - 1 or m′<msuperscript𝑚′𝑚m^{\prime}<mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m and i𝑖iitalic\_i did not receive a message from any agent saying that j𝑗jitalic\_j crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT; or (c) i𝑖iitalic\_i lies about the numbers zjim−1superscriptsubscript𝑧𝑗𝑖𝑚1z\_{ji}^{m-1}italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 1 end\_POSTSUPERSCRIPT sent by j𝑗jitalic\_j or about which agent reported that j𝑗jitalic\_j crashed. Again we consider each of these cases in turn. We can assume without loss of generality that i𝑖iitalic\_i did not pretend to crash in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, since otherwise the arguments for deviations of type 8 would apply. 1. (a) Suppose that i𝑖iitalic\_i lies by saying that j𝑗jitalic\_j did not crash before m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT even though i𝑖iitalic\_i knows that j𝑗jitalic\_j did in fact crash earlier. This means that i𝑖iitalic\_i is claiming to have received a message from j𝑗jitalic\_j in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. Clearly, it cannot be the case that i𝑖iitalic\_i knows that j𝑗jitalic\_j crashed before m′−1superscript𝑚′1m^{\prime}-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT - 1, because then i𝑖iitalic\_i would know that no agent would get a message from j𝑗jitalic\_j in round m′−1superscript𝑚′1m^{\prime}-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT - 1, and an inconsistency would be detected by j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT if the deviation had any impact on the outcome. Thus, we can assume that j𝑗jitalic\_j in fact crashed in round m′−1superscript𝑚′1m^{\prime}-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT - 1. Since we are assuming that i𝑖iitalic\_i first deviates in round m𝑚mitalic\_m, i𝑖iitalic\_i must have learned in round m−1𝑚1m-1italic\_m - 1 about j𝑗jitalic\_j’s crash in round m′−1superscript𝑚′1m^{\prime}-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT - 1. That means that either (i) m′=msuperscript𝑚′𝑚m^{\prime}=mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_m and i𝑖iitalic\_i did not receive a message from j𝑗jitalic\_j in round m−1𝑚1m-1italic\_m - 1 or (ii) m′<msuperscript𝑚′𝑚m^{\prime}<mitalic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m and i𝑖iitalic\_i must have received a message from some agent j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT with this information in round m−1𝑚1m-1italic\_m - 1. We can assume without loss of generality that i𝑖iitalic\_i gets a message from j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT in round m−1𝑚1m-1italic\_m - 1, for otherwise i𝑖iitalic\_i would do at least as well by not lying to j𝑗jitalic\_j, and ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) would hold by the induction hypothesis. Consider case (i). If m=2𝑚2m=2italic\_m = 2, then i𝑖iitalic\_i pretending that j𝑗jitalic\_j did not crash in round 1 can help only if this leads to round 1111 being viewed as clean. But this is the case only if j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT received a message from j𝑗jitalic\_j in round 1 (although i𝑖iitalic\_i did not). According to σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT, i𝑖iitalic\_i’s round m𝑚mitalic\_m message includes the status report SRim𝑆superscriptsubscript𝑅𝑖𝑚SR\_{i}^{m}italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT. Agent i𝑖iitalic\_i must send such a status report even with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, otherwise an inconsistency is detected and clearly i𝑖iitalic\_i is worse off. Since i𝑖iitalic\_i claims to have received a message from j𝑗jitalic\_j in round 1, SRim[j]𝑆superscriptsubscript𝑅𝑖𝑚delimited-[]𝑗SR\_{i}^{m}[j]italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT [ italic\_j ] has the form (∞,zji1)superscriptsubscript𝑧𝑗𝑖1(\infty,z\_{ji}^{1})( ∞ , italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT ), where zjim−1[i]superscriptsubscript𝑧𝑗𝑖𝑚1delimited-[]𝑖z\_{ji}^{m-1}[i]italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 1 end\_POSTSUPERSCRIPT [ italic\_i ] is the random number sent in round 1111 to all agents. Given that we have assumed that j𝑗jitalic\_j also sent a round 1 message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT also received zji1[i]=zjj′1[j′]superscriptsubscript𝑧𝑗𝑖1delimited-[]𝑖superscriptsubscript𝑧𝑗superscript𝑗′1delimited-[]superscript𝑗′z\_{ji}^{1}[i]=z\_{jj^{\prime}}^{1}[j^{\prime}]italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_i ] = italic\_z start\_POSTSUBSCRIPT italic\_j italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ]. Thus, j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT will detect an inconsistency and decide ⊥bottom\bot⊥ unless i𝑖iitalic\_i correctly guesses zji1[i]superscriptsubscript𝑧𝑗𝑖1delimited-[]𝑖z\_{ji}^{1}[i]italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_i ]. The probability of i𝑖iitalic\_i guessing zji1[i]superscriptsubscript𝑧𝑗𝑖1delimited-[]𝑖z\_{ji}^{1}[i]italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT [ italic\_i ] correctly is at most 1n1𝑛\frac{1}{n}divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG. We now partition ℱℱ\mathcal{F}caligraphic\_F into three sets of failure patterns ℱ1subscriptℱ1\mathcal{F}\_{1}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, ℱ2subscriptℱ2\mathcal{F}\_{2}caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, and ℱ3subscriptℱ3\mathcal{F}\_{3}caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT, and show that for j=1,2,3𝑗 123j=1,2,3italic\_j = 1 , 2 , 3, i𝑖iitalic\_i does at least as well with σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT as with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ). Again, ℱ1subscriptℱ1\mathcal{F}\_{1}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT consists of the failure patterns in ℱℱ\mathcal{F}caligraphic\_F where with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖subscriptsuperscript→𝜎𝑐𝑜𝑛𝑠𝑖(\sigma\_{i},\vec{\sigma}^{\mathit{cons}}\_{-i})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ), f+1𝑓1f+1italic\_f + 1 failures are detected. Clearly the claim holds in this case. ℱ2subscriptℱ2\mathcal{F}\_{2}caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT consists of the failure patterns F′superscript𝐹′F^{\prime}italic\_F start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT in ℱ−ℱ1ℱsubscriptℱ1\mathcal{F}-\mathcal{F}\_{1}caligraphic\_F - caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT where the message that i𝑖iitalic\_i sent in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT has no impact on the outcome; that is, either i𝑖iitalic\_i crashes before sending the message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT or no nonfaulty agent is reachable from j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT without i𝑖iitalic\_i between round m+1𝑚1m+1italic\_m + 1 and f+1𝑓1f+1italic\_f + 1. Let σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT be identical to σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT except that, at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, i𝑖iitalic\_i replaces the reports relative to j𝑗jitalic\_j with SRi𝑆subscript𝑅𝑖SR\_{i}italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT (the correct report) in messages sent to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, while sending the same messages to other agents. Thus, i𝑖iitalic\_i has fewer deviations with σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT than with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Clearly, conditional on ℛ(Ii)∩ℛ(ℱ2)ℛsubscript𝐼𝑖ℛsubscriptℱ2\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{2})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ), i𝑖iitalic\_i gets the same expected utility with (σi,σ→−i𝑐𝑜𝑛𝑠)subscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) as with (σi′,σ→−i𝑐𝑜𝑛𝑠)superscriptsubscript𝜎𝑖′superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠(\sigma\_{i}^{\prime},\vec{\sigma}\_{-i}^{\mathit{cons}})( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ). It is easy to check that ℱ2subscriptℱ2\mathcal{F}\_{2}caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT satisfies the permutation assumption with respect to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, so by the induction hypothesis, ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds conditional on ℛ(Ii)∩ℛ(ℱ2)ℛsubscript𝐼𝑖ℛsubscriptℱ2\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{2})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ). Let ℱ3subscriptℱ3\mathcal{F}\_{3}caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT consist of the remaining failure patterns in ℱℱ\mathcal{F}caligraphic\_F. In runs of ℛ(σi,σ→−i𝑐𝑜𝑛𝑠)∩ℛ(Ii)∩ℛ(ℱ3)ℛsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsubscript𝐼𝑖ℛsubscriptℱ3\mathcal{R}(\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})\cap\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3})caligraphic\_R ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∩ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ), j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT detects an inconsistency and decides ⊥bottom\bot⊥ unless i𝑖iitalic\_i guesses the random number correctly. Again, it is not hard to check that ℱ3subscriptℱ3\mathcal{F}\_{3}caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT satisfies the permutation assumption with respect to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Since the largest utility that i𝑖iitalic\_i can get if no inconsistency is detected is β0isubscript𝛽0𝑖\beta\_{0i}italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT, \| \| \| \| \| --- \| --- \| --- \| \| \| ui((σi,σ→−i𝑐𝑜𝑛𝑠)∣ℛ(Ii)∩ℛ(ℱ3))≤1nβ0i+n−1nβ2i.subscript𝑢𝑖conditionalsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsubscript𝐼𝑖ℛsubscriptℱ31𝑛subscript𝛽0𝑖𝑛1𝑛subscript𝛽2𝑖u\_{i}((\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})\mid\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3}))\leq\frac{1}{n}\beta\_{0i}+\frac{n-1}{n}\beta\_{2i}.italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ) ) ≤ divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT + divide start\_ARG italic\_n - 1 end\_ARG start\_ARG italic\_n end\_ARG italic\_β start\_POSTSUBSCRIPT 2 italic\_i end\_POSTSUBSCRIPT . \| \| On the other hand, by Lemma [1](#Thmlemma1 "Lemma 1. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"), \| \| \| \| \| --- \| --- \| --- \| \| \| ui(σ→𝑐𝑜𝑛𝑠∣ℛ(Ii)∩ℛ(ℱ3))≥1nβ0i+n−1nβ1i.subscript𝑢𝑖conditionalsuperscript→𝜎𝑐𝑜𝑛𝑠ℛsubscript𝐼𝑖ℛsubscriptℱ31𝑛subscript𝛽0𝑖𝑛1𝑛subscript𝛽1𝑖u\_{i}(\vec{\sigma}^{\mathit{cons}}\mid\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3}))\geq\frac{1}{n}\beta\_{0i}+\frac{n-1}{n}\beta\_{1i}.italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ) ) ≥ divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG italic\_β start\_POSTSUBSCRIPT 0 italic\_i end\_POSTSUBSCRIPT + divide start\_ARG italic\_n - 1 end\_ARG start\_ARG italic\_n end\_ARG italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT . \| \| Since β1i>β2isubscript𝛽1𝑖subscript𝛽2𝑖\beta\_{1i}>\beta\_{2i}italic\_β start\_POSTSUBSCRIPT 1 italic\_i end\_POSTSUBSCRIPT > italic\_β start\_POSTSUBSCRIPT 2 italic\_i end\_POSTSUBSCRIPT, we have \| \| \| \| \| --- \| --- \| --- \| \| \| ui(σ→𝑐𝑜𝑛𝑠∣ℛ(Ii)∩ℛ(ℱ3))≥ui((σi,σ→−i𝑐𝑜𝑛𝑠)∣ℛ(Ii)∩ℛ(ℱ3)).subscript𝑢𝑖conditionalsuperscript→𝜎𝑐𝑜𝑛𝑠ℛsubscript𝐼𝑖ℛsubscriptℱ3subscript𝑢𝑖conditionalsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑐𝑜𝑛𝑠ℛsubscript𝐼𝑖ℛsubscriptℱ3u\_{i}(\vec{\sigma}^{\mathit{cons}}\mid\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3}))\geq u\_{i}((\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{cons}})\mid\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3})).italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ) ) ≥ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ) ) . \| \| Therefore, ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds if m=2𝑚2m=2italic\_m = 2. Continuing with case (i), suppose that m>2𝑚2m>2italic\_m > 2. Now it is possible that i𝑖iitalic\_i pretending that j𝑗jitalic\_j did not crash can help even if j𝑗jitalic\_j did not send a message to j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. Nevertheless, essentially the same argument will work. This is because now SRi𝑆subscript𝑅𝑖SR\_{i}italic\_S italic\_R start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT would have to include zjim−1superscriptsubscript𝑧𝑗𝑖𝑚1z\_{ji}^{m-1}italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 1 end\_POSTSUPERSCRIPT. Moreover, zjim−1[j′]=zj′jm−2[j′]superscriptsubscript𝑧𝑗𝑖𝑚1delimited-[]superscript𝑗′superscriptsubscript𝑧superscript𝑗′𝑗𝑚2delimited-[]superscript𝑗′z\_{ji}^{m-1}[j^{\prime}]=z\_{j^{\prime}j}^{m-2}[j^{\prime}]italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 1 end\_POSTSUPERSCRIPT [ italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ] = italic\_z start\_POSTSUBSCRIPT italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT italic\_j end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 2 end\_POSTSUPERSCRIPT [ italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ], the random number in {0,…,n−1}0…𝑛1\{0,\ldots,n-1\}{ 0 , … , italic\_n - 1 } sent by j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT to j𝑗jitalic\_j in round m−2𝑚2m-2italic\_m - 2. Clearly, j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT knows this number, so i𝑖iitalic\_i would have to guess it correctly. The argument now proceeds as above. Now consider case (ii). There are two ways in which i𝑖iitalic\_i can ignore the information that j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT sent about j𝑗jitalic\_j in round m−1𝑚1m-1italic\_m - 1. The first is to pretend that j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT crashed in round m−1𝑚1m-1italic\_m - 1; the second is for i𝑖iitalic\_i to lie about the message that it received from j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT (but to say that it did get a message from j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT). In the first case, as with deviations of type 8, we can assume without loss of generality that i𝑖iitalic\_i does not know that j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT is faulty at the beginning round m𝑚mitalic\_m. We partition ℱℱ\mathcal{F}caligraphic\_F into three sets much as in the argument for case (i): ℱ1subscriptℱ1\mathcal{F}\_{1}caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, the failure patterns in which more than f+1𝑓1f+1italic\_f + 1 failures are detected with σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT; ℱ2subscriptℱ2\mathcal{F}\_{2}caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, the failure patterns where i𝑖iitalic\_i’s lie has no impact on the outcome; and ℱ3subscriptℱ3\mathcal{F}\_{3}caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT, the remaining failure patterns. Again, it is easy to see that ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds conditional on ℛ(Ii)∩ℛ(ℱ1)ℛsubscript𝐼𝑖ℛsubscriptℱ1\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{1})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) and ℛ(Ii)∩ℛ(ℱ2)ℛsubscript𝐼𝑖ℛsubscriptℱ2\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{2})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ). To see that ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds conditional on ℛ(Ii)∩ℛ(ℱ3)ℛsubscript𝐼𝑖ℛsubscriptℱ3\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ), we use the reachability assumption, much as we did for as in (d) of the argument for deviations of type 1. By part 1 of the reachability assumption, if i𝑖iitalic\_i pretends that j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT crashed in round m−1𝑚1m-1italic\_m - 1, an inconsistency will be detected with probability at least (2M−1)/2M2𝑀12𝑀(2M-1)/2M( 2 italic\_M - 1 ) / 2 italic\_M. Thus, the same argument as that used in part (e) of the argument for deviations of type 1 shows that ([3](#S3.E3 "3 ‣ Proof. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus")) holds conditional on ℛ(Ii)∩ℛ(ℱ3)ℛsubscript𝐼𝑖ℛsubscriptℱ3\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ). The analysis is essentially the same if i𝑖iitalic\_i lies about the message it received from j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT, except that, conditional on ℛ(Ii)∩ℛ(ℱ3)ℛsubscript𝐼𝑖ℛsubscriptℱ3\mathcal{R}(I\_{i})\cap\mathcal{R}(\mathcal{F}\_{3})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∩ caligraphic\_R ( caligraphic\_F start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT ), by the reachability assumption, j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT receives the round m−1𝑚1m-1italic\_m - 1 message from j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT with probability at least (2M−1)/2M2𝑀12𝑀(2M-1)/2M( 2 italic\_M - 1 ) / 2 italic\_M, so j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT receives inconsistent reports about j𝑗jitalic\_j’s status in round m−1𝑚1m-1italic\_m - 1, and decides ⊥bottom\bot⊥. 2. (b) Suppose that i𝑖iitalic\_i lies to some j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT in round m𝑚mitalic\_m by saying that j𝑗jitalic\_j crashed at or before round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT although i𝑖iitalic\_i received a message from j𝑗jitalic\_j in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT and either m′=m−1superscript𝑚′𝑚1m^{\prime}=m-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_m - 1 or m′<m−1superscript𝑚′𝑚1m^{\prime}<m-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT < italic\_m - 1 and i𝑖iitalic\_i did not receive a message from any agent saying that j𝑗jitalic\_j crashed in round m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. If m′=m−1superscript𝑚′𝑚1m^{\prime}=m-1italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_m - 1, then we can proceed as in part (a). Specifically, we can use the reachability assumption to show that i𝑖iitalic\_i is better off if i𝑖iitalic\_i does not lie. The analysis is similar if i𝑖iitalic\_i pretends to have received a message in round m−1𝑚1m-1italic\_m - 1 from some agent j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT saying that j𝑗jitalic\_j crashed in an earlier round. If i𝑖iitalic\_i did not receive a message from j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT in round m−1𝑚1m-1italic\_m - 1 saying that j𝑗jitalic\_j crashed before m′superscript𝑚′m^{\prime}italic\_m start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT but is claiming to have done so, then we can again use the same arguments as in part (a) where either i𝑖iitalic\_i must guess the random number zj′j′′m−2[j′]superscriptsubscript𝑧superscript𝑗′superscript𝑗′′𝑚2delimited-[]superscript𝑗′z\_{j^{\prime}j^{\prime\prime}}^{m-2}[j^{\prime}]italic\_z start\_POSTSUBSCRIPT italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 2 end\_POSTSUPERSCRIPT [ italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ] known by j′superscript𝑗′j^{\prime}italic\_j start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT (if j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT did not send a round m−1𝑚1m-1italic\_m - 1 message to i𝑖iitalic\_i) or i𝑖iitalic\_i has to lie about the round m−1𝑚1m-1italic\_m - 1 report of j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT. 3. (c) It is easy to see that i𝑖iitalic\_i does not gain if i𝑖iitalic\_i lies about which agent told him that j𝑗jitalic\_j crashed or about the values zjim−1superscriptsubscript𝑧𝑗𝑖𝑚1z\_{ji}^{m-1}italic\_z start\_POSTSUBSCRIPT italic\_j italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m - 1 end\_POSTSUPERSCRIPT sent by j𝑗jitalic\_j to i𝑖iitalic\_i in round m−1𝑚1m-1italic\_m - 1 (and may be worse off, if an inconsistency is detected). This completes the proof of the inductive step and, with it, the proof of the theorem. ∎ ### 3.3 A π𝜋\piitalic\_π-Sequential Equilibrium for Fair Consensus Our π𝜋\piitalic\_π-Nash equilibrium requires an agent i𝑖iitalic\_i to decide on ⊥bottom\bot⊥ whenever i𝑖iitalic\_i detects a problem. While this punishes the agent that causes the problem, it also punishes i𝑖iitalic\_i. Would a rational agent actually play such a punishment strategy? Note that the need to punish occurs only off the equilibrium path; if all agents follow σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT, agents never decide ⊥bottom\bot⊥. But to get agents to play according to σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT requires the threat of playing ⊥bottom\bot⊥. There might be a concern that this is an empty threat; a rational agent might not be willing to play ⊥bottom\bot⊥ if it detects a deviation. The solution concept of sequential equilibrium [[15](#bib.bib15)] is a refinement of Nash equilibrium that, roughly speaking, requires that agents also make best responses not only on the equilibrium path, but off the equilibrium path as well. We now define π𝜋\piitalic\_π-sequential equilibrium, a generalization of sequential equilibrium that allows for faulty agents (where, as before, π𝜋\piitalic\_π is a distribution on failure contexts). We then show that σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT is essentially a π𝜋\piitalic\_π-sequential equilibrium. #### 3.3.1 Defining π𝜋\piitalic\_π-sequential equilibrium Roughly speaking, a strategy profile σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG is a sequential equilibrium if, for each agent i𝑖iitalic\_i and information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for agent i𝑖iitalic\_i, σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is a best response to σ→−isubscript→𝜎𝑖\vec{\sigma}\_{-i}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT conditional on reaching Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT (i.e. conditional on ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT )). The problem is that the probability of ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) is 0 if Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is not on the equilibrium path, so we cannot condition on ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ). Define a belief system μ𝜇\muitalic\_μ to be a function that associates with each agent i𝑖iitalic\_i and information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for agent i𝑖iitalic\_i a probability μIisubscript𝜇subscript𝐼𝑖\mu\_{I\_{i}}italic\_μ start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT on histories in Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Say that a belief system μ𝜇\muitalic\_μ is consistent with σ→normal-→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG and π𝜋\piitalic\_π if there exists a sequence of completely mixed strategy profiles σ→1,σ→2,…superscript→𝜎1superscript→𝜎2…\vec{\sigma}^{1},\vec{\sigma}^{2},\ldotsover→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT , … (where a strategy profile is completely mixed if it gives positive positive probability to every action at every information set) converging to σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG such that \| \| \| \| \| --- \| --- \| --- \| \| \| μIi(h)=limM→∞πσ→M(h)πσ→M(Ii).subscript𝜇subscript𝐼𝑖ℎsubscript→𝑀subscript𝜋superscript→𝜎𝑀ℎsubscript𝜋superscript→𝜎𝑀subscript𝐼𝑖\mu\_{I\_{i}}(h)=\lim\_{M\to\infty}\frac{\pi\_{\vec{\sigma}^{M}}(h)}{\pi\_{\vec{\sigma}^{M}}(I\_{i})}.italic\_μ start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_h ) = roman\_lim start\_POSTSUBSCRIPT italic\_M → ∞ end\_POSTSUBSCRIPT divide start\_ARG italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_M end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_h ) end\_ARG start\_ARG italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_M end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) end\_ARG . \| \| Note that μIisubscript𝜇subscript𝐼𝑖\mu\_{I\_{i}}italic\_μ start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT, π𝜋\piitalic\_π, and σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG together define a probability distribution over runs in ℛ(Ii)ℛsubscript𝐼𝑖\mathcal{R}(I\_{i})caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ). Let μIi,π,σ→subscript𝜇subscript𝐼𝑖𝜋→𝜎\mu\_{I\_{i},\pi,\vec{\sigma}}italic\_μ start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_π , over→ start\_ARG italic\_σ end\_ARG end\_POSTSUBSCRIPT denote this probability distribution. A pair (σ→,μ)→𝜎𝜇(\vec{\sigma},\mu)( over→ start\_ARG italic\_σ end\_ARG , italic\_μ ) is a π𝜋\piitalic\_π-sequential equilibrium if μ𝜇\muitalic\_μ is a belief system consistent with σ→→𝜎\vec{\sigma}over→ start\_ARG italic\_σ end\_ARG and π𝜋\piitalic\_π such that, for every agent i𝑖iitalic\_i, information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, and strategy σi′superscriptsubscript𝜎𝑖′\sigma\_{i}^{\prime}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, ui((σi,σ→−i)∣ℛ(Ii))≥ui((σi′,σ→−i)∣ℛ(Ii))subscript𝑢𝑖conditionalsubscript𝜎𝑖subscript→𝜎𝑖ℛsubscript𝐼𝑖subscript𝑢𝑖conditionalsuperscriptsubscript𝜎𝑖′subscript→𝜎𝑖ℛsubscript𝐼𝑖u\_{i}((\sigma\_{i},\vec{\sigma}\_{-i})\mid\mathcal{R}(I\_{i}))\geq u\_{i}((\sigma\_{i}^{\prime},\vec{\sigma}\_{-i})\mid\mathcal{R}(I\_{i}))italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ) ≥ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ), where now the expected utility is taken with respect to μIi,π,σ→subscript𝜇subscript𝐼𝑖𝜋→𝜎\mu\_{I\_{i},\pi,\vec{\sigma}}italic\_μ start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_π , over→ start\_ARG italic\_σ end\_ARG end\_POSTSUBSCRIPT. (Kreps and Wilson’s \citeyearKW82 definition of sequential equilibrium is identical, except that there is no distribution π𝜋\piitalic\_π on failure contexts.) #### 3.3.2 Extending σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT to a π𝜋\piitalic\_π-sequential equilibrium We now show that the protocol σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT can be extended to a π𝜋\piitalic\_π-sequential equilibrium with minimal changes. In the proof of Theorem [3](#Thmtheorem3 "Theorem 3. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus"), we showed that i𝑖iitalic\_i could not gain by deviating at an information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT where there were no deviations of type 1–9 prior to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. We did not show that i𝑖iitalic\_i does not gain from deviating at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT if an inconsistency is detected at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, so that i𝑖iitalic\_i is expected to decide ⊥bottom\bot⊥. In fact, if i𝑖iitalic\_i believes that the inconsistency may go unnoticed by other agents due to crashes and consensus may still be reached on some value in {0,1}01\{0,1\}{ 0 , 1 }, then i𝑖iitalic\_i always gains by not deciding ⊥bottom\bot⊥. However, suppose that μ𝑠𝑒superscript𝜇𝑠𝑒\mu^{\mathit{se}}italic\_μ start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT is a belief system such that at an information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT for i𝑖iitalic\_i that is off the equilibrium path due to a deviation (or multiple deviations) from σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT by agents other than i𝑖iitalic\_i, i𝑖iitalic\_i believes that these agents decided ⊥bottom\bot⊥ when they deviated. (Intuitively, i𝑖iitalic\_i believes that if the agents were crazy enough to deviate in the first place, then they were also crazy enough to decide ⊥bottom\bot⊥.) In that case, deciding ⊥bottom\bot⊥ is also a best response for i𝑖iitalic\_i. The belief system μ𝑠𝑒superscript𝜇𝑠𝑒\mu^{\mathit{se}}italic\_μ start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT is not enough to deal with information sets Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT off the equilibrium path due to i𝑖iitalic\_i himself having deviated. Agent i𝑖iitalic\_i cannot believe that it played ⊥bottom\bot⊥ when it in fact did not. To get a sequential equilibrium, we modify σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT at information sets off the equilibrium path that are reached due only to agent i𝑖iitalic\_i’s deviations. Define the strategy σi𝑠𝑒subscriptsuperscript𝜎𝑠𝑒𝑖\sigma^{\mathit{se}}\_{i}italic\_σ start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT so that it agrees with σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT at every information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT where agent i𝑖iitalic\_i has not deviated in the past. Thus, in particular, i𝑖iitalic\_i decides ⊥bottom\bot⊥ with σi𝑠𝑒superscriptsubscript𝜎𝑖𝑠𝑒\sigma\_{i}^{\mathit{se}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT if i𝑖iitalic\_i detects an inconsistency at one of these information sets. More generally, say that an information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is unsalvageable if i𝑖iitalic\_i knows at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT that another agent j𝑗jitalic\_j deviated or detected an inconsistency at a point when j𝑗jitalic\_j had not crashed, and thus decided ⊥bottom\bot⊥. Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is certainly unsalvageable if reaching Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT requires deviations by agents other than i𝑖iitalic\_i (for then the agent that performed that deviation decided ⊥bottom\bot⊥). But even if i𝑖iitalic\_i is the only agent who deviates at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT may be unsalvageable. For example, i𝑖iitalic\_i does not send a message to j𝑗jitalic\_j in round m1subscript𝑚1m\_{1}italic\_m start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, i𝑖iitalic\_i sends a message to j𝑗jitalic\_j in round m2>m1subscript𝑚2subscript𝑚1m\_{2}>m\_{1}italic\_m start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT > italic\_m start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, and then j𝑗jitalic\_j sent a message to i𝑖iitalic\_i in round m2+1subscript𝑚21m\_{2}+1italic\_m start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT + 1, the round-(m2+2)subscript𝑚22(m\_{2}+2)( italic\_m start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT + 2 ) information set where i𝑖iitalic\_i receives j𝑗jitalic\_j’s message is also unsalvageable. If Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is unsalvageable, i𝑖iitalic\_i decides ⊥bottom\bot⊥. Finally, if Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is salvageable, then at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT agent i𝑖iitalic\_i acts in a way that is most likely to have the other agents think that there has been no inconsistency. In general, there may be more than one failure pattern that will prevent a nonfaulty agent from realizing that there is an inconsistency. For example, if f=1𝑓1f=1italic\_f = 1, n=3𝑛3n=3italic\_n = 3, and agent 1111 did not send a message to agent 2222 in round m𝑚mitalic\_m, but did send a message to agent 3, then i𝑖iitalic\_i can either not send a message to any agent in round m+1𝑚1m+1italic\_m + 1, or it can send a message to agent 3. If it is more likely that neither 2 nor 3 failed in round m𝑚mitalic\_m than agent 2 failed before telling agent 3 that it did not hear from 1, then it would be better for i𝑖iitalic\_i not to send a message to 2 or 3 in round m+1𝑚1m+1italic\_m + 1. If there is more than one best response, then i𝑖iitalic\_i chooses a fixed one according to some ordering on actions. (Note that this means that, unlike σ→𝑐𝑜𝑛𝑠superscript→𝜎𝑐𝑜𝑛𝑠\vec{\sigma}^{\mathit{cons}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT, the behavior of σ→𝑠𝑒superscript→𝜎𝑠𝑒\vec{\sigma}^{\mathit{se}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT may depend on π𝜋\piitalic\_π.) Having defined σ→𝑠𝑒superscript→𝜎𝑠𝑒\vec{\sigma}^{\mathit{se}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT, we can now define μ𝑠𝑒superscript𝜇𝑠𝑒\mu^{\mathit{se}}italic\_μ start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT formally. We assume that there are only finitely many actions that i𝑖iitalic\_i can play at each of its information sets Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT: it can send one of KIisubscript𝐾subscript𝐼𝑖K\_{I\_{i}}italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT possible messages and/or decide one of ⊥bottom\bot⊥, 0, or 1 if it has not yet made a decision, or do nothing. Given an integer M>0𝑀0M>0italic\_M > 0, let σ→Msuperscript→𝜎𝑀\vec{\sigma}^{M}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_M end\_POSTSUPERSCRIPT be the strategy profile where at each information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, agent i𝑖iitalic\_i plays σi𝑠𝑒(Ii)superscriptsubscript𝜎𝑖𝑠𝑒subscript𝐼𝑖\sigma\_{i}^{\mathit{se}}(I\_{i})italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) with probability 1−1/M11𝑀1-1/M1 - 1 / italic\_M, and divides the remaining probability 1/M1𝑀1/M1 / italic\_M over all the actions that can be played at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT as follows: if i𝑖iitalic\_i has already decided before, then i𝑖iitalic\_i sends each of the KIisubscript𝐾subscript𝐼𝑖K\_{I\_{i}}italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT possible messages with equal probability 1M(KIi+1)1𝑀subscript𝐾subscript𝐼𝑖1\frac{1}{M(K\_{I\_{i}}+1)}divide start\_ARG 1 end\_ARG start\_ARG italic\_M ( italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT + 1 ) end\_ARG and does nothing with probability 1M(KIi+1)1𝑀subscript𝐾subscript𝐼𝑖1\frac{1}{M(K\_{I\_{i}}+1)}divide start\_ARG 1 end\_ARG start\_ARG italic\_M ( italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT + 1 ) end\_ARG; if i𝑖iitalic\_i has not yet decided at Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, then for each of the KIisubscript𝐾subscript𝐼𝑖K\_{I\_{i}}italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT messages m that it can send, it decides ⊥bottom\bot⊥ and sends m with probability 1M(KIi+1)−1M2(KIi+1)1𝑀subscript𝐾subscript𝐼𝑖11superscript𝑀2subscript𝐾subscript𝐼𝑖1\frac{1}{M(K\_{I\_{i}}+1)}-\frac{1}{M^{2}(K\_{I\_{i}}+1)}divide start\_ARG 1 end\_ARG start\_ARG italic\_M ( italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT + 1 ) end\_ARG - divide start\_ARG 1 end\_ARG start\_ARG italic\_M start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ( italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT + 1 ) end\_ARG, decides ⊥bottom\bot⊥ and sends no message with probability 1M(KIi+1)−1M2(KIi+1)1𝑀subscript𝐾subscript𝐼𝑖11superscript𝑀2subscript𝐾subscript𝐼𝑖1\frac{1}{M(K\_{I\_{i}}+1)}-\frac{1}{M^{2}(K\_{I\_{i}}+1)}divide start\_ARG 1 end\_ARG start\_ARG italic\_M ( italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT + 1 ) end\_ARG - divide start\_ARG 1 end\_ARG start\_ARG italic\_M start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ( italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT + 1 ) end\_ARG, and performs each of the remaining 3(KIi+1)3subscript𝐾subscript𝐼𝑖13(K\_{I\_{i}}+1)3 ( italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT + 1 ) possible actions with equal probability 13M2(KIi+1)13superscript𝑀2subscript𝐾subscript𝐼𝑖1\frac{1}{3M^{2}(K\_{I\_{i}}+1)}divide start\_ARG 1 end\_ARG start\_ARG 3 italic\_M start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ( italic\_K start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT + 1 ) end\_ARG. Clearly σ→Msuperscript→𝜎𝑀\vec{\sigma}^{M}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_M end\_POSTSUPERSCRIPT is completely mixed and the sequence σ→Msuperscript→𝜎𝑀\vec{\sigma}^{M}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_M end\_POSTSUPERSCRIPT converges to σ→𝑠𝑒superscript→𝜎𝑠𝑒\vec{\sigma}^{\mathit{se}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT. Given a round-m𝑚mitalic\_m information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT and global history h∈Iiℎsubscript𝐼𝑖h\in I\_{i}italic\_h ∈ italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, let \| \| \| \| \| --- \| --- \| --- \| \| \| μIi𝑠𝑒(h)=limM→∞πσ→M(h)πσ→M(Ii).superscriptsubscript𝜇subscript𝐼𝑖𝑠𝑒ℎsubscript→𝑀subscript𝜋superscript→𝜎𝑀ℎsubscript𝜋superscript→𝜎𝑀subscript𝐼𝑖\mu\_{I\_{i}}^{\mathit{se}}(h)=\lim\_{M\to\infty}\frac{\pi\_{\vec{\sigma}^{M}}(h)}{\pi\_{\vec{\sigma}^{M}}(I\_{i})}.italic\_μ start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT ( italic\_h ) = roman\_lim start\_POSTSUBSCRIPT italic\_M → ∞ end\_POSTSUBSCRIPT divide start\_ARG italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_M end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_h ) end\_ARG start\_ARG italic\_π start\_POSTSUBSCRIPT over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_M end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) end\_ARG . \| \| The effect of this definition of μIisesuperscriptsubscript𝜇subscript𝐼𝑖𝑠𝑒\mu\_{I\_{i}}^{se}italic\_μ start\_POSTSUBSCRIPT italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_s italic\_e end\_POSTSUPERSCRIPT beliefs is that if Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is off the equilibrium path as a result of some other agent j𝑗jitalic\_j’s deviation, then i𝑖iitalic\_i believes that j𝑗jitalic\_j played ⊥bottom\bot⊥. Moreover, i𝑖iitalic\_i believes that other agents j𝑗jitalic\_j have similar beliefs. Theorem [4](#Thmtheorem4 "Theorem 4. ‣ 3.3.2 Extending 𝜎⃗^𝑐𝑜𝑛𝑠 to a 𝜋-sequential equilibrium ‣ 3.3 A 𝜋-Sequential Equilibrium for Fair Consensus ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") shows that σ→𝑠𝑒superscript→𝜎𝑠𝑒\vec{\sigma}^{\mathit{se}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT is a π𝜋\piitalic\_π-sequential equilibrium for a reasonable and uniform π𝜋\piitalic\_π. ###### Theorem 4. If f+1<n𝑓1𝑛f+1<nitalic\_f + 1 < italic\_n, π𝜋\piitalic\_π is a distribution that supports reachability, is uniform, and allows up to f𝑓fitalic\_f failures, and agents care only about consensus, then (σ→𝑠𝑒,μ𝑠𝑒)superscriptnormal-→𝜎𝑠𝑒superscript𝜇𝑠𝑒(\vec{\sigma}^{\mathit{se}},\mu^{\mathit{se}})( over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT , italic\_μ start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT ) is a π𝜋\piitalic\_π-sequential equilibrium. ###### Proof. Fix an agent i𝑖iitalic\_i, a round-m𝑚mitalic\_m information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, and strategy σisubscript𝜎𝑖\sigma\_{i}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. It is easy to see that μ𝑠𝑒superscript𝜇𝑠𝑒\mu^{\mathit{se}}italic\_μ start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT is consistent. Thus, it suffices to show that \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ui((σi𝑠𝑒,σ→−i𝑠𝑒)∣ℛ(Ii))≥ui((σi,σ→−i𝑠𝑒)∣ℛ(Ii)).subscript𝑢𝑖conditionalsuperscriptsubscript𝜎𝑖𝑠𝑒subscriptsuperscript→𝜎𝑠𝑒𝑖ℛsubscript𝐼𝑖subscript𝑢𝑖conditionalsubscript𝜎𝑖superscriptsubscript→𝜎𝑖𝑠𝑒ℛsubscript𝐼𝑖u\_{i}((\sigma\_{i}^{\mathit{se}},\vec{\sigma}^{\mathit{se}}\_{-i})\mid\mathcal{R}(I\_{i}))\geq u\_{i}((\sigma\_{i},\vec{\sigma}\_{-i}^{\mathit{se}})\mid\mathcal{R}(I\_{i})).italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ) ≥ italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( ( italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , over→ start\_ARG italic\_σ end\_ARG start\_POSTSUBSCRIPT - italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT ) ∣ caligraphic\_R ( italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ) . \| \| (6) \| We need to consider the cases where (a) Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is consistent with σ→𝑠𝑒superscript→𝜎𝑠𝑒\vec{\sigma}^{\mathit{se}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT, (b) Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is inconsistent with σ→𝑠𝑒superscript→𝜎𝑠𝑒\vec{\sigma}^{\mathit{se}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT and unsalvageable, and (c) Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is inconsistent with σ→𝑠𝑒superscript→𝜎𝑠𝑒\vec{\sigma}^{\mathit{se}}over→ start\_ARG italic\_σ end\_ARG start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT and salvageable. In case (a), σi𝑠𝑒subscriptsuperscript𝜎𝑠𝑒𝑖\sigma^{\mathit{se}}\_{i}italic\_σ start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT agrees with σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT; the argument of the proof of Theorem [3](#Thmtheorem3 "Theorem 3. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") shows that it is a best response. In case (b), the definition of μ𝑠𝑒superscript𝜇𝑠𝑒\mu^{\mathit{se}}italic\_μ start\_POSTSUPERSCRIPT italic\_se end\_POSTSUPERSCRIPT guarantees that i𝑖iitalic\_i ascribes probability 1 to whichever agent has deviated or detected a deviation playing ⊥bottom\bot⊥, so it is a best response for i𝑖iitalic\_i to play ⊥bottom\bot⊥. Finally, in case (c), for failure patterns where some other agent j𝑗jitalic\_j detects i𝑖iitalic\_i’s deviation, i𝑖iitalic\_i ascribes probability 1 to j𝑗jitalic\_j playing ⊥bottom\bot⊥, so it does not matter what i𝑖iitalic\_i does. On the other hand, for failure patterns where all the nonfaulty agents will consider it possible that there are no deviations, the proof of Theorem [3](#Thmtheorem3 "Theorem 3. ‣ 3.2.2 A 𝜋-Nash equilibrium ‣ 3.2 Obtaining a 𝜋-Nash equilibrium ‣ 3 Possibility and Impossibility Results for Consensus ‣ Rational Consensus") shows that i𝑖iitalic\_i should continue to play in a way consistent with σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT. If there are several choices of how to play that might be consistent with σi𝑐𝑜𝑛𝑠superscriptsubscript𝜎𝑖𝑐𝑜𝑛𝑠\sigma\_{i}^{\mathit{cons}}italic\_σ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_cons end\_POSTSUPERSCRIPT, then i𝑖iitalic\_i should clearly play one that is best. ∎ 4 Discussion ------------- We have provided a strategy for consensus that is a π𝜋\piitalic\_π-Nash equilibrium and can be extended to a π𝜋\piitalic\_π-sequential equilibrium, where π𝜋\piitalic\_π is a distribution on contexts that allows up to f𝑓fitalic\_f failures and satisfies minimal conditions, as long as n>f+1𝑛𝑓1n>f+1italic\_n > italic\_f + 1. Although our argument is surprisingly complicated, we have considered only the simplest possible case: synchronous systems, crash failures, and only one player deviating (i.e., no coalitions). A small variant of our strategy also gives a Nash and sequential equilibrium even if coalitions are allowed, but proving this seems significantly more complicated. We are currently writing up the details carefully. Of course, things will get even worse once we allow more general types of failures, such as omission failures and Byzantine failures. But such failure types, combined with rational agents, are certainly of interest if we want to apply consensus in, for example, financial settings of the type considered by Mazières \citeyearMaz15. Consensus is known to be impossible in an asynchronous setting, even with just one failure [[9](#bib.bib9)], but algorithms that attain consensus with high probability are well known (e.g., [[5](#bib.bib5)]). We may thus hope to get an ϵitalic-ϵ\epsilonitalic\_ϵ–π𝜋\piitalic\_π-Nash equilibrium in the asynchronous setting if we also allow rational agents. We believe that the techniques developed in this paper will be applicable to these more difficult problems. It is also worth examining our assumptions regarding distributions in more detail. The uniformity assumption implies that no agent is more likely to fail than any other. If all agents can be identified with identical computers, then this seems quite reasonable. But if one agent can be identified with a computer that is known to be more prone to failure, then the uniformity assumption no long holds. Note that the uniformity assumption does allow for correlated failures, just as long as the permutation of a correlated failure is just as likely as the unpermuted version. Now consider the assumption that π𝜋\piitalic\_π supports reachability. If we are considering Nash equilibrium (where there is only one deviating agent), the assumption says that the probability, conditional on an information set Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT (and some assumptions about failures), that some information (about a message sent by an agent that crashes or about the fact that an agent crashed in a particular round) is quite high, where “quite high” is a function of the number of agents M𝑀Mitalic\_M that are nonfaulty according to Iisubscript𝐼𝑖I\_{i}italic\_I start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. Since the more nonfaulty agents there are, the more likely it is that an agent l≠i𝑙𝑖l\neq iitalic\_l ≠ italic\_i is reachable from j𝑗jitalic\_j without i𝑖iitalic\_i. However, once we allow coalitions K𝐾Kitalic\_K of agents, it becomes less likely that l∉K𝑙𝐾l\notin Kitalic\_l ∉ italic\_K is reachable from j𝑗jitalic\_j without K𝐾Kitalic\_K with probability 1/2M12𝑀1/2M1 / 2 italic\_M, not taking K𝐾Kitalic\_K into account. To take an extreme example, suppose that \|K\|=k𝐾𝑘\|K\|=k\| italic\_K \| = italic\_k, f=1𝑓1f=1italic\_f = 1, and n=k+2𝑛𝑘2n=k+2italic\_n = italic\_k + 2. Now suppose that i𝑖iitalic\_i receives a message from agent j′′superscript𝑗′′j^{\prime\prime}italic\_j start\_POSTSUPERSCRIPT ′ ′ end\_POSTSUPERSCRIPT in round 3 that some other agent j𝑗jitalic\_j, from whom i𝑖iitalic\_i got a round 1 message, crashed in round 1. Further suppose that round 1 would be considered clean if j𝑗jitalic\_j did not crash in round 1 and that i𝑖iitalic\_i’s utility would be higher if round 1 is considered clean rather than a later round. Thus, it may be to i𝑖iitalic\_i’s benefit not to forward j𝑗jitalic\_j’s message; if j𝑗jitalic\_j in fact crashes without any nonfaulty agent hearing j𝑗jitalic\_j’s message, i𝑖iitalic\_i will be better off. Since all the agents in K𝐾Kitalic\_K can coordinate in not forwarding j𝑗jitalic\_j’s message, j𝑗jitalic\_j’s message will reach a nonfaulty agent only if either j𝑗jitalic\_j is nonfaulty, or j𝑗jitalic\_j crashes either after round 2 or crashes at round 2, but still sends a message to the nonfaulty agent that is not in K𝐾Kitalic\_K before crashing. Since M=n𝑀𝑛M=nitalic\_M = italic\_n in this case, this means that j𝑗jitalic\_j’s message must reach a nonfaulty agent with probability at least 2n−12n=4k+34k+42𝑛12𝑛4𝑘34𝑘4\frac{2n-1}{2n}=\frac{4k+3}{4k+4}divide start\_ARG 2 italic\_n - 1 end\_ARG start\_ARG 2 italic\_n end\_ARG = divide start\_ARG 4 italic\_k + 3 end\_ARG start\_ARG 4 italic\_k + 4 end\_ARG, independent of k𝑘kitalic\_k. For small k𝑘kitalic\_k, this seems quite reasonable; for large k𝑘kitalic\_k, it does not. This suggests that this assumption is appropriate if k+f𝑘𝑓k+fitalic\_k + italic\_f is not a large fraction of n𝑛nitalic\_n. Our final comment concerns the fairness assumption. While this assumption distinguishes our work from some of the other related work (e.g., [[3](#bib.bib3), [6](#bib.bib6)]), since, as we observed above, a consensus protocol must essentially implement a randomized dictatorship, achieving fairness once we get consensus in the presence of rational and faulty agents is not that difficult; we must simply ensure that the rational agents cannot affect the probability of a particular agent being selected as dictator. We enforce this using appropriate randomization in our protocol. The requirement in [[6](#bib.bib6)] that consensus must be achieved no matter what the deviating agents do turns out to have far more impact on the technical results than the fairness requirement. In any case, we believe that the need for dealing with both rational and faulty agents in consensus protocols is compelling. There is clearly much more to be done on this problem.
12535c37-bd7a-4891-984e-29ae72a1bb59	trentmkelly/LessWrong-43k	LessWrong	Hassa Deega Ebowai or the paradox of religiousness in front of adversity Hasa Diga Ebowai (["Does it mean "no worries for the rest of our lives?"" "Kinda"](http://www.youtube.com/watch?feature=endscreen&NR=1&v=AhxChl9bGl0)) is a song from Trey Parker and Matt Stone's "The Book of Mormon", an affectionate parody of religion in general. A lot of the comedy in that song is drawn from the unexpectedness of the reaction to adversity displayed within. Do listen to it before proceeding. The stereotype is that, when troubled and in a position of weakness, where they have no power over their fates, humans tend to turn towards the LORD for consolation. Especially if the religion promises a good afterlife to the patient, meek and submissive, and a bad one to the defiant and insolent. Even when it doesn't (such as in most denominations of Judaism, AFAIK), people are encouraged to not "curse His rotten" name when everything goes wrong for them and they can't do anything about it (see book of Job). The other side of the stereotype is that, the more powerful, confident and knowledgeable humans become, the less religious they become. This can also be seen on the time axis of a single individual's existence when, young, they care little about sin and the afterlife, and, old, they do nothing but pray all day to make up for all the awful stuff they did (and there might be some genuinely awful behavior in there). So I've been trolling Wikipedia for examples of demographics and populations that would have commonly practiced the cursing of the LORD, but I only found reference to vikings doing that, in a "I won't believe in you, but I will believe in me, and live by my own strength" kind of way, which isn't exactly what I'm looking for. Does anyone here know anything about these different ways people react to adversity, and what they mean from a rationalistic standpoint?
07537ba9-b4c9-48a7-ab11-5f31151d79a5	trentmkelly/LessWrong-43k	LessWrong	A sketch of acausal trade in practice One implication of functional decision theory is that it might be possible to coordinate across causally-disconnected regions of the universe. This possibility has been variously known as acausal trade, multiverse-wide cooperation via superrationality (MSR), or evidential cooperation across large worlds (ECL). I’ll use the MSR abbreviation (since I don’t want to assume that the cooperation needs to be motivated by evidential decision theory). I broadly believe that the assumptions behind MSR make sense. But I think the standard presentations of it don’t give very good intuitions for how it might actually work in practice. In this post I present an alternative framing of MSR centered around the concept of “psychohistory”. Background Assume that humanity colonizes the universe at a large scale (e.g. controlling many galaxies), and also discovers that we live in some kind of much-larger multiverse (e.g. any of Tegmark's four types of multiverse). Assume that humanity becomes a highly coordinated entity capable of making large-scale commitments, and also converges towards believing that some version of superrationality or functional decision theory is normatively correct. Reasoning about psychohistory In order to make accurate predictions about other causally-disconnected civilizations, we would need to develop a scientific understanding of the dynamics of civilization development, and in particular the values and governance structures that other civilizations are likely to end up with. Call this science “psychohistory”. Why should we think that psychohistory is even possible? If we build intergalactic civilizations of astronomical complexity, then they might become more difficult to understand in proportion to our increased collective intelligence. But in many domains that we study, there are a few core principles which provide a great deal of insight (e.g. atoms, DNA, relativity). Surprisingly, this also occurs in domains which involve complex interactions betw
8b9f7278-c291-467f-86d4-c525517d4b02	trentmkelly/LessWrong-43k	LessWrong	Meetup : Atlanta Discussion article for the meetup : Atlanta WHEN: 17 March 2012 06:30:00PM (-0500) WHERE: 2094 North Decatur Road, Decatur, GA 30033-5367 The next meetup will be Saturday, March 17th at 6:30pm at Chocolate Coffee in Decatur: http://www.mychocolatecoffee.com/ 2094 North Decatur Road, Decatur, GA 30033-5367 (404) 982-0790 We will be finishing up the "Mysterious Answers to Mysterious Questions" sequence at the next meeting. As always, any other topics you want to bring up are fair game! Here is the official agenda of our next meeting: http://wiki.lesswrong.com/wiki/Mysterious_Answers_to_Mysterious_Questions * * 1.26 "Science" as Curiosity-Stopper * 1.27 Applause Lights * 1.28 Truly Part of You * 1.29 Chaotic Inversion Please let me know if you have any questions or comments! I look forward to seeing everyone there! ps. join the mailing list! http://groups.google.com/group/atlanta-less-wrong-meetup-group Discussion article for the meetup : Atlanta
0b657e1c-00fb-4f55-9f0d-0277574a8d1b	trentmkelly/LessWrong-43k	LessWrong	Superintelligence 24: Morality models and "do what I mean" This is part of a weekly reading group on Nick Bostrom's book, Superintelligence. For more information about the group, and an index of posts so far see the announcement post. For the schedule of future topics, see MIRI's reading guide. ---------------------------------------- Welcome. This week we discuss the twenty-fourth section in the reading guide: Morality models and "Do what I mean". This post summarizes the section, and offers a few relevant notes, and ideas for further investigation. Some of my own thoughts and questions for discussion are in the comments. There is no need to proceed in order through this post, or to look at everything. Feel free to jump straight to the discussion. Where applicable and I remember, page numbers indicate the rough part of the chapter that is most related (not necessarily that the chapter is being cited for the specific claim). Reading: “Morality models” and “Do what I mean” from Chapter 13. ---------------------------------------- Summary 1. Moral rightness (MR) AI: AI which seeks to do what is morally right 1. Another form of 'indirect normativity' 2. Requires moral realism to be true to do anything, but we could ask the AI to evaluate that and do something else if moral realism is false 3. Avoids some complications of CEV 4. If moral realism is true, is better than CEV (though may be terrible for us) 2. We often want to say 'do what I mean' with respect to goals we try to specify. This is doing a lot of the work sometimes, so if we could specify that well perhaps it could also just stand alone: do what I want. This is much like CEV again. Another view Olle Häggström again, on Bostrom's 'Milky Way Preserve': > The idea [of a Moral Rightness AI] is that a superintelligence might be successful at the task (where we humans have so far failed) of figuring out what is objectively morally right. It should then take objective morality to heart as its own values.1,2 > > > Bostrom sees a number of pros
e70badd7-8d67-4aae-9cef-d7d9a12a1e80	trentmkelly/LessWrong-43k	LessWrong	Is Optimization Correct? 1. Is Optimization Correct? 　The risk of "optimization" has long been recognized in AI alignment, such as the problems of instrumental convergence (Bostrom 2014). Nevertheless, it is difficult for AI designers to escape from the engineering concept of "optimization" because the concept of "optimization" is so strongly rooted in engineering design including AI design. 　In value alignment, AI can be designed to optimize a certain value. Such AI alignment may seem harmless from intuition especially when the value is universally accepted as a good value, such as wellbeing, truth, justice, etc. However, such intuition may be wrong regarding advanced Artificial General Intelligence (AGI). This article introduces the “Optimization Prohibition Theorem” as a concept that AI designers can refer, as an easy-to-understand design guideline of AGI (Okamoto 2024). 　The Optimization Prohibition Theorem is a theorem that prohibits optimization targets based on engineering design principles in AI alignment of advanced AGI. The Optimization Prohibition Theorem is proven, under certain assumptions, as follows: (Proof) (1). (Assumption 1) AI alignment is done so that a plurality of AIs have optimization targets based on engineering design principles. (2). (Assumption 2) A plurality of AIs are powerful AGIs that have sufficient resources and can achieve optimization goals using all means. (3). (Assumption 3) The optimization targets for a plurality of AGIs are different and are not satisfied simultaneously. (4). Under the above assumptions, if AGI1 and AGI2 try to achieve different optimization targets, only AGI2 is the obstacle of AGI1’s optimization. Both AGI1 and AGI2 have sufficient resources to achieve the optimization targets and they use any means to achieve the optimization targets, which will result in conflicts between AGIs, which damage AGIs and infringe human rights by collateral damages. (5). Therefore, the Optimization Prohibition Theorem is established as a
d964b4d1-ff7d-4cf4-9de7-655d8db8f175	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	England & Wales & Windfalls Another huge thanks to Holly Scott, Aryan Yadav, Jide Alaga, Cullen O’Keefe, Haydn Bellfield and Peter Wills. This post has been greatly improved thanks to all your helpful feedback. --- This is the third post in ‘[Towards a Worldwide, Watertight Windfall Clause](https://forum.effectivealtruism.org/s/68dCXfuvykT3RmYy4)’, a sequence I’m writing on the legal viability of the Windfall Clause in seven important common law jurisdictions. Today, I’m discussing the viability of the Clause in England & Wales. How to read this post: ========================== This is a [very long post](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Appendix___Why_is_this_post_so_long_), so you may not want to read everything I have written. If you want to get maximum value from this post in the shortest space of time, I have four recommendations: * Read [my advice](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#How_to_read_this_post_) on how to read these posts - I’ve already given some guidance in an earlier post. You should definitely check that out before you start reading. * (Skim)read [my first post](https://forum.effectivealtruism.org/posts/DJuhFbtJLJ92pCsKW/towards-a-worldwide-watertight-windfall-clause) - This provides a summary of why I think this question is important and why I’ve chosen these specific [issues](https://forum.effectivealtruism.org/posts/DJuhFbtJLJ92pCsKW/towards-a-worldwide-watertight-windfall-clause#Which_Issues_) and [jurisdictions](https://forum.effectivealtruism.org/posts/DJuhFbtJLJ92pCsKW/towards-a-worldwide-watertight-windfall-clause#Which_Jurisdictions_). It also lays out the [defined terms](https://forum.effectivealtruism.org/s/68dCXfuvykT3RmYy4/p/DJuhFbtJLJ92pCsKW#Defined_Terms) that I’ve used throughout the sequence. * Check out ‘[The Windfall Clause has a remedies problem](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem)’ - I’m concerned enough about the lack of remedies for the Developer’s breach of the Clause that I’ve written a separate post addressing this topic. I strongly recommend that you read that, too. * Start at the top, stop reading when it feels irrelevant - I’ve structured this post so that the crucial stuff comes first, with later sections being progressively less important to read. If you’re a non-lawyer, I suggest you just focus on the [takeaways](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Key_takeaways_) and [recommendations](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Recommendations_). If you’re a lawyer, you may want to stop reading after the [good faith](http://www.www) section, or after my discussion of a [shares-based Clause](http://discussion). Summary of issues considered: ================================= \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| Legal question: \| Jurisdiction or Governing Law? \| Is this a problem for the Clause? \| Is this easily fixed? \| \| [Are valuable remedies available for breach of the Clause?](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Are_valuable_remedies_available_for_breach_of_the_Clause_) \| Governing Law \| Yes \| No \| \| [Is a duty of good faith implied into the Agreement?](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Will_a_duty_of_good_faith_be_implied_into_the_Agreement__If_not__can_such_a_duty_be_expressly_introduced_) \| Governing Law \| Maybe \| No \| \| [Is the Clause a breach of national competition law?](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Would_industry_wide_adoption_of_the_Clause_constitute_a_breach_of_UK_competition_law_) \| (Usually) Jurisdiction \| Maybe \| Yes \| \| [Can the Developer easily issue new shares?](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#If_the_Clause_allows_the_Developer_to_pay_the_Counterparty_by_issuing_share_options__rather_than_paying_cash__what_steps_must_be_taken_to_create_and_issue_these_shares_) \| Jurisdiction \| Maybe \| Maybe \| \| [Is agreeing to the Clause a breach of directors’ duties?](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Could_signing_the_Agreement_be_a_breach_of_a_directors__duty_to_act_in_the_best_interests_of_the_company_) \| Jurisdiction \| No \| N/A \| \| [Can the Counterparty give good consideration for the Agreement?](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#If_the_Agreement_is_a_binding_contract__what_would_constitute_good_consideration_) \| Governing Law \| No \| N/A \| \| [Can a purely donative Agreement be enforced?](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#If_the_Agreement_is_purely_donative__could_it_still_be_enforced_) \| Governing Law \| Yes \| Yes \| \| [Are there legal restrictions on size or nature of donations?](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#What_are_the_legal_restrictions_on_the_size_and_nature_of_charitable_donations__if_any_) \| Jurisdiction \| Maybe \| Yes \| \| [Can dividends be paid to avoid triggering the obligation?](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Can_a_company_pay_out_dividends_to_shareholders_before_meeting_its_obligations_under_the_clause__If_so__is_it_possible_to_contract_around_this_) \| Jurisdiction \| No \| N/A \| Key takeaways: ================== Although English[[1]](#fnscbs51yfydq) and Delaware contract and company law are similar, there are several differences which could make it difficult to enforce a Windfall Clause in the United Kingdom. I am very concerned about: * The de facto absence of remedies for breaches of the Agreement - The common law approach to calculating damages for breach of contract is unlikely to adequately compensate the Counterparty and there are a wide range of scenarios where equitable remedies for the Developer’s breach will be unavailable. This could render the Agreement practically unenforceable. This is a big enough issue that I’ve made a separate post about the topic, which you can find [here](https://docs.google.com/document/d/1r0-T8AzPpSicWzVICnL-UUmjgpHZWT4SSie7i48eINw/edit). I am somewhat concerned about: * The enforceability of good faith obligations - English courts are notoriously hostile to good faith clauses and might be unwilling to construe good faith obligations widely enough to prevent the Developer from going against the spirit of the Clause. I am mildly concerned about: * The risk that industry-wide adoption of the Windfall Clause is declared a breach of competition law - EU and UK case law suggests that the Competition and Markets Authority might find widely-publicised adoption of the Clause to distort competition and so declare it void. This seems unlikely, but the serious repercussions of breaching competition law make it worth guarding against. * Constitutional restrictions which may prevent the implementation of a shares-based Clause - If a Developer and its parent company have unamended Articles of Association and unsupportive shareholders, it would be practically impossible to implement a shares-based Clause. Thankfully, this won’t be a problem for DeepMind, which is a wholly-owned Alphabet subsidiary. Nonetheless, the possibility of changes to the structure of the UK AI industry makes it worth considering how to ensure the option of a shares-based Clause remains available across the sector. I am not concerned about: * The risk that signing the Agreement is a breach of directors’ duties. * Issues with enforceability relating to the absence of consideration. * Possible restrictions on the size and nature of charitable donations. * The risk that the Developer will pay its shareholders dividends to avoid triggering the Clause. Recommendations: ==================== Drafters of an English Windfall Clause should take the following steps: 1. Set out the purpose of the Agreement in the recitals - This will support a finding of bad faith against a Developer who deliberately acts against the spirit of the Agreement. 2. Include general and specific good faith provisions in the contract, along with express prohibitions on the Developer’s behaviour - Taken together, these should substantially reduce the risk that the Developer can deliberately frustrate the contract. 3. Refrain from loudly publicising a Director’s signatory status - To avoid violating competition law, neither the Developer nor any associated industry bodies should loudly champion the Windfall Clause or encourage other Developers to sign up, or require the Counterparty to do either of these things. 4. Make the promotion of economic growth an objective of the Agreement and a charitable purpose of the Counterparty - This may encourage a favourable interpretation of the Agreement by the UK competition authority. 5. Contact the CMA/DMU for a short-form opinion on the Agreement - This could help the drafters make sure that the Agreement is not in breach of competition regulations before it is implemented. 6. (*In the event of an industry shakeup*) Encourage Developers to adopt constitutions which allow for a shares-based Clause - It will be easier to implement a shares-based Clause if the Articles already support this. This can be supplemented by outreach to investors concerning the benefits of the Clause. 7. Ensure the Counterparty gives consideration and/or draft the Agreement as a deed - Either option will avoid the risk that the court finds the agreement unenforceable for want of consideration. Drafters might include both options to provide helpful redundancy. 8. Ensure the Counterparty remains apolitical - So long as the Counterparty does not promote political causes, it should not encounter issues with charity regulations. I have also written recommendations for improving the availability of remedies for breach of the terms of the Agreement. You can find those [here](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Takeaways_and_recommendations_). Core Assumption - Governing Law and Jurisdiction are the same: ================================================================== A core assumption of this post is that the Developer will choose the law of the state in which it is domiciled as the governing law of the Agreement. This is a reasonable assumption because a Developer will likely seek to benefit from home-court advantage - the fact that its legal team and domestic courts will have a deeper understanding of their domestic law and so be better placed to predict outcomes. This is a big simplification nonetheless as it's still common for multinational corporations to enter into contracts governed by a foreign law. Relaxing this assumption should lead to a modest positive update in favour of the viability of the Windfall Clause because, in reality, it will be possible for a Developer to shop around for the most appropriate governing law for the Agreement. This will be a particularly relevant consideration in later posts, as companies domiciled in developing countries are far likelier to choose a foreign governing law. To help anyone who would like to explore this issue further, I have highlighted which issues relate to governing law and which relate to jurisdiction in the table above. The Legal Issues: ===================== Are valuable remedies available for breach of the Clause? ------------------------------------------------------------- In short: no, they are not. [Check out my post on this topic here](https://docs.google.com/document/d/1r0-T8AzPpSicWzVICnL-UUmjgpHZWT4SSie7i48eINw/edit#). Will a duty of good faith be implied into the Agreement? If not, can such a duty be expressly introduced? ------------------------------------------------------------------------------------------------------------- ### Recommendations: English courts are characteristically hostile to good faith duties, believing them to erode contractual certainty. As such, the drafters of an English Agreement must be aware of the limitations of good faith and take special care when drafting good faith provisions to ensure that they are interpreted widely enough by the courts. My recommendations here are as follows: 1. Explain the purpose of the Clause in the recitals - To ensure that the parties’ goals in signing the Agreement are unambiguous and to encourage a court interpretation which is favourable to the Counterparty, drafters of an English Windfall Clause should include lengthy recitals at the beginning of the Agreement explaining its purpose and objectives. 2. Include both general and specific good faith provisions - To encourage an expansive interpretation of good faith duties, a more general duty of good faith between the parties should be accompanied by provisions outlining specific situations in which the Developer and Counterparty would need to act bona fides. 3. Include express prohibitions on the Developer’s behaviour which do not rely on good faith - Given the risk that courts will discard or narrowly interpret even the broadest good faith provisions, drafters should include as many prohibitions as the Developer will permit on specific harmful behaviours which the drafters perceive as potential failure modes for the Agreement. ### The Law: English courts will rarely imply good faith into a contract and have been traditionally hostile even towards express duties, preferring to exclude such provisions or interpret them as narrowly as possible. However, this is a rapidly evolving area of the law, and the traditional approach of English courts may be changing.[[2]](#fnw9u8ahcketc) Lower courts will now interpret an express good faith clause as requiring parties to: 1. act honestly; 2. be faithful to the parties’ agreed common purpose; 3. not use any contractual discretion for an ‘ulterior purpose’; 4. deal fairly and openly with one another; and, 5. ‘have regard’ to one another’s interest when making decisions.[[3]](#fn79t5owrscf9) 2. is perhaps the most important of these in the context of the Windfall Clause. If enforceable, it would allow the Counterparty to prevent the Developer from deliberately frustrating the purpose of the Agreement by, for instance, trading away WGAI to another company in its group structure.[[4]](#fnnpn2i6hx8kq) Unfortunately, recent judicial approval of expanded good faith obligations comes almost entirely from the High Court - the Court of Appeal has considered the topic of express good faith clauses just once in the last decade. In that case, the Court of Appeal aggressively narrowed the scope of an express good faith clause, stating that it applied only to two specific areas of the contract and was not an overriding duty applicable to all the parties' obligations under the contract.[[5]](#fnw1c0leb7oos) Given the changing state of the law since this ruling, it is difficult to say how broadly the higher courts will construe express good faith clauses in the future. ### How does this affect the viability of the Clause? The authors of the original Windfall Clause report indicate that some of the Developer’s potential escape routes from the Clause can be foreclosed by the duty of good faith implied into American contract law.[[6]](#fngexplwo9hy8) I would echo this assessment. Given the colossal sums at stake in a windfall-generating scenario, the possibility of a get-out would strongly incentivise the Developer to try its luck with a carefully calculated breach of contract.[[7]](#fnitrwrpnufoh) Meanwhile, given the unusual nature of the Agreement, the drafters are unlikely to anticipate all possible failure modes, increasing the likelihood that such a get out exists. Even setting aside the legal loopholes which might be discovered by an advanced AI system, it seems risky to assume that a team of highly-skilled lawyers couldn’t find a way for the Developer to limit its obligations. This makes the inclusion of robust good faith duties essential to enforce performance. Unfortunately, uncertainty as to the scope of good faith obligations in English law could present a serious problem for the Clause - if the higher courts construed the Developer’s good faith duties too narrowly, it would become practically impossible to enforce the terms of the Agreement. For example, if the Supreme Court decided that a duty to act in good faith did not prevent the Developer from selling WGAI if such a sale was financially desirable for the Developer’s shareholders, the shareholders may be able to force a sale even if this would substantially frustrate the Agreement.[[8]](#fn0bnvjt2jkngc) A similar outcome would result if the Supreme Court rejected a general duty of good faith altogether. The risk of narrow interpretation motivates [recommendation 1.](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Recommendations_1) By explaining the purpose and objectives of the Clause in the recitals, the court will be encouraged to find bad faith where the Developer has clearly acted against that purpose. Recitals are not binding in English law, however, when construing the terms in a contract, the court will have regard to ‘all the background knowledge’ which both parties would have known at the time of the contract,[[9]](#fnamwne140l2c) and the inclusion of the Clause’s purpose at the head of the Agreement will make it difficult for the Developer to deny its obligations. To encourage this interpretation, the recitals might also expressly link the requirements of good faith with the incompleteness of the contract, outlining that the novelty of the Agreement means the parties will not have considered all possible contingencies and intend to rely heavily on good faith for its enforceability. This risk also motivates [recommendation 2.](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Recommendations_1)The reticence of English courts to interpret good faith duties expansively might be counteracted by the explicit separation of a general duty of good faith, expressed to be an organising principle in the interpretation of the contract,[[10]](#fnl28ze6qi4nl) from a set of more specific good faith clauses.[[11]](#fnvxz1w3yxn3k) For example, the parties might agree that the Developer should always act with good faith when negotiating the sale or licensing of intellectual property in any of its AI systems - if the Developer went on to knowingly sell WGAI, this would then prevent the Developer from claiming that it was free to act in its commercial interest in doing so. Specific clauses such as this could be strengthened by including a description of what good faith would actual entail in this particular context. For example, the parties could stipulate that a Developer acting bona fide would always notify the Counterparty of its intention to agree to a licence or sale and inform the Counterparty of the general structure of the contract, making it hard for a Developer to claim good faith if it sold WGAI off in secret. Specific duties of good faith such as this would therefore make it hard for a Developer to argue that that it was not subject to good faith duties in possible breach scenarios. Meanwhile, the existence of a further general duty to act bona fide - distinct from these specific requirements - would encourage the court to interpret all behaviour by both parties as being subject to requirement of good faith. ### Alternatives to English good faith: Unfortunately, the above two recommendations are not magic bullets. Firstly, there remains the risk that hostile English courts will simply refuse to accept such expansive duties, as they have repeatedly done in the past. Secondly, these recommendations do little to prevent inadvertent frustration of the Agreement,[[12]](#fn22qd7mmf94w) where the Developer unknowingly places (pre-)WGAI and any related windfall profits beyond the Counterparty’s reach.[[13]](#fnwylx2219sfb) Thirdly, even where the court is willing in principle to accept that the Developer could have acted mala fide, the [evidential burden remains high](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Issue_4___The_evidential_burden_to_be_discharged_by_the_Counterparty_remains_high_), and it will be a challenge for the Counterparty to demonstrate bad faith, particularly if the Developer has taken steps to obfuscate its intentions. A recognition of the outstanding issues with English good faith duties motivates [recommendation 3.](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Recommendations_1) Express prohibitions are helpful as they act as a backstop, allowing for a remedy even where a court is unwilling or unable to find bad faith. For example, the drafters might insert a requirement that the Developer must give the Counterparty a day’s notice before it grants any IPR to a third party as consideration for the issue of shares.[[14]](#fnoj0j1bzgdh) If the Developer then attempted to [transfer pre-WGAI to a third party](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Issue_3___There_is_no_obvious_way_to_stop_a_third_party_from_acquiring_rights_in_the_relevant_IP_) to sidestep its obligations under the Agreement, this would allow the Counterparty to seek an injunction to prevent it from doing so. Considering possible failure modes for the Agreement and then expressly prohibiting them in this way will thus help to make the contract more robustly enforceable than it would be relying on good faith alone. Before moving on, I should also note that issues with English good faith may justify choosing a foreign governing law for the Agreement, even if England remains the chosen jurisdiction.[[15]](#fnms7imgsslpa) I will not consider this further here, but I intend to discuss the topic later in the sequence. Would industry-wide adoption of the Clause constitute a breach of UK competition law? ----------------------------------------------------------------------------------------- Note - Haydn Bellfield and Shin-Shin Hua also have a [paper](https://yjolt.org/sites/default/files/23_yale_j.l._tech._415_ai_antitrust_nov_0.pdf) on competition law and the Windfall Clause. Much of this section is a repeat of their findings, but they reach slightly different conclusions to mine. I encourage you to read both analyses and form your own opinions. ### Recommendations: To avoid issues with UK competition law, the parties to an English Agreement should take the following steps: 1. Ensure that neither signatories nor industry bodies publicly encourage other Developers to agree to the Clause - To avoid the risk that industry-wide adoption is seen as a prohibited ‘concerted practice’, signatory Developers, industry bodies and the Counterparty should neither encourage competitors to sign an Agreement nor loudly publicise a Developer’s signatory status. 2. Make the promotion of economic growth an objective of the Counterparty - If the Counterparty were a charity, it might adopt as one of its charitable objectives the promotion of free markets and economic progress. This would make it more difficult for the CMA to claim that industry-wide adoption of the Clause was anti-competitive or failed to promote economic progress in society at large. 3. Include the promotion of economic growth as one of the objectives of the Agreement as stated in the recitals - The justification here is the same as for recommendation 3. 4. Contact the CMA/DMU for a short-form opinion - The CMA offers free, non-binding advice on novel or unresolved questions about the application of competition law. A short-form opinion could help Developers avoid accidental infringements. ### The Law: Under UK competition law, agreements or ‘concerted practices’ between organisations which have a distortionary effect on trade within the UK are prohibited.[[16]](#fnegdupi9fltv) This prohibition covers more than signed contracts - competition rules are interpreted teleologically, meaning the CMA will stretch the meaning of terms in legislation if this is needed to promote competitive markets.[[17]](#fnnpz2mzp9vjo) Consequently, terms like ‘agreement’ and ‘concerted practice’ can include policies promoted by representative industry bodies or unspoken arrangements which involve no more than a ‘meeting of minds’. Importantly for our purposes, this will even include a unilateral public announcement by an organisation, provided that such an announcement reduces competitors’ uncertainty about that organisation’s future commercial behavior.[[18]](#fnvis6tlk515) If the CMA finds that such an arrangement constitutes a violation of competition law, the arrangement is automatically void, the guilty parties become liable to pay fines of up to 10% of total worldwide turnover,[[19]](#fnxtlj1xo8mor) and the directors of the Developer may be disqualified or receive criminal penalties.[[20]](#fnef0n8y7dvd6) ### How does this affect the viability of the Clause? Issues with competition law might arise if the Windfall Clause is adopted by multiple AI labs and widely publicised, either by industry bodies like the Partnership on AI[[21]](#fn0zny3qn8i5xa) or by the labs themselves. If this behaviour was considered a concerted practice and did not qualify for any exemptions it would be void in UK law, allowing Developers to abandon their obligations under the Agreement with impunity.[[22]](#fn0ab8i5eox4hm) Windfall Clauses as a concerted practice: The Windfall Clause is ostensibly a bilateral agreement between the Counterparty and the Developer. However, case law indicates that widespread adoption of the Clause might be considered part of an industry-wide concerted practice between Developers if each signatory to the Agreement went on to widely publicise their signatory status and encourage other Developers to sign up. The risk in such a scenario is that the CMA would consider Developers to be deliberately reducing uncertainty about their future commercial behaviour as an implicit invitation to other Developers to engage in similar anti-competitive practices. More speculatively, a concerted practice might be found for similar reasons if an industry body like PAI heavily promoted the adoption of the Clause.[[23]](#fnjp44bcf3zm) If the CMA took such an interpretation this would be catastrophic for the Clause because every Agreement made within or outside the UK would be void and unenforceable in English law. Admittedly, this interpretation is a stretch. Signatories can point to immediate justifications for signing the Agreement such as improved employee relations and public goodwill, making it hard to claim that the Clause is a cynical attempt to restrict competition. Furthermore, it is not clear that industry-wide adoption would have anticompetitive effects. However, as I will discuss shortly, both the Developer and CMA might face private incentives to interpret industry-wide adoption uncharitably following the achievement of windfall profits. Furthermore, certain scenarios present a greater risk of such an interpretation. For example, industry-wide adoption following the onset of an AI race might reasonably be interpreted as anticompetitive,[[24]](#fn745lr2lvz6g) because one Developer’s decision to sign up to a Windfall Clause could reduce competitive pressures on other Developers by reducing the funds available to the signatory for reinvestment in R&D.[[25]](#fnbttw7g7c2ve) This provides a reasonable justification for taking at least some steps to mitigate the risk that industry-wide adoption of the Clause is considered anti-competitive. The risk of such an interpretation motivates [recommendation 1.](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Recommendations_2)The CMA offers free, non-binding advice to companies on novel questions about the application of competition law.[[26]](#fnzxs83kui44f) It may be worth requesting a short-form opinion from them on the circumstances in which industry-wide adoption of the Clause would violate competition law. This would allow industry bodies and the parties to the Agreement to take steps to avoid any infringement of the regulations with a better understanding of how the CMA would interpret their behaviour. This risk also motivates [recommendation 2.](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Recommendations_2) Provided that industry bodies and Developers themselves steer clear of actively promoting the Clause, it seems unlikely that industry-wide adoption will be considered a concerted practice. Drafters should also take care to exclude any provisions requiring the Counterparty to promote a Developer’s signatory status, as the CMA might interpret this as a roundabout attempt by a Developer to encourage industry-wide adoption. As I highlight in [the post on remedies](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Sidestepping_Specific_Performance_), excluding such provisions presents a risk that the Agreement will be unenforceable for want of consideration, if the Counterparty intended to offer consideration in the form of publicity. Nonetheless, the risk that the Agreement is rendered void seems important enough that the Counterparty should explore alternative forms of consideration. The ‘technical or economic progress’ exception: Another possible solution which could be used to justify public promotion of the industry-wide adoption of the Clause is that, under UK competition law, an arrangement between organisations which otherwise breaches competition law is exempt provided it satisfies four specific criteria. These are: 1. It promotes technical or economic progress; 2. It allows consumers a ‘fair share’ of the resulting benefit; 3. It only imposes restrictions which are ‘not indispensable’ to attaining these objectives; and, 4. It doesn’t help the parties to the arrangement eliminate their competition. Although industry-wide adoption of the Clause likely meets criteria 2.-4., it is not clear whether the Windfall Clause promotes technical or economic progress. The issue is that the CMA’s definition of these concepts is surprisingly narrow, only including progress which is purely economic in nature and discounting even pseudo-economic benefits like improved working conditions or better public health.[[27]](#fnjtue9evvp8h) Recent pronouncements by the CMA suggest that it might broaden this definition in the future, but there have yet to be any enforcement decisions by the authority reflecting a change of tack.[[28]](#fnwrzu79nkn58) Even assuming the CMA retains its current definition, it is certainly possible to argue that the Windfall Clause promotes economic progress. For instance, its proponents might claim that the redirection of windfall profits towards scientific research organisations could spur innovation, further expanding the British economy. Yet there are also reasonable arguments to the contrary. A skeptical CMA might argue that industry-wide adoption of the Clause reduces investment capital available to AI firms, slowing the rate of technological advancement. Alternatively, they might claim that increased expenditure by top firms due to windfall distributions indirectly increases prices for AI-powered goods and services, ultimately harming the consumer. Belfield and Shua raise a further concern that signatory Developers would have an incentive to reduce output as they approach profit levels that would trigger the Clause to avoid their more onerous distribution obligations.[[29]](#fnqd4yeuj2mb) All this makes it highly unclear whether or not the CMA would uphold the Clause on economic grounds. My uncertainty here is increased when considering the private incentives of the Developer and the CMA. Setting aside the merits of each argument, as a government body the CMA might be incentivised to take the interpretation which best responded to political exigencies following the Developer’s achievement of windfall profits. Therefore, if the UK government felt it could extract greater rents by directly taxing or nationalising a Developer with WGAI, or if it believed that an unencumbered Developer would be better positioned to promote British national interests, policymakers might pressure the CMA to deliberately misinterpret the Clause. At the same time, a successful Developer may have little incentive to fight this uncharitable interpretation. Provided the risk of criminal sanctions for its directors appears minimal, the Developers may benefit from accepting a CMA fine in return for permanent release from their Agreement obligations.[[30]](#fnho2j73qq4kc) In light of these risks, [recommendations 3. and 4.](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Recommendations_2) provide a simple, low-cost way of avoiding an infringement. They do so by improving the odds that the Clause qualifies for the economic progress exemption: if the stated purpose of the Agreement is the promotion of economic progress and the Counterparty can point to projects it has already undertaken to promote a healthy British economy then it will be far more difficult for the CMA to claim that the adoption of the Clause has anticompetitive effects. ### Why I don't think there's a serious risk of infringement: Ultimately, I find it unlikely that UK competition law will present a problem for the Windfall Clause.[[31]](#fnjyiox85r7m) It’s true that if industry-wide public adoption of the Windfall Clause was considered a concerted practice, and if the CMA did interpret the Clause as failing to promote economic progress, then the Agreement would be unenforceable. However, both of these arguments are somewhat speculative, and a voiding of the Agreement would require the [conjunction](https://www.lesswrong.com/posts/QAK43nNCTQQycAcYe/conjunction-fallacy#:~:text=The%20conjunction%20rule%20of%20probability,is%20called%20a%20conjunction%20fallacy.) of them both. Furthermore, there appear to be cheap and effective ways to both prevent a breach of competition law and ensure that the economic progress exemption applies. In any event, it seems worthwhile taking steps such as these to avoid even small risks of an infringement, given the serious implications for the enforceability of the Clause of any violation. Such steps will also offer comfort to the Developer and its directors, who may be averse to enter an Agreement which presents even a small risk of fines or criminal penalties. If the Clause allows the Developer to pay the Counterparty by issuing share options, rather than paying cash, what steps must be taken to create and issue these shares? ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Note - this won’t create any problems for instituting a shares-based Clause at DeepMind. If you believe DeepMind is and always will be the sole realistic English candidate to develop WGAI, you may want to skip this section. ### Recommendations Before implementing a shares-based Clause, proponents of an Agreement should consider the following steps: 1. Investigate the Articles of Association of top Developers - If the Articles are unfavourable to a shares-based Clause, proponents of the Clause may need to consider cash-based solutions.[[32]](#fnzr0mz4t0mhs) 2. Engage in investor outreach at top Developers on the benefits of the Windfall Clause - Strong investor support for the Agreement could prove essential to implementing a shares-based formulation of the Clause. Strong support could also bring an added benefit of encouraging Developers’ directors to agree to the contract. 3. Encourage founders to amend their Articles to support a shares-based Clause - Founders typically own most of the shares in their business, so it will be easier to implement a shares-based Clause before additional investors come on board.[[33]](#fn20gjdq8ercp) If proponents of the Clause are convinced of the benefits of a shares-based clause, it could be worth putting some time into early outreach to founders. ### The Law: The current procedure for issuing new shares in a limited company is simple but requires supermajority shareholder consent. Specifically, a simple majority of shareholders must vote in favour of a fresh issue of shares and a subsequent supermajority (75%) is needed to disapply shareholders’ pre-emption rights to purchase the shares.[[34]](#fnr78pr0u2at) If the company wants to create and issue a new class of shares, it will also need supermajority support to create those shares.[[35]](#fndjw15g8h08c) Note that these are only the default rules. All of the above procedures can be varied by special resolution, by the court, or on the company’s incorporation. ### How does this affect the viability of the Clause? Under the status quo: Provided that DeepMind remains England’s top contender to develop WGAI, it’s highly unlikely that procedural requirements will present any barriers for the implementation of a shares-based Clause. To understand why, it’s worth briefly outlining the key obstacles to implementation: 1. The attitudes of the Developer’s shareholders towards the Clause - If the requisite plurality / majority of shareholders oppose the Clause, it will be impossible to use a shares-based Clause in the Agreement. 2. The organisation’s voting rules - As I’ve noted earlier, the procedures for creating and issuing a new class can be varied. If the voting thresholds are different in any given Developer, the viability of the shares-based Clause will vary accordingly. 3. The current classes of share in existence - The specifics of any share issuance to the Counterparty remains an open question. However, a class of shares might already exist which meets the requirements of a shares-based Clause.[[36]](#fnf8qkwdbkxi9) Subject to certain additional conditions, this could obviate the need for supermajority support for the Agreement by removing the requirement to approve an amendment to the Articles of Association.[[37]](#fngdmobse39i) As a [wholly-owned Alphabet subsidiary](https://find-and-update.company-information.service.gov.uk/company/12181850/persons-with-significant-control), DeepMind is not held back by any of these obstacles. Their subsidiary status means that there’s minimal risk of shareholder conflict, which in turn makes organisational voting rules and current classes of share largely irrelevant, as the Developer will always be able to achieve the requisite supermajority needed to implement the Clause. This should provide comfort to proponents of a shares-based Clause. Following an industry shakeup: Unfortunately, the absence of obstacles to implementation outlined above is contingent on the current structure of the UK’s AI industry. There are two reasons to believe that this structure may change: 1. DeepMind may not always be a Google subsidiary or have the same corporate structure - DeepMind has made repeatedly sought [greater independence from Google](https://www.wsj.com/articles/google-unit-deepmind-triedand-failedto-win-ai-autonomy-from-parent-11621592951), and it seems plausible that the company will one day achieve it. Investors’ attitudes and constitutional considerations could be much more important to achieving the relevant supermajorities at an autonomous organisation. 2. DeepMind may not always be England’s top contender to develop transformative AI - Right now, DeepMind has no true competitors in the UK; however, there is a small chance that a competitor firm could form in the UK, particularly for those with longer timelines.[[38]](#fnbh70i8fwig) If a new Developer lacked the constitutional framework or shareholder support to implement a shares-based Clause, it might be practically impossible to do so. These possibilities motivate [recommendations 1.-3.](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Recommendations) Engagement with founders and a greater awareness of DeepMind’s (and other Developers’) constitutional structure would help ensure that it is possible to implement a shares-based Clause even if industry circumstances change. At the same time, engagement with investors would increase the likelihood of achieving sufficient support for the Clause to pass the resolutions needed to implement it. Whilst DeepMind retains its status as a wholly-controlled Alphabet subsidiary and England’s AI hegemon, there is little value in taking these steps. However, these steps may prove extremely important if the country’s AI industry experiences any significant structural shifts. A note on the cash-based Clause: One final thing to note is that, if a shares-based Clause was equally as viable as a cash-based Clause the above recommendations might be unnecessary, as a Developer unable to implement the former could simply opt for the latter. However, as I discussed [earlier in the sequence](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Sidestepping_Specific_Performance_), a shares-based Clause may offer significant advantages over a cash-based Clause in terms of available remedies. This provides additional justification for the recommendations I outlined above, as the parties may not have the option of falling back on cash distributions.[[39]](#fnty66voqjt4p) Could signing the Agreement be a breach of a directors' duty to act in the best interests of the company? ------------------------------------------------------------------------------------------------------------- It’s highly unlikely that signing the Agreement will be a breach of the directors’ duty to act in the best interests of the Developer, because English courts are generally reluctant to interfere with businesses’ freely-made commercial decisions. The original report dedicates considerable space to discussing the topic of business judgment,[[40]](#fnud2d1uk3cy) so it seems worth explaining why the same issues are unlikely to apply in English law. This is because English courts take a passive approach to questions of business governance,[[41]](#fnd7sog8n0uxg) simply requiring directors to act in the way they honestly believe will promote the success of the company.[[42]](#fn94zcb1mkz27) The decisions of a board cannot be appealed ‘on merits’,[[43]](#fnksncbrllov) and a court will only overrule a decision by a director if it is clearly taken in bad faith or it falls below the ‘Wednesdbury unreasonableness’ standard, meaning it was so unreasonable that no reasonable director could have considered it to be in the company’s best interests.[[44]](#fn61d3p78dq18) For the directors of a Developer to be in breach of their duty in signing the Agreement, they would need to fall below this standard. This appears highly unlikely because there are good commercial reasons why a Developer might agree to the Clause, already outlined in FHI’s report.[[45]](#fnyldcj3bv7e) To reproduce their argument in brief: a board can justify agreeing to the Clause on the grounds that it generates goodwill, attracts talented researchers with ethical concerns about advanced AI, and reduces political risk, in exchange for a very small chance of a payout. Despite the large ex post costs of the Clause for a successful Developer, this is a reasonable tradeoff for a board to make ex ante. Directors of a successful Developer who signed the Agreement can thus claim that they were acting in the best interests of the company at the time and, as such, did not breach their duties.[[46]](#fniqpwt2tumj) This argument is particularly compelling in the English context, given the low bar for business judgment set by Wednesbury. Even if signing the Agreement is not an ideal business decision, it seems unlikely that a court would determine that no reasonable director could have agreed to it. It’s also worth highlighting the advantages of the English definition of ‘success of the company’, which doesn’t just consider shareholder value but requires directors to have regard to issues like the long-term consequences of their decisions and their company’s environmental impact.[[47]](#fnitobi6hp9al) This broad definition indicates that directors of a Developer could justify signing an Agreement for pro-social reasons, even if they could not do so on a purely economic justification. It tandem with the low requirements set by Wednesdbury, this strongly indicates that the directors of a Developer will not breach their duties to the company in signing the Agreement. If the Agreement is a binding contract, what would constitute good consideration? ------------------------------------------------------------------------------------- English law is famously hands-off when it comes to questions of consideration, and courts will readily uphold contracts so long as something of value has been provided by both sides.[[48]](#fn1g3brsi8n9t) This is true no matter how one-sided the contract turns out to be.[[49]](#fnwvpmq289sfj) Given this, there is no reason to be concerned that the Windfall Clause will fail for want of consideration. Assuming the Counterparty provides something of value in return, such as a small payment or grantmaking support for other charitable donations made by the Developer, then it will be able to enforce the terms of the Agreement.[[50]](#fnwxqtsyd7e08) If the Agreement is purely donative, could it still be enforced? -------------------------------------------------------------------- Short answer: yes. Slightly longer answer: although promissory estoppel is not an independent cause of action in English law, a valid deed is enforceable in the absence of consideration.[[51]](#fn3st3wxoo8t3) An Agreement could thus be executed as a deed if the drafters wanted to avoid issues with consideration.[[52]](#fn71rc85wrjwi) What are the legal restrictions on the size and nature of charitable donations, if any? ------------------------------------------------------------------------------------------- Note - this is the least-researched part of this post, because it didn’t seem that important. Still, it’s possible I’m misinterpreting the rules around corporate donations or charity law so I’d appreciate feedback from anyone with more experience of the topic.[[53]](#fnt8aw589yyel) Corporate donations are largely unregulated in the UK. Most importantly for our purposes, there do not appear to be any limitations on the size of donations which a company may make, although it’s plausible that absurdly large donations could be a breach of a board’s duty to promote the success of the company. As such, I do not anticipate that English charity regulations will be an issue for the viability of the Clause. That said, there are two issues which are worth flagging here. I have outlined each one in turn, and each is accompanied by a simple solution: 1. English companies cannot make political donations without majority shareholder support - This is construed widely and includes donations to support activities ‘intended to affect public support for a political party, organisation or candidate, or to influence voters’.[[54]](#fn4knmx7wed12) If a company does so without shareholder consent, its directors can be liable to the company for the full amount donated. It is unlikely, although possible, that the Counterparty could engage in political activity - for example, the Counterparty might inadvertently become involved in politics by supporting politicians with strong stances on climate change or lobbying a populist party to institute democratic reforms.[[55]](#fn1fudcxv8odx) To minimise the legal risks of any such activity, the Counterparty should refrain from any overtly political donations. 2. English charities may not have a political purpose - If an organisation’s purpose is chiefly political, it will not be recognised as a charity in English law. What counts as ‘political’ is not clearly defined but it is narrower than one might expect - it has previously included organisations like Amnesty International and the Anti-Vivisection Society.[[56]](#fnlx0pplys6s) The Charity Commission has become more permissive in recent years, allowing political action by charities so long as it is only ‘incidental’ to the objectives of the organisation.[[57]](#fnxzusaqh27w) Nonetheless, if the Counterparty is to be domiciled in the UK, it should take care to avoid having expressly political objectives. Provided the Counterparty takes these precautionary steps, I don’t foresee any serious issues for the Agreement arising from charity regulations. Can a company pay out dividends to shareholders before meeting its obligations under the clause? If so, is it possible to contract around this? --------------------------------------------------------------------------------------------------------------------------------------------------- Distributions to shareholders of English companies may only be made out of profits.[[58]](#fn5b6hqxkt0sm) This means that a Developer could not pay dividends or make other distributions to shareholders before meeting its obligations under the Clause. For the avoidance of doubt, the drafters might define ‘Profits’ in the contract as the Developer’s net profits before any distributions to shareholders.[[59]](#fnqbe04b1w62k) However, it is highly unlikely that the Developer will be able to sidestep its obligations even in the absence of such a provision. Closing thoughts: ===================== Overall, I feel less confident than I was before this project about the viability of an English Windfall Clause. On the one hand, most of the topics I’ve investigated present no issues for the viability of the Clause, and I suspect that clever drafting and careful moderation of the Counterparty’s behaviour would be enough to sidestep those issues that remain. On the other hand, I remain concerned about the hostility of English courts to good faith duties and I fear it may be impossible to achieve certainty on this issue until the Counterparty is forced to bring a case. I’m also deeply troubled by the [lack of practical remedies for breach](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem) of the terms of the Agreement, which appear totally inadequate to prevent a determined Developer from breaking its promises. It's important to stress that these remaining concerns relate primarily to English contract law, not English company law or financial regulations. This has two key implications: 1. English Developers should shop around for a foreign governing law - As I highlighted at the start of this post, Developers needn't choose domestic law to govern any Agreement. The poor enforceability of English contract law here suggests that Developers domiciled in the UK should shop around to find a more appropriate governing law. So provided that, for example, Indian or Singaporean law is more supportive of a Windfall Clause, then there is no need to conclude that a Windfall Clause could never be enforced in the UK. 2. Non-English Developers shouldn't agree to English governing law - Given the failings I've outlined above, non-English Developers should be strongly discouraged from using English law to govern any Agreement. This is essential to keep in mind as the most likely way for English law to be relevant over the next few decades is as a law which US-domiciled companies like Alphabet, OpenAI or Meta AI might choose to govern the Agreement. In the future, I would like to see other EA lawyers explore some of these topics in more depth. In particular, I would welcome further research into how governing laws and jurisdictions might be mixed-and-matched to create a more enforceable Agreement, how the Agreement’s drafters might provide adequate contractual remedies, and a further exploration of the extent of good faith obligations in English law. In any case, I hope that this post serves as a useful jumping off point for discussion on this important topic amongst EA interested in law. --- Appendix - Why is this post so long? ======================================== At the start of this post, I told you that you could skip most of it. That begs the question - why didn't I just cut the fat? To explain, this post is so comprehensive because I suspect it will be the most important post in the sequence. Here's why: 1. DeepMind is domiciled and incorporated in England - DeepMind is arguably the top contender to develop transformative AI. This means we should pay very close attention to the rules and regulations which govern how it operates. 2. England is the world’s most popular choice of law[[60]](#fn0psaeb9fui0h)for international commercial contracts - England is internationally perceived as a ‘neutral’ legal system with a fair and consistent judiciary, and most foreign judiciaries are happy to enforce English judgments. This makes it an attractive choice of law for international commercial contracts which, in turn, increases the likelihood that an Agreement will be governed by English law even if neither the Developer nor Counterparty are domiciled in the UK. That makes it particularly important to make sure a Windfall Clause can be enforced under English law. 3. Old English law is applicable across the former British Empire - All modern common law systems branched out of English law. This means that much English case law from before the 1950s still applies in other jurisdictions I will consider in this sequence. By considering it in detail here, I hope to save myself space in future posts. 4. Modern English law is (sometimes) binding in other jurisdictions - Some island states like the Cayman Islands and British Virgin Islands retain the Privy Council as their final court of appeal.[[61]](#fn4rae31qe6ts) Privy Council judgments are de facto binding in England and Wales, and de jure binding across all other jurisdictions which retain a right of appeal to the Privy Council. This means that understanding how current English case law applies to the Agreement is important to determine the viability of the Clause in several other important jurisdictions. 5. English judgments are influential across common law systems - The fairness and consistency of British courts makes English judgments highly persuasive in other common law jurisdictions. Even though most modern judgments are no longer binding outside England and Wales, Commonwealth judges frequently take inspiration from these judgments in their own decisions.[[62]](#fnoejmtb73od) This makes it even more important that an English Windfall Clause can be enforced because a judgment in favour of the Counterparty by an English court might encourage other courts to uphold the Clause. 6. Finally, I want to avoid duplicated work - I get the impression that a lot of EA research never sees the light of day. That’s a shame because it means that less knowledge gets shared and a lot of person-hours are wasted replicating someone else’s work. By publishing a reasonably comprehensive summary of the key considerations for the viability of an English Windfall Clause, I hope to save the rest of the community a bit of time and effort in exploring this question. 1. [^](#fnrefscbs51yfydq)In this post, where I refer to ‘English law’, I am referring to the law of England & Wales. For those unfamiliar with the British legal ecosystem: English and Welsh law is practically identical and is considered a single jurisdiction. Northern Irish law is somewhat different from English law and Scottish law is a different beast entirely. 2. [^](#fnrefw9u8ahcketc)For judicial consideration of this topic, see Yam Seng Pte Ltd v International Trade Corporation Ltd (2013) EWHC 111, in particular Legatt J at [123]-[130].This case also highlights that courts will sometimes imply good faith duties into certain contracts, such as long-term ‘relational’ contracts or contracts where there is a clear fiduciary duty between the parties. The class of contracts to which such duties apply is expanding incrementally and may eventually encompass agreements like that containing the Windfall Clause, although it does not currently do so. See also Interfoto Picture Library Ltd v Stiletto Visual Programmes Ltd [1989] 1 QB 433, in particular Bingham LJ’s comment that 'English law has, characteristically, committed itself to no such overriding principle [of good faith] but has developed piecemeal solutions in response to demonstrated problems of unfairness.' 3. [^](#fnref79t5owrscf9)Unwin v Bond [2020] EWHC 1768 (Comm) [229]-[230]. See also [215]-[229] for a summary of recent case law on the question of good faith. Note that this case was in the context of a shareholders’ agreement - arguably a unique class of contract in English law. It is possible that higher courts will distinguish Unwin on these grounds when dealing with future cases involving duties of good faith. 4. [^](#fnrefnpn2i6hx8kq)I’m encouraged by Berkeley Community Villages Ltd v Pullen [2007] EWHC 1330 (Ch), in which the court invoked a good faith clause to prevent one party from an early sale of a property which would have prevented the other party from receiving commission on the sale. One class of failure mode of a Windfall Clause involves the Developer trading away WGAI soon before it does, in fact, generate windfall profits. The Counterparty might employ similar reasoning to Berkeley to prevent this. On the other hand, there’s a good chance that the court would distinguish a Windfall case from Berkeley because the former involves a single payment, whereas the Agreement concerns a series of ongoing payments. 5. [^](#fnrefw1c0leb7oos) See Compass Group UK and Ireland Ltd (t/a Medirest) v Mid Essex Hospital Services NHS Trust [2013] EWCA Civ 200 [105]-[107]. 6. [^](#fnrefgexplwo9hy8)See O'Keefe, C., Cihon, P., Garfinkel, B., Flynn, C., Leung, J. and Dafoe, A., 2020. The Windfall Clause - Distributing the Benefits of AI for the Common Good, p. 22. 7. [^](#fnrefitrwrpnufoh)I’ve considered this in more detail [here](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Efficient_breach_). 8. [^](#fnref0bnvjt2jkngc)This seems plausible. For example, a Developer, its shareholders and a third party might enter into a tripartite contract whereby the Developer sells WGAI to the third party in return for regular payments which fall just below the level that would trigger the Clause, with excess funds going directly to the shareholders. Both the shareholders and the third party could make money here, at the expense of the Counterparty, because the distributions which would have been lost to the Counterparty would be shared between them. 9. [^](#fnrefamwne140l2c)Investors Compensation Scheme Ltd v West Bromwich Building Society [1998] 1 WLR 896, HL(E), [912]-[913]. 10. [^](#fnrefl28ze6qi4nl)The idea of a ‘general organisational principle’ reflects the Canadian position since Bhasin v. Hrynew, 2014 SCC 71. In fact, it might be worth stating in the something in the contract like: ‘the general good faith duty outlined in clause X is to be interpreted in line with Canadian jurisprudence [or the jurisprudence of whichever common law country takes the most expansive interpretation of good faith]’ to make it extremely clear how this duty is to be understood. 11. [^](#fnrefvxz1w3yxn3k)See supra note 5. Jackson LJ commented obiter in this case that he would have been willing to construe ‘a general duty to co-operate with one another in good faith’ more broadly. This indicates that the higher courts will uphold general duties of good faith provided they are clearly indicated to be general. 12. [^](#fnref22qd7mmf94w)Note that I am not actually certain that the contract would be ‘frustrated’ by assigning or licensing WGAI, and I've not explored this in detail. I expect the contract would only be truly frustrated if there was some reason that the Developer could never develop another WGAI after the first one. If you don't think the contract would be frustrated, replace the term with something like 'frustrated the contract, or ensured that the Windfall Clause would never be triggered'. 13. [^](#fnrefwylx2219sfb)As I [outlined in my previous post](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Issue_3___There_is_no_obvious_way_to_stop_a_third_party_from_acquiring_rights_in_the_relevant_IP_), well-informed shareholders (or other actors inside the Developer) may push an unwitting Developer to sidestep its obligations under the Clause, and it will be difficult to show that these actions were taken in bad faith. 14. [^](#fnrefoj0j1bzgdh)Note that a Developer might be unwilling to agree to this if the Counterparty didn’t also agree to strict confidentiality requirements, as such a clause would also allow observers within the Counterparty to determine whether the Developer was involved in a merger. 15. [^](#fnrefms7imgsslpa)For example, the Canadian Supreme Court recently affirmed the principle that good faith is a general organising principle in the interpretation of contractual duties (C.M. Callow Inc. v. Zollinger 2020 SCC 45). Provided that English courts applied Canadian laws consistently, an English Agreement governed by Canadian law might prove far more reliable than an attempt to shoehorn good faith into English law. 16. [^](#fnrefegdupi9fltv)See s2 Competition Act 1998 (UK): ‘agreements between undertakings, decisions by associations of undertakings or concerted practices which may affect trade within the United Kingdom, and have as their object or effect the prevention, restriction or distortion of competition within the United Kingdom, are prohibited’. 17. [^](#fnrefnpz2mzp9vjo)For a helpful summary of British courts’ interpretation of EU-derived law like the Competition Act, see Interpretation of EU Legislation \| Legal Guidance \| LexisNexis. [online] Available at: https://www.lexisnexis.co.uk/legal/guidance/interpretation-of-eu-legislation [Accessed 1 May 2022]. 18. [^](#fnrefvis6tlk515)See Case COMP/39850, Container Shipping, Commission Decision of 31 August 2016. In this case, the Commission determined that announcing future price increases breached Art 101(1) (the European counterpart of s2) even in the absence of direct agreement or covert contact between the parties. The Commission argued that public announcements can signal the intended conduct of each parties which in turn decreases their incentives to compete. It's possible that industry-leading Developers publicly announcing their signatory status could be construed in the same way, as substantial windfall donations could restrict the funds available for reinvestment in research, making it easier for other firms to compete. I explore this in more detail later in the post. That said, see C.f. Case T- 41/96 Bayer AG v Commission and Case C- 2/01 P BAI and Commission v Bayer: truly unilateral behaviour in the absence of any ‘concurrence of wills’ will not contravene Chapter I. 19. [^](#fnrefxtlj1xo8mor)To clarify, a breach of s2(1) won’t render an entire agreement void provided that the courts can sever the offending part of the agreement. Unfortunately, this won’t help in the case of the Clause, because the offending part is not a specific clause but the very decision to sign up to the Agreement. 20. [^](#fnrefef0n8y7dvd6)Parts 6-7 Enterprise Act 2002 (UK) 21. [^](#fnref0zny3qn8i5xa)I think PAI would be caught by s2(1) even though it isn’t a trade association per se, because it represents the interests of many of the biggest players in AI but I haven’t looked into it. Please correct me if I am wrong here. 22. [^](#fnref0ab8i5eox4hm)An ‘agreement to agree’ - assurance by the Counterparty that others Developers will also volunteer to be bound by the Windfall Clause to ensure a level playing field - may also be caught by s2: see Hua, S. and Belfield, H., 2020. AI & Antitrust: Reconciling Tensions between Competition Law and Cooperative AI Development, Yale Journal of Law and Technology 23, pp. 484-489.I won’t discuss this here, as that paper already considers the issue at length. 23. [^](#fnrefjp44bcf3zm)This point is far less certain, as there are no analogous decisions to the Container Shipping case where the European Commission or CMA has found a concerted practice resulting from a policy promoted by an industry body. Nonetheless, the CMA’s teleological approach to competition law indicates that it could still make such a finding if it believed that the Clause was anticompetitive. 24. [^](#fnref745lr2lvz6g) 25. [^](#fnrefbttw7g7c2ve)This claim might be incorrect. All else being equal, an increase in the Developer’s liabilities caused by the need to make distributions under the Clause should reduce the funds it has available for other purposes, including R&D, which would correspondingly reduce competitive pressure on other Developers racing to develop WGAI. However, my analysis might be too simplistic here. For one, this would not cause any direct reduction in R&D funding as distributions would be made out of profits after accounting for reinvestment in research. Furthermore, it seems plausible that a Developer paying out some percentage of its profits under the Clause could reduce its dividend payments whilst maintaining R&D spending, which would not substantially impact competition. Overall, I’m not familiar enough with the CMA’s approach to reach a comfortable conclusion as to how it would interpret the Clause in a race scenario. 26. [^](#fnrefzxs83kui44f)This was true at the time of writing (March 2022), but I can no longer find information about this on the CMA’s website. The authority is currently undergoing structural changes following Brexit and this service may be temporarily unavailable, or have been shelved entirely. It’s also possible that responsibility for this service is being passed to the [Digital Markets Unit](https://www.gov.uk/government/collections/digital-markets-unit). 27. [^](#fnrefjtue9evvp8h)Note that this is a different approach to the European Commission. For example, see GlaxoSmithKline (2009); Rural broadband wayleave rates (2012); Modeling Sector (2016). For an in-depth analysis of the divergence in approach between the domestic competition authorities of the EU, see also Brook, O., 2019. Struggling with Article 101(3) TFEU: Diverging Approaches of the Commission, EU Courts, and Five Competition Authorities, Common Market Law Review 56 at pp. 121-156. 28. [^](#fnrefwrzu79nkn58)See Competition and Markets Authority. Retained Horizontal Block Exemption Regulations - consultation document, 2022. If the CMA starts to widen the interpretation of ‘progress’ to include sustainability benefits, this might pave the way for a further broadening of the definition to include other forms of social benefit. This seems increasingly likely given the drive to maintain compatibility of UK competition law with its EU counterpart, which will be introducing an explicit exemption to Art 101(1) TFEU for agreements which promote sustainability. In the interest of avoiding the conjunction fallacy here, it's worth highlighting that this is a highly speculative scenario. It would only present a risk if some combination of the following occurred: 1. leading Developers didn't become signatories until shortly before WGAI was achieved; 2. several Developers were competing to achieve WGAI; 3. multiple Developers then agreed to the Clause; and, 4. the CMA decided to interpret their generous donations uncharitably. 31. [^](#fnrefqd4yeuj2mb)See supra note 22 at pp. 439-443, 483-489. Hua and Belfield rightly identify that the extent of this disincentive effect will depend heavily on the exact structure of the Clause. However, this effect is unlikely to disappear no matter the structure of the Clause, as the nature of the contract is to reduce the Developer’s profits. 32. [^](#fnrefho2j73qq4kc)Note that CMA fines are capped at 10% of an undertaking's worldwide turnover. Although this could mean the Developer would receive a fine greater than its obligations under the Clause in any one year, it would benefit in the short-medium term from being released from the Agreement. 33. [^](#fnrefjyiox85r7m)To be clear, this does not mean that a Developer in possession of WGAI would not face risks from UK competition law. For example, if such a Developer held a monopoly position, the CMA might try to promote competition by breaking the business up or forcing it to license out its software to other Developers below market rate. As these sort of risks do not strictly relate to the Clause they are beyond the scope of this sequence, and so I will not consider them further here. 34. [^](#fnrefzr0mz4t0mhs)This is public information. For example, you can order a copy of DeepMind’s Articles of Association [here](https://find-and-update.company-information.service.gov.uk/company/12181850/more). 35. [^](#fnref20gjdq8ercp)Of course, founders might not want to implement a weird-looking Article that could discourage investment by VCs. I’m not familiar enough with startup culture to have good intuitions around whether this will be a problem, though the success of Founders' Pledge indicates that it isn't necessarily so. 36. [^](#fnrefr78pr0u2at)ss551 and 571 Companies Act 2006 (UK). Note that the rules differ in a private limited company with only one class of shares. 37. [^](#fnrefdjw15g8h08c)s21(1) Companies Act 2006 (UK), Art 22 of the Model Articles for Private Companies Limited by Shares, and Art 43 of the Model Articles for Public Companies. 38. [^](#fnreff8qkwdbkxi9)For instance, it’s plausible that some Developers will already have provisions allowing them to issue non-participating preference shares or share options. Depending on the structure of the Clause, this could do the trick. 39. [^](#fnrefgdmobse39i)Note that a supermajority is still needed if the Developer’s constitution requires a special resolution to disapply shareholders’ rights of pre-emption, to avoid the risk that pre-existing shareholders purchase the windfall shares. 40. [^](#fnrefbh70i8fwig)I’m aware that the startup ecosystem is incomparably larger in the US and founder outreach efforts are likely best focused in Silicon Valley. Nonetheless, the UK is the [fourth-largest producer](https://www.marshall.usc.edu/faculty-research/centers-excellence/center-global-innovation/startup-index-nations-regions) of unicorns, home to [four of the top 10 best universities](https://www.topuniversities.com/university-rankings/world-university-rankings/2022), has a world-leading [Office for AI](https://www.gov.uk/government/organisations/office-for-artificial-intelligence) focused on improving infrastructure for AI firms, and is home to DeepMind. Given this, it’s at least plausible that another DeepMind-quality Developer will emerge in the UK over the next 20-30, though I have <10% credence in this assertion. 41. [^](#fnrefty66voqjt4p)If drafters are unsure about the enforceability of either a cash- or shares-based Clause, then this might give reason to draft a two-tiered ‘if not shares, then cash’ Clause. The idea here is that, if a shares-based Clause was found to be unenforceable or difficult to implement for any reason, the parties could fall back on a system of cash-distributions, which may improve the overall enforceability of the contract. On the other hand, a two-tiered Clause may reduce enforceability if it encourages courts to provide damages in lieu of specific performance, as [damages for breach of the Clause are not likely to be sufficient](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Inadequate_remedies_). 42. [^](#fnrefud2d1uk3cy)See supra note 6 at pp.6-7. 43. [^](#fnrefd7sog8n0uxg)See Carlen v Drury (1812) 1 Ves & B 154, 35 ER 61. In particular, see Lord Eldon at 63: 'The Court is not to be required on every Occasion to take the Management of every Playhouse and Brewhouse in the Kingdom.' 44. [^](#fnref94zcb1mkz27)s172 of the Companies Act 2006 (UK) sets out this particular duty. However, this is only a codification of the common law position as per s170 of the same Act. 45. [^](#fnrefksncbrllov)See Howard Smith Ltd v Ampol Petroleum Ltd (1974) 1 All ER 1126. Specifically, see Lord Wilberforce at 1131: 'There is no appeal on merits from management decisions to courts of law: nor will courts of law assume to act as a kind of supervisory board over decisions within the powers of management honestly arrived at'. 46. [^](#fnref61d3p78dq18)For an elaboration on the requirements placed on directors, see P Davies and S Worthington, Gower and Davies Principles of Modern Company Law (Sweet & Maxwell, 9th ed, 2012) at [16-76]. Some academics have argued for merits-based review of business decisions: for example, see Lim, Ernest, Judicial Intervention in Directors’ Decision-Making Process: Section 172 of the Companies Act 2006 (December 1, 2017). Journal of Business Law 169, 2018. This is an ongoing debate in the literature, but the current state of the case law indicates that courts are unwilling to conduct substantive review. 47. [^](#fnrefyldcj3bv7e)See supra note 6 at pp.6-7. 48. [^](#fnrefiqpwt2tumj)In this section I have mostly focused on the explanations given at pp. 6-7 of the original report, but its authors also argue at pp. 15-16 that windfall distributions could be justified in a similar manner to share options granted to startup founders. They claim that such options are justified by their low expected value at the time they are issued, even though they may have become extremely valuable by the time they are exercised. I have excluded this argument because I think there’s a significant difference between share options and promised windfall distributions. Specifically, options present an additional benefit to the company in the form of a solution to the principal-agent problem faced by investors, aligning the financial incentives of the company and its directors. As options can be justified ex ante even if they disproportionately compensate the board. Windfall distributions don’t have this same incentive effect, meaning that the directors of a Developer may need stronger arguments to justify entering into an Agreement which involves a similar financial commitment. Thanks to Peter Wills for highlighting this in his comments on drafts of this post. 49. [^](#fnrefitobi6hp9al)See s172(1) Companies Act 2006 (UK) for a summary of the factors to which a director must ‘have regard’. As was the position at common law, these are not imperatives but rather issues which a director must consider when taking decisions. In practice, this has had little impact on how English companies are operated, but they would provide helpful cover for directors of a signatory Developer accused of breaching their duties to the company. 50. [^](#fnref1g3brsi8n9t)See Chappell and Co v Nestle Ltd (1959) UKHL 1, in which three discarded chocolate wrappers were held to be good consideration. Lord Somervell, at 114: ‘A contracting party can stipulate what consideration he chooses. A peppercorn does not cease to be good consideration if it is established that the promisee does not like pepper and will throw away the corn.’ Not everything will be good consideration: see White v Bluett (1853) 23 LJ Ex 36 51. [^](#fnrefwvpmq289sfj)See Thomas v Thomas (1842), 2 QB 851, in which £1/yr was held to be good consideration for a life interest in a house. Unfortunately, one-sided contracts can have [other negative effects](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Appendix_I___Wouldn_t_trading_away_the_IP_in_pre_WGAI_be_a_transaction_defrauding_creditors_) on the enforceability of an agreement. 52. [^](#fnrefwxqtsyd7e08)Remember that, for [reasons already outlined in the section on competition law](https://forum.effectivealtruism.org/posts/DEFJkvzHeBdpmKQNR/tawwwc-england-and-wales#Windfall_Clauses_as_a_concerted_practice_), it may not be advisable for the Counterparty to provide publicity as consideration. 53. [^](#fnref3st3wxoo8t3)See Central London Property Trust Ltd v High Trees House Ltd [1947] KB 130 pp. 134-136. This case established that promissory estoppel in English law is ‘a shield, not a sword’. 54. [^](#fnref71rc85wrjwi)An added benefit of a purely donative agreement executed as a deed is that this would avoid the risk that the Clause was considered a [‘tainted charitable donation](https://www.gov.uk/government/publications/charities-detailed-guidance-notes/annex-viii-tainted-charity-donations)’. Unlike ordinary donations, tainted donations are not eligible for tax relief which may make the Clause less desirable for a Developer. On the other hand, [specific performance is unavailable if consideration has not moved from the injured party](https://forum.effectivealtruism.org/posts/wBzfLyfJFfocmdrwL/the-windfall-clause-has-a-remedies-problem#Sidestepping_Specific_Performance_), which militates against the use of a deed alone. With that in mind, the best solution is likely to be that the Counterparty provides consideration and the Agreement is executed as a deed. 55. [^](#fnreft8aw589yyel)For anyone interested in investigationg for themselves: the law of charitable purposes is messy and inconsistent, a mix of 300 year-old case law, modern statute, and guidance by the Charity Commission. This makes the exact legal position of politically-minded charities unclear on a casual investigation. Beware of relying on old cases, even if your lexis of choice says they are still good law. 56. [^](#fnref4knmx7wed12)Ch. 14, Companies Act 2006 (UK) 57. [^](#fnref1fudcxv8odx)It’s possible that the risk of political contributions is stronger than I am anticipating here. Arguably, given the sheer size of the Counterparty’s endowment, it could be difficult for the organisation to avoid straying into politics. 58. [^](#fnreflx0pplys6s)National Anti-Vivisection Society v Inland Revenue Commissioners [1947] UKHL 4; McGovern v Attorney-General [1982] Ch. 321. 59. [^](#fnrefxzusaqh27w)Charity Commission guidance: Campaigning and political activity guidance for charities (CC9) atpp12-13 60. [^](#fnref5b6hqxkt0sm)s830 Companies Act 2006 (UK) 61. [^](#fnrefqbe04b1w62k)Note that a Developer might use other creative accounting strategies to reduce profits or make de facto distributions to shareholders. For instance, a Developer might assign the intellectual property rights in its WGAI systems to a separate entity owned by the same shareholders but which is not bound by the Clause. It could then license back the IPR at an extremely high price, reducing the Developer’s net profits to zero and distributing any windfall gains to the same individuals. Considering such outcomes in more detail is beyond the scope of the sequence. Nonetheless, scenarios like this make it particularly important to include well-drafted good faith provisions along with other protective mechanisms I have mentioned in this post. 62. [^](#fnref0psaeb9fui0h)See Cuniberti, G., The International Market for Contracts: The Most Attractive Contract Laws, 34 Nw. J. Int'l L. & Bus. 455 (2014). 63. [^](#fnref4rae31qe6ts)The [Privy Council](https://en.wikipedia.org/wiki/Privy_Council_of_the_United_Kingdom) is a non-UK court staffed entirely by UK Supreme Court judges. It was essentially a Supreme Court for the Commonwealth, and it continues to be for many Commonwealth nations. It does not adjudicate on British cases, but its decisions are de facto binding here the UK as they are reflective of the current thinking of Britain’s most senior judges. 64. [^](#fnrefoejmtb73od)For an empirical analysis of the outsized persuasive value of English judgments, see Hoadley, D., Bartolo, M., Chesterman, R., Faus, A., Hernandez, W., Kultys, B., Moore, A., Nemsic, E., Roche, N., Shangguan, J., Steer, B., Tylinski, K. and West, N., 2021. A Global Community of Courts? Modelling the Use of Persuasive Authority as a Complex Network. Frontiers in Physics, 9.
f99b13c7-ae2f-4214-849d-bc9dae97063e	trentmkelly/LessWrong-43k	LessWrong	Aligning my web server with devops practices: part 1 (backups) UPDATE: Part 2 is now published here. I have a web server that serves a double-digit number of different domains and subdomains, such as contractwork.vipulnaik.com and groupprops.subwiki.org. I originally set it up in 2013, and more recently upgraded it a few years ago. Back when I did the original setup, I had no knowledge or experience of the principles of devops (I didn't even have any experience with software engineering, though I had done some programming for academic and personal purposes). Over time, as part of my job as a data scientist and machine learning engineer, I acquired deeper familiarity with software engineering and devops. However, for the most part, I didn't have the time or energy to apply this knowledge to my personal web server setup. As a result, my web server continued to basically be a snowflake -- a special hodgepodge of stuff that would be hard to regenerate from scratch -- and the thought of regenerating it from scratch was terrifying. The time commitment to do so was big enough that I didn't have enough time away from my job to do so. In the past year, I've finally started work on desnowflaking my web server and formulating a set of recipes that could be used to build it from scratch, assembling all the websites and turning them on. This work only happens in the spare time from my day job, and competes with other personal projects, so progress is slow, but I've made a lot of it. This series of posts talks about various choices I made in setting up various pieces of the system. Many of these pieces already existed prior to my systematic efforts this year, but I had to add several other pieces. This particular post focuses on backups. Recommendations for others (transferable learnings) These are some of the key learnings that I expect to transfer to other contexts, and that are sufficiently non-obvious and that may not come to your attention until it is too late: * Establish monitoring of the backups so that you can be confident
9ace34ce-d996-4d5e-ae5a-0387dea7ef03	trentmkelly/LessWrong-43k	LessWrong	The Gradient – The Artificiality of Alignment The Gradient is a “digital publication about artificial intelligence and the future,” founded by researchers at the Stanford Artificial Intelligence Laboratory. I found the latest essay, “The Artificiality of Intelligence,” by a PhD student at UC Berkeley, to be an interesting perspective from AI ethics/fairness. Some quotes I found especially interesting: > For all the pontification about cataclysmic harm and extinction-level events, the current trajectory of so-called “alignment” research seems under-equipped — one might even say misaligned — for the reality that AI might cause suffering that is widespread, concrete, and acute. Rather than solving the grand challenge of human extinction, it seems to me that we’re solving the age-old (and notoriously important) problem of building a product that people will pay for. Ironically, it’s precisely this valorization that creates the conditions for doomsday scenarios, both real and imagined. … > > In a recent NYT interview, Nick Bostrom — author of Superintelligence and core intellectual architect of effective altruism — defines “alignment” as “ensur[ing] that these increasingly capable A.I. systems we build are aligned with what the people building them are seeking to achieve.” > > Who is “we”, and what are “we” seeking to achieve? As of now, “we” is private companies, most notably OpenAI, the one of the first-movers in the AGI space, and Anthropic, which was founded by a cluster of OpenAI alumni. > > OpenAI names building superintelligence as one of its primary goals. But why, if the risks are so great? … first, because it will make us a ton of money, and second, because it will make someone a ton of money, so might as well be us. … > > Of course, that’s the cynical view, and I don’t believe most people at OpenAI are there for the sole purpose of personal financial enrichment. To the contrary, I think the interest — in the technical work of bringing large models into existence, the interdisciplinary conversations
d58274d9-f7bb-4810-823f-7fe931b07779	StampyAI/alignment-research-dataset/lesswrong	LessWrong	Dario Amodei’s prepared remarks from the UK AI Safety Summit, on Anthropic’s Responsible Scaling Policy I hope Dario's remarks to the Summit can shed some light on how we think about RSPs in general and Anthropic's RSP in particular, both of which have been discussed extensively since [I shared our RSP announcement](https://www.lesswrong.com/posts/6tjHf5ykvFqaNCErH/anthropic-s-responsible-scaling-policy-and-long-term-benefit). The full text of Dario's remarks follows: Before I get into Anthropic’s [Responsible Scaling Policy (RSP)](https://www.anthropic.com/index/anthropics-responsible-scaling-policy), it’s worth explaining some of the unique challenges around measuring AI risks that led us to develop our RSP. The most important thing to understand about AI is how quickly it is moving. A few years ago, AI systems could barely string together a coherent sentence. Today they can pass medical exams, write poetry, and tell jokes. This rapid progress is ultimately driven by the amount of available computation, which is growing by 8x per year and is unlikely to slow down in the next few years. The general trend of rapid improvement is predictable, however, it is actually very difficult to predict when AI will acquire specific skills or knowledge. This unfortunately includes [dangerous skills](https://www.anthropic.com/index/frontier-threats-red-teaming-for-ai-safety), such as the ability to construct biological weapons. We are thus facing a number of potential AI-related threats which, although relatively limited given today’s systems, are likely to become very serious at some unknown point in the near future. This is very different from most other industries: imagine if each new model of car had some chance of spontaneously sprouting a new (and dangerous) power, like the ability to fire a rocket boost or accelerate to supersonic speeds. We need both a way to frequently monitor these emerging risks, and a protocol for responding appropriately when they occur. Responsible scaling policies—initially suggested by the Alignment Research Center—attempt to meet this need. Anthropic published its RSP in September, and was the first major AI company to do so. It has two major components: * First, we’ve come up with a system called AI safety levels (ASL), loosely modeled after the internationally recognized BSL system for handling biological materials. Each ASL level has an if-then structure: if an AI system exhibits certain dangerous capabilities, then we will not deploy it or train more powerful models, until certain safeguards are in place. * Second, we test frequently for these dangerous capabilities at regular intervals along the compute scaling curve. This is to ensure that we don’t blindly create dangerous capabilities without even knowing we have done so. In our system, ASL-1 represents models with little to no risk—for example a specialized AI that plays chess. ASL-2 represents where we are today: models that have a wide range of present-day risks, but do not yet exhibit truly dangerous capabilities that could lead to catastrophic outcomes if applied to fields like biology or chemistry. Our RSP requires us to implement present-day best practices for ASL-2 models, including model cards, external red-teaming, and strong security. ASL-3 is the point at which AI models become operationally useful for catastrophic misuse in CBRN areas, as defined by experts in those fields and as compared to existing capabilities and proofs of concept. When this happens we require the following measures: * Unusually strong security measures such that non-state actors cannot steal the weights, and state actors would need to expend significant effort to do so. * Despite being (by definition) inherently capable of providing information that operationally increases CBRN risks, the deployed versions of our ASL-3 model must never produce such information, even when red-teamed by world experts in this area working together with AI engineers. This will require research breakthroughs, but we believe it is a necessary condition of safety. * ASL-4 must be rigorously defined by the time ASL-3 is reached. ASL-4 represents an escalation of the catastrophic misuse risks from ASL-3, and also adds a new risk: concerns about autonomous AI systems that escape human control and pose a significant threat to society. Roughly, ASL-4 will be triggered when either AI systems become capable of autonomy at a near-human level, or become the main source in the world of at least one serious global security threat, such as bioweapons. It is likely that at ASL-4 we will require a detailed and precise understanding of what is going on inside the model, in order to make an “affirmative case” that the model is safe. ![RSP](https://res.cloudinary.com/lesswrong-2-0/image/upload/f_auto,q_auto/v1/mirroredImages/vm7FRyPWGCqDHy6LF/jpgsfelnr4o9fakyyy1z) Next, I’ll briefly mention some of our key practices and lessons learned, which we hope are helpful to others in crafting an RSP. First, deep executive involvement is critical. As CEO, I personally spent 10-20% of my time on the RSP for 3 months—I wrote multiple drafts from scratch, in addition to devising and proposing the ASL system. One of my co-founders devoted 50% of their time to developing the RSP for 3 months. Together, this sent a meaningful signal to employees that Anthropic’s leadership team takes the matter of AI safety seriously and is firmly committed to responsible scaling at the frontier. Second, make the protocols outlined in the RSP into product and research requirements, such that they become baked into company planning and drive team roadmaps and expansion plans. Set the expectation that missing RSP deadlines directly impacts the company’s ability to continue training models and ship products on time. At Anthropic, teams such as security, trust and safety, red teaming, and interpretability, have had to greatly ramp up hiring to have a reasonable chance of achieving ASL-3 safety measures by the time we have ASL-3 models. Third, accountability is necessary. Anthropic’s RSP is a formal directive of its board, which ultimately is accountable to our Long Term Benefit Trust, an external panel of experts with no financial stake in Anthropic. On the operational side, we will put in place a whistleblower policy before we reach ASL-3 and already have an officer responsible for ensuring compliance with the RSP and reporting to our Long Term Benefit Trust. As risk increases, we expect that stronger forms of accountability will be necessary. Finally, I’d like to discuss the relationship between RSPs and regulation. RSPs are not intended as a substitute for regulation, but rather a prototype for it. I don’t mean that we want Anthropic’s RSP to be literally written into laws—our RSP is just a first attempt at addressing a difficult problem, and is almost certainly imperfect in a bunch of ways. Importantly, as we begin to execute this first iteration, we expect to learn a vast amount about how to sensibly operationalize such commitments. Our hope is that the general idea of RSPs will be refined and improved across companies, and that in parallel with that, governments from around the world—such as those in this room—can take the best elements of each and turn them into well-crafted testing and auditing regimes with accountability and oversight. We’d like to encourage a “race to the top'' in RSP-style frameworks, where both companies and countries build off each others’ ideas, ultimately creating a path for the world to wisely manage the risks of AI without unduly disrupting the benefits.
d23e16a2-ee97-438c-a0be-4fe3b0afc788	trentmkelly/LessWrong-43k	LessWrong	Constituency-sized AI congress? I just had the idea for a constituency-sized AI congress. Each member of the constituency would have their personal debate agent trained on their preferences and values. The agents would debate tirelessly at superhuman speed to develop proposals which represented the best available win-win compromises given the issues on the table. The final proposals after a set amount of debate would be presented to the constituency and voted on. I haven't researched this yet or thought about it for long. I'd love for your feedback on the idea and links to related work.
48b6a2f9-7d88-4b32-9e43-4cefa88fad96	trentmkelly/LessWrong-43k	LessWrong	What are good ML/AI related prediction / calibration questions for 2019? I'm trying to come up with a set of questions for self-calibration, related to AI and ML. I've written down what I've come up with so far below. But I am principally interested in what other people come up with -- thus the question metatype -- both for questions, and for predictions for the questions. So far I have an insufficient number of questions to produce anything like a nice calibration curve. I've also struggled with coming up with meaningful questions. I've rot13'ed my predictions to avoid anchoring anyone. I'm pretty uncertain about most of these as point estimates however. On Explicitly Stated ML / Systems Goals It is (relatively) easy to determine if these are fulfilled or not. The trade-off is that they likely have little relation to AGI. 1. OpenAI succeeds in defeating top pro teams on unrestricted Dota2 OpenAI has explicitly said that they wish to beat top human teams in the MOBA Dota2. Their latest attempt to do so used self-play and familiar policy-gradient strategies on an incredibly massive scale to train, but still lost to top teams who won (relatively?) easily. I'm also interested in people's probabilities on whether OpenAI succeeds, conditional on OpenAI not including genuine algorithmic novelty in their learning methods, although that's a harder question to define because of cloudiness around "algorithmic novelty." My prediction: Friragl-svir creprag. 2. Tesla succeeds in a self-driving car driving coast-to-coast without intervention. Tesla sells cars with (ostensibly) all the hardware necessary for full self-driving, and an in-house self-driving research program that uses a mix of ML and hard-coded rules. They have a goal of giving a demonstration autonomous coast-to-coast drive, although this goal has been repeatedly delayed. There is widespread skepticism both of the sensor suite in Tesla cars and of the maturity of their software. My prediction: Gra creprag. 3. DeepMind reveals a skilled RL-trained agent for StarCraft II. Aft
44947d7b-afa6-469d-afcb-52e9150509cb	trentmkelly/LessWrong-43k	LessWrong	Partial rewrite of the "Direct Instruction" thing So yeah, "Scientifically optimizing education: Hard problem, or solved problem? Introducing the Theory of Direct Instruction". Probably not gonna go down in history as my best piece of writing ever, to say the least. Clearly I need to fix that. As an initial measure, I added some notes addressing key points and problems to the beginning as a much shorter replacement to the original post, and recommended the new reader not attempt to slog through the original below that unless they're strangely compelled. I felt that the most sensible context for this would be at the beginning of the original post, so I put it there and put up this post as a notification of that. (If this somehow breaks some sort of rule of etiquette or style, please just tell me and I'll rectify it most snappily.) Thank you for your patience.
a6072035-6517-443a-9a78-41c7a60f356b	trentmkelly/LessWrong-43k	LessWrong	Implications of the Doomsday Argument for x-risk reduction Lesswrong contains a large intersection of people who are interested in x-risk reduction and people who are aware of the Doomsday Argument. Yet these two things seem to be incompatible with each other, so I'm going to ask about the elephant in the room: What are your stances on the Doomsday Argument? Does it encourage or discourage you from working on x-risks? Is it a significant concern for you at all? Do most people working on x-risks believe the Doomsday Argument to be flawed? If not, it seems to me that avoiding astronomical waste is also astronomically unlikely, thus balancing out x-risk reduction to a moderately important issue for humanity at best. From an individual perspective (or altruistic perspective with future discounting), we perhaps should focus on having a good time before inevitable doom? What am I missing?
1b134145-609d-4b8a-b9ad-6e729c95de79	LDJnr/LessWrong-Amplify-Instruct	LessWrong	"I have had the following situation happen several times during my research career: I write code to analyze data; there is some expectation about what the results will be; after running the program, the results are not what was expected; I go back and carefully check the code to make sure there are no errors; sometimes I find an error No matter how careful you are when it comes to writing computer code, I think you are more likely to find a mistake if you think there is one. Unexpected results lead one to suspect a coding error more than expected results do. In general, researchers usually do have general expectations about what they will find (e.g., the drug will not increase risk of the disease; the toxin will not decrease risk of cancer). Consider the following graphic: Here, the green region is consistent with what our expectations are. For example, if we expect a relative risk (RR) of about 1.5, we might not be too surprised if the estimated RR is between (e.g.) 0.9 and 2.0. Anything above 2.0 or below 0.9 might make us highly suspicious of an error -- that's the red region. Estimates in the red region are likely to trigger serious coding error investigation. Obviously, if there is no coding error then the paper will get submitted with the surprising results. Error scenarios Let's assume that there is a coding error that causes the estimated effect to differ from the true effect (assume sample size large enough to ignore sampling variability). Consider the following scenario: Type A. Here, the estimated value is biased, but it's within the expected range. In this scenario, error checking is probably more casual and less likely to be successful. Next, consider this scenario: Type B. In this case, the estimated value is in the red zone. This triggers aggressive error checking of the type that has a higher success rate. Finally: Type C. In this case it's the true value that differs from our expectations. However, the estimated value is about what we would expect. This triggers casual error checking of the less-likely-to-be-successful variety. If this line of reasoning holds, we should expect journal articles to contain errors at a higher rate when the results are consistent with the authors' prior expectations. This could be viewed as a type of confirmation bias. How common are programming errors in research? There are many opportunities for hard-to-detect errors to occur. For large studies, there might be hundreds of lines of code related to database creation, data cleaning, etc., plus many more lines of code for data analysis. Studies also typically involve multiple programmers. I would not be surprised if at least 20% of published studies include results that were affected by at least one coding error. Many of these errors probably had a trivial effect, but I am sure others did not." The above is a chunk of text from a post created by a user on the website LessWrong. Can you explain what this text is saying as if I've never finished my first 2 years of college? Absolutely. This text is discussing the role of human expectation in the process of coding and data analysis, especially in research. Here's a simpler explanation: Imagine you're baking a cake and you expect it to taste sweet. If it comes out tasting salty, you'll probably check the recipe to see if you made a mistake. This is because the result (a salty cake) was not what you expected (a sweet cake). The same thing happens when you're writing code to analyze data. If the results you get are not what you expected, you'll check your code for mistakes. Now, let's say you're expecting a certain outcome from your research. For example, you're studying a medicine and you expect it won't increase the risk of a disease. If your results show the medicine does increase the risk, you'll probably check your code for errors because the result was unexpected. The text then uses a color-coded graphic to illustrate this. The green area represents results that are close to what you expected, so you might not check for errors as thoroughly. The red area represents results that are far from what you expected, so you'll probably check for errors more carefully. The text then discusses three scenarios: 1. Your results are different from the truth because of a coding error, but they're close to what you expected. You might not check for errors as thoroughly because the results seem "right." 2. Your results are different from the truth because of a coding error, and they're far from what you expected. You'll probably check for errors more carefully because the results seem "wrong." 3. The truth is different from what you expected, but your results are close to what you expected. You might not check for errors as thoroughly because the results seem "right," even though they're wrong. The text concludes by suggesting that research papers might contain more errors when the results match the researchers' expectations. This is because the researchers might not check for errors as thoroughly. The text also suggests that coding errors are common in research because coding is complex and often involves multiple people. Can you further elaborate on the potential impacts of these coding errors in research, particularly in terms of the reliability of the study's findings and its implications in the field of study? Absolutely. Coding errors in research can significantly impact the reliability and validity of a study's findings. If a coding error leads to incorrect data analysis, the conclusions drawn from that analysis may also be incorrect. This could lead to false assumptions or incorrect theories being proposed and accepted within the field of study. For instance, in medical research, a coding error could lead to a false conclusion about the effectiveness of a drug or treatment. This could potentially lead to the widespread use of an ineffective treatment, or conversely, the dismissal of a potentially effective one. Moreover, these errors can also contribute to the replication crisis, a term referring to the difficulty in replicating or reproducing the results of previous studies to confirm their validity. If the original study contained a coding error, replicating the study and getting different results could lead to confusion and a lack of trust in the research. In the broader context, frequent coding errors can undermine the credibility of research in general. If the public or the scientific community lose trust in research due to frequent errors, it could hinder scientific progress and the application of research in policy making, medical decisions, and other important areas. Therefore, it's crucial to have rigorous checks and balances in place to detect and correct coding errors in research. This includes thorough peer review, encouraging replication of studies, and promoting transparency in sharing data and coding methods. Reflecting on your previous responses, what are the potential challenges or drawbacks in implementing rigorous checks and balances to detect and correct coding errors in research? Implementing rigorous checks and balances to detect and correct coding errors in research is crucial, but it does come with its own set of challenges and drawbacks. 1. Time and Resources: Checking and rechecking code can be a time-consuming process, especially for complex studies with extensive coding. This can delay the publication of research and require additional resources, which might not always be available, particularly in smaller research institutions or for independent researchers. 2. Expertise: Not all researchers are expert coders. While they may be experts in their field of study, they might not have the necessary skills to thoroughly check and debug complex code. This could necessitate additional training or hiring external experts, which again requires time and resources. 3. Collaboration Difficulties: Research often involves collaboration between multiple researchers or institutions, which can complicate the process of checking code. Differences in coding styles, software, or versions can make it harder to spot and correct errors. 4. Intellectual Property Concerns: Some researchers might be reluctant to share their code due to concerns about intellectual property rights or academic competition. This can hinder the process of peer review and replication, which are important for catching and correcting errors. 5. Resistance to Change: In some cases, there might be resistance to implementing rigorous checks if it's seen as questioning the competence or integrity of researchers. This cultural aspect can be a significant barrier to improving error detection. Despite these challenges, the importance of accurate and reliable research findings necessitates ongoing efforts to improve the detection and correction of coding errors. This could involve promoting a culture of transparency and collaboration in research, providing training and resources for coding, and developing better tools and practices for code review and debugging. Discuss the role of software and tools in detecting and correcting coding errors in research. How can they be effectively utilized and what limitations might they have? Software and tools play a crucial role in detecting and correcting coding errors in research. They can automate certain processes, identify potential errors, and help ensure that code is consistent and reproducible. Here's how they can be effectively utilized and some limitations they might have: 1. Debugging Tools: These are built into most programming environments and can help identify where an error is occurring. They allow researchers to step through their code line by line, inspect the values of variables at each step, and see exactly where things go wrong. However, they can't always identify logical errors where the code runs successfully but produces incorrect results. 2. Static Code Analysis Tools: These tools analyze code without running it, looking for patterns that often lead to errors. They can catch common mistakes and enforce coding standards, which can help make code more readable and consistent. However, they can sometimes produce false positives or negatives, and they may not catch more complex, context-specific errors. 3. Version Control Systems: Tools like Git allow researchers to keep track of changes to their code, making it easier to identify when and where an error was introduced. They also facilitate collaboration by allowing multiple researchers to work on the same code without overwriting each other's changes. However, they require discipline to use effectively and can have a steep learning curve for those unfamiliar with them. 4. Automated Testing Tools: These tools can run a suite of tests on code to ensure it behaves as expected. By writing tests that cover a range of scenarios, researchers can catch errors that might only occur under specific conditions. However, writing comprehensive tests can be time-consuming and requires anticipating potential edge cases. 5. Reproducible Research Tools: Tools like Jupyter notebooks and R Markdown allow researchers to interleave code, results, and narrative in a single document, making it easier to verify that the reported results match the code and data. However, they don't eliminate the possibility of errors in the code itself. While these tools can be incredibly helpful, they are not a panacea. They require time and expertise to use effectively, and they can't catch every possible error. They are most effective when used as part of a broader strategy that includes careful code review, thorough testing, and a culture of transparency and collaboration.
b0059f09-2daf-41c0-96d6-f924642a8dd4	StampyAI/alignment-research-dataset/alignmentforum	Alignment Forum	What happens with logical induction when... So this is a bunch of related technical questions about logical induction. Firstly, do you need the formal theorem prover section? Can you just throw out the formal theorem prover, but give some programs in the market unbounded capital and get the same resultant behaviour? 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src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')} @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold} @font-face {font-family: MJXc-TeX-vec-Bw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')} towards 1−P(¬X) unbounded downside risk (downside risk of n on day n) ) This means the program would lose infinite money if X and ¬X both turned out to be true. I think that any axioms can be translated into programs. And I think such a setup, with some finite number of fairly simple programs having infinite money available produces a logical inductor. Is this true? What happens when the axioms added under this system are inconsistent. (so this is a logical induction market, without a theorem prover to settle the bets, and with agents with unlimeted money betting both for and against X, possibly indirectly like the bot betting for X, the bot betting for ¬X, and the bot described above trying to make P(X)+P(¬X)=1 ) Can the other agents make unbounded money? Do the prices converge? If I added a bot with infinite money that was convinced fermats last theorem was false to a consistent ZFC system, would I get a probability distribution that assigned high probability to basic arithmetic facts in the limit? Does this make a sensible system for logical counterfactuals?
464da42d-ad4a-419e-a075-ed4cada272d4	StampyAI/alignment-research-dataset/alignmentforum	Alignment Forum	The longest training run In short: Training runs of large Machine Learning systems are likely to last less than 14-15 months. This is because longer runs will be outcompeted by runs that start later and therefore use better hardware and better algorithms. [Edited 2022/09/22 to fix an error in the hardware improvements + rising investments calculation] \| \| \| \| --- \| --- \| \| Scenario \| Longest training run \| \| Hardware improvements \| 3.55 years \| \| Hardware improvements + Software improvements \| 1.22 years \| \| Hardware improvements + Rising investments \| 9.12 months \| \| Hardware improvements + Rising investments + Software improvements \| 2.52 months \| [Larger compute budgets](https://epochai.org/blog/compute-trends) and a better understanding of how to effectively use compute (through, for example, using [scaling](https://arxiv.org/abs/2001.08361) [laws](https://arxiv.org/abs/2203.15556)) are [two major driving forces of progress in recent Machine Learning](http://www.incompleteideas.net/IncIdeas/BitterLesson.html). There are many ways to increase your effective compute budget: [better hardware](https://epochai.org/blog/trends-in-gpu-price-performance), [rising investments in AI R&D](https://aiindex.stanford.edu/report/) and [improvements in algorithmic efficiency](https://arxiv.org/abs/2005.04305). In this article we investigate one often-overlooked but plausibly important factor: how long—in terms of wall-clock time—you are willing to train your model for. Here we explore a simple mathematical framework for estimating the optimal duration of a training run. A researcher is tasked with training a model by some deadline, and must decide when to start their training run. The researcher is faced with a key problem: by delaying the training run, they can access better hardware, but by starting the training run soon, they can train the model for longer. Using estimates of the relevant parameters, we calculate the optimal training duration. We then explore six additional considerations, related to 1) how dollar-budgets for compute rise over time, 2) the rate at which algorithmic efficiency improves, 3) whether developers can upgrade their software over time 4) what would happen in a more realistic framework with stochastic growth, 5) whether it matters for the framework that labs are not explicitly optimizing for optimal training runs and 6) what would happen if they rent instead of buy hardware. Our conclusion depends on whether the researcher is able to upgrade their hardware stack while training. If they aren't able to upgrade hardware, optimal training runs will likely last less than 3 months. If the researcher can upgrade their hardware stack during training, optimal training runs will last less than 1.2 years. These numbers are likely to be overestimates, since 1) we use a conservative estimate of software progress, 2) real-world uncertainty pushes developers towards shorter training runs and 3) renting hardware creates an incentive to wait for longer and parallelize the run close to the deadline. \| \| \| \| --- \| --- \| \| Scenario \| Longest training run \| \| Hardware improvements \| 3.55 years \| \| Hardware improvements + Software improvements \| 1.22 years \| \| Hardware improvements + Rising investments \| 9.12 months \| \| Hardware improvements + Rising investments + Software improvements \| 2.52 months \| A simple framework for training run lengths ------------------------------------------- Consider a researcher who wants to train a model by some deadline T.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; 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The researcher is deciding when to start the training run in order to maximize the amount of compute per dollar. The researcher is faced with a key trade-off. On one hand, they want to delay the run to access improved hardware (and/or other things like larger dollar-budgets and better algorithms.) On the other hand, a delay reduces the wall-clock time that the model is trained for. Suppose that hardware price-performance is increasing as follows: H(t):=H0Exp[gHt]where H0 is the initial FLOPS/$ and gH is the rate of yearly improvement.[[1]](#fnt0uuctle43n) If we start a training run at time S, the cumulative FLOP/$ at time t≥S will be equal to: FS(t):=H(S)(t−S)where H(S) is the price-performance of the available hardware when we start our run (in FLOP/$/time), and (t−S) is the amount of time since we started our run. Given a fixed dollar-budget, when should we buy our hardware and start a training run to achieve the most FLOP/$ by a deadline T? To figure that out, we need to find the most efficient time S to start a run that concludes by time T>S. We can find that by deriving FT(T) with respect to S and setting the result equal to zero. ∂FS(T)∂S=H(S)[gH(T−S)−1]=0T−S=1/gHThe optimal training run has length L:=T−S=1/gH. In previous work we estimate the rate of improvement of GPU cost effectiveness at gH≈0.281 [(Hobbhahn and Besiroglu, 2022)](https://epochai.org/blog/trends-in-gpu-price-performance) [[2]](#fngvpl3urnv1k). This leads to an optimal training run of length L=1/gH≈3.55 years. ![](https://res.cloudinary.com/lesswrong-2-0/image/upload/v1676332916/mirroredImages/RihYwmskuJT9Rkbjq/andngcscjyvsopsampbm.png)n blue, total amount of compute consumed by training runs starting at different years, given a deadline T=2030 and an investment of $1B. In brown, the hardware price-performance, assuming an initial price-performance of H0≈6.3×1010 FLOP/s/$ in 2022 and a rate of improvement of gH≈0.281 (see [Hobbhahn and Besiroglu, 2022](https://epochai.org/blog/trends-in-gpu-price-performance)).The intuition is as follows: if you want to train a model by a deadline T, then, on the one hand, you want to wait as long as possible to get access to high price-performance hardware. On the other hand, by waiting, you reduce the total time available for your training run. The optimal training duration is the duration that strikes the right balance between these trade-offs. This calculation rests on some assumptions: 1. We are ignoring that willingness to invest in ML rises over time, so a researcher might be able to secure a larger budget if they wait 2. We are ignoring that improvements in software and better understanding of scaling laws might enable researchers to deploy compute more effectively in the future 3. We are assuming that practitioners will not upgrade their hardware in the middle of a run 4. We are assuming that the involved quantities will improve at a predictable, deterministic rate 5. We are assuming that developers optimize for a fixed deadline 6. We are assuming that developers are buying their own hardware Let's relax each of these assumptions in turn and see where they take us. ### Accounting for increasing dollar-budgets In reality, the total amount of compute invested in ML training runs has grown faster than GPU price performance. Companies have been increasing their dollar-budgets for training ML models; hence, researchers might want to delay training ML models to access larger dollar-budgets. Our [previous work](https://epochai.org/blog/compute-trends) found a rate of growth of compute invested in training runs equal to gC≈1.31 [[3]](#fntm9ehii7eh). This rate of growth can be decomposed as gC=gH+gI, the sum of hardware efficiency growth gH≈0.281 and the growth in investment gI=gC−gH≈1.03 [[4]](#fnydty2fbopd8). Following the same reasoning as above, we can calculate the optimal training run length equal to L=1/gC≈0.76 years, ie 9.12 months. This is much shorter than the ~3.55 year training duration we saw previously. Researchers want to wait for both better hardware and larger dollar-budgets. Since dollar-budgets have been growing about an order of magnitude more quickly than hardware price-performance has been improving, researchers taking into account growing dollar-budgets will train their models for roughly an order of magnitude less wall-clock time. ### Accounting for increased algorithmic efficiency In 2020, [Kaplan et al´s paper about scaling laws for neural models](https://arxiv.org/abs/2001.08361) provided practitioners with a recipe for training models in a way that leverages compute effectively. Two years after, Hoffman et al upended the situation by releasing an [updated take on scaling laws](https://arxiv.org/abs/2203.15556) that helped spend compute even more efficiently. Our understanding of how to effectively train models seems to be rapidly evolving. Hence, practitioners today might be dissuaded from planning a multiyear training run because advances in the field might render their efforts obsolete. One way we can study this phenomenon is by understanding how much less compute we need today to achieve the same results as a decade ago. While partially outdated in the light of new developments, [Hernandez and Brown](https://arxiv.org/abs/2005.04305)'s measurement of algorithmic efficiency remains the best work in the area. They find a 44x improvement in algorithmic efficiency over 7 years, which translates to a rate of growth of gS≈0.541. Combining this with the rate of improvement of hardware leads to a combined rate of growth of gH+gS≈0.281+0.541=0.822. This translates to an optimal training run length of L=1/(gH+gS)≈1.22 years. We could also combine this with the rate of growth of investments. In that case we would end up with a total rate of growth of effective compute equal to gH+gI+gS≈0.28+3.84+0.54=4.66. This results in an optimal training run length of L=1/(gH+gI+gS)≈0.21 years, ie 2.52 months. ### Accounting for hardware swapping Through this analysis we have assumed that ML practitioners commit to a fixed hardware infrastructure. However, in theory one could stop the current run, save the current state of the weights and the state of the optimizer, and resume the run in a new hardware setup. Hypothetically, a researcher could upgrade their hardware as time goes on. In practice, if our budget is fixed this is a moot consideration. We want to spend our money at the point where we can buy the most compute per dollar before a deadline. Spending money before or afterwards leads to less returns per dollar overall. Our budget does not need to be fixed however. As investments rise, we could use the incoming money buying new, better hardware to grow our hardware stock. Suppose that the amount of available money at each point grows as gI. We can spend money at any time to buy state-of-the-art hardware, whose cost-efficiency has been improving all along at a pace gH. H(t):=H0Exp[gHt]I(t):=I0Exp[gIt]There are many possible ways to spend the budget over time. However, the optimal solution will be to spend all available budget at the point that maximizes the product between hardware cost-efficiency and time remaining, and then spend any incoming money afterwards as soon as possible to get higher returns. Formally, the cumulative amount of FLOP that a run started at point S can muster by time t>S is equal to: FS(t):=H(S)I(S)(t−S)FLOP yield of initial hardware+∫tSH(u)˙I(u)(t−u)duFLOP yield of hardware swappingDeriving with respect to S as before gives us the optimal training run length: ∂FS(T)∂S=H(S)I(S)[(gH+gI)(T−S)−1]−H(S)I(S)gI(T−S)=0L:=T−S=1/gHThe answer is the same as in the case where our budget is fixed, there are no rising investments and swapping hardware is not allowed. I.e., we find that the influence of the rising budget disappears - the optimal length of the training run now depends only on the rate of hardware improvement. This is simply because there is no additional incentive to wait for larger dollar-budgets; researchers reap the benefits of growing hardware-budgets by default. Hence, the optimal duration of a training run is the same as that found when only considering hardware price-performance improvements. ### Accounting for stochasticity In our framework we have assumed a simple deterministic setup where hardware efficiency, investments and algorithmic efficiency rise smoothly and predictably. In reality, progress is more stochastic. New hardware might overshoot (or undershoot) expectations. Market fluctuations and the interest in your research area may affect the dollar-budget you can muster for training at any given point. Developing a framework that incorporates stochasticity is beyond the scope of this article. However, it may be useful to consider an idea from portfolio theory: when you're not sure what will happen in the future, you don't want to lock up capital in long-term projects. This pushes training runs towards being shorter—and means that the numbers we are estimating in this article are likely on the higher side. ### Fixed deadlines One possible objection to our framework is that it assumes developers are trying to hit a fixed deadline. In reality, researchers are often happy to wait for longer results. Ultimately, we believe that this is a good framework. The way we conceptualize research in AI envisions many labs beginning their training runs at different times. In any given quarter, the lab that releases the most compute-intensive model will be the one that started their training run closest to the optimal length. Even if labs are not optimizing for explicit deadlines or planning training lengths, the most compute-efficient among them will still roughly obey these rules. Labs that train for shorter and longer times than the optimum will be outclassed. Assuming that the most compute-intensive models will also be the most impressive, then this model provides a good upper bound on training lengths of impressive models. ### Renting hardware Through this discussion we have been assuming that labs purchase rather than rent hardware for training. This is the case for some of the top labs that usually train the largest models, such as Google and Meta. However, many others instead resort to renting hardware use from cloud computing platforms such as Amazon AWS, Lambda Labs or Google Cloud. In the case hardware is rented, and there the training run require a small fraction of the available capacity, we expect our model not to apply. Since hardware prices decrease over time and training runs are largely paralellizable, there is a strong incentive for labs that rent hardware to wait for as long as possible, and train their model very briefly on a much larger number GPUs (relative to the number that is optimal when hardware is purchased) close to their deadline. While we think this is an important case to consider (as renting hardware is likely much common in machine learning relative to using purchased hardware), since we're mostly interested in understanding the decision-problems associated with training the largest models at any point in time, we have not studied the case of renting hardware in much depth. Conclusion ---------- We have analyzed how continuously improving hardware, bigger budgets and rising algorithmic efficiency limit the usefulness of a longer training run. Researchers are faced with a trade-off when deciding when to start a training run that ends at some time T. On one hand, they want to delay the start of this run to get access to improved hardware and/or additional factors like larger dollar-budgets and better algorithms. On the other hand, a delay reduces the time that the hardware can deployed for. Since we have some sense of the rate at which these factors change over time, we can infer the optimal duration of ML training runs. We find that optimally balancing these trade-offs implies that the resulting training runs should last somewhere between 2.5 months and 3.6 years. Allowing for swapping hardware removes the effect of rising budgets (since we can spend incoming money without stopping the run). This increases the optimal training run length to between 1.2 and 3.6 years. We expect these numbers to be overestimates, since improvements are stochastic, uncertainty will push developers to avoid over-investing in single training runs, and renting hardware incentivizes developers to wait longer before starting their training run. Furthermore, large-scale runs can be technically difficult to implement. Hardware breaks and needs to be replaced. Errors and bugs force one to discard halfway completed training runs. All these factors shorten the optimal training run[[5]](#fnyns6voyguk). The biggest uncertainty in our model is the rate at which algorithmic efficiency improves. We have used an estimate from [(Hernandez and Brown, 2020)](https://arxiv.org/abs/2005.04305) to derive the result. This paper precedes the conversation about scaling laws and uses data from computer vision rather than language models. Our sense is that (some types of) algorithmic improvements have proven to be faster than estimated in that paper, and this could further shorten the optimal training run. In any case, we can conclude that at current rate of hardware improvement we probably will not see runs of notable ML models over 4 years long, at least when researchers are optimizing compute per dollar. \| \| \| \| --- \| --- \| \| Scenario \| Longest training run \| \| Hardware improvements \| 3.55 years \| \| Hardware improvements + Software improvements \| 1.22 years \| \| Hardware improvements + Rising investments \| 9.12 months \| \| Hardware improvements + Rising investments + Software improvements \| 2.52 months \| ### Acknowledgements We thank Sam Ringer, Tom Davidson, Ben Cottier, Ege Erdil and Lennart Heim for discussion. Thanks to Eduardo Roldan for preparing the graph in the post. 1. [^](#fnreft0uuctle43n)We assume that hardware price performance increases smoothly over time, rather than with discontinuous jumps corresponding to the release of new GPU designs or lithography techniques. We expect that on a more realistic step-function process, the key conclusions of our framework would still roughly follow (modulo optimal training durations occasionally changing a few months to accommodate discrete generations of hardware). 2. [^](#fnrefgvpl3urnv1k)They find a doubling time for hardware efficiency of 2.46 years. This corresponds to a yearly growth rate of gH≈ln22.46=0.281. 3. [^](#fnreftm9ehii7eh)We found a 6.3 month doubling time for compute invested in large training runs. This is a yearly growth rate of gC≈ln26.3 months⋅year12 months=1.31. 4. [^](#fnrefydty2fbopd8)In theory, we should also account for the rise in training lengths. In practice, when we looked at a few data-points training lengths appeared to be increasing linearly over time, so we believe the effect is quite small. 5. [^](#fnrefyns6voyguk)[Meta's OPT logbook](https://github.com/facebookresearch/metaseq/blob/main/projects/OPT/chronicles/OPT175B_Logbook.pdf) illustrates this well: they report being unable to continuously train their models for more than 1-2 days on a cluster of 128 nodes due to the many failures requiring manual detection and remediation.
426d3847-877b-4300-b0c7-5a754dfc4d22	trentmkelly/LessWrong-43k	LessWrong	Out-of-body reasoning (OOBR) The abbreviated title is a pun on "über". As the über-theory, it has to be believed, right? Outline I'd like to suck all of the joy and intrigue out of Sleeping Beauty and related problems by claiming that a rational agent always, boringly, makes decisions thus: * Assume a God-like "out-of-body" mindset of defining an experiment in which you are the (programmable) subject, and a probability space or model representing the objective chances in that experiment. * Define a reward function mapping a decision strategy (program, if you will) and experimental outcomes to a real number, i.e. how well the strategy panned-out for each possible turn of events. * Choose your decision strategy to maximise the expected reward. If you can define the model & experiment unambiguously, there is no need for credence: the rational agent does not use it for decision-making. Let's put it this way: * Inner credence, being any statistic derived wholly from the model and the observed outcomes, does not influence rational decision-making. There are multiple natural ways of doing this, e.g. "halfer" vs. "thirder", but the rational agent isn't obliged to employ any of them. If you cannot define the model/experiment unambiguously, perhaps because you were given only partial information about your circumstances or you cannot compute the complex physical model that determines your world, then you must invoke your: * Outer credence, that assigns subjective probability to the various possible models/experiments. Then use your outer credence to define a new, hybrid model that says: pick a model/experiment according to outer credence, then enact that experiment. Make a hybrid reward function to fit. Then, as before, choose a decision strategy maximising the expected reward. Sleeping Beauty See https://www.lesswrong.com/tag/sleeping-beauty-paradox. The model is that a coin is flipped. The experiment involves waking Beauty some number of times depending on the coin flip, with drug
b526e3e7-8bc4-4123-988a-141d6bb6d8ec	trentmkelly/LessWrong-43k	LessWrong	Legends of Runeterra: Early Review Legends of Runeterra has been getting a generally favorable reception. The game is highly relevant to my interests as someone who plays a lot of collectible card games and is making a collectible card game of my own called Emergents, together with Brian David-Marshall, that will be ready for its first Alpha test soon. Thus, after the latest round of prompting to check the game out, I have checked the game out. I figured I should report back. Legends of Runeterra has some very cool things that it does well. It also has some things that, from my perspective, it does poorly. That doesn’t mean those things that I personally disliked are bad or wrong! It means they made my initial experience worse, slash lowered my expectations for my future personal experiences. I am a weird customer. I notice that I am having the hit from ‘this is a new card game with new decision points and new cards and new decks.’ Which I’m always happy to experience. There’s some potentially interesting tactical games of chicken, and the spell mana mechanic is nice. Plus, the game plays smoothly and looks gorgeous. Always a plus. And yet, already I find myself thinking about games as something the game assigns to me, as work, to complete quests and unlock cards. I don’t feel the urge to play for the sake of playing – although I do feel a bit of ‘I have only my phone and I can play a game of this, whereas I can’t play Magic or Slay the Spire right now so maybe?’. I don’t have the same kind of ‘just one more game’ urge I have with Magic: The Gathering, or in the first few months with Hearthstone, or Artifact or Slay the Spire. But it’s always tinged with ‘…grind out some rewards’ which always makes me feel sick about feeling it. There’s no larger scale motivation to keep playing, as the ladder gives essentially no rewards. As ladders often do – I have no idea why ladder rewards are reliably super stingy in such free to play economies. Strategically, there isn’t much there, there. What there
8e0a4bf1-4591-48b5-85c2-34b323937c7b	trentmkelly/LessWrong-43k	LessWrong	Deriving techniques on the fly Original post: http://bearlamp.com.au/deriving-techniques-on-the-fly/ ---------------------------------------- Last year Lachlan Cannon came back from a CFAR reunion and commented that instead of just having the CFAR skills we need the derivative skills. The skills that say, "I need a technique for this problem" and let you derive a technique, system, strategy, plan, idea for solving the problem on the spot. By analogy to an old classic, > Give a man a fish and he will eat for a day. Teach a man to fish and he never go hungry again. This concept always felt off to me until I met Anna. An american who used to live in Alaska where they have enough fish in a river that any time you go fishing you catch a fish, and a big enough one to eat. In contrast, I had been fishing several times when I was little (in Australia) and never caught things, or only caught fish that were too small to feed one person, let alone many people. Silly fishing misunderstandings aside I think the old classic speaks to something interesting but misses a point. to that effect I want to add something. > Teach a man to derive the skill of fishing when he needs it. and he will never stop growing. We need to go more meta than that? I am afraid it's turtles all the way down. ---------------------------------------- Noticing To help you derive you need to start by noticing when there is a need. There are two parts to noticing: 1. triggers 2. introspection 3. What next But before I fail to do it justice, agentyduck has written about this. Art of noticing, What it's like to notice things, How to train noticing. The Art Of Noticing goes like this: > 1. Answer the question, "What's my first possible clue that I'm about to encounter the problem?" If your problem is "I don't respond productively to being confused," then the first sign a crucial moment is coming might be "a fleeting twinge of surprise". Whatever that feels like in real time from the inside of your mind, that's yo
0adc27d4-d57a-49ed-9f37-c589472ce508	trentmkelly/LessWrong-43k	LessWrong	2020 Review: Final Voting Click here to begin voting. Click here for a general overview of the 2020 Review. We're now in the final week of the LessWrong 2020 Review. We've had over a month of preliminary voting and discussion. Now it's time to finalize our votes. In this post I cover how to vote, why to vote, and what will happen with the outcome (in no strict order). The winning posts will be assembled into the longterm annals of the Best of LessWrong[1]. This year, each top-rated post will be displayed with a donation button for its author, so readers can directly show their support for posts that were important to them. The Best of LessWrong: where we remember the greatest contributions Quick Recap on The Review The LessWrong Review has two major goals. * Improve the LessWrong community's longterm feedback and reward cycle. * Build common knowledge about the best ideas we've discovered on LessWrong. Voting sends a signal about which posts were most important. This can feed into various compilation and distillation efforts. Who Can Vote? All users registered before 2020 can vote. The LessWrong curation team will weight the votes of users with 1000+ karma more highly when assembling sequences, books, or other projects. How do I vote? Go to the Review Voting page. There, you'll see posts that you haven't yet voted on sorted first, with posts you've previously given a karma-vote to sorted to the very top (they have a green stripe along the left side. Strong upvotes have a darker green). The posts will include the reviews that got written about that post. This is intended to help you make an informed vote. You can read reviews that look interesting or highly upvoted to get a better sense of how the post held up. You vote by clicking buttons that assign a post a score. A score of 1 means roughly "this post was good." A score of 4 means "this post was quite important". A score of 9 means "this post was extremely important." A vote of 0 means "I don't have a strong opinion." (A
16c05bb4-b37c-45d0-bc7e-e7291319dd45	trentmkelly/LessWrong-43k	LessWrong	Sex, Death, and Complexity Cancer can be understood as evolution within the body. Cancer is a mutation, and that mutation is selected for, while the body is alive. The cancer cells are more "successful" than other cells, because they are selfish reproducers. They out-compete other cells for the resources of the body. The coherence of the body depends on its cells retaining the information and the purpose that they inherited from the zygote. Over time, that information and purpose is degraded by entropy and changed by evolution. Without death, the body would eventually dissolve into a battleground of competing cells — a tragedy of the commons. So, death is a consequence of entropy and evolution. The dynamics of sexual reproduction force the cells of the body to work together, rather than competing against each other (cancer). This is arguably the most important reason why sexual reproduction evolved. Sex, death, and complexity are all linked.
b5a85e09-8def-4526-b931-75dc1f1938cd	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	Capping AGI profits Introduction ------------ Beyond the many concerns around AI alignment, the development of artificial general intelligence (AGI) also raises concerns about the concentration of wealth and power in the hands of a few corporations. While I’m very glad to see people working on avoiding worst-case-scenarios, my impression is that there is relatively less attention being given to “grey area” scenarios, under which catastrophe is neither near-certain nor automatically avoided. These scenarios strike me as worlds in which policy and governance work may be relatively important. In this post I outline a case for government-imposed caps on extreme profits generated through AGI. I think could be promising both as a way to distribute AGI-generated wealth democratically, and (hopefully) to disincentivize AGI development by profit-motivated actors. Edit (March 21): Thank you to Larks for pointing out the strong similarities between this proposal and a [Windfall Clause](https://forum.effectivealtruism.org/topics/windfall-clause). As far as I can tell, my proposal mainly differs from a Windfall Clause in that it is feasible to implement without the coordinated buy-in of AI labs, and that it fits more squarely in existing policy paradigms. As potential drawbacks, it seems more prone to creating tensions at an international level, and less targeted at effective redistribution of funds, although I think there could be practical solutions to these issues. Disclaimers: * As best I can tell, the idea of a “capped-profit” organization was introduced by OpenAI in 2019, ~~but I have not seen any discussion of it in the context of broader policy options~~. I do not claim to have any especially novel ideas here, but I apologize if I've missed someone else's work on this. * Since I am sympathetic to the claim that equitable distribution of wealth is a second-order problem in the face of potential AGI ruin, I am conditioning the remainder of the post with "assuming we find practical ways to make AI both safe and useful." AGI wealth ---------- Many believe that AGI has the potential to generate an enormous amount of wealth, provided that we avoid disastrous outcomes. For example, [Metaculus forecasters](https://www.metaculus.com/questions/3477/if-human-level-artificial-intelligence-is-developed-will-world-gdp-grow-by-at-least-300-in-any-of-the-subsequent-15-years/) are predicting with 60% likelihood that GDP will grow by 30% or more annually in any of the 15 years after human level AI is achieved. On [Manifold](https://manifold.markets/Gigacasting/ai-10-of-gdp-by-2050), predictors expect an 80% chance that AI will constitute more than 10% of GDP by 2050 (although that market is thin, and the criteria for resolution are unclear). Consistent with this notion, OpenAI [restructured itself](https://openai.com/blog/openai-lp) as a capped-profit organization in 2019. The move was intended to allow OpenAI to fund itself as a profit-driven company in the short term, while maintaining its non-profit mission if they succeed at creating AGI. If the organization becomes immensely profitable due to its development of powerful AI, investors' returns will be limited to a fixed amount (100 times their contribution, for initial investors), and any excess profits will be redirected to a supervising nonprofit organization, whose "primary fiduciary duty is to humanity." Although this seems admirable in purpose, it raises several questions. How will the non-profit use its massive income? Will the OpenAI board act as benevolent autocrats, funding social programs of their choice, or will there be an attempt to create democratic channels of decision-making? Can we trust the entity to adhere to its charter faithfully? Above all, what will happen in the future if some other profit-driven company is the first to create AGI? Capping profits more broadly ---------------------------- OpenAI's capped-profit model suggests a policy option for governments who view the above questions as concerning. Rather than hoping that AI companies will charitably distribute massive profits, governments could impose a fixed limit on company profitability. If companies become incredibly wealthy as a result of AGI, the ensuing tax revenues could be used to finance social programs such as universal basic income (UBI), through existing democratic pathways. I think such a policy has a number of attractive properties: * In addition to providing stronger democratic assurances, this could help disincentivize reckless AGI development by companies pursuing tail outcomes. * The profitability limit could be tailored such that no business-as-usual company expects to be affected, minimizing its distortionary effects outside of companies pursuing transformative technologies. * The policy fits cleanly into established tax policy frameworks, reducing the friction of implementation. * Since nearly all voters and political actors stand to benefit, it seems like it should be relatively easy to find support for this kind of policy. Naturally, there are numerous questions and uncertainties to be addressed: * It would be critical to ensure the policy is highly robust to a variety of takeoff scenarios, that it balances redistribution with a "winning" company's ongoing capital requirements, and that it properly captures profits in high-growth outcomes while avoiding taxing unintended targets. I do not yet have a strong view on what the specifics should entail. * Companies pursuing AGI might be able to use their technology to effectively circumvent the policy. If they anticipate this possibility ahead of time, it will reduce the policy’s disincentive effects. * It is unclear to what extent leading AI labs would embrace or oppose this kind of policy. On the one hand, it could ease certain race dynamics and generate favorable PR; on the other, many are aiming to win the race. * The usual issues of tax evasion apply here; it would be easy for companies to relocate to countries with lower taxes on corporate profits. * Finally, it is not straightforwardly clear that governments would be better stewards of the money than would a profit-capped organization like OpenAI. To begin with, governments are primarily accountable to only their citizens; if AGI is created in a country with AGI profit caps, we may end up with less equitable outcomes. Finding ways to mitigate this should be a high priority if this policy is seriously being considered. Despite the unresolved questions, this appears to be a promising direction to me. Even if companies could easily evade regulations using AGI, it seems plausible that such a policy could create a Schelling point for cooperation and contribution to the social good. On the last bullet point, I suspect that there are solutions to the problem of distributing tax revenues globally, which at least outperform corporate disbursements in expectation. I also think that starting with a legible and broadly popular policy would be a very good way to initiate public discussion of AI governance. While there is likely a lot of behind-the-scene work that I am unaware of, my impression is that existing momentum in public AI policy is not heading in the most useful direction. It strikes me that taking a positive first step, especially one which recognizes surprising claims like "AGI companies may grow 100 or 1000X," would be helpful for shifting the Overton window on policy in the right direction.
f22bd0a8-d098-49cb-841d-666cfc7cdb28	trentmkelly/LessWrong-43k	LessWrong	Meetup : Melbourne, practical rationality Discussion article for the meetup : Melbourne, practical rationality WHEN: 07 June 2013 07:00:00PM (+1000) WHERE: 491 King Street, West Melbourne VIC 3003, Australia NOTE: We've moved a stone's throw from our old office. Note the new address (which is unchanged from May, but changed from April). Practical rationality. This meetup repeats on the 1st Friday of each month and is distinct from our social meetup on the 3rd Friday of each month. Topic for June: Hypothetical Apostasies - bring a written down belief you think it likely that at least several other attendees will share but that just might be wrong http://lesswrongmelbourne.uservoice.com/forums/203428-general/suggestions/3881488-small-groups-hypothetical-apostasies (http://wiki.lesswrong.com/mediawiki/images/c/ca/How_to_Run_a_Successful_Less_Wrong_Meetup_Group.pdf p23) Discussion: http://groups.google.com/group/melbourne-less-wrong All welcome from 6:30pm. Call the phone number on the door and I'll let you in. Discussion article for the meetup : Melbourne, practical rationality
c48b624d-0c4f-4ca6-99cb-35fff9d124c1	trentmkelly/LessWrong-43k	LessWrong	Post AGI effect prediction What differentiate us from AI? Human's sense of meaning usually are derived from feeling special, on an individual or collective basis. We want to prove ourselves different through tastes, hobbies, and career choices. Difference also show through collective culture and individual values. Biologically, we are all different from our gene sequence. Our tiny difference creates huge leverage that lead some people to enormous wealth while others stay in poverty. It's important to answer how AI is fundamentally different from us, and what's our value proposition longterm? Emotion, physical intelligence are not what makes us unique I used to assume that AI can't innovate, nor is conscious and have true understanding. After months of inquiries and research in the AI community and neuroscience community, I have completely shifted my mind. Human is not unique biologically, and a different neural network substrate could lead to equivalent results. I have been trying to answer an important question: what is the essence of intelligence, and will AI have emotions, and be self-conscious like we do? The vision of robotics would automate our society has aspired many people, including my mentor who is a founder devoted all his life engineering robotic hands. Physical intelligence will be taken over by humanoid longterm, and the midterm solutions of specialized robotic system will be eliminated. Math is the basis of intelligence, and Vitalik gave me a few pointers on his opinion on intelligence. It was quite inspiring he thinks human intelligence is no different from LLM. It feels more human than human if intuition is the marker. RLHF could generate outputs that we find agreeable by having emotions. Bio tissues surely are not the only substrate for emotions/intelligence, and Vitalik believed that emotions are algorithmic. A fun example is to have something running on a clock speed of 1 cycle per minute that's implemented by people passing cards around with their hands to have emer
8e5bed4b-da9b-46f8-97fa-d5766edd83bc	trentmkelly/LessWrong-43k	LessWrong	Ethical and incentive-compatible way to share finances with partner when you both work? My partner and I have been living together for a year. We currently share our finances to some extent (we have our own bank accounts, but have a shared credit card). I've been thinking about how our current system can be made more optimal. We mostly buy food, travel expenses, and other shared goods on the credit card. Obviously, any system of sharing finances should leave both partners feeling like the arrangement is fair. That's what I mean by ethical. I think that a relationship should be a team effort, and if you can give your partner 1 utilon at the cost of 0.5 utilons to yourself, you should do it every time. If they do the same thing, I think it's very likely that you're both better off in expectation. Also, I think it's better if a method of splitting finances as incentive-compatible as possible. By that, I mean, both partners should be incentivized to spend money in a way that optimizes total utility between both partners. Money is the one of the most common causes of arguments in relationships, and it seems to me that more incentive-compatible ways of sharing income might be a way to prevent those arguments. (I think it was William Vickrey who said something like (paraphrasing) "We should get the part of our economy to that can be made to run off self-interest running as smoothly as possible, so more of our attention can be spent on areas that truly cannot be solved except through altruism." By the way, if anyone can track down the source of this quote, I'd be very grateful.) Brainstorming (trying to say as much as I can about the problem before proposing a solution): 1. Diminishing marginal utility means that inequality of consumption is a sign that your relationship could be better. In my case, I make about 4x more than my partner. It would be very bad for aggregate utility if I lived large while my partner had to pinch pennies. 2. A strategy like "one partner pays for rent, the other pays for food" gives the partner who pays for rent no incentive
1f90e7c7-c621-4170-b20d-108e082c8824	StampyAI/alignment-research-dataset/arxiv	Arxiv	Hierarchical Optimal Transport for Comparing Histopathology Datasets 1 Introduction --------------- Histopathology images are routinely used in the diagnostic workup of many cancers. Beyond the standard identification of tumor grade and subtype, histopathology images also contain an abundance of visual information that may have predictive and prognostic value. Recent advances in deep learning for histopathology are facilitating the extraction of this information, enabling prediction of genetic alterations, treatment response and survival ([Coudray et al.(2018)Coudray, Ocampo, Sakellaropoulos, Narula, Snuderl, Fenyö, Moreira, Razavian, and Tsirigos](#bib.bibx6); [Muhammad et al.(2021)Muhammad, Xie, Sigel, Doukas, Alpert, Simpson, and Fuchs](#bib.bibx16); [Wulczyn et al.(2021)Wulczyn, Steiner, Moran, Plass, Reihs, Tan, Flament-Auvigne, Brown, Regitnig, Chen, Hegde, Sadhwani, MacDonald, Ayalew, Corrado, Peng, Tse, Müller, Xu, Liu, Stumpe, Zatloukal, and Mermel](#bib.bibx26); [Echle et al.(2020)Echle, Rindtorff, Brinker, Luedde, Pearson, and Kather](#bib.bibx9)). Despite the promise of supervised deep learning, the large, labeled datasets required to train complex networks on histopathology images are scarce, especially in less common cancers and label types. To overcome data scarcity during model training, transfer learning techniques can be utilized by pre-training models on a larger—ideally similar—dataset. The model can then be fine-tuned on the small target dataset of interest. Until now, determining which similar dataset to use for pre-training in histopathology has been driven by intuition or trial and error ([Srinidhi et al.(2021)Srinidhi, Ciga, and Martel](#bib.bibx21)). In this paper, we introduce a principled approach to measuring similarities between histopathology datasets. Specifically, we propose a novel distance between histopathology datasets which we call Hierarchical Histopathology Optimal Transport (HHOT) based on Optimal Transport (OT), a method to compare distributions. Our method uses OT to compute the distance between individual slides and to compute the distances between entire datasets (\figurereffig:Figure1). We show that HHOT is highly predictive of transfer learning accuracy in a tumor vs. normal prediction setting, and that it is significantly faster than a naive (non-hierarchical) Optimal Transport approach. Figure 1: Schematic of how HHOT distances are calculated at the slide (a) and dataset (b) level. Tiling of individual slides, done to overcome memory limits, introduces hierarchical structure to the calculation of OT between different datasets. fig:Figure1 [For a pair of slides, the OT distance is calculated using the OT distance between all pairs of tiles. ][centred] ![Refer to caption](/html/2204.08324/assets/images/Fig2_spatial_rev.png) \subfigure[For a pair of datasets, the OT distance is calculated using the OT distance between all pairs of slides][centred] ![Refer to caption](/html/2204.08324/assets/images/Fig1_OT_rev_Cslides.png) Figure 1: Schematic of how HHOT distances are calculated at the slide (a) and dataset (b) level. Tiling of individual slides, done to overcome memory limits, introduces hierarchical structure to the calculation of OT between different datasets. 2 Background ------------- Optimal Transport (OT) is a mathematical framework centered around the goal of comparing probability distributions, with deep theory ([Villani(2003)](#bib.bibx23); [Villani(2008)](#bib.bibx24); [Santambrogio(2015)](#bib.bibx20)) and applications to various fields, ranging from economics ([Galichon(2016)](#bib.bibx12)) to meteorology ([Cullen and Maroofi(2003)](#bib.bibx7)). Although it can be formulated in more general settings, in this work we are interested in its discrete Euclidean formulation, which considers two finite collections of points {𝐱(i)}i=1nsuperscriptsubscriptsuperscript𝐱𝑖𝑖1𝑛\{\textbf{x}^{(i)}\}\_{i=1}^{n}{ x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT, {𝐲(j)}j=1msuperscriptsubscriptsuperscript𝐲𝑗𝑗1𝑚\{\textbf{y}^{(j)}\}\_{j=1}^{m}{ y start\_POSTSUPERSCRIPT ( italic\_j ) end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_j = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT, 𝐱(i),𝐲(j)∈ℝdsuperscript𝐱𝑖superscript𝐲𝑗 superscriptℝ𝑑\textbf{x}^{(i)},\textbf{y}^{(j)}\in\mathbb{R}^{d}x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT , y start\_POSTSUPERSCRIPT ( italic\_j ) end\_POSTSUPERSCRIPT ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_d end\_POSTSUPERSCRIPT, represented as empirical distributions: μ=∑i=1n𝐩iδ𝐱(i)𝜇superscriptsubscript𝑖1𝑛subscript𝐩𝑖subscript𝛿superscript𝐱𝑖\mu=\sum\_{i=1}^{n}\textbf{p}\_{i}\delta\_{\textbf{x}^{(i)}}italic\_μ = ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT italic\_δ start\_POSTSUBSCRIPT x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT, ν=∑j=1m𝐪jδ𝐲(j)𝜈superscriptsubscript𝑗1𝑚subscript𝐪𝑗subscript𝛿superscript𝐲𝑗\nu=\sum\_{j=1}^{m}\textbf{q}\_{j}\delta\_{\textbf{y}^{(j)}}italic\_ν = ∑ start\_POSTSUBSCRIPT italic\_j = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT q start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT italic\_δ start\_POSTSUBSCRIPT y start\_POSTSUPERSCRIPT ( italic\_j ) end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT, where p and q are probability vectors (non-negative and adding up to one). At a high level, the goal of OT is to find an optimal correspondence between these distributions, and in doing so, define a notion of similarity between them. Given a cost function (often called the ground metric) between pairs of points c:ℝd×ℝd→ℝ+:𝑐→superscriptℝ𝑑superscriptℝ𝑑limit-fromℝc:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}+italic\_c : blackboard\_R start\_POSTSUPERSCRIPT italic\_d end\_POSTSUPERSCRIPT × blackboard\_R start\_POSTSUPERSCRIPT italic\_d end\_POSTSUPERSCRIPT → blackboard\_R +, the goal of OT is to find a correspondence between μ𝜇\muitalic\_μ and ν𝜈\nuitalic\_ν with minimal cost. Formally, the Kantorovich formulation of discrete OT seeks a coupling matrix Γ∈ℝn×mΓsuperscriptℝ𝑛𝑚\Gamma\in\mathbb{R}^{n\times m}roman\_Γ ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_n × italic\_m end\_POSTSUPERSCRIPT that solves: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| OTc(μ,ν)=def.minΓ∈Π(μ,ν)⁡⟨Γ,𝐂⟩=∑ijΓij𝐂ij,superscriptdef.subscriptOT𝑐𝜇𝜈subscriptΓΠ𝜇𝜈Γ𝐂subscript𝑖𝑗subscriptΓ𝑖𝑗subscript𝐂𝑖𝑗\mathrm{OT}\_{c}(\mu,\nu)\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def.}}}}{{=}}\min\_{\Gamma\in\Pi(\mu,\nu)}\langle\Gamma,\textbf{C}\rangle=\sum\_{ij}\Gamma\_{ij}\textbf{C}\_{ij},roman\_OT start\_POSTSUBSCRIPT italic\_c end\_POSTSUBSCRIPT ( italic\_μ , italic\_ν ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG def. end\_ARG end\_RELOP roman\_min start\_POSTSUBSCRIPT roman\_Γ ∈ roman\_Π ( italic\_μ , italic\_ν ) end\_POSTSUBSCRIPT ⟨ roman\_Γ , C ⟩ = ∑ start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT roman\_Γ start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT C start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT , \| \| (1) \| where 𝐂ij=def.c(𝐱(i),𝐲(j))superscriptdef.subscript𝐂𝑖𝑗𝑐superscript𝐱𝑖superscript𝐲𝑗\textbf{C}\_{ij}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def.}}}}{{=}}c(\textbf{x}^{(i)},\textbf{y}^{(j)})C start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG def. end\_ARG end\_RELOP italic\_c ( x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT , y start\_POSTSUPERSCRIPT ( italic\_j ) end\_POSTSUPERSCRIPT ). The constraint set Π(μ,ν)Π𝜇𝜈\Pi(\mu,\nu)roman\_Π ( italic\_μ , italic\_ν ) enforces ΓΓ\Gammaroman\_Γ to be measure-preserving, i.e., to have μ𝜇\muitalic\_μ and ν𝜈\nuitalic\_ν as its marginals: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Π(μ,ν)=def.{Γ∈ℝ+n×m\|Γ𝟏=𝐩,Γ⊤𝟏=𝐪}.superscriptdef.Π𝜇𝜈conditional-setΓsubscriptsuperscriptℝ𝑛𝑚formulae-sequenceΓ𝟏𝐩superscriptΓtop𝟏𝐪\Pi(\mu,\nu)\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def.}}}}{{=}}\{\Gamma\in\mathbb{R}^{n\times m}\_{+}\medspace\|\medspace\Gamma\textbf{1}=\textbf{p},\medspace\Gamma^{\top}\textbf{1}=\textbf{q}\}.roman\_Π ( italic\_μ , italic\_ν ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG def. end\_ARG end\_RELOP { roman\_Γ ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_n × italic\_m end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT + end\_POSTSUBSCRIPT \| roman\_Γ 1 = p , roman\_Γ start\_POSTSUPERSCRIPT ⊤ end\_POSTSUPERSCRIPT 1 = q } . \| \| (2) \| It can be shown that for c(𝐱,𝐲)=‖𝐱−𝐲‖p𝑐𝐱𝐲superscriptnorm𝐱𝐲𝑝c(\textbf{x},\textbf{y})=\\|\textbf{x}-\textbf{y}\\|^{p}italic\_c ( x , y ) = ∥ x - y ∥ start\_POSTSUPERSCRIPT italic\_p end\_POSTSUPERSCRIPT, OTc(μ,ν)1/psubscriptOT𝑐superscript𝜇𝜈1𝑝\mathrm{OT}\_{c}(\mu,\nu)^{1/p}roman\_OT start\_POSTSUBSCRIPT italic\_c end\_POSTSUBSCRIPT ( italic\_μ , italic\_ν ) start\_POSTSUPERSCRIPT 1 / italic\_p end\_POSTSUPERSCRIPT is a proper distance metric between distributions (i.e., satisfies all metric axioms) ([Peyré and Cuturi(2019)](#bib.bibx18)). As noted above, the coupling matrix ΓΓ\Gammaroman\_Γ can be interpreted as a soft matching or probabilistic correspondence between the elements of μ𝜇\muitalic\_μ and ν𝜈\nuitalic\_ν, in the sense that ΓijsubscriptΓ𝑖𝑗\Gamma\_{ij}roman\_Γ start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT is high if 𝐱(i),𝐲(j)superscript𝐱𝑖superscript𝐲𝑗\textbf{x}^{(i)},\textbf{y}^{(j)}x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT , y start\_POSTSUPERSCRIPT ( italic\_j ) end\_POSTSUPERSCRIPT are in ‘correspondence’, and low otherwise. In some cases (such as the common case n=m𝑛𝑚n=mitalic\_n = italic\_m and 𝐩,𝐪𝐩𝐪\textbf{p},\textbf{q}p , q uniform), the optimal coupling turns out to be sparse, in which case ΓΓ\Gammaroman\_Γ defines a deterministic one-to-one mapping between the 𝐱(i)superscript𝐱𝑖\textbf{x}^{(i)}x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT’s and 𝐲(j)superscript𝐲𝑗\textbf{y}^{(j)}y start\_POSTSUPERSCRIPT ( italic\_j ) end\_POSTSUPERSCRIPT’s. Problem ([1](#S2.E1 "1 ‣ 2 Background ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets")) is a linear programming problem and thus solvable exactly in O(n3)𝑂superscript𝑛3O(n^{3})italic\_O ( italic\_n start\_POSTSUPERSCRIPT 3 end\_POSTSUPERSCRIPT ) time ([Peyré and Cuturi(2019)](#bib.bibx18)). This makes it impractical even for moderately sized collections of points. However, seminal work by [Cuturi(2013)](#bib.bibx8) showed that a regularized version of this problem can be solved much more efficiently. The regularization consists of adding a entropy term on the objective: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| OTc,ε(μ,ν)=def.minΓ∈Π(μ,ν)⁡⟨Γ,𝐂⟩+εH(Γ).superscriptdef.subscriptOT𝑐𝜀𝜇𝜈subscriptΓΠ𝜇𝜈Γ𝐂𝜀HΓ\mathrm{OT}\_{c,\varepsilon}(\mu,\nu)\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def.}}}}{{=}}\min\_{\Gamma\in\Pi(\mu,\nu)}\langle\Gamma,\textbf{C}\rangle+\varepsilon\mathrm{H}(\Gamma).roman\_OT start\_POSTSUBSCRIPT italic\_c , italic\_ε end\_POSTSUBSCRIPT ( italic\_μ , italic\_ν ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG def. end\_ARG end\_RELOP roman\_min start\_POSTSUBSCRIPT roman\_Γ ∈ roman\_Π ( italic\_μ , italic\_ν ) end\_POSTSUBSCRIPT ⟨ roman\_Γ , C ⟩ + italic\_ε roman\_H ( roman\_Γ ) . \| \| (3) \| This entropy-regularized OT problem can be solved very efficiently using the Sinkhorn-Knopp algorithm ([Cuturi(2013)](#bib.bibx8)). One downside of this regularization is that OTc,ε(μ,μ)≠0subscriptOT𝑐𝜀𝜇𝜇0\mathrm{OT}\_{c,\varepsilon}(\mu,\mu)\neq 0roman\_OT start\_POSTSUBSCRIPT italic\_c , italic\_ε end\_POSTSUBSCRIPT ( italic\_μ , italic\_μ ) ≠ 0, which in particular implies this quantity is no longer a valid distance. To alleviate this, prior work ([Genevay et al.(2018)Genevay, Peyré, and Cuturi](#bib.bibx14); [Salimans et al.(2018)Salimans, Zhang, Radford, and Metaxas](#bib.bibx19)) has considered a debiased version this quantity, also known as the Sinkhorn divergence: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| SDc,ε(μ,ν)=def.OTc,ε(μ,ν)−12(OTc,ε(μ,μ)+OTc,ε(ν,ν))superscriptdef.subscriptSD𝑐𝜀𝜇𝜈subscriptOT𝑐𝜀𝜇𝜈12subscriptOT𝑐𝜀𝜇𝜇subscriptOT𝑐𝜀𝜈𝜈\mathrm{SD}\_{c,\varepsilon}(\mu,\nu)\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def.}}}}{{=}}\mathrm{OT}\_{c,\varepsilon}(\mu,\nu)-\tfrac{1}{2}\bigl{(}\mathrm{OT}\_{c,\varepsilon}(\mu,\mu)+\mathrm{OT}\_{c,\varepsilon}(\nu,\nu)\bigr{)}roman\_SD start\_POSTSUBSCRIPT italic\_c , italic\_ε end\_POSTSUBSCRIPT ( italic\_μ , italic\_ν ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG def. end\_ARG end\_RELOP roman\_OT start\_POSTSUBSCRIPT italic\_c , italic\_ε end\_POSTSUBSCRIPT ( italic\_μ , italic\_ν ) - divide start\_ARG 1 end\_ARG start\_ARG 2 end\_ARG ( roman\_OT start\_POSTSUBSCRIPT italic\_c , italic\_ε end\_POSTSUBSCRIPT ( italic\_μ , italic\_μ ) + roman\_OT start\_POSTSUBSCRIPT italic\_c , italic\_ε end\_POSTSUBSCRIPT ( italic\_ν , italic\_ν ) ) \| \| (4) \| In addition to satisfying SDc,ε(μ,ν)≥0subscriptSD𝑐𝜀𝜇𝜈0\mathrm{SD}\_{c,\varepsilon}(\mu,\nu)\geq 0roman\_SD start\_POSTSUBSCRIPT italic\_c , italic\_ε end\_POSTSUBSCRIPT ( italic\_μ , italic\_ν ) ≥ 0, with equality if and only if μ=ν𝜇𝜈\mu=\nuitalic\_μ = italic\_ν, this divergence comes with many other desirable theoretical properties: it is positive, convex, metrizes weak converge in distribution ([Feydy et al.(2019)Feydy, Séjourné, Vialard, Amari, Trouvé, and Peyré](#bib.bibx10)), and leads to faster statistical rates of estimation of the exact OT problem ([Chizat et al.(2020)Chizat, Roussillon, Léger, Vialard, and Peyré](#bib.bibx5)). In Section [4.1](#S4.SS1 "4.1 Hierarchical Histopathology Optimal Transport ‣ 4 Optimal transport between histopathology datasets ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets"), we will introduce our method using OTc,ε(μ,ν)subscriptOT𝑐𝜀𝜇𝜈\mathrm{OT}\_{c,\varepsilon}(\mu,\nu)roman\_OT start\_POSTSUBSCRIPT italic\_c , italic\_ε end\_POSTSUBSCRIPT ( italic\_μ , italic\_ν ) for notational simplicity, noting that it can naturally use SDc,ε(μ,ν)subscriptSD𝑐𝜀𝜇𝜈\mathrm{SD}\_{c,\varepsilon}(\mu,\nu)roman\_SD start\_POSTSUBSCRIPT italic\_c , italic\_ε end\_POSTSUBSCRIPT ( italic\_μ , italic\_ν ) instead. In addition, we will drop c𝑐citalic\_c and ϵitalic-ϵ\epsilonitalic\_ϵ from the notation for OTOT\mathrm{OT}roman\_OT when these are clear from the context. 3 Related work --------------- Our HHOT method for histopathology builds on previous work in hierarchical OT for other domains and previous work in non-hierarchical OT within histopathology. For example, [Yurochkin et al.(2019)Yurochkin, Claici, Chien, Mirzazadeh, and Solomon](#bib.bibx27) have described a hierarchical OT method for measuring distances between documents, using words and topics as the hierarchical levels. In addition, specifically for histopathology images, non-hierarchical OT has been used to compare individual cell morphology ([Basu et al.(2014)Basu, Kolouri, and Rohde](#bib.bibx4); [Wang et al.(2011)Wang, Ozolek, Slepčev, Lee, Chen, and Rohde](#bib.bibx25)) and to quantify domain shift at the tile level ([Stacke et al.(2021)Stacke, Eilertsen, Unger, and Lundström](#bib.bibx22)). In addition, the relationship between OT-calculated dataset distances and the difficulty of transferability has been previously described by ([Alvarez-Melis and Fusi(2020)](#bib.bibx3); [Gao and Chaudhari(2021)](#bib.bibx13); [Achille et al.(2021)Achille, Paolini, Mbeng, and Soatto](#bib.bibx1)), although the notion of distance they define is generic and thus does not leverage the hierarchical nature of individual datasets. 4 Optimal transport between histopathology datasets ---------------------------------------------------- ### 4.1 Hierarchical Histopathology Optimal Transport We consider a pair of histopathology datasets, 𝖣asubscript𝖣𝑎\mathsf{D}\_{a}sansserif\_D start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT and 𝖣bsubscript𝖣𝑏\mathsf{D}\_{b}sansserif\_D start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT, collected from different tissues, centers, or populations. We seek a notion of distance that lets us assess their similarity. Each dataset consists of slides s, which in turn are subdivided into tiles t. Let n𝑛nitalic\_n and m𝑚mitalic\_m denote, respectively, the number of slides in each dataset. In addition, we denote by nisubscript𝑛𝑖n\_{i}italic\_n start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT the number of tiles in the i𝑖iitalic\_i-th slide of the first dataset, and analogously for mjsubscript𝑚𝑗m\_{j}italic\_m start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT. We can view 𝖣asubscript𝖣𝑎\mathsf{D}\_{a}sansserif\_D start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT and 𝖣bsubscript𝖣𝑏\mathsf{D}\_{b}sansserif\_D start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT as point clouds or, more formally, empirical distributions as described in section [2](#S2 "2 Background ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets"). From this viewpoint, we can think of these datasets as samples of slide images sampled from two different underlying distributions. Unless additional information is provided, we can simply take the weights associated to each slide (p and q) to be uniform, as is typically done in practical application of OT to point clouds or images ([Peyré and Cuturi(2019)](#bib.bibx18)). After defining a suitable notion of distance between pairs of slides, one could in principle use problem ([3](#S2.E3 "3 ‣ 2 Background ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets")) (or its debiased counterpart, eq. ([4](#S2.E4 "4 ‣ 2 Background ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets"))) to obtain a notion of similarity between DasubscriptD𝑎\mathrm{D}\_{a}roman\_D start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT and DbsubscriptD𝑏\mathrm{D}\_{b}roman\_D start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT. However, this naive approach would require loading entire slides into memory, which is computationally infeasible—precisely the reason why these images are tiled in the first place. Our proposed solution to this computational hurdle is to interpret the slides themselves as collections (formally, distributions) of tiles, and use OT once more, now to define a notion of distance between these. To this end, we first define the cost between individual tiles as the distance between them. Although we could in principle use the Euclidean distance between their raw pixel representations, a more meaningful comparison can be obtained by first embedding these images in some lower dimensional space (e.g., using a neural network pre-trained on a large image dataset), and then computing a distance between them. Hence, we define c(𝐭u,𝐭v)=‖ϕ(𝐭u)−ϕ(𝐭v)‖2𝑐subscript𝐭𝑢subscript𝐭𝑣superscriptnormitalic-ϕsubscript𝐭𝑢italic-ϕsubscript𝐭𝑣2c(\textbf{t}\_{u},\textbf{t}\_{v})=\\|\phi(\textbf{t}\_{u})-\phi(\textbf{t}\_{v})\\|^{2}italic\_c ( t start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT , t start\_POSTSUBSCRIPT italic\_v end\_POSTSUBSCRIPT ) = ∥ italic\_ϕ ( t start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT ) - italic\_ϕ ( t start\_POSTSUBSCRIPT italic\_v end\_POSTSUBSCRIPT ) ∥ start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT, where ϕitalic-ϕ\phiitalic\_ϕ is a pre-trained encoder. We collect all such pairwise costs in a matrix 𝐂tilesubscript𝐂tile\textbf{C}\_{\text{tile}}C start\_POSTSUBSCRIPT tile end\_POSTSUBSCRIPT of size ni×mjsubscript𝑛𝑖subscript𝑚𝑗n\_{i}\times m\_{j}italic\_n start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT × italic\_m start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT, and solve the corresponding OT problem: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| OTϕ,ε(𝐬i,𝐬j)=minΓ∈Π(𝐩i,𝐪j)⁡⟨Γtile,𝐂tileij⟩+εH(Γ).subscriptOTitalic-ϕ𝜀subscript𝐬𝑖subscript𝐬𝑗subscriptΓΠsubscript𝐩𝑖subscript𝐪𝑗subscriptΓtilesubscriptsuperscript𝐂𝑖𝑗tile𝜀HΓ\mathrm{OT}\_{\phi,\varepsilon}(\textbf{s}\_{i},\textbf{s}\_{j})=\min\_{\Gamma\in\Pi(\textbf{p}\_{i},\textbf{q}\_{j})}\langle\Gamma\_{\text{tile}},\textbf{C}^{ij}\_{\text{tile}}\rangle+\varepsilon\textup{H}(\Gamma).roman\_OT start\_POSTSUBSCRIPT italic\_ϕ , italic\_ε end\_POSTSUBSCRIPT ( s start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) = roman\_min start\_POSTSUBSCRIPT roman\_Γ ∈ roman\_Π ( p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , q start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) end\_POSTSUBSCRIPT ⟨ roman\_Γ start\_POSTSUBSCRIPT tile end\_POSTSUBSCRIPT , C start\_POSTSUPERSCRIPT italic\_i italic\_j end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT tile end\_POSTSUBSCRIPT ⟩ + italic\_ε H ( roman\_Γ ) . \| \| (5) \| As in section [2](#S2 "2 Background ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets"), we can also define a debiased version of this slide-to-slide distance: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| SDϕ,ε(𝐬i,𝐬j)=OTϕ,ε(𝐬i,𝐬j)−12(OTϕ,ε(𝐬i,𝐬i)+OTϕ,ε(𝐬j,𝐬j))subscriptSDitalic-ϕ𝜀subscript𝐬𝑖subscript𝐬𝑗subscriptOTitalic-ϕ𝜀subscript𝐬𝑖subscript𝐬𝑗12subscriptOTitalic-ϕ𝜀subscript𝐬𝑖subscript𝐬𝑖subscriptOTitalic-ϕ𝜀subscript𝐬𝑗subscript𝐬𝑗\mathrm{SD}\_{\phi,\varepsilon}(\textbf{s}\_{i},\textbf{s}\_{j})=\mathrm{OT}\_{\phi,\varepsilon}(\textbf{s}\_{i},\textbf{s}\_{j})-\tfrac{1}{2}\bigl{(}\mathrm{OT}\_{\phi,\varepsilon}(\textbf{s}\_{i},\textbf{s}\_{i})+\mathrm{OT}\_{\phi,\varepsilon}(\textbf{s}\_{j},\textbf{s}\_{j})\bigr{)}roman\_SD start\_POSTSUBSCRIPT italic\_ϕ , italic\_ε end\_POSTSUBSCRIPT ( s start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) = roman\_OT start\_POSTSUBSCRIPT italic\_ϕ , italic\_ε end\_POSTSUBSCRIPT ( s start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) - divide start\_ARG 1 end\_ARG start\_ARG 2 end\_ARG ( roman\_OT start\_POSTSUBSCRIPT italic\_ϕ , italic\_ε end\_POSTSUBSCRIPT ( s start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , s start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) + roman\_OT start\_POSTSUBSCRIPT italic\_ϕ , italic\_ε end\_POSTSUBSCRIPT ( s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT , s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) ) \| \| (6) \| Compared to other possible ways to compare slides using their tiles (like using a mean or centroid tile), this OT-based approach is appealing because (i) it does not lose information by aggregating the tiles, and (ii) it recovers some of the global structure that it lost by tiling, i.e., the relation between the tiles in the context of the slide. Specifically, in operationalizing similarity through matching, problem ([5](#S4.E5 "5 ‣ 4.1 Hierarchical Histopathology Optimal Transport ‣ 4 Optimal transport between histopathology datasets ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets")) seeks to find corresponding tiles across slides, and does it coherently as a result of the marginal constraint (e.g., a single tile cannot be matched to all tiles in the other slide). Once we have computed ([5](#S4.E5 "5 ‣ 4.1 Hierarchical Histopathology Optimal Transport ‣ 4 Optimal transport between histopathology datasets ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets")) for every pair of slides (𝐬i,𝐬j)subscript𝐬𝑖subscript𝐬𝑗(\textbf{s}\_{i},\textbf{s}\_{j})( s start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) from the two datasets, we collect them in a matrix 𝐂slidesubscript𝐂slide\textbf{C}\_{\text{slide}}C start\_POSTSUBSCRIPT slide end\_POSTSUBSCRIPT with entries [𝐂slide]ij=OTε(𝐬i,𝐬j)subscriptdelimited-[]subscript𝐂slide𝑖𝑗subscriptOT𝜀subscript𝐬𝑖subscript𝐬𝑗[\textbf{C}\_{\text{slide}}]\_{ij}=\mathrm{OT}\_{\varepsilon}(\textbf{s}\_{i},\textbf{s}\_{j})[ C start\_POSTSUBSCRIPT slide end\_POSTSUBSCRIPT ] start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT = roman\_OT start\_POSTSUBSCRIPT italic\_ε end\_POSTSUBSCRIPT ( s start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ). With this, finally we have a ground cost between slides, which we can use to compute the sought-after distance between datasets using OT once more: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| OTε(𝖣a,𝖣b)=minΓ∈Π(𝐩,𝐪)⁡⟨Γ,𝐂slide⟩+εH(Γ),subscriptOT𝜀subscript𝖣𝑎subscript𝖣𝑏subscriptΓΠ𝐩𝐪Γsubscript𝐂slide𝜀HΓ\mathrm{OT}\_{\varepsilon}(\mathsf{D}\_{a},\mathsf{D}\_{b})=\min\_{\Gamma\in\Pi(\textbf{p},\textbf{q})}\langle\Gamma,\textbf{C}\_{\text{slide}}\rangle+\varepsilon\textup{H}(\Gamma),roman\_OT start\_POSTSUBSCRIPT italic\_ε end\_POSTSUBSCRIPT ( sansserif\_D start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT , sansserif\_D start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ) = roman\_min start\_POSTSUBSCRIPT roman\_Γ ∈ roman\_Π ( p , q ) end\_POSTSUBSCRIPT ⟨ roman\_Γ , C start\_POSTSUBSCRIPT slide end\_POSTSUBSCRIPT ⟩ + italic\_ε H ( roman\_Γ ) , \| \| (7) \| and its debiased counterpart: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| SDϕ,ε(𝖣a,𝖣b)=OTϕ,ε(𝖣a,𝖣b)−12(OTϕ,ε(𝖣a,𝖣b)+OTϕ,ε(𝖣a,𝖣b)).subscriptSDitalic-ϕ𝜀subscript𝖣𝑎subscript𝖣𝑏subscriptOTitalic-ϕ𝜀subscript𝖣𝑎subscript𝖣𝑏12subscriptOTitalic-ϕ𝜀subscript𝖣𝑎subscript𝖣𝑏subscriptOTitalic-ϕ𝜀subscript𝖣𝑎subscript𝖣𝑏\mathrm{SD}\_{\phi,\varepsilon}(\mathsf{D}\_{a},\mathsf{D}\_{b})=\mathrm{OT}\_{\phi,\varepsilon}(\mathsf{D}\_{a},\mathsf{D}\_{b})-\tfrac{1}{2}\bigl{(}\mathrm{OT}\_{\phi,\varepsilon}(\mathsf{D}\_{a},\mathsf{D}\_{b})+\mathrm{OT}\_{\phi,\varepsilon}(\mathsf{D}\_{a},\mathsf{D}\_{b})\bigr{)}.roman\_SD start\_POSTSUBSCRIPT italic\_ϕ , italic\_ε end\_POSTSUBSCRIPT ( sansserif\_D start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT , sansserif\_D start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ) = roman\_OT start\_POSTSUBSCRIPT italic\_ϕ , italic\_ε end\_POSTSUBSCRIPT ( sansserif\_D start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT , sansserif\_D start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ) - divide start\_ARG 1 end\_ARG start\_ARG 2 end\_ARG ( roman\_OT start\_POSTSUBSCRIPT italic\_ϕ , italic\_ε end\_POSTSUBSCRIPT ( sansserif\_D start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT , sansserif\_D start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ) + roman\_OT start\_POSTSUBSCRIPT italic\_ϕ , italic\_ε end\_POSTSUBSCRIPT ( sansserif\_D start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT , sansserif\_D start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ) ) . \| \| (8) \| ### 4.2 Computational Implementation We use the python optimal transport (POT) ([Flamary et al.(2021)Flamary, Courty, Gramfort, Alaya, Boisbunon, Chambon, Chapel, Corenflos, Fatras, Fournier, Gautheron, Gayraud, Janati, Rakotomamonjy, Redko, Rolet, Schutz, Seguy, Sutherland, Tavenard, Tong, and Vayer](#bib.bibx11)) and geomloss ([Feydy et al.(2019)Feydy, Séjourné, Vialard, Amari, Trouvé, and Peyré](#bib.bibx10)) libraries to solve the individual OT problems ([5](#S4.E5 "5 ‣ 4.1 Hierarchical Histopathology Optimal Transport ‣ 4 Optimal transport between histopathology datasets ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets")) and ([7](#S4.E7 "7 ‣ 4.1 Hierarchical Histopathology Optimal Transport ‣ 4 Optimal transport between histopathology datasets ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets")). Using the vanilla Sinkhorn algorithm, solving the first of these to δ1subscript𝛿1\delta\_{1}italic\_δ start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT-accuracy has O(nimj/δ1)𝑂subscript𝑛𝑖subscript𝑚𝑗subscript𝛿1O(n\_{i}m\_{j}/\delta\_{1})italic\_O ( italic\_n start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT / italic\_δ start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) computational complexity ([Altschuler et al.(2017)Altschuler, Niles-Weed, and Rigollet](#bib.bibx2)), and analogously O(nm/δ2)𝑂𝑛𝑚subscript𝛿2O(nm/\delta\_{2})italic\_O ( italic\_n italic\_m / italic\_δ start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) for the latter. Taking δ1=δ2subscript𝛿1subscript𝛿2\delta\_{1}=\delta\_{2}italic\_δ start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = italic\_δ start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, the total complexity scales as O((nm+∑ijnimj)/δ)𝑂𝑛𝑚subscript𝑖𝑗subscript𝑛𝑖subscript𝑚𝑗𝛿O((nm+\sum\_{ij}n\_{i}m\_{j})/\delta)italic\_O ( ( italic\_n italic\_m + ∑ start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT italic\_n start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT italic\_m start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) / italic\_δ ). \figurereffig:Figure4c shows empirical runtimes for our method. Our implementation of HHOT can be found here: \urlhttps://github.com/ayeaton/HHOT 5 Methodology and Results -------------------------- \floatconts fig:Figure2 \subfigure[Source tile visualizing cartilage within a LUAD slide.] ![Refer to caption](/html/2204.08324/assets/images/Fig2_rev.jpg) \subfigure[Target slide of LUAD. The orange highlighted region is cartilaginous.] ![Refer to caption](/html/2204.08324/assets/images/Fig2_rev_c.jpg) \subfigure[Heatmap of coupling between the source tile and target slide. High coupling is observed in the cartilaginous region.] ![Refer to caption](/html/2204.08324/assets/images/Fig2_rev_b.png) Figure 2: Representative example of a source tile tightly coupling to target tiles of the same tissue type, suggesting that HHOT between tiles implicitly incorporates biological information. We used whole slide images retrieved from the TCGA (https://portal.gdc.cancer.gov/) for six common cancer types, including both primary tumor samples and matched normal samples. Specifically we focused on Stomach Adenocarcinoma (STAD), Bladder Urothelial Carcinoma (BLCA), Lung Adenocarcinoma (LUAD), Lung Squamous-cell Carcinoma (LUSC), Colon Adenocarcinoma (COAD), and Pancreatic Adenocarcinoma (PAAD). Slide images were tiled into 512x512 pixel non-overlapping images at 20x magnification, and tiles with more than 50 percent background were discarded as described in [Coudray et al.(2018)Coudray, Ocampo, Sakellaropoulos, Narula, Snuderl, Fenyö, Moreira, Razavian, and Tsirigos](#bib.bibx6); [Noorbakhsh et al.(2020)Noorbakhsh, Farahmand, Foroughi pour, Namburi, Caruana, Rimm, Zarringhalam, and Chuang](#bib.bibx17). We extracted a median of 474 tiles for BLCA, 777 for COAD, 445 for LUAD, 448 for LUSC, 278 for PAAD and 622 for STAD. We then used Inception-V3 pre-trained on ImageNet to compress 512x512 pixel images to 2048 length vectors. For comparison of collections of tiles, we set the regularization parameter ε𝜀\varepsilonitalic\_ε to 0.25. We calculated OT between slides, and found that tiles with the highest coupling were those that were visually similar. As an example, we show the OT coupling matrix between tiles from two representative slides in \figurereffig:Figure2. We chose an example tile from the source slide which displays cartilage tissue. We then show the coupling of this source tile to all tiles in the target slide. We visualize the strength of coupling in \figurereffig:Figure2c; the highest coupling to the cartilage source tile is tightly localized and corresponds to a cartilaginous region in the target. These results demonstrate that an OT solution implicitly incorporates biological structure. Figure 3: HHOT enables better clustering of slide images by cancer type, as compared to the centroid tile distance method. HHOT also reflects the diversity of images within a single cancer type. This suggests that HHOT distances better preserve morphological information. fig:Figure3 \subfigure [Centroid UMAP.]![Refer to caption](/html/2204.08324/assets/images/Fig3_rev_a.png) \subfigure [OT UMAP.]![Refer to caption](/html/2204.08324/assets/images/Fig3_rev_b.png) [KNN performance. ]![Refer to caption](/html/2204.08324/assets/images/Fig3_rev.png) \subfigure[HHOT distance retains visual diversity within cancer datasets. ]![Refer to caption](/html/2204.08324/assets/images/fig3_rev_e_nolab.png) Figure 3: HHOT enables better clustering of slide images by cancer type, as compared to the centroid tile distance method. HHOT also reflects the diversity of images within a single cancer type. This suggests that HHOT distances better preserve morphological information. In slide-to-slide comparisons, HHOT preserves more relevant histomorphological information than other methods, such as using a mean or centroid tile. We demonstrate this with a task for clustering the slides by cancer type. We created a matrix 𝐂slidesubscript𝐂slide\textbf{C}\_{\text{slide}}C start\_POSTSUBSCRIPT slide end\_POSTSUBSCRIPT with entries [𝐂slide]ij=OTε(𝐬i,𝐬j)subscriptdelimited-[]subscript𝐂slide𝑖𝑗subscriptOT𝜀subscript𝐬𝑖subscript𝐬𝑗[\textbf{C}\_{\text{slide}}]\_{ij}=\mathrm{OT}\_{\varepsilon}(\textbf{s}\_{i},\textbf{s}\_{j})[ C start\_POSTSUBSCRIPT slide end\_POSTSUBSCRIPT ] start\_POSTSUBSCRIPT italic\_i italic\_j end\_POSTSUBSCRIPT = roman\_OT start\_POSTSUBSCRIPT italic\_ε end\_POSTSUBSCRIPT ( s start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , s start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) as described above including all pairs of slides. For reference, we also calculated a distance matrix between slides using the centroid tile of a slide as described in [Howard et al.(2021)Howard, Dolezal, Kochanny, Schulte, Chen, Heij, Huo, Nanda, Olopade, Kather, Cipriani, L., and Pearson](#bib.bibx15). We visualized the relationship between slides using UMAP for both OT and centroid-tile distance (\figurereffig:Figure3a, \figurereffig:Figure3b). Visually, we observe that HHOT distance enables better clustering of the slide images by cancer type. Quantitatively, using K-nearest neighbors over K ranging from two to eight, we performed a cancer-type classification task with our two distance matrices as input. We show that OT retains similarity within cancer-types and dissimilarity across cancer-types as expected, and does so better than the centroid-tile distance baseline. \figurereffig:Figure3e shows representative examples of how HHOT distances reflect the diversity of images with a single cancer type. HHOT tightly groups PAAD slides, and these slide images are visually very similar. In contrast, HHOT shows more variability in the LUSC slides, and these images are more variable. We observed that target datasets that are similar to the source dataset based on HHOT distance are more improved by pre-training, i.e. a negative correlation between HHOT distance and transferability. In this paper, we focus on a tumor vs. normal task. For our pre-training tasks, we standardized the total dataset size to 209 slides, and used a cross validation scheme to create four datasets per cancer type with 169 slides for training and 40 for validation. For our fine-tuning task, we standardized the dataset to 65 slides, and used a cross validation scheme to create four datasets per cancer type with 25 slides for training and 40 for validation. We conducted our task over all the pre-training datasets and all the fine-tuning datasets for 16 experiments per cancer type comparison. For each dataset pair, we pre-trained a single-layer perceptron to predict tumor or normal status, using feature vectors ϕ(𝐭)italic-ϕ𝐭\phi(\textbf{t})italic\_ϕ ( t ) and a learning rate of 1e-2. We then fine-tuned this pre-trained model for each of the other cancer types, using only 25 target slides and a learning rate of 1e-10. We quantify transferability across tasks using the relative improvement in AUC obtained by transfer learning: 𝖳𝗋𝖺𝗇𝗌𝖿𝖾𝗋𝖺𝖻𝗂𝗅𝗂𝗍𝗒(DT,DS)=(𝖠𝖴𝖢(DS→DT)−𝖠𝖴𝖢(DT))/𝖠𝖴𝖢(DT)𝖳𝗋𝖺𝗇𝗌𝖿𝖾𝗋𝖺𝖻𝗂𝗅𝗂𝗍𝗒subscript𝐷𝑇subscript𝐷𝑆𝖠𝖴𝖢→subscript𝐷𝑆subscript𝐷𝑇𝖠𝖴𝖢subscript𝐷𝑇𝖠𝖴𝖢subscript𝐷𝑇\textsf{Transferability}(D\_{T},D\_{S})=(\textsf{AUC}(D\_{S}\rightarrow D\_{T})-\textsf{AUC}(D\_{T}))/\textsf{AUC}(D\_{T})Transferability ( italic\_D start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT , italic\_D start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT ) = ( AUC ( italic\_D start\_POSTSUBSCRIPT italic\_S end\_POSTSUBSCRIPT → italic\_D start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT ) - AUC ( italic\_D start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT ) ) / AUC ( italic\_D start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT ). We observed a negative correlation between the HHOT distance and transferability most strongly for PAAD, LUAD, LUSC, and STAD (\figurereffig:Figure4a). Consistently, visually similar datasets, such as the two lung cancer datasets, show the smallest HHOT distance and the highest transferability (green and purple squares and plus in \figurereffig:Figure4a). For BLCA and COAD, we observe that the AUC of a model without pre-training is already very high (\figurereffig:Figure4b); thus, pre-training on other datasets cannot improve on this already high AUC. This is explicitly quantified for these two cancers by the nearly zero slope of the best fit line (\figurereffig:Figure4b, boxplot of no-pretaining AUCs aggregating four datasets of 25 target slides). Figure 4: HHOT is predictive of model transferability across cancer types. Datasets with large HHOT distances, representing more visually different data, have worse transferability. The regression coefficients are [β𝛽\betaitalic\_β, β0subscript𝛽0\beta\_{0}italic\_β start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT] [0.0024, −5.47×10−5-5.47E-5-5.47\text{\times}{10}^{-5}start\_ARG - 5.47 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 5 end\_ARG end\_ARG] (COAD), [0.0016, −6×10−5-6E-5-6\text{\times}{10}^{-5}start\_ARG - 6 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 5 end\_ARG end\_ARG] (BLCA), [0.022, −6.8×10−4-6.8E-4-6.8\text{\times}{10}^{-4}start\_ARG - 6.8 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 4 end\_ARG end\_ARG] (PAAD), [0.042, −1.7×10−3-1.7E-3-1.7\text{\times}{10}^{-3}start\_ARG - 1.7 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 3 end\_ARG end\_ARG] (LUAD), [0.043, −2×10−3-2E-3-2\text{\times}{10}^{-3}start\_ARG - 2 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 3 end\_ARG end\_ARG] (LUSC), [0.049, −2.7×10−3-2.7E-3-2.7\text{\times}{10}^{-3}start\_ARG - 2.7 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 3 end\_ARG end\_ARG] (STAD). For COAD and BLCA, the baseline AUC is already quite high (b), so transfer learning with other cancer types results in little improvement. HHOT is faster than non-hierarchical OT between slides (c). fig:example2 [HHOT vs. transferability. ] ![Refer to caption](/html/2204.08324/assets/x1.png) \subfigure [No-transfer AUC.]![Refer to caption](/html/2204.08324/assets/images/Fig5_rev_auc.png) \subfigure [HHOT reduces runtime.]![Refer to caption](/html/2204.08324/assets/images/Fig5_rev.png) Figure 4: HHOT is predictive of model transferability across cancer types. Datasets with large HHOT distances, representing more visually different data, have worse transferability. The regression coefficients are [β𝛽\betaitalic\_β, β0subscript𝛽0\beta\_{0}italic\_β start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT] [0.0024, −5.47×10−5-5.47E-5-5.47\text{\times}{10}^{-5}start\_ARG - 5.47 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 5 end\_ARG end\_ARG] (COAD), [0.0016, −6×10−5-6E-5-6\text{\times}{10}^{-5}start\_ARG - 6 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 5 end\_ARG end\_ARG] (BLCA), [0.022, −6.8×10−4-6.8E-4-6.8\text{\times}{10}^{-4}start\_ARG - 6.8 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 4 end\_ARG end\_ARG] (PAAD), [0.042, −1.7×10−3-1.7E-3-1.7\text{\times}{10}^{-3}start\_ARG - 1.7 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 3 end\_ARG end\_ARG] (LUAD), [0.043, −2×10−3-2E-3-2\text{\times}{10}^{-3}start\_ARG - 2 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 3 end\_ARG end\_ARG] (LUSC), [0.049, −2.7×10−3-2.7E-3-2.7\text{\times}{10}^{-3}start\_ARG - 2.7 end\_ARG start\_ARG times end\_ARG start\_ARG power start\_ARG 10 end\_ARG start\_ARG - 3 end\_ARG end\_ARG] (STAD). For COAD and BLCA, the baseline AUC is already quite high (b), so transfer learning with other cancer types results in little improvement. HHOT is faster than non-hierarchical OT between slides (c). Finally, we show that not only does HHOT preserve the hierarchical structure of the data and correlate with difficulty of transfer learning, it is also much faster than a flat (non-hierarchical) OT approach. We compare the time it took to compute OT distance between four to eighteen slides, with 100 tiles each and observed that the time it took to calculate OT distance between slides increased in a polynomial order in the flat case (Figure [4](#S5.F4 "Figure 4 ‣ 5 Methodology and Results ‣ Hierarchical Optimal Transport for Comparing Histopathology Datasets")). 6 Discussion ------------- In this paper we introduced a principled approach to compare histopathology datasets called HHOT. Our work adds to the OT and histopathology literature by proposing a method to compare datasets while also preserving the structure lost by standard tiling approaches. Specifically, we propose to solve first an inner tile-to-tile OT problem for all pairs of slides, and then solve the outer slide-to-slide OT problem between datasets. We first show that correspondences in the OT coupling matrix ΓtilesubscriptΓ𝑡𝑖𝑙𝑒\Gamma\_{tile}roman\_Γ start\_POSTSUBSCRIPT italic\_t italic\_i italic\_l italic\_e end\_POSTSUBSCRIPT map the same type of tissue across slides. We then show that OT distance performs better than a naive, centroid-tile distance based method in a cancer-type prediction task. We also show that HHOT distance between histopathology datasets correlates with transferability. Furthermore, we find that the degree of correlation between HHOT distance and transferability is associated with baseline AUC. Simply, if the task is already close to optimal, it is difficult to improve using pre-training. Finally, we show that HHOT distance is much faster than a naive, flat approach to comparing histopathology datasets. In addition to applications presented in this paper, promising applications of HHOT include outlier detection, clustering analysis, dataset visualization, and facilitating multi-modal integration of whole slide images and molecular data. In conclusion, we have demonstrated that HHOT has many benefits for researchers working with tiled histopathology data. Future work may focus on using OT to direct dataset creation to optimize transferability, tune tile size and normalization hyper-parameters, and compare feature vectors learned from different models. \acksR.G.K., R.M., D.A.M, and G.H were employed by Microsoft corporation while performing this work. A.Y. was employed by Microsoft corporation for a duration of this work. Part of this work has used computing resources at the NYU School of Medicine High Performance Computing Facility.
1209eec5-4375-46e6-984d-dc65bb702052	trentmkelly/LessWrong-43k	LessWrong	Principles for Alignment/Agency Projects "John, what do you think of this idea for an alignment research project?" I get questions like that fairly regularly. How do I go about answering? What principles guide my evaluation? Not all of my intuitions for what makes a project valuable can easily be made legible, but I think the principles in this post capture about 80% of the value. Tackle the Hamming Problems, Don't Avoid Them Far and away the most common failure mode among self-identifying alignment researchers is to look for Clever Ways To Avoid Doing Hard Things (or Clever Reasons To Ignore The Hard Things), rather than just Directly Tackling The Hard Things. The most common pattern along these lines is to propose outsourcing the Hard Parts to some future AI, and "just" try to align that AI without understanding the Hard Parts of alignment ourselves. The next most common pattern is to argue that, since Hard Parts are Hard, we definitely don't have enough time to solve them and should therefore pretend that we're going to solve alignment while ignoring them. Third most common is to go into field building, in hopes of getting someone else to solve the Hard Parts. (Admittedly these are not the most charitable summaries.) There is value in seeing how dumb ideas fail. Most of that value is figuring out what the Hard Parts of the problem are - the taut constraints which we run into over and over again, which we have no idea how to solve. (If it seems pretty solvable, it's probably not a Hard Part.) Once you can recognize the Hard Parts well enough to try to avoid them, you're already past the point where trying dumb ideas has much value. On a sufficiently new problem, there is also value in checking dumb ideas just in case the problem happens to be easy. Alignment is already past that point; it's not easy. You can save yourself several years of time and effort by actively trying to identify the Hard Parts and focus on them, rather than avoid them. Otherwise, you'll end up burning several years on ideas
fd060e8b-0914-4c84-bead-78425316eaaa	trentmkelly/LessWrong-43k	LessWrong	[link] Speed is the New Intelligence From Scott Adams Blog The article really is about speeding up government, but the key point is speed as a component of smart: > A smart friend told me recently that speed is the new intelligence, at least for some types of technology jobs. If you are hiring an interface designer, for example, the one that can generate and test several designs gets you further than the “genius” who takes months to produce the first design to test. When you can easily test alternatives, the ability to quickly generate new things to test is a substitute for intelligence. This shifts the focus from the ability to grasp and think through very complex topics (includes good working memory and memory recall in general) to the ability new topics quickly (includes quick learning and unlearning, creativity). > Smart people in the technology world no long believe they can think their way to success. Now the smart folks try whatever plan looks promising, test it, tweak it, and reiterate. In that environment, speed matters more than intelligence because no one has the psychic ability to pick a winner in advance. All you can do is try things that make sense and see what happens. Obviously this is easier to do when your product is software based. This also changes the type of grit needed. The grit to push through a long topic versus the grit try lots of new things and to learn from failures.
57a4b5a2-8ca8-42f4-877a-38e1582dbe1e	trentmkelly/LessWrong-43k	LessWrong	Have we really forsaken natural selection? Natural selection is often charged with having goals for humanity, and humanity is often charged with falling down on them. The big accusation, I think, is of sub-maximal procreation. If we cared at all about the genetic proliferation that natural selection wanted for us, then this time of riches would be a time of fifty-child families, not one of coddled dogs and state-of-the-art sitting rooms. But (the story goes) our failure is excusable, because instead of a deep-seated loyalty to genetic fitness, natural selection merely fitted humans out with a system of suggestive urges: hungers, fears, loves, lusts. Which all worked well together to bring about children in the prehistoric years of our forebears, but no more. In part because all sorts of things are different, and in part because we specifically made things different in that way on purpose: bringing about children gets in the way of the further satisfaction of those urges, so we avoid it (the story goes). This is generally floated as an illustrative warning about artificial intelligence. The moral is that if you make a system by first making multitudinous random systems and then systematically destroying all the ones that don’t do the thing you want, then the system you are left with might only do what you want while current circumstances persist, rather than being endowed with a consistent desire for the thing you actually had in mind. Observing acquaintences dispute this point recently, it struck me that humans are actually weirdly aligned with natural selection, more than I could easily account for. Natural selection, in its broadest, truest, (most idiolectic?) sense, doesn’t care about genes. Genes are a nice substrate on which natural selection famously makes particularly pretty patterns by driving a sensical evolution of lifeforms through interesting intricacies. But natural selection’s real love is existence. Natural selection just favors things that tend to exist. Things that start existing: great.
cf370eb4-55e2-4567-9b80-348a3e6e9dd6	trentmkelly/LessWrong-43k	LessWrong	How to solve deception and still fail. A mostly finished post I'm kicking out the door. You'll get the gist. I There's a tempting picture of alignment that centers on the feeling of "As long as humans stay in control, it will be okay." Humans staying in control, in this picture, is something like humans giving lots of detailed feedback to powerful AI, staying honestly apprised of the consequences of its plans, and having the final say on how plans made by an AI get implemented.[1] Of course, this requires the AI to be generating promising plans to begin with, or else the humans are just stuck rejecting bad plans all day. But conveniently, in this picture we don't have to solve the alignment problem the hard way. We could train the AI on human approval to get it to generate in-some-sense-good plans. As long as humans stay in control, it will be okay. Normally, training even indirectly for human approval teaches AIs to learn to deceive humans to more reliably maximize approval. Which is why, in order for humans to stay in control, we need to be able to solve deception - not just detecting it after the fact, but producing AIs that actually don't try to deceive the humans. An AI deceiving the human (from Christiano et al. 2017). The sort of thing we'd like to understand how to categorically avoid. The hope goes that human's un-deceived approval is a sufficient quality check, and that we'll pick good AI-generated plans and make the future go well. This post is about how that picture is flawed - how an AI can generate a plan that humans approve of, based on lots of human feedback, that isn't deceptive in the narrow sense, but that's still bad. Put a certain way, my thesis sounds nuts. What's supposed to count as a valid human preference if human-guided plans, honestly approved of, aren't sufficient? Is this just paranoid essentialism, insisting that there's some essence of rightness that I magically know AIs risk missing, no matter their relationship with humans? II I'm not being essentialist. Humans
f1e9e5c2-1270-4eb5-a6c5-56c4220b013d	trentmkelly/LessWrong-43k	LessWrong	Difficulty classes for alignment properties I don't think this idea is particularly novel, but it comes up often in conversations I have, so I figured it'd be good to write it up. How do you prevent deception from AI systems? One obvious thing to try would be to make sure that your model never thinks deceptive thoughts. There are several ways you could go about this, to varying degrees of effectiveness. For instance, you could check for various precursors to deceptive alignment. More prosaically, you could try identifying deceptive circuits within your model, or looking at what features your model is keeping track of and identifying any suspicious ones. I think these are pretty reasonable approaches. However, I think they fail to address failure modes like deep deceptiveness. A system can be deceptive even if no individual part looks deceptive, due to complex interactions between the system and the environment. More generally, cognition and optimization power can be externalized to the environment. One could make the argument that focusing on more salient and dangerous failure modes like deceptive alignment makes a lot more sense. However - especially if you’re interested in approaches that work in the worst case and don’t rely on reality turning out optimistically one way or the other - you probably want approaches that prevent things from going badly at all. So, how do you prevent any failure modes that route through deception? In an earlier post, I wrote about robust intent alignment as the solution, and one research direction I think is feasible to get there. But here I want to make a different point, about what it would look like to interface with deception in that setup. Start from the intuition that deception in a system is a property of the person being deceived more than it is the deceiver. It follows pretty naturally that deception is better viewed as a property of the composite system that is the agent and its environment. So, if you wanted to interface with the general thing that is deceptio
3954b5d2-de19-4c3f-b768-c874dc23e6b6	trentmkelly/LessWrong-43k	LessWrong	SIA fears (expected) infinity It's well known that the Self-Indication Assumption (SIA) has problems with infinite populations (one of the reasons I strongly recommend not using the probability as the fundamental object of interest, but instead the decision, as in anthropic decision theory). SIA also has problems with arbitrarily large finite populations, at least in some cases. What cases are these? Imagine that we had these (non-anthropic) probabilities for various populations: p0, p1, p2, p3, p4... Now let us apply the anthropic correction from SIA; before renormalising, we have these weights for different population levels: 0, p1, 2p2, 3p3, 4p4... To renormalise, we need to divide by the sum 0 + p1 + 2p2 + 3p3 + 4p4... This is actually the expected population! (note: we are using the population as a proxy for the size of the reference class of agents who are subjectively indistinguishable from us; see this post for more details) So using SIA is possible if and only if the (non-anthropic) expected population is finite (and non-zero). Note that it is possible for the anthropic expected population to be infinite! For instance if pj is C/j3, for some constant C, then the non-anthropic expected population is finite (being the infinite sum of C/j2). However once we have done the SIA correction, we can see that the SIA-corrected expected population is infinite (being the infinite sum of some constant times 1/j).
3a82a0ea-62a0-4260-8c26-a2f86d6005aa	StampyAI/alignment-research-dataset/blogs	Blogs	New paper: “Asymptotic logical uncertainty and the Benford test” [![Asymptotic Logical Uncertainty and The Benford Test](http://intelligence.org/wp-content/uploads/2015/09/Benford.png)](http://arxiv.org/abs/1510.03370)We have released a new paper on [logical uncertainty](https://intelligence.org/2015/01/09/new-report-questions-reasoning-logical-uncertainty/), co-authored by Scott Garrabrant, Siddharth Bhaskar, Abram Demski, Joanna Garrabrant, George Koleszarik, and Evan Lloyd: “[Asymptotic logical uncertainty and the Benford test](http://arxiv.org/abs/1510.03370)[.”](http://arxiv.org/abs/1510.03370) Garrabrant gives some background on his approach to logical uncertainty [on the Intelligent Agent Foundations Forum](https://agentfoundations.org/item?id=270): > The main goal of logical uncertainty is to learn how to assign probabilities to logical sentences which have not yet been proven true or false. > > > One common approach is to change the question, assume logical omniscience and only try to assign probabilities to the sentences that are independent of your axioms (in hopes that this gives insight to the other problem). Another approach is to limit yourself to a finite set of sentences or deductive rules, and assume logical omniscience on them. Yet another approach is to try to define and understand logical counterfactuals, so you can try to assign probabilities to inconsistent counterfactual worlds. > > > One thing all three of these approaches have in common is they try to allow (a limited form of) logical omniscience. This makes a lot of sense. We want a system that not only assigns decent probabilities, but which we can formally prove has decent behavior. By giving the system a type of logical omniscience, you make it predictable, which allows you to prove things about it. > > > However, there is another way to make it possible to prove things about a logical uncertainty system. We can take a program which assigns probabilities to sentences, and let it run forever. We can then ask about whether or not the system eventually gives good probabilities. > > > At first, it seems like this approach cannot work for logical uncertainty. Any machine which searches through all possible proofs will eventually give a good probability (1 or 0) to any provable or disprovable sentence. To counter this, as we give the machine more and more time to think, we have to ask it harder and harder questions. > > > We therefore have to analyze the machine’s behavior not on individual sentences, but on infinite sequences of sentences. For example, instead of asking whether or not the machine quickly assigns 1/10 to the probability that the 3↑↑↑↑3rd digit of π is a 5 we look at the sequence: > > > an:= the probability the machine assigns at timestep 2n to the n↑↑↑↑nth digit of π being 5, > > > and ask whether or not this sequence converges to 1/10. > > [Benford’s law](https://en.wikipedia.org/wiki/Benford%27s_law) is the observation that the first digit in base 10 of various random numbers (e.g., random powers of 3) is likely to be small: the digit 1 comes first about 30% of the time, 2 about 18% of the time, and so on; 9 is the leading digit only 5% of the time. In their paper, Garrabrant et al. pick the Benford test as a concrete example of logically uncertain reasoning, similar to the π example: a machine passes the test iff it consistently assigns the correct subjective probability to “The first digit is a 1.” for the number 3 to the power f(n), where f is a fast-growing function and f(n) cannot be quickly computed. Garrabrant et al.’s new paper describes an algorithm that passes the Benford test in a nontrivial way by searching for infinite sequences of sentences whose truth-values cannot be distinguished from the output of a weighted coin. In other news, the papers “[Toward idealized decision theory](http://arxiv.org/abs/1507.01986)” and “[Reflective oracles: A foundation for classical game theory](http://arxiv.org/abs/1508.04145)” are now available on arXiv. We’ll be presenting a version of the latter paper with a slightly altered title (“Reflective oracles: A foundation for game theory in artificial intelligence”) at [LORI-V](https://www.yoursaas.cc/websites/36224472513387025486/) next month. Update June 12, 2016: “Asymptotic logical uncertainty and the Benford test” has been accepted to AGI-16. #### Sign up to get updates on new MIRI technical results Get notified every time a new technical paper is published. * * × The post [New paper: “Asymptotic logical uncertainty and the Benford test”](https://intelligence.org/2015/09/30/new-paper-asymptotic-logical-uncertainty-and-the-benford-test/) appeared first on [Machine Intelligence Research Institute](https://intelligence.org).
ddc60ef1-350a-4d27-bacd-d0842085acb2	trentmkelly/LessWrong-43k	LessWrong	Hedonistic Isotopes: Abstract Simple scaling of hedonistic values can become fairly imprecise when given scales, especially when comparing elevations of euphoria or emptiness. Hedonistic Values may be more precise and specific than other methods of description. If there is more variation in describing experiences, it can fill gaps in tacit knowledge that fails to be articulated into its comprehensible constituent, if any at all. Firstly, the person of experience should list every emotion they experience in x situation. Such that we converge the plethora of words into an underlying topic. This can be done by scaling different temperaments in relation to each other on a scale, which then chooses the most ideal “feeling” closest to all the emotions the user had mentioned. For example if we have the word distraught and annoyed we may converge them to the word “temperamental”. This may be done heuristically, but to avoid major biases, it’s best to plot the vast variation of hedonistic traits on a graph (Which is time consuming, at the expense of precision). Lastly, each trait in itself that was in result of the users experiences would be scaled out of 10 or 20, but this value can be endlessly adjusted towards the overall hedonistic framework of the individual which is far more accurate. Note you may utilize a value for each semantic that comprises an experience, but to get obtain the isotope/value of the output trait, you take the semantic list the user mentioned and obtain its mean (Sum of each scale value / Total amount of words(description) ).
1c2ac878-ece6-4dcd-b401-c37883aa1f93	StampyAI/alignment-research-dataset/blogs	Blogs	What counts as death? When imagining a world of [digital people](https://www.cold-takes.com/how-digital-people-could-change-the-world/) - as in some of the [utopia links](https://www.cold-takes.com/utopia-links/) from last week (as well as my [digital people sketches](https://www.cold-takes.com/imagining-yourself-as-a-digital-person-two-sketches/) from a while ago) - it's common to bump into some classic questions in philosophy of personal identity, like:* Would a [duplicate](https://www.cold-takes.com/how-digital-people-could-change-the-world/#productivity) of you be "you?" * If you got physically destroyed and replaced with an exact duplicate of yourself, did you die? (This question could connect directly to [whether "converting yourself to a digital person"](https://www.cold-takes.com/imagining-yourself-as-a-digital-person-two-sketches/) is equivalent to dying.) My answers are "sort of" and "no." My philosophy on "what counts as death" is simple, though unconventional, and it seems to resolve most otherwise mind-bending [paradoxical thought experiments about personal identity](https://waitbutwhy.com/2014/12/what-makes-you-you.html). It is the same basic idea as the one advanced by Derek Parfit in [Reasons and Persons](https://www.amazon.com/Reasons-Persons-Derek-Parfit/dp/019824908X);[1](https://www.cold-takes.com/p/e7b0fda7-1c46-4ab4-a826-798c4f4dbd6c#fn1) Parfit also claims it is similar to Buddha's view[2](https://www.cold-takes.com/p/e7b0fda7-1c46-4ab4-a826-798c4f4dbd6c#fn2) (so it's got that going for it). I haven't been able to find a simple, compact statement of this philosophy, and I think I can lay it out in about a page. So here it is, presented simply and without much in the way of caveats (this is "how things feel to me" rather than "something I'm confident in regardless of others' opinions"): Constant replacement. In an important sense, I stop existing and am replaced by a new person each moment (second or minute or whatever). The sense in which it feels like I "continue to exist, as one unified thread through time" is just an illusion, created by the fact that I have memories of my past. The only thing that is truly "me" is this moment; next moment, it will be someone else. Kinship with past and future selves. My future self is a different person from me, but he has an awful lot in common with me: personality, relationships, ongoing projects, and more. Things like my relationships and projects are most of what give my current moment meaning, so it's very important to me whether my future selves are around to continue them. So although my future self is a different person, I care about him a lot, for the same sorts of reasons I care about friends and loved ones (and their future selves).[3](https://www.cold-takes.com/p/e7b0fda7-1c46-4ab4-a826-798c4f4dbd6c#fn3) If I were to "die" in the common-usage (e.g., medical) sense, that would be bad for all those future selves that I care about a lot.[4](https://www.cold-takes.com/p/e7b0fda7-1c46-4ab4-a826-798c4f4dbd6c#fn4) (I do of course refer to past and future Holdens in the first person. When I refer to someone as "me," that means that they are a past or future self, which generally means that they have an awful lot in common with me. But in a deeper philosophical sense, my past and future selves are other people.) And that's all. I'm constantly being replaced by other Holdens, and I care about the other Holdens, and that's all that's going on. * I don't care how quickly the cells in my body die and get replaced (if it were once per second, that wouldn't bother me). My self is already getting replaced all the time, and replacing my cells wouldn't add anything to that. * I don't care about "continuity of consciousness" (if I were constantly losing consciousness while all my cells got replaced, that wouldn't bother me). * If you vaporized me and created a copy of me somewhere else, that would just be totally fine. I would think of it as teleporting. It'd be chill. * If you made a bunch of copies of me, I would be all of them in one sense (I care about them a lot, in the same way that I normally care about future selves) and none of them in another sense (just as I am not my future selves). * If you did something really weird like splitting my brain in half and combining each half with someone else's brain, that would create two people that I care about more than a stranger and less than "Holden an hour from now." * I don't really find any [thought experiments](https://waitbutwhy.com/2014/12/what-makes-you-you.html) on this topic trippy or mind bending. They're all just cases where I get replaced with some other people who have some things in common with me, and that's already happening all the time. Pros and cons of this view -------------------------- (This isn't going to feel very balanced, because this view "feels right" to me, but if I get good additional cons [in the comments](https://forum.effectivealtruism.org/posts/87AugxNfxAgAqR3mv/comments-for-shorter-cold-takes-pieces?commentId=NnqpqzyrdhFWPy8oE) I might run them in a future post.) The main con I see is that "constant replacement" is a pretty unusual way of thinking about things. I think many people think they would find it kind of horrifying to imagine that they wink out of existence every second and get replaced by someone else. To those people, though, I would suggest "trying it on": try to imagine, for let's say a full week, that you're fully convinced of constant replacement, and see whether it feels as impossible to live with as it seems at first. You might initially expect to find yourself constantly terrified of your impending death, but my guess is you won't be able to keep that up, and you'll soon be feeling and acting pretty normal. You won't make any weird decisions, because "concern for future selves" provides pretty much the same functional value as "concern for oneself" in normal circumstances (I just think it works better in exotic circumstances). If that's right, "constant replacement" could join a number of other ideas that feel so radically alien (for many) that they must be "impossible to live with," but actually are just fine to live with. (E.g., atheism; [physicalism](https://plato.stanford.edu/entries/physicalism/); weird things about physics. I think many proponents of these views would characterize them as having fairly normal day-to-day implications while handling some otherwise confusing questions and situations better.) As for the pros: * Having sat with it a while, the view now feels very intuitive to me. + Constant replacement isn't some novel or radical idea. E.g., it's similar to the idea that [now is all there ever is](https://www.brainyquote.com/quotes/eckhart_tolle_523615#:~:text=Eckhart%20Tolle%20Quotes&text=People%20don't%20realize%20that%20now%20is%20all%20there%20ever,or%20anticipation%20in%20your%20mind.). (And as noted above, Derek Parfit claims that Buddha took a similar view.) A lot of people live in this headspace. + Constant replacement seems sort of obviously true when I think about my relationship to my far-past self: the me of 10 years ago really feels like a different person that I happen to have memories of. And the me of 10 years from now is probably the same kind of deal. So my relationship to the me of 1 minute from now should be qualitatively the same kind of thing, just much less so, and that seems about right. + Once you accept constant replacement, the rest of the view seems like common sense. + To be clear, this isn't always how I've thought. I used to stare at some random object and think "Is this moment of me staring at this object the only me that has ever existed? (How would I know if it weren't?)" and feel sort of freaked out. But at a certain point I just started answering "Yeah" and it started feeling correct, and chill.* It seems good that when I think about questions like "Would situation \_\_ count as dying?", I don't have to give answers that are dependent on stuff like how fast the atoms in my body turn over - stuff I have basically never thought about and that doesn't feel deeply relevant to what I care about. Instead, when I think about whether I'd be comfortable with something like teleportation, I find myself thinking about things I actually do care about, like my life projects and relationships, and the future interactions between me and the world. * All of the [paradoxical thought experiments](https://waitbutwhy.com/2014/12/what-makes-you-you.html) about teleportation, brain transplants, etc. stop feeling confusing or mind-bending. I feel like I could make sense of things even in a potential [radically unfamiliar future](https://www.cold-takes.com/how-digital-people-could-change-the-world/). * I probably don't have the same kind of fear of death that most people have. I figure my identity has already changed dramatically enough to count as most of the way toward death at least a few times so far, so it doesn't feel like a totally unprecedented thing that's going to happen to me. Anyway, if you think this is crazy, have at it in the comments. --- Footnotes --------- 1. For key quotes from [Reasons and Persons](https://smile.amazon.com/Reasons-Persons-Derek-Parfit-ebook/dp/B006QV7ZMS), see pages 223-224; 251; 279-282; 284-285; 292; 340-341. For explanations of "psychological continuity" and "psychological connectedness" (which Parfit frequently uses in discussing what matters for what counts as death), see page 206. "Psychological connectedness" is a fairly general idea that seems consistent with what I say here; "psychological continuity" is a more specific idea that is less important on my view (though also see pages 288-289, where Parfit appears to equivocate on how much, and how, psychological continuity matters). [↩](#fnref1)- "As Appendix J shows, Buddha would have agreed. The Reductionist View [the view Parfit defends] is not merely part of one cultural tradition. It may be, as I have claimed, the true view about all people at all times." [Reasons and Persons](https://smile.amazon.com/Reasons-Persons-Derek-Parfit-ebook/dp/B006QV7ZMS) page 273. Emphasis in original. [↩](#fnref2)- There's the additional matter that he's held responsible for my actions, which makes sense if only because my actions are predictive of his actions. [↩](#fnref3)- I don't personally care all that much about these future selves' getting to "exist," as an end in itself. I care more about the fact that their disappearance would mean the end of the stories, projects, relationships, etc. that I'm in. But you could easily take my view of personal identity while caring a lot intrinsically about whether your future selves get to exist. [↩](#fnref4)
80975aab-c890-4fba-94ae-1fa3bdc44f6b	trentmkelly/LessWrong-43k	LessWrong	Geometric Rationality is Not VNM Rational One elephant in the room throughout my geometric rationality sequence, is that it is sometimes advocating for randomizing between actions, and so geometrically rational agents cannot possibly satisfy the Von Neumann–Morgenstern axioms. That is correct: I am rejecting the VNM axioms. In this post, I will say more about why I am making such a bold move. A Model of Geometric Rationality I have been rather vague on what I mean by geometric rationality. I still want to be vague in general, but for the purposes of this post, I will give a concrete definition, and I will use the type signature of the VNM utility theorem. (I do not think this definition is good enough, and want it to restrict its scope to this post.) A preference ordering on lotteries over outcomes is called geometrically rational if there exists some probability distribution P over interval valued utility functions on outcomes such that L⪯M if and only if GU∼PEO∼LU(O)≤GU∼PEO∼MU(O). For comparison, an agent is VNM rational there exists a single utility function U, such that L⪯M if and only if EO∼LU(O)≤EO∼MU(O). Geometric Rationality is weaker than VNM rationality, since under reasonable assumptions, we can assume the utility function of a VNM rational agent is interval valued, and then we can always take the probability distribution that assigns probability 1 to this utility function. Geometric Rationality is strictly weaker, because it sometimes strictly prefers lotteries over any of the deterministic outcomes, and VNM rational agents never do this. The VNM utility theorem says that any preference ordering on lotteries that satisfies some simple axioms must be VNM rational (i.e. have a utility function as above). Since I am advocating for a weaker notion of rationality, I must reject some of these axioms. Against Independence The VNM axiom that I am rejecting is the independence axiom. It states that given lotteries A, B, and C, and probability p, A⪯B if and only if pC+(1−p)A⪯pC+(1−p)B. Thus, mi
dde6ca77-acb9-4f76-81f2-e2413f7cc44b	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	XPT forecasts on (some) biological anchors inputs This post was co-authored by the Forecasting Research Institute and Rose Hadshar. Thanks to Josh Rosenberg for managing this work, Zachary Jacobs and Molly Hickman for the underlying data analysis, Bridget Williams for fact-checking and copy-editing, the whole FRI XPT team for all their work on this project, and our external reviewers. TL;DR ===== * As part of the Existential Risk Persuasion Tournament (XPT) we asked participants to forecast several questions that allowed us to infer inputs to Ajeya Cotra’s [biological anchors model](https://docs.google.com/document/d/1IJ6Sr-gPeXdSJugFulwIpvavc0atjHGM82QjIfUSBGQ/edit#). The XPT superforecasters’ predictions differ substantially from Cotra’s on hardware costs, willingness to spend and algorithmic efficiency: ![](https://res.cloudinary.com/cea/image/upload/f_auto,q_auto/v1/mirroredImages/ccw9v9giKxg8nyLhp/qyiyiidtg7pwcdcecllj)* There are no XPT forecasts relating to other inputs to Cotra’s model, most notably the 2020 training computation requirements distribution. * Taking Cotra’s model and 2020 training computation requirements distribution as given, and using relevant XPT superforecaster forecasts as inputs, leads to substantial differences in model output: ![](https://res.cloudinary.com/cea/image/upload/f_auto,q_auto/v1/mirroredImages/ccw9v9giKxg8nyLhp/sttjuyrdpvhgv1v3cg8d)\The most aggressive and most conservative estimates can be considered equivalent to 90% confidence interval for the median estimate.[[1]](#fn736fhvw6wlo) Using median XPT inputs implies median transformative AI (TAI) timelines of around ~2090, compared to Cotra’s 2050 median timeline in 2020, and her [2040 median timeline](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines%23Making_a_one_time_upward_adjustment_for__2020_FLOP_____) in 2022. * Using 90% confidence interval (CI) XPT inputs: + Even the most aggressive XPT superforecaster inputs imply a lower probability that the compute required for training TAI is available than Cotra predicts, but the most conservative XPT superforecaster inputs predict TAI by 2100 as less likely than not. * Most of the difference in outputs comes down to differences in forecasts on: + Compute price halving time from 2025 to 2100 + Doubling time of spending on compute for the most expensive training run from 2025 onwards * Note that: + Both Cotra and XPT forecasts on FLOP/$ are already inaccurate. However, Cotra's will necessarily prove more accurate and the current estimate is outside the XPT 90% CI. + The XPT forecast for the most expensive training run (by 2024) is already inaccurate, but it's not yet clear whether this forecast is more or less accurate than Cotra's forecast for 2025, which remains much higher than current estimates. * XPT superforecasters’ all-things-considered TAI timelines are longer than those suggested by using Cotra’s model with XPT inputs. When asked about AI timelines in a survey at the end of the XPT, the median superforecaster put a probability of 3.75% on TAI by 2070. In contrast, Cotra’s model with superforecaster XPT inputs suggests a ~35%probability of TAI by 2070. * To the extent that timeline beliefs are based on the biological anchors model, *and* to the extent that these beliefs are based on a training requirements distribution similar to Cotra’s, then the actual value of the inputs on compute price halving time and doubling time of compute spending could have a significant bearing on expected timelines. Introduction ============ This post: * Compares estimates made by Ajeya Cotra and XPT [forecasts](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#The_forecasts) on questions relating to timelines until the compute required for TAI is attainable, and shows how the differences in forecasts impact the outputs of Cotra’s biological anchors model * Discusses [why](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#What_drives_the_differences_between_Cotra_and_XPT_forecasters_) Cotra and XPT forecasters disagree, and [which forecasts are more accurate](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Which_forecasts_are_more_accurate_) * Notes that [XPT forecasters’ all-things-considered TAI timelines](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#XPT_superforecasters__all_things_considered_view_on_TAI_timelines) are longer than those implied by using XPT forecasts as inputs to Cotra’s model * Includes appendices on: + The [arguments](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Appendix_A__Arguments_made_for_different_forecasts) given by Cotra and the XPT forecasters for their respective forecasts + [XPT expert](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Appendix_B__XPT_expert_forecasts_related_to_when_the_compute_required_for_TAI_will_be_attainable) (as opposed to superforecaster) forecasts relating to the biological anchors model Background on the Forecasting TAI with biological anchors report ---------------------------------------------------------------- In 2020, Ajeya Cotra at Open Philanthropy published her [Forecasting TAI with biological anchors report](https://docs.google.com/document/d/1IJ6Sr-gPeXdSJugFulwIpvavc0atjHGM82QjIfUSBGQ/edit#). The report modeled the probability that the compute required for building transformative AI (TAI) would be attainable in a given year, using: * An estimate of the amount of compute required to train a TAI model that uses machine learning architectures available in 2020. This was developed using various biological anchors[[2]](#fnx9yquvymqx) and is referred to as the “2020 training computation requirements distribution”. * An estimate of when the amount of compute required for TAI would be obtainable, which was developed from forecasts on hardware prices, willingness to spend, and algorithmic efficiency. Cotra’s ‘best guess’ model outputted a probability of ~46% that the compute required for TAI would be attainable by 2050. Cotra gave her overall median TAI timeline as 2050. In August 2022, Cotra [published some updates](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines) to her model, and shifted her median TAI timeline forward to 2040. Background on the Existential Risk Persuasion Tournament (XPT) -------------------------------------------------------------- In 2022, the [Forecasting Research Institute](https://forecastingresearch.org/) (FRI) ran the Existential Risk Persuasion Tournament (XPT). From June through October 2022, 169 forecasters, including 80 superforecasters and 89 experts in topics related to existential risk, developed forecasts on questions related to existential and catastrophic risk. Forecasters stopped updating their forecasts on 31st October 2022. FRI hopes to run future iterations of the tournament. You can see the results from the tournament overall [here](https://static1.squarespace.com/static/635693acf15a3e2a14a56a4a/t/64abffe3f024747dd0e38d71/1688993798938/XPT.pdf), results relating to AI risk [here](https://forum.effectivealtruism.org/posts/K2xQrrXn5ZSgtntuT/what-do-xpt-forecasts-tell-us-about-ai-risk-1), and to AI timelines in general [here](https://forum.effectivealtruism.org/posts/KGGDduXSwZQTQJ9xc/what-do-xpt-forecasts-tell-us-about-ai-timelines). Comparing Cotra and XPT forecasts --------------------------------- Some XPT questions relate directly to some of the inputs to Cotra’s biological anchors model. Specifically, there are XPT questions that relate to some of Cotra’s forecasts on hardware prices, willingness to spend, and algorithmic efficiency:[[3]](#fnrrryf8sfjaf) \| \| \| \| \| --- \| --- \| --- \| \| XPT question \| Comparison \| [Input to Cotra's model](https://docs.google.com/spreadsheets/d/1TjNQyVHvHlC-sZbcA7CRKcCp0NxV6MkkqBvL408xrJw/edit#gid=505210495) \| \| [47. What will be the lowest price, in 2021 US dollars, of 1 GFLOPS with a widely-used processor by the end of 2024, 2030, 2050?](https://docs.google.com/document/d/1rNN0L7vGZJeviMe_vtS2HQrVZompiYZ8tSnHp8-Q1IE/edit) \| Median XPT superforecaster forecast for 2024 converted from petaFLOPS-days to FLOP per $ and compared with Cotra's forecast for 2025 \| FLOP per $ at the start of period (2025) \| \| Inferred doubling time between median XPT superforecaster forecasts for 2024 and 2050, compared with Cotra's doubling time from 2025 to 2100 \| Compute price halving time in this period (2025–2100), in years \| \| [46. How much will be spent on compute in the largest AI experiment by the end of 2024, 2030, 2050?](https://docs.google.com/document/d/1IsKwKlD1jO8VRVuPEEzVHHTrmtx4xyDybNypWmIDZ8A/edit) \| Comparison of median XPT superforecaster 2024 forecast with Cotra 2025 forecast \| Compute cost for the most expensive training run at the start of period (2025), in 2020 USD \| \| Inferred doubling time between median XPT superforecaster forecasts for 2024 and 2050, compared with Cotra's doubling time from 2025 to 2100 \| Doubling time of spending on compute for the most expensive training run at start of period (2025), in years. \| \| [48. By what factor will training efficiency on ImageNet classification have improved over AlexNet by the end of 2024, 2030?](https://docs.google.com/document/d/1zwnvDF-7rQYW6R45am0RLGLxas15TLH-NJpA-XyDVRg/edit) \| Inferred doubling time between median XPT superforecaster forecasts for 2024 and 2030, compared with Cotra's doubling time from 2025 to 2100 \| Halving time of compute requirements per path over this period (2025–2100), in years \| Caveats and notes ----------------- It is important to note that there are several limitations to this analysis: * Outputs from Cotra’s model using some XPT inputs do not reflect the overall views of XPT forecasters on TAI timelines. + Based on commentary during the XPT, it’s unlikely that XPT forecasters would accept the assumptions of Cotra’s model, or agree with all of Cotra’s forecasts where there were no relevant XPT forecasts (most notably, the 2020 training computation requirements distribution). + In a survey we ran at the end of the XPT, superforecasters predicted a 3.8% chance of TAI by 2070, which is much lower than the corresponding ~35% outputted by Cotra’s model using relevant XPT forecasts as inputs. * Cotra’s model is very sensitive to changes in the training requirements distribution, so inputs on hardware prices and willingness to spend will not be significant for all values of that distribution. + In particular, for lower estimates of training requirements, XPT inputs would remain consistent with very short timelines. * None of the XPT forecasts are of exactly the same questions that Cotra uses as inputs. And some notes: * In this post, we focus on the forecasts of XPT superforecasters, as opposed to experts, when comparing with Cotra’s forecasts. + Analysis of the XPT differentiated experts into those with expertise in the specific domain of a question (in this case, AI), those with expertise in other domains related to existential risk (biosecurity, nuclear weapons, and climate change), and those with general expertise in existential risk studies. Too few AI domain experts answered the questions relevant to Cotra’s model to allow for analysis, so the expert figures provided here include all types of experts in the XPT. + Compared to superforecasters, XPT experts’ forecasts tended to be closer to Cotra by around an order of magnitude, but when inputted into Cotra’s model they produced similar outputs to those drawing on XPT superforecaster forecasts. - The exception to this is that the most aggressive XPT expert forecasts produced a probability of ~51% that the compute required for TAI is available by 2050, compared with 32% using the most aggressive XPT superforecaster forecasts. - See [Appendix B](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Appendix_B__XPT_expert_forecasts_related_to_when_the_compute_required_for_TAI_will_be_attainable) for more details. * The number of superforecasters who provided forecasts for each of the three key input questions ranged from 31 to 32. The forecasts ============= ![](https://res.cloudinary.com/cea/image/upload/f_auto,q_auto/v1/mirroredImages/ccw9v9giKxg8nyLhp/mnb4bz8kisoik60lh9ik)See workings [here](https://docs.google.com/spreadsheets/d/1tw2B1okJUdLrTIeDzooMPP16yduxZPzgLPHafD6Q6_8/edit#gid=0) and [here](https://docs.google.com/spreadsheets/d/1ZW4j1DbOYnFSGj0WjzNMEBCSN6daTKAg63ZO_B2tcws/edit#gid=505210495)[[4]](#fnzjuoyohk3ui). \The 'most aggressive' and 'most conservative' forecasts can be considered equivalent to 90% confidence intervals for the median estimate.[[5]](#fnree5mvhcuue) What drives the differences between Cotra and XPT forecasters? ============================================================== Differences in inputs --------------------- Relevant XPT forecasts differ substantially from Cotra’s. ### Hardware costs FLOP per $ in 2025* * Cotra ([2022](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines)): 3.8E+18 * XPT: 7E+17 (for 2024) * Cotra factors in that big companies get cheaper rates on GPUs.[[6]](#fnzk3i3ubxvco) XPT forecasters were explicitly asked to forecast the initial retail price of the chip on its release. That would explain most of the difference, assuming rates for big companies are 2–3x cheaper (which is what Cotra claims). Compute price halving time from 2025 to 2100 (years) * Cotra: 2.5 * XPT: 4.1 (for 2024–2050) * In the short run, this difference is driven by Cotra factoring in efficiency improvements specific to machine learning, such as increasing arithmetic-to-communication ratios via e.g. memory locality, further reductions in precision.[[7]](#fnq5jzsylhl6e) These improvements were not relevant to the question XPT forecasters were asked, so they didn’t take them into account. * In the long run, this difference seems to mostly come down to the likelihood that novel technologies like optical computing substantially reduce compute prices in the future. * Cotra flags that this is her least robust forecast and that after 2040 the forecast is particularly unreliable.[[8]](#fn9toxsxc0mx) ### Willingness to spend Compute cost for most expensive training run to 2025 * Cotra: $1bn * XPT: $35m (for 2024) * XPT forecasters are predicting spending for a year earlier than Cotra. * XPT forecasters made their predictions three years after Cotra made hers. + An influential [blog post by OpenAI in 2018](https://openai.com/research/ai-and-compute) noted rapid increases in the compute cost of the most expensive training runs, with a doubling time of 3.6 months. However, [more recent analysis](https://epochai.org/blog/compute-trends) using more data suggests a longer doubling time of ~6 months. + In [2022](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines%23Making_a_one_time_upward_adjustment_for__2020_FLOP_____), Cotra updated downwards on the likelihood of a $1bn training run by 2025. * Cotra expects that the most expensive training run is likely to be unreleased.[[9]](#fnpltlpmivcm) XPT forecasters do not highlight this possibility in their forecasts even though unreleased models are included in the relevant resolution criteria. It is unclear whether XPT forecasters disagree substantively with Cotra, missed this consideration in their analysis, or were confused about resolution criteria. If Cotra's expectation is correct, and we accept her claim that unreleased runs are likely to be 2–8 times more expensive, XPT forecasts would still be an order of magnitude lower than Cotra’s forecast, but more comparable with her conservative forecast of $300 million. Doubling time of spending on compute for the most expensive training run from 2025 onwards (years) * Cotra: 2.5[[10]](#fnu9c3xxsfmwg) * XPT: 8.4 (for 2024–2050) * Cotra’s reasoning for her 2.5 years doubling time rests on estimating various anchors and working backwards from them: + She assumes a compute cost of $1bn in 2025 and the incentive to build a transformative model. + She arrives at a doubling time from 2025–2040 of 2 years by estimating how much companies would be willing to spend on a project overall, and the ratio of overall spending to spending on compute for final training runs. + Then assuming that the doubling times lengthens, hitting a cap at 1% of GDP and eventually syncing up with the GDP growth rates of the largest national economy, which Cotra estimates at 3%. * Many XPT forecasters approached the question differently, by estimating current costs and then adding a modest multiplier. ### Algorithmic progress Halving time of compute requirements from 2025 to 2100 (years) * Cotra: 2–3.5 * XPT: 1.6 (for 2024–2030) * Cotra notes that she spent very little time on this forecast.[[11]](#fnwnozgaaqfvs) * XPT forecasts tend to be more conservative than Cotra and here they are more aggressive. * Comparability of the Cotra and XPT forecasts is particularly low here: + XPT forecasters were asked to forecast expected improvements on a specific narrow application (image classification on ImageNet). Cotra expects improvements on narrow applications to be easier than improvements on a general and poorly defined metric like TAI.[[12]](#fnag1a5j2ga6k) This likely explains much of the difference in forecasts here. - Cotra also draws on data from narrow applications, but then applies an upwards adjustment factor. We haven’t applied an adjustment factor to the XPT forecasts in our main analysis, as Cotra isn’t explicit about her methodology and we didn’t want to introduce more subjectivity.[[13]](#fnrq3znbe25wp) * We did a robustness check using an estimated upwards adjustment factor, and found that adjusting XPT forecasts on compute requirement halving times does not significantly shift model outputs. (See [appendix](https://docs.google.com/document/d/e/2PACX-1vSYEQlpDHDmJG1q_A_79o7Ya2qMCjb4l8UY_3-jYFqLmqFoKzogA6RwBztQjjmfpoerwMoWkvlYfQCt/pub#h.3qx40vp62x4v) for details.) + The XPT forecasts were only for 2024 and 2030, whereas Cotra’s estimate was for 2025–2100. + Cotra estimates different halving times for each of her six biological anchors.[[14]](#fnvyw066je3as) We haven’t attempted to extrapolate this from XPT forecasts, because Cotra’s methodology isn’t very transparent and we didn’t want to introduce more subjectivity. Differences in outputs ---------------------- Taking XPT forecasts as inputs to Cotra’s model leads to differences in outputs. * Taking the median forecasts from XPT superforecasters as inputs to Cotra’s model produces a probability that the compute required for TAI is attainable by 2050 of ~20% (~3% by 2030, ~60% by 2100). * The most aggressive[[15]](#fnl84026ya6b) forecasts from XPT superforecasters produce a probability of ~32% by 2050 (~6% by 2030, ~71% by 2100). * The most conservative[[16]](#fnonnqacsxrvf) forecasts from XPT superforecasters produce a probability of ~7% by 2050 (~1% by 2030, ~31% by 2100). * Cotra’s best guess inputs produce a probability of ~46% by 2050 (~8% by 2030, ~78% by 2100). + In 2020, Cotra gave an overall median TAI timeline of 2050. + In [2022](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines), she updated her overall median to 2040. + Using the XPT forecasts as inputs to the model would translate to overall median TAI timelines of: - Median: ~2090 - Most aggressive: ~2065 - Most conservative: >2100 Most of the difference in outputs comes down to differences in forecasts on: * Compute price halving time from 2025 to 2100. * Doubling time of spending on compute for the most expensive training run from 2025 onwards. + This is the single biggest driver of difference among the inputs we have XPT forecasts for. Which forecasts are more accurate? ================================== It’s not possible yet to determine which forecasts are more accurate across the board; in some cases we’d need to wait until 2100 to find out, and the earliest resolution date for final comparison is 2025. That said, since Cotra and the XPT forecasters made their predictions, relevant new data has been released which already gives some indication of accuracy on some inputs. Epoch have developed estimates of the current FLOP per $ and the compute cost for the most expensive training run to date. We can compare these to the Cotra and XPT estimates: ![](https://res.cloudinary.com/cea/image/upload/f_auto,q_auto/v1/mirroredImages/ccw9v9giKxg8nyLhp/wcbspeydiud9yyspropq)\* The 'most aggressive' and 'most conservative' forecasts can be considered equivalent to 90% confidence intervals for the median estimate.[[17]](#fntocoiurtgkl) \\Note that these Epoch estimates are not forecasts of what these inputs will be in future, but estimates of the current value of the inputs at a given point in time (dates in brackets in the table). See [here](https://epochai.org/trends#hardware-trends-section) for the FLOP/$ estimate and [here](https://colab.research.google.com/drive/1O99z9b1I5O66bT78r9ScslE_nOj5irN9?usp%3Dsharing%23scrollTo%3DPqkx-E3NQocI) for the estimate of compute cost for most expensive training run. If we accept the Epoch estimates, then this suggests that as of 2023: * Both Cotra and XPT forecasts on FLOP/$ are already inaccurate, although Cotra's 2022 estimate will necessarily prove more accurate (and the current estimate is outside the XPT 90% CI). * The XPT forecast for the most expensive training run (by 2024) is already inaccurate (though it's not yet clear whether this forecast is more or less accurate than Cotra's forecast for 2025, which remains much higher than Epoch’s current estimate). It remains to be seen how inaccurate the XPT (and Cotra’s) forecasts will prove, but it is striking that these XPT forecasts are already inaccurate even though they were made after Cotra’s and for an earlier resolution date. The forecasts for which it’s not yet clear whether Cotra or XPT forecasters will prove more accurate are: * Halving time of compute requirements from 2025 to 2100 + There is a recent Epoch estimate of this using historical data ([0.75](https://epochai.org/blog/revisiting-algorithmic-progress) years, analysis published Dec 2022), but it won’t be clear until much closer to 2100 which forecasts are on track to be more accurate. * Compute price halving time from 2025 to 2100 + Though we note that both XPT and Cotra forecasts on FLOP/$ for 2024/2025 are already inaccurate, and that our inferred XPT halving time is based on XPT forecasts on FLOP/$ for 2024 and 2050. * Doubling time of spending on compute for the most expensive training run from 2025 onwards, though note that: + Our inferred XPT doubling time is based on XPT forecasts on most expensive training runs by 2024 and 2050, and the 2024 forecast is already inaccurate. + There is a substantial difference (greater than one order of magnitude) between Epoch’s estimate of the most expensive training run to date, and Cotra’s 2025 forecast. XPT superforecasters’ all-things-considered view on TAI timelines ================================================================= As we mentioned above, this analysis takes Cotra’s model and many of her inputs as a given, and uses XPT forecasts for particular inputs. It cannot be read as a statement of XPT forecasters’ all-things-considered view on TAI timelines. In fact, from questions in a postmortem survey conducted at the end of the XPT, we know thatXPT forecasters’ all-things-considered TAI timelines are longer than this analysis of Cotra’s model suggests. Superforecasters made the following explicit predictions in the postmortem survey: * Probability of AGI by 2070: 13% + “Artificial general intelligence is defined here as any scenario in which cheap AI systems are fully substitutable for human labor, or if AI systems power a comparably profound transformation (in economic terms or otherwise) as would be achieved in such a world.” * Probability of TAI by 2070: 3.75% + “Transformative AI is defined here as any scenario in which global real GDP during a year exceeds 115% of the highest GDP reported in any full prior year.” * Probability of >15% GWP growth by 2100: 3% + “By 2100, will the global real GDP in a year ever exceed 115% of the highest GDP reported in any full prior year?” The output of Cotra’s model using superforecaster XPT inputs is more aggressive than XPT superforecasters’ overall views. Using the XPT superforecaster inputs in Cotra’s model outputs 35% by 2070, and 60% by 2100.[[18]](#fny19nkgqtdpm) Note that: * XPT superforecasters think AGI is considerably more likely than TAI by 2070. * XPT forecasters' views appear inconsistent. + ~26% of superforecasters predicted AGI by 2070 as 50% likely or more, but ~38% agree or strongly agree that AGI will arise by the end of 2072. ~36% of experts predicted AGI by 2070 as 50% likely or more, but ~61% agree or strongly agree that AGI will arise by the end of 2072. + Superforecasters predict a 3% chance of >15% growth by 2100,[[19]](#fnq8480uoqlbo) and a 3.75% chance of TAI (defined as >15% growth) by 2070. - Experts predict a 10% chance of >15% growth by 2100,[[20]](#fn2fjhfiti01s) and a 16% chance of TAI by 2070, so their views are even less coherent on this question. Appendix A: Arguments made for different forecasts ================================================== Both Cotra and the XPT forecasters gave arguments for their forecasts. In Cotra’s case, she puts forward arguments directly in the relevant [section](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#) of her report and in [appendices](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#). In the XPT case: * During the tournament, forecasters were assigned to teams. * Within teams, forecasters discussed and exchanged arguments in writing. * Each team was asked to produce a ‘rationale’ summarizing the arguments raised in team discussion. * The rationales from different teams on each XPT question were summarized by the FRI team. This appendix contains direct quotes from: * Cotra’s report, appendices and 2022 update * XPT team rationales Note that we haven't made any edits to these quotes, including where there are grammatical errors. Hardware costs -------------- ![](https://res.cloudinary.com/cea/image/upload/f_auto,q_auto/v1/mirroredImages/ccw9v9giKxg8nyLhp/llgegkj3n1ud0x4tx6ky)### Meta points * Cotra thinks these numbers are the least robust in her report.[[21]](#fn7l90xxd3ls9) * She also thinks the forecast is more reliable till 2040 and then less reliable.[[22]](#fnb7yik8au95a) ### Cotra’s arguments In 2020: * Recent trends have been slower, are probably more informative, and probably reflect diminishing returns.[[23]](#fntcd9uv850ye) * The older, faster trend held for a long time and over multiple hardware transitions. Extrapolating the recent trend for several times longer than the older trend seems wrong.[[24]](#fnqiimvg78byi) * NVIDIA A100 is a big improvement on the V100.[[25]](#fnwa98esiuxr) * Specializing chips for deep learning applications will create a one off improvement in the next 5–10 years.[[26]](#fn6dof7uvxcbj) * In the longer term, unknown unknowns and new technologies will probably lead to further improvements.[[27]](#fnk4d4a3477mi) + Technologies noted: optical computing, three-dimensional circuits, reversible computing, quantum computing. In 2022: * The 2020 forecast used V100 as its reference machine, but the A100 was 2–3x more powerful.[[28]](#fnbw2j45fu8aa) * The 2020 forecast was based on rental prices, but big companies get 2–3x cheaper prices.[[29]](#fns4agwzqg6wi) * The 2020 forecast assumes ⅓ utilization of FLOP/s, but utilization then improved to around 50%.[[30]](#fnc26a00cvb37) ### XPT arguments Arguments for lower hardware costs (closer to Cotra’s forecasts): * Some XPT forecasters used outdated data to form their base rates, and so unknowingly predicted future lowest costs as being higher than present costs.[[31]](#fni6f273fqkr) This is an argument for lower forecasts than Cotra or XPT. * Covid inflated costs of electricity and hardware, but efficiencies in development and falling energy prices will drive costs down again.[[32]](#fnl38f8tktkq) * Recent price-performance trends have been slower than usual, and there could be a return to the older order of magnitude improvements every 8 or 4 years.[[33]](#fnk9tchtr0bp) * Novel technologies might lead to a discontinuous drop in prices.[[34]](#fnm319n7vve2) + Possible technologies cited are optical computing, quantum computing, reversible and three-dimensional circuits, and unknown advances. * Historical trends show an order of magnitude improvement in price-performance every decade.[[35]](#fngi0yeduewvm) Arguments for higher hardware costs than Cotra forecasts: * Since 2010 the rate of price decline has slowed.[[36]](#fnieunsj2qvqr) One team cited the IEEE report, ‘[More Moore](https://irds.ieee.org/images/files/pdf/2021/2021IRDS_MM.pdf)’. * War, particularly over Taiwan, could raise prices.[[37]](#fnr54exi4l6nl) * Global economic decline could slow technological advances.[[38]](#fnuuntszjpqoh) * Progress may be getting harder.[[39]](#fnvvqi9u1adji) * We may reach fundamental physical limits.[[40]](#fnakupzs8hdho) * Demand for more efficient chips may be low.[[41]](#fnqpu8rjdnwkn) * Future technological developments are uncertain and could raise prices.[[42]](#fngxsk2ag6pkp) * FLOP rates might stabilize in the future and optimization might shift to memory architectures.[[43]](#fnb9yab4mx67q) * Materials for chips are rare and have other uses.[[44]](#fnwstb6iaj9fk) * A catastrophe or extinction event could halt price decreases.[[45]](#fn4w6519lrusw) Willingness to spend -------------------- ![](https://res.cloudinary.com/cea/image/upload/f_auto,q_auto/v1/mirroredImages/ccw9v9giKxg8nyLhp/xprxfbhij6qwwssb2364)### Cotra’s arguments * On compute cost for most expensive training run to 2025: + $1bn by 2025 is consistent with recent spending scaling according to [this](https://openai.com/blog/ai-and-compute/) 2018 OpenAI blog.[[46]](#fnjcg29m30g) + It is also consistent with the existing resources of AI companies like Google.[[47]](#fn3igsee5vlnx) + Excitement around deep learning is sufficient such that several companies will be willing to spend a few hundred million dollars on experiments which don’t generate much revenue at the moment.[[48]](#fnd8pml2evi6e) + Spending on the most compute intensive unreleased/proprietary model is likely 2–8x larger than AlphaStar already.[[49]](#fnjji9p2haoy) + “It also seems quite likely that by the end of 2020, a single ML training run costing at least $20M will have been completed, and that by the end of 2021, a single training run costing at least $80M will have been completed.”[[50]](#fniit8rlrsmh) * On doubling time of spending on compute 2025–2040: + [Note that for these arguments Cotra assumes that a company has spent $1bn on a training run in 2025 and has the incentive of building a transformative model.][[51]](#fnoz8765uouig) + By 2040 an AI company could spend hundreds of billions on a project to train a transformative model.[[52]](#fnp42zmh193s) - Current cash in hand of relevant companies is ~$50–100bn.[[53]](#fnza495v8sl4g) - These companies’ market capitalization tends to be 10x as large as cash in hand, close to $1 trillion.[[54]](#fnpswcyp1u5e8) * They could probably borrow up to 50% of this.[[55]](#fnscpkxkjecf) - If AI progress continues, these companies’ share of the economy will grow.[[56]](#fnhcedwnmdkbb) + The ratio of overall project spend to spend on compute for final training runs may be 2–10x.[[57]](#fn9z035kl63u4) + This suggests that in 2040 an AI project would be willing to spend $100bn on compute for a final training run.[[58]](#fnsq4ytfzcjjg) + This implies a doubling time of 2 years.[[59]](#fnmuqvnvxj50o) * On long-run willingness to spend: + Eventually growth in spending on compute will keep pace with the GDP growth of the largest national economy.[[60]](#fniujw8p1ugwb) - 3% is in keeping with average US growth over the past few decades.[[61]](#fnoocwg4lvde) + Anchoring to the Manhattan and Apollo projects would suggest that the maximum spend would be around 1% of the GDP of the largest country.[[62]](#fnrjcs2p9hg6) * [2022 update](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines%23Making_a_one_time_upward_adjustment_for__2020_FLOP_____): “[There’s been a major market downturn that hit tech companies especially hard](https://www.cbsnews.com/news/tech-companies-layoffs-stock-market-cryptocurrency/); it seems a little less likely to me now than it did when writing the report that there will be a billion dollar training run by 2025.” ### XPT arguments General comments: * Low forecasts are derived from applying a modest multiplier to current costs. Higher forecasts identify anchors (such as company budgets or costs of previous mega-projects) and assume fast scaling up to those anchors.[[63]](#fnmhiw3fnxyta) * Lower forecasts assume current manufacturing processes will continue. Higher forecasts imagine novel technology.[[64]](#fnl9b1tv6dqun) Arguments for lower spending than Cotra forecasts: * Training costs have been stable at around $10m for the last few years.[[65]](#fnetmsawbnk8) * Current trend increases are not sustainable for many more years.[[66]](#fnizgiixeqjsm) One team cited [this](https://aiimpacts.org/trends-in-the-cost-of-computing/) AI Impacts blog post. * Major companies are cutting costs.[[67]](#fnz53jukboly) * Increases in model size and complexity will be offset by a combination of falling compute costs, pre-training, and algorithmic improvements.[[68]](#fnejfq54kjbyf) * Large language models will probably see most attention in the near future, and these are bottlenecked by availability of data, which will lead to smaller models and lower compute requirements.[[69]](#fnfvkqh3q6u3) * Growth may already be slowing down.[[70]](#fnl9mc89we9kn) * In the future, AI systems may be more modular, such that single experiments remain small even if total spending on compute increases drastically.[[71]](#fnamxjodiun8) * Recent spending on compute may have been status driven.[[72]](#fn0ix7mtmb6mlg) * There seems to be general agreement that experiments of more than a few months are unwise, which might place an upper bound on how much compute can cost for a single experiment.[[73]](#fngysbfx3oe5k) Arguments for higher spending (closer to Cotra’s forecasts): * As AI creates more value, more money will be spent on development.[[74]](#fnfafjllwoial) * A mega-project could be launched nationally or internationally which leads to this level of spending.[[75]](#fnsqw8i6ormgq) * There is strong competition between actors with lots of resources and incentives to develop AI.[[76]](#fno2a7vpo5uh) * The impact of AI on AI development or the economy at large might raise the spending ceiling arbitrarily high.[[77]](#fnfwql072onar) Algorithmic progress -------------------- ![](https://res.cloudinary.com/cea/image/upload/f_auto,q_auto/v1/mirroredImages/ccw9v9giKxg8nyLhp/x0ngjafdjtl0c2duwtx7)### Meta points * Cotra says she’s spent very little time on this: the least time of any of the major components of her model.[[78]](#fnew8ze3kr47) * The comparability is low: + The XPT question only covers 2024 and 2030, which is a small part of the time from 2025 to 2100. + The XPT question was specifically about efficiency improvements on ImageNet, and the relationship between that and the most relevant kinds of algorithmic progress for TAI is unclear. - Cotra notes that we should expect efficiency improvements on narrow, well-defined tasks to be faster than those most relevant to TAI.[[79]](#fnamojubiv4t) + Cotra breaks her forecasts down for each of the biological anchors she considers; the XPT question generates only one overall number. ### Cotra’s arguments * [Hernandez and Brown 2020](https://arxiv.org/pdf/2005.04305.pdf) show halving every 13-16 months.[[80]](#fnmo2eyygb6aq) * But these are on narrow well-defined tasks which researchers can directly optimize for, so forecasts should be adjusted up.[[81]](#fn8ct49j3vr5d) * There might also be breakthrough progress at some point.[[82]](#fnofy638opskn) ### XPT arguments General comments: * Extrapolating current growth rates leads to above median forecasts, and median and below median forecasts assume that current growth rates will slow.[[83]](#fn7z8f6ze1278) Arguments for slower algorithmic progress (closer to Cotra’s forecast): * It’s possible no further work will be done in this area such that no further improvements are made.[[84]](#fn8uypddzxh5s) * Recently the focus has been on building very large models rather than increasing efficiency.[[85]](#fn7w68fm2va6j) * There may be hard limits on how much computation is required to train a strong image classifier.[[86]](#fnt02npjjf2zi) * Accuracy may be more important for models given what AI is used for, such that leading researchers target accuracy rather than efficiency gains.[[87]](#fn3gcw5wtj1ry) * If there is a shift towards explainable AI, this may require more compute and so slow efficiency growth rates.[[88]](#fnu5lg1oldi9) * Improvements may not be linear, especially as past improvements have been “lumpy” (i.e. improvements have come inconsistently) and the reference source is only rarely updated.[[89]](#fnefls1wfvo5f) * Very high growth rates are hard to sustain and tend to revert to the mean.[[90]](#fn7vs2ikh3xl3) Arguments for faster algorithmic progress: * Pure extrapolation of improvements to date would result in fast progress.[[91]](#fnaygwmerb4wf) * Quantum computing might increase compute power and speed.[[92]](#fnsl68iqurcv) * As AI models grow and become limited by available compute, efficiency will become increasingly important and necessary for improving accuracy.[[93]](#fnevo36u0up86) * “The [Papers with Code ImageNet benchmark sorted by GFLOPs](https://paperswithcode.com/sota/image-classification-on-imagenet?metric%3DTop%25205%2520Accuracy%26dimension%3DGFLOPs) shows several more recent models with good top 5 accuracy and a much lower GFLOPs used than the current leader, EfficientNet.” If GFLOPS is a good indicator of training efficiency, then large efficiency increases may already have been made.[[94]](#fn38gxe0yakea) * This technology is in its infancy so there may still be great improvements to be made.[[95]](#fnq5gtiwextll) Appendix B: XPT expert forecasts related to when the compute required for TAI will be attainable ================================================================================================ ![](https://res.cloudinary.com/cea/image/upload/f_auto,q_auto/v1/mirroredImages/ccw9v9giKxg8nyLhp/e88nk4qwxunw4nfpehqb)\* The 'most aggressive' and 'most conservative' forecasts can be considered equivalent to 90% confidence intervals for the median estimate.[[96]](#fnpi55hwql9bo) Notes: * Here, experts includes all experts in the XPT, including experts in fields other than AI which are relevant to existential risk. + We chose to present results for all experts because the sample sizes were bigger than those of domain experts (14-21 compared to 5-6 domain experts). However, we expect readers will vary in how much weight they want to put on the forecasts of domain experts vs. general x-risk experts vs. non-domain experts on these questions. For details on each subgroup's forecasts on these questions, see Appendix 5 [here](https://static1.squarespace.com/static/635693acf15a3e2a14a56a4a/t/64abffe3f024747dd0e38d71/1688993798938/XPT.pdf), where you can navigate to each question to see each subgroup's forecast. * The range in expert inputs tends to be larger than the range in superforecaster inputs. * The most aggressive expert inputs are the only XPT inputs which produce probabilities higher than Cotra’s. Appendix C: Applying an upwards adjustment factor to the XPT compute halving time forecasts =========================================================================================== Cotra bases her forecast for compute requirement halving times on data about algorithmic progress on narrow applications, but then applies an upwards adjustment factor, to account for her belief that algorithmic progress will be slower for general applications, than it is for narrow applications. We didn’t apply an adjustment factor to the XPT forecasts in our main analysis, as Cotra isn’t explicit about her methodology and we didn’t want to introduce more subjectivity.[[97]](#fnwhj1pui5tyc) But it is possible to do a robustness check using an estimated upwards adjustment factor, as follows: * The Hernandez and Brown paper Cotra cites shows halving times of 13–16 months.[[98]](#fn9u7wlabe4oc) * Christiano’s summary of Grace’s paper has halving times of 13–36 months.[[99]](#fn6bmtulxqjiv) * Let's guess that the bottom of Cotra’s unadjusted range is a halving time of 14 months. * Her final estimates range from 24–36 months, which is an adjustment of 1.7 at the bottom end. * This would mean the top of her unadjusted range is 21, which is consistent with Cotra putting more weight on Hernandez and Brown.[[100]](#fnaoyi4t53r0g) * If we apply an upward adjustment factor of 1.7 to all of the XPT figures, we end up with a median halving time of 2.72 years (90% CI: 2.55 years to 5.78 years). * If we input this adjusted figure into the model, we get the following outputs: ![](https://res.cloudinary.com/cea/image/upload/f_auto,q_auto/v1/mirroredImages/ccw9v9giKxg8nyLhp/qh3dhotyfunwn6a3szzm)Workings [here](https://docs.google.com/spreadsheets/d/1tw2B1okJUdLrTIeDzooMPP16yduxZPzgLPHafD6Q6_8/edit#gid=610909524). \The 'most aggressive' and 'most conservative' forecasts can be considered equivalent to 90% confidence intervals for the median estimate.[[101]](#fn05naqbi3hvgi) So applying a rough upwards adjustment factor to the XPT forecasts on compute requirement halving times does not significantly shift model outputs. 1. [^](#fnref736fhvw6wlo)* For the relevant questions in the XPT, forecasters were asked to provide their 5th, 25th, 50th, 75th, and 95th percentile forecasts. In this analysis we use the term, ‘median’ to refer to analyses using the group’s median forecast for the 50th percentile of each question. We use the term ‘most aggressive’ to refer to analyses using the group medians for the 5th percentile estimate of the question relating to hardware costs, and the 95th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the lowest plausible hardware costs and the highest plausible willingness to spend and algorithmic efficiency to give the highest plausible likelihood of TAI.) We use the term ‘most conservative’ to refer to analyses using the group medians for the 95th percentile estimate of the question relating to hardware costs, and the 5th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the highest plausible hardware costs and the lowest plausible willingness to spend and algorithmic efficiency to give the lowest plausible likelihood of TAI.) The most aggressive and most conservative estimates can be considered equivalent to 90% confidence interval for the median estimate. See [here](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Comparing_Cotra_and_XPT_forecasts) for context on which XPT questions map to which biological anchors inputs. 2. [^](#fnrefx9yquvymqx) Biological anchors refers to four hypotheses for the amount of computation that would be required to train a transformative model using 2020 architectures and algorithms: total computation done over evolution, total computation done over a human lifetime, the computational power of the human brain, and the amount of information in the human genome. All four anchors rely on an estimate of the amount of computation performed by the human brain, measured in floating point operations per second (FLOP/s). See [here](https://docs.google.com/document/d/1IJ6Sr-gPeXdSJugFulwIpvavc0atjHGM82QjIfUSBGQ/edit%23heading%3Dh.yrpq5m3bxdvh) for an introduction to the framework. 3. [^](#fnrefrrryf8sfjaf) More detail on the XPT forecasts on these questions can be found in pages 655 to 676 of the [XPT report](https://static1.squarespace.com/static/635693acf15a3e2a14a56a4a/t/64abffe3f024747dd0e38d71/1688993798938/XPT.pdf). 4. [^](#fnrefzjuoyohk3ui) This spreadsheet uses as a template Cotra's publicly available [spreadsheet](https://docs.google.com/spreadsheets/d/1TjNQyVHvHlC-sZbcA7CRKcCp0NxV6MkkqBvL408xrJw/edit#gid=505210495), linked to from her [report](https://docs.google.com/document/d/1IJ6Sr-gPeXdSJugFulwIpvavc0atjHGM82QjIfUSBGQ/edit#heading=h.c5pt0lvk9kkw). 5. [^](#fnrefree5mvhcuue) For the relevant questions in the XPT, forecasters were asked to provide their 5th, 25th, 50th, 75th, and 95th percentile forecasts. In this analysis we use the term, ‘median’ to refer to analyses using the group’s median forecast for the 50th percentile of each question. We use the term ‘most aggressive’ to refer to analyses using the group medians for the 5th percentile estimate of the question relating to hardware costs, and the 95th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the lowest plausible hardware costs and the highest plausible willingness to spend and algorithmic efficiency to give the highest plausible likelihood of TAI.) We use the term ‘most conservative’ to refer to analyses using the group medians for the 95th percentile estimate of the question relating to hardware costs, and the 5th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the highest plausible hardware costs and the lowest plausible willingness to spend and algorithmic efficiency to give the lowest plausible likelihood of TAI.) The most aggressive and most conservative estimates can be considered equivalent to 90% confidence interval for the median estimate. See [here](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Comparing_Cotra_and_XPT_forecasts) for context on which XPT questions map to which biological anchors inputs. 6. [^](#fnrefzk3i3ubxvco) “I was using the rental price of a V100 (~$1/hour), but big companies get better deals on compute than that, by about another 2-3x.” [here](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines%23Making_a_one_time_upward_adjustment_for__2020_FLOP_____) 7. [^](#fnrefq5jzsylhl6e) “Communication costs currently account for roughly ~70%-80% of the cost of a GPU, and Paul’s understanding is that the recent trend in ML chips has been toward increasing arithmetic-to-communication ratios. Pushing further in that direction (e.g. switching to chips with more localized memory) could bring communication costs more in-line with arithmetic costs and reduce total costs by a factor of ~3. Deep learning applications could also gain a factor of ~2 from switching to 8-bit precision computations (rather than 16-bit).” [p. 30](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#). 8. [^](#fnref9toxsxc0mx) “Because they have not been the primary focus of my research, I consider these estimates unusually unstable, and expect that talking to a hardware expert could easily change my mind.” [p. 26](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.nmcod2jynsy4). “This forecast feels most solid and plausible out to ~2040 or so, beyond which it feels substantially more murky and likely incorrect.” [p. 4](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.fmq8f6whj003). “Of all the quantitative estimates in this document, I consider these forecasts the most likely to be knowably mistaken. While most of the other quantitative estimates in this document have a lot more absolute uncertainty associated with them, there is a lot more low-hanging fruit left in improving short- and medium-term hardware price forecasts. For example, my understanding is that semiconductor industry professionals regularly write highly detailed technical reports forecasting a number of hardware cost-efficiency metrics, and I have neither read any of this literature nor interviewed any hardware experts on this question.” [p. 30](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.xi6z3buznjb7) 9. [^](#fnrefpltlpmivcm) “I would guess that the most compute-intensive training run for an unreleased and/or proprietary model (e.g., a language model powering Google Assistant or Google Translate) is already ~2-8x larger than AlphaStar’s ~1.3e23, costing ~$2-8M.” [p. 36](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.nc2d14p8i1pd) “[N]ote that there will probably be a non-trivial delay between the first time a training run of size X is completed and the first time such a training run is published, and my forecasts are about the former”. [p. 37](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.nc2d14p8i1pd) 10. [^](#fnrefu9c3xxsfmwg) In Cotra’s model, this number is a point estimate for ‘Doubling time of spending on compute for the most expensive training run at start of period (2025)’. When she reviewed this post, Cotra confirmed that it made sense to treat this as the doubling time from 2025 onwards. 11. [^](#fnrefwnozgaaqfvs) “I have done very little research into algorithmic progress trends. Of the four main components of my model (2020 compute requirements, algorithmic progress, compute price trends, and spending on computation) I have spent the least time thinking about algorithmic progress.” [p. 5](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 12. [^](#fnrefag1a5j2ga6k) “Additionally, it seems plausible to me that both sets of results would overestimate the pace of algorithmic progress on a transformative task, because they are both focusing on relatively narrow problems with simple, well-defined benchmarks that large groups of researchers could directly optimize.[] Because no one has trained a transformative model yet, to the extent that the computation required to train one is falling over time, it would have to happen via proxies rather than researchers directly optimizing that metric (e.g. perhaps architectural innovations that improve training efficiency for image classifiers or language models would translate to a transformative model). Additionally, it may be that halving the amount of computation required to train a transformative model would require making progress on multiple partially-independent sub-problems (e.g. vision and language and motor control).” [p. 6](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 13. [^](#fnrefrq3znbe25wp) “I have attempted to take the Hernandez and Brown 2020 halving times (and Paul’s summary of the Grace 2013 halving times) as anchoring points and shade them upward to account for the considerations raised above. There is massive room for judgment in whether and how much to shade upward; I expect many readers will want to change my assumptions here, and some will believe it is more reasonable to shade downward." [p. 6](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 14. [^](#fnrefvyw066je3as) “I chose to break down the algorithmic progress forecast by hypothesis rather than use a single value describing how the 2020 compute requirements distribution shifts to the left in future years. This is because hypotheses which predict that the amount of computation required to train a transformative model is already very low (such as the Lifetime Anchor hypothesis) seems like they should also predict that further algorithmic progress would be difficult and there is not as much room to reduce compute requirements even further.” [p. 7](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#) 15. [^](#fnrefl84026ya6b) For the relevant questions in the XPT, forecasters were asked to provide their 5th, 25th, 50th, 75th, and 95th percentile forecasts. In this analysis we use the term, ‘median’ to refer to analyses using the group’s median forecast for the 50th percentile of each question. We use the term ‘most aggressive’ to refer to analyses using the group medians for the 5th percentile estimate of the question relating to hardware costs, and the 95th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the lowest plausible hardware costs and the highest plausible willingness to spend and algorithmic efficiency to give the highest plausible likelihood of TAI.) We use the term ‘most conservative’ to refer to analyses using the group medians for the 95th percentile estimate of the question relating to hardware costs, and the 5th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the highest plausible hardware costs and the lowest plausible willingness to spend and algorithmic efficiency to give the lowest plausible likelihood of TAI.) The most aggressive and most conservative estimates can be considered equivalent to 90% confidence interval for the median estimate. See [here](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Comparing_Cotra_and_XPT_forecasts) for context on which XPT questions map to which biological anchors inputs. 16. [^](#fnrefonnqacsxrvf) For the relevant questions in the XPT, forecasters were asked to provide their 5th, 25th, 50th, 75th, and 95th percentile forecasts. In this analysis we use the term, ‘median’ to refer to analyses using the group’s median forecast for the 50th percentile of each question. We use the term ‘most aggressive’ to refer to analyses using the group medians for the 5th percentile estimate of the question relating to hardware costs, and the 95th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the lowest plausible hardware costs and the highest plausible willingness to spend and algorithmic efficiency to give the highest plausible likelihood of TAI.) We use the term ‘most conservative’ to refer to analyses using the group medians for the 95th percentile estimate of the question relating to hardware costs, and the 5th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the highest plausible hardware costs and the lowest plausible willingness to spend and algorithmic efficiency to give the lowest plausible likelihood of TAI.) The most aggressive and most conservative estimates can be considered equivalent to 90% confidence interval for the median estimate. See [here](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Comparing_Cotra_and_XPT_forecasts) for context on which XPT questions map to which biological anchors inputs. 17. [^](#fnreftocoiurtgkl)For the relevant questions in the XPT, forecasters were asked to provide their 5th, 25th, 50th, 75th, and 95th percentile forecasts. In this analysis we use the term, ‘median’ to refer to analyses using the group’s median forecast for the 50th percentile of each question. We use the term ‘most aggressive’ to refer to analyses using the group medians for the 5th percentile estimate of the question relating to hardware costs, and the 95th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the lowest plausible hardware costs and the highest plausible willingness to spend and algorithmic efficiency to give the highest plausible likelihood of TAI.) We use the term ‘most conservative’ to refer to analyses using the group medians for the 95th percentile estimate of the question relating to hardware costs, and the 5th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the highest plausible hardware costs and the lowest plausible willingness to spend and algorithmic efficiency to give the lowest plausible likelihood of TAI.) The most aggressive and most conservative estimates can be considered equivalent to 90% confidence interval for the median estimate. See [here](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Comparing_Cotra_and_XPT_forecasts) for context on which XPT questions map to which biological anchors inputs. 18. [^](#fnrefy19nkgqtdpm) Not all superforecasters completed the end-of-tournament survey. However, using the forecasts from only the subset of superforecasters who did complete the survey does not change the results. Using this subset’s forecasts as inputs to Cotra’s model outputs the same probability of TAI by 2070 and 2100 (35% and 60%, respectively). 19. [^](#fnrefq8480uoqlbo) The probability of >15% growth by 2100 was asked about in both the main component of the XPT and the postmortem survey. The results here are from the postmortem survey. The superforecaster median estimate for this question in the main component of the XPT was 2.75% (for all superforecaster participants and the subset that completed the postmortem survey). 20. [^](#fnref2fjhfiti01s) The probability of >15% growth by 2100 was asked about in both the main component of the XPT and the postmortem survey. The results here are from the postmortem survey. The experts median estimate for this question in the main component of the XPT was 19% for all expert participants and 16.9% for the subset that completed the postmortem survey. 21. [^](#fnref7l90xxd3ls9) “Because they have not been the primary focus of my research, I consider these estimates unusually unstable, and expect that talking to a hardware expert could easily change my mind.” [p. 26](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.nmcod2jynsy4) “Of all the quantitative estimates in this document, I consider these forecasts the most likely to be knowably mistaken. While most of the other quantitative estimates in this document have a lot more absolute uncertainty associated with them, there is a lot more low-hanging fruit left in improving short- and medium-term hardware price forecasts. For example, my understanding is that semiconductor industry professionals regularly write highly detailed technical reports forecasting a number of hardware cost-efficiency metrics, and I have neither read any of this literature nor interviewed any hardware experts on this question.” [p. 30](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.xi6z3buznjb7) 22. [^](#fnrefb7yik8au95a) “This forecast feels most solid and plausible out to ~2040 or so, beyond which it feels substantially more murky and likely incorrect.” [p. 4](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.fmq8f6whj003) 23. [^](#fnreftcd9uv850ye) “Other things being equal, the recent slower trend is probably more informative than older data, and is fairly likely to reflect diminishing returns in the silicon chip manufacturing industry.” [p. 2](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.96w8mskhfp5l) 24. [^](#fnrefqiimvg78byi) “However, the older trend of faster growth has held for a much longer period of time and through more than one change in “hardware paradigms.” I don’t think it makes sense to extrapolate the relatively slower growth from 2008 to 2018 over a period of time several times longer than that” [p. 2](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit%23heading%3Dh.96w8mskhfp5l) 25. [^](#fnrefwa98esiuxr) “Additionally, a technical advisor informs me that the [NVIDIA A100 GPU](https://www.nvidia.com/en-us/data-center/a100/) (released in 2020) is substantially more powerful than the V100 that it replaced, which could be more consistent with a ~2-2.5 year doubling time than a ~3.5 year doubling time.” [p. 3](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.96w8mskhfp5l) 26. [^](#fnref6dof7uvxcbj) “On top of that, it seems that we can expect a one-time ~6x improvement in the next ~5-10 years from specializing chips for deep learning applications.” [p. 29](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.xi6z3buznjb7) 27. [^](#fnrefk4d4a3477mi) “The above reasoning was focused on listing all the foreseeable improvements on the horizon for silicon-based chips, but I believe there is substantial possibility for both a) “unknown unknown” sources of improvements to silicon chips and b) transition to an exotic form of hardware. For example, at least some companies are actively working on [optical computing](https://en.wikipedia.org/wiki/Optical_computing) in particular -- I would bet that effective FLOP per dollar will eventually move past the plateau, potentially reaching values multiple orders of magnitude higher. Possibilities that seem somewhat more distant include [three-dimensional circuits](https://en.wikipedia.org/wiki/Three-dimensional_integrated_circuit), [reversible computing](https://en.wikipedia.org/wiki/Reversible_computing), and [quantum computing](https://en.wikipedia.org/wiki/Quantum_computing%23:~:text%3DQuantum%2520computing%2520is%2520the%2520use,are%2520known%2520as%2520quantum%2520computers.).” [p. 32](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.jx25381jyv09) 28. [^](#fnrefbw2j45fu8aa) “I was using [the V100](https://www.nvidia.com/en-us/data-center/v100/) as my reference machine; this was in fact the most advanced publicly available chip on the market as of 2020, but it was released in 2018 and on its way out, so it was better as an estimate for 2018 or 2019 compute than 2020 compute. The more advanced [A100](https://www.nvidia.com/en-us/data-center/a100/) was 2-3x more powerful per dollar and released in late 2020 almost immediately after my report was published.” [here](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines%23Making_a_one_time_upward_adjustment_for__2020_FLOP_____) 29. [^](#fnrefs4agwzqg6wi) “I was using the rental price of a V100 (~$1/hour), but big companies get better deals on compute than that, by about another 2-3x.” [here](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines%23Making_a_one_time_upward_adjustment_for__2020_FLOP_____) 30. [^](#fnrefc26a00cvb37) “I was assuming ~⅓ utilization of FLOP/s, which was in line with what people were achieving then, but utilization seems to have improved, maybe to ~50% or so.” [here](https://www.lesswrong.com/posts/AfH2oPHCApdKicM4m/two-year-update-on-my-personal-ai-timelines%23Making_a_one_time_upward_adjustment_for__2020_FLOP_____) 31. [^](#fnrefi6f273fqkr) Question 47: 337, “Given that five out of eight team forecasters used faulty data, we should conclude that the team forecast is also faulty for all dates and percentiles”, “many forecasters only used the outdated Wikipedia article referenced in the question description. That article was specifically the price/performance data for the more recent models of GPUs. (The article was updated recently, though it still doesn't cover the dedicated AI infrastructure hardware sold by Nvidia like their new H100 line.) This led to most forecasters using obsolete data for their baselines and predicting future GFLOPS prices that are worse than the already achieved results. The difference in the source data quality fully explains the widely divergent forecasts for 2024, which should normally be simple - and numerically similar - extrapolations of the status quo.” 344, “This question has a shallow pool of forecasters with limited arguments given for the estimates and erroneous inputs.” 32. [^](#fnrefl38f8tktkq) Question 47: 336, “‘The biggest price is not hardware itself but electricity, data-center usage and human AI-scientists salaries.’ The COVID pandemic inflated costs for electricity and hardware but efficiencies in development, and energy costs, will drive this down again.” 33. [^](#fnrefk9tchtr0bp) Question 47: 336, “recent performance/$ trend is slower than long-run (there could be a return to the longer run trends of OOM every 8 or 4 years.)” 34. [^](#fnrefm319n7vve2) Question 47: 336, “uncertainty regarding future technological improvements”; “potential for discovering new modes of computing leading to discontinuous improvements”. 340, “The strongest argument for lower extreme forecasts is that some novel technology precipitates discontinuous progress in the trend of the cost of computation for training AI models. Optical neural networks are a promising technology with the potential to improve AI model training in this way.” See also 341, “Potential prospects for a revolutionary technology (e.g. optical computing, quantum computing, reversible and three-dimensional circuits) as per Cotra's report. This could break the foreseen plateau and lead to continued doubling every 3-4 years past 2040 and go back to a 1-2 year doubling.” See also 343, “Application of advanced AI or AGI to the problem could transformatively decrease prices in an unpredictable way.” See also 344, “Quantum computing seems to be accelerating progress - it's going to get much cheaper much quicker imho”. 35. [^](#fnrefgi0yeduewvm) Question 47: 336, “trend of order of magnitude improvement in price-performance every 10 years”. 36. [^](#fnrefieunsj2qvqr) Question 47: 336, “advancement may have been slowing since 2010 and rate of decline in prices could continue to slow”. 341, “Faltering of Moore's Law. See the IEEE's 2021 IRDS report, More Moore, Table MM for challenges.” See also 339, “Unstable world and a decline in Moore's law limit the factors that drove down costs in previous years. 37. [^](#fnrefr54exi4l6nl) Question 47: 336, “war, especially over Taiwan, could raise prices and/or slow advancement”. See also 339, “Unstable world and a decline in Moore's law limit the factors that drove down costs in previous years. It could take decades for the US to reshore semiconductor manufacturing to the US (and to China). This means Taiwan tensions could throw wrenches into cost dropping.” 38. [^](#fnrefuuntszjpqoh) Question 47: 336, “global economic decline could lead to slower advancement”. 39. [^](#fnrefvvqi9u1adji) Question 47: 341, “If early technological progress can be seen as a low-hanging fruit, further progress inherently becomes harder. Many experts (as quoted in Cotra, 2020) expect much less improvement over the next century than we have seen in the past century.” 40. [^](#fnrefakupzs8hdho) Question 47: 336, “potential for hard/impossible to surpass fundamental physical limits”. 340, “The strongest argument for higher extreme forecasts is that Moore’s law slows due to physical limitations in manufacturing, GPU cost per compute slows because of limits to parallelization, and there is are no new technologies to pick up the flattening S-curve and continue the trend.” 341, “Known limitations of specific technologies. The existence of fundamental physical limits.” 41. [^](#fnrefqpu8rjdnwkn) Question 47: 341, “ Lack of high demand (or diminished urgency) for ever more efficient chips.” 42. [^](#fnrefgxsk2ag6pkp) Question 47: 336, “uncertainty regarding future technological development - potential for new tech to lead to higher prices.” 43. [^](#fnrefb9yab4mx67q) Question 47: 339, “Processors in the future may not necessarily have greater FLOP rates, which hit limits of Moore's law, but superior memory architecture (e.g. Apple's M1/m2 chips did this by being better suited to scientific computing workloads). Apple's success: access a distributed RAM with almost no latency: [Apple M1 destroys Intel and AMD in newly-released benchmarks \| TechRadar](https://www.techradar.com/news/apple-m1-destroys-intel-and-amd-in-newly-released-benchmarks%23:~:text%3DApple%2520M1%2520destroys%2520Intel%2520and%2520AMD%2520in%2520newly%252Dreleased%2520benchmarks,-By%2520Jess%2520Weatherbed%26text%3DIt%2527s%2520now%2520been%2520revealed%2520through,the%2520Intel%2520Core%2520i9%252D11900K). FLOP rate may become static at one point, meaning memory optimisations will rule. There may be another metric, such as effective FLOP rate, that might emerge instead.” 44. [^](#fnrefwstb6iaj9fk) Question 47: 339, “Building processors requires rare earth minerals that will not be as abundant and have other uses (solar cells, Li-ion batteries)”. 45. [^](#fnref4w6519lrusw) Question 47: 343, “Realization of catastrophic or existential risks could halt or reverse price decreases (or otherwise make them irrelevant).” See also 337, “The effect of catastrophic risk could be important for 2050 (as per questions 1 to 12): a few of the scenarios could imply a temporal reversion to previous and more expensive forms of computing, such as mechanical computing or paper and pen. This could increase the price of one GFLOPS to values not seen in decades. However, since the forecasters' predictions of such catastrophes are relatively low (around 5%), only the 95th percentile forecasts should be affected by this consideration.” 46. [^](#fnrefjcg29m30g) “This would require doubling spending on the most expensive training run about once every 6 months, which is consistent with what I understand of [the recent pace of spending scaleup](https://openai.com/blog/ai-and-compute/) and the existing resources of AI companies such as Google.” [pp. 4-5](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.z7u133pzed6k) 47. [^](#fnref3igsee5vlnx) “This would require doubling spending on the most expensive training run about once every 6 months, which is consistent with what I understand of [the recent pace of spending scaleup](https://openai.com/blog/ai-and-compute/) and the existing resources of AI companies such as Google.” [pp. 4-5](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.z7u133pzed6k) 48. [^](#fnrefd8pml2evi6e) “However, it does appear that there is enough short-term excitement about deep learning that several companies will have the budget to scale up to training runs costing a few hundred million dollars while only having to demonstrate promising research results and/or very modest value-added for now.” [p. 36](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.nc2d14p8i1pd) 49. [^](#fnrefjji9p2haoy) “I would guess that the most compute-intensive training run for an unreleased and/or proprietary model (e.g., a language model powering Google Assistant or Google Translate) is already ~2-8x larger than AlphaStar’s ~1.3e23, costing ~$2-8M.” [p. 36](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.nc2d14p8i1pd) “[N]ote that there will probably be a non-trivial delay between the first time a training run of size X is completed and the first time such a training run is published, and my forecasts are about the former”. [p. 37](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.nc2d14p8i1pd) 50. [^](#fnrefiit8rlrsmh) [P. 36](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.nc2d14p8i1pd) 51. [^](#fnrefoz8765uouig) “The possibility of training a transformative model would provide an enormous incentive. Given this incentive, how much additional money would an AI company be willing and able to spend on a training run over the next couple of decades (if they had already ramped up to ~$1B training runs)?” [p. 37](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.ce0olvo3jfpb) 52. [^](#fnrefp42zmh193s) “I would guess that an AI company could spend hundreds of billions on a project to train a transformative model by ~2040.” [p. 38](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.ce0olvo3jfpb) 53. [^](#fnrefza495v8sl4g) “The largest AI companies already have enough [cash on hand](https://www.upcounsel.com/cash-on-hand) that they could relatively quickly deploy tens of billions for a lucrative enough project. As of Q4 2019, both [Microsoft](https://www.microsoft.com/) and [Alphabet](https://abc.xyz/) (the parent company of Google and [DeepMind](https://deepmind.com/)) had more than $100B in cash on hand, and Facebook and Amazon each have more than $50B;[] this could theoretically be spent given buy-in from only a small number of people in leadership positions at each of those companies. Those four companies have already invested heavily in AI research and relevant infrastructure such as data centers; other large tech companies have not made a large investment into AI but also have large amounts of cash on hand (e.g. Apple has over $100B)[] and could imaginably make that transition over ~5-10 years if AI continues to look like a lucrative field.” [p. 38](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.ce0olvo3jfpb) 54. [^](#fnrefpswcyp1u5e8) “Large tech companies’ [market capitalization](https://en.wikipedia.org/wiki/Market_capitalization) tends to be ~10x as large as their cash on hand (close to $1 trillion).” [p. 38](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.ce0olvo3jfpb) 55. [^](#fnrefscpkxkjecf) “It seems unlikely that a company could borrow money much past its market capitalization -- particularly for a single risky venture -- but seems possible that it could borrow something in the range of ~10%-50% of market cap for a project like training a potentially transformative model; this could make $100-500B in additional funds available.” [p. 38](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.ce0olvo3jfpb) 56. [^](#fnrefhcedwnmdkbb) “I would expect such companies to grow significantly as a share of the economy over the next 20 years in the worlds where AI progress continues, and increase in their borrowing power and ability to attract investment.” [p. 38](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.ce0olvo3jfpb) 57. [^](#fnref9z035kl63u4) See [here](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.1iu8tiytml42), particularly, “My overall intuition based on the above information is that all-in costs for a large project to train an ML model -- including the cost of salaries, data and environments, and all the compute used to experiment at smaller scales -- could get to within ~2-10x the cost of the compute for the single final training run in the medium term.” [p. 42](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.1iu8tiytml42) 58. [^](#fnrefsq4ytfzcjjg) “This suggests that by 2040, an AI project would be willing and able to spend about $100B on computation to train a transformative model.” [p. 42](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.1iu8tiytml42) 59. [^](#fnrefmuqvnvxj50o) “If willingness to spend in 2040 is $100B and willingness to spend in 2025 is $1B, this suggests a doubling time of about two years in that period.” [p. 42](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.1iu8tiytml42) 60. [^](#fnrefiujw8p1ugwb) “Eventually, I expect that growth in spending on computation will keep pace with growth in the GDP of the largest national economy.” [p. 44](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.d3f6hblbagva) 61. [^](#fnrefoocwg4lvde) “I will assume that the GDP of the largest national economy will grow at ~3% annually, which is similar to the average growth rate of the United States (the current largest national economy) over the last few decades.” [p. 44](https://docs.google.com/document/d/1qjgBkoHO_kDuUYqy_Vws0fpf-dG5pTU4b8Uej6ff2Fg/edit#heading=h.d3f6hblbagva) 62. [^](#fnrefrjcs2p9hg6) “Anchoring to the costs of major technological megaprojects such as the Manhattan Project (which cost about ~1.7% of a year of GDP over five years) and the Apollo Project (which cost about ~3.6% of a year of GDP over its four peak years), I assumed that the maximum level of spending on computation for a single training run that could be reached is ~1% of the GDP of the largest country.” [p. 5](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.z7u133pzed6k) 63. [^](#fnrefmhiw3fnxyta) Question 46: 338, “The main split between predictions is between lower estimates (including the team median) that anchor on present project costs with a modest multiplier, and higher estimates that follow Cotra in predicting pretty fast scaling will continue up to anchors set by demonstrated value-added, tech company budgets, and megaproject percentages of GDP." 64. [^](#fnrefl9b1tv6dqun) Question 46: 340, “Presumably much of these disagreement[s] stem from different ways of looking at recent AI progress. Some see the growth of computing power as range bound by current manufacturing processes and others expect dramatic changes in the very basis of how processors function leading to continued price decreases.” 65. [^](#fnrefetmsawbnk8) Question 46: 337, “training cost seems to have been stuck in the $10M figure for the last few years.”; “we have not seen such a large increase in the estimated training cost of the largest AI model during the last few years: AlphaZero and PALM are on the same ballpark.” 341, “For 2024, the costs seem to have flattened out and will be similar to now. To be on trend in 2021, the largest experiment would need to be at $0.2-1.5bn. GPT-3 was only $4.6mn” 66. [^](#fnrefizgiixeqjsm) Question 46: 341, “The AI impacts note also states that the trend would only be sustainable for a few more years. 5-6 years from 2018, i.e. 2023-24, we would be at $200bn, where we are already past the total budgets for even the biggest companies.” 67. [^](#fnrefz53jukboly) Question 46: 336, “The days of 'easy money' may be over. There's some serious belt-tightening going on in the industry (Meta, Google) that could have a negative impact on money spent.” 68. [^](#fnrefejfq54kjbyf) Question 46: 337, “It also puts more weight on the reduced cost of compute and maybe even in the improved efficiency of minimization algorithms, see question 48 for instance.” 336, “After 2030, we expect increased size and complexity to be offset by falling cost of compute, better pre-trained models and better algorithms. This will lead to a plateau and possible even a reduction in costs.”; “In the near term, falling cost of compute, pre-trained models, and better algorithms will reduce the expense of training a large language model (which is the architecture which will likely see the most attention and investment in the short term).” See also 343, “$/FLOPs is likely to be driven down by new technologies and better chips. Better algorithm design may also improve project performance without requiring as much spend on raw compute.” See also 339, “The low end scenarios could happen if we were to discover more efficient training methods (eg take a trained model from today and somehow augment it incrementally each year rather than a single batch retrain or perhaps some new research paradigm which makes training much cheaper).” 69. [^](#fnreffvkqh3q6u3) Question 46: 336, “Additionally, large language models are currently bottlenecked by available data. Recent results from DeepMind suggest that models over ~100 billion parameters would not have enough data to optimally train. This will lead to smaller models and less compute used in the near term. For example, GPT-4 will likely not be significantly larger than Chinchilla. <https://arxiv.org/abs/2203.15556>”. 341, “The data availability is limited.” See also 340, “The evidence from Chinchilla says that researchers overestimated the value of adding parameters (see <https://www.lesswrong.com/posts/6Fpvch8RR29qLEWNH/chinchilla-s-wild-implications>). That is probably discouraging researchers from adding more parameters for a while. Combined with the difficulty of getting bigger text datasets, that might mean text-oriented systems are hitting a wall. (I'm unsure why this lasts long - I think other datasets such as video are able to expand more).” 70. [^](#fnrefl9mc89we9kn) Question 46: 340, “The growth might be slowing down now.”; “Or maybe companies were foolishly spending too little a few years ago, but are now reaching diminishing returns, with the result that declining hardware costs mostly offset the desire for bigger models.” 71. [^](#fnrefamxjodiun8) Question 46: 340, “Later on, growth might slow a lot due to a shift to modular systems. I.e. total spending on AI training might increase a good deal. Each single experiment could stay small, producing parts that are coordinated to produce increasingly powerful results.” See also 339, “2050 At this point I'm not sure it will be coherent to talk about a single AI experiment, models will probably be long lived things which are improved incrementally rather than in a single massive go. But they'll also be responsible for a large fraction of the global GDP so large expenditures will make sense, either at the state level or corporation.” 72. [^](#fnref0ix7mtmb6mlg) Question 46: 340, :Some forecasters don't expect much profit from increased spending on AI training. Maybe the recent spending spree was just researchers showing mpanies are about to come to their senses and stop spending so much money.” 73. [^](#fnrefgysbfx3oe5k) Question 46: 340; “There may some limits resulting from training time. There seems to be agreement that it's unwise to attempt experiments that take more than a few months. Maybe that translates into a limit on overall spending on a single experiment, due to limits on how much can be done in parallel, or datacenter size, or supercomputer size?” 74. [^](#fnreffafjllwoial) Question 46: 343, “Monetization of AGI is in its early stages. As AI creates new value, it's likely that additional money will be spent on increasingly more complex projects.” Note that this argument refers to forecasts higher than the team median forecasts, and the team median for 2024 was $25m. 75. [^](#fnrefsqw8i6ormgq) Question 46: 337, “This will make very much sense in the event that a great public project or international collaboration will be assembled for researching a particular aspect of AI (a bit in the line of project Manhattan for the atomic bomb, the LHC for collider physics or ITER for fusion). The probability of such a collaboration eventually appearing is not small. Other scenario is great power competition between China and the US, with a focus on AI capabilities.” 76. [^](#fnrefo2a7vpo5uh) Question 46: 336, “There is strong competition between players with deep pockets and strong incentives to develop and commercialize 'AI-solutions'.” 77. [^](#fnreffwql072onar) Question 46: 344, “Automatic experiments run by AI are beyond valuation”. 337, “One forecast suggest astronomical numbers for the largest project in the future, where the basis of this particular forecast is the possibility of an AI-driven economic explosion (allowing for the allocation of arbitrarily large resources in AI).” 78. [^](#fnrefew8ze3kr47) “I have done very little research into algorithmic progress trends. Of the four main components of my model (2020 compute requirements, algorithmic progress, compute price trends, and spending on computation) I have spent the least time thinking about algorithmic progress.” [p. 5](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 79. [^](#fnrefamojubiv4t) “Additionally, it seems plausible to me that both sets of results would overestimate the pace of algorithmic progress on a transformative task, because they are both focusing on relatively narrow problems with simple, well-defined benchmarks that large groups of researchers could directly optimize.# Because no one has trained a transformative model yet, to the extent that the computation required to train one is falling over time, it would have to happen via proxies rather than researchers directly optimizing that metric (e.g. perhaps architectural innovations that improve training efficiency for image classifiers or language models would translate to a transformative model). Additionally, it may be that halving the amount of computation required to train a transformative model would require making progress on multiple partially-independent sub-problems (e.g. vision and language and motor control). I have attempted to take the Hernandez and Brown 2020 halving times (and Paul’s summary of the Grace 2013 halving times) as anchoring points and shade them upward to account for the considerations raised above.” [p. 6](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 80. [^](#fnrefmo2eyygb6aq) “For incremental progress,the main source I used was [Hernandez and Brown 2020](https://arxiv.org/pdf/2005.04305.pdf), “Measuring the Algorithmic Efficiency of Neural Networks.” The authors reimplemented open source state-of-the-art (SOTA) ImageNet models between 2012 and 2019 (six models in total). They trained each model up to the point that it achieved the same performance as AlexNet achieved in 2012, and recorded the total FLOP that required. They found that the SOTA model in 2019, [EfficientNet B0](https://arxiv.org/pdf/1905.11946.pdf), required ~44 times fewer training FLOP to achieve AlexNet performance than AlexNet did; the six data points fit a power law curve with the amount of computation required to match AlexNet halving every ~16 months over the seven years in the dataset.# They also show that [linear programming](https://en.wikipedia.org/wiki/Linear_programming) displayed a similar trend over a longer period of time: when hardware is held fixed, the time in seconds taken to solve a standard basket of mixed integer programs by SOTA commercial software packages halved every ~13 months over the 21 years from 1996 to 2017.” [p. 6](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 81. [^](#fnref8ct49j3vr5d) “Additionally, it seems plausible to me that both sets of results would overestimate the pace of algorithmic progress on a transformative task, because they are both focusing on relatively narrow problems with simple, well-defined benchmarks that large groups of researchers could directly optimize.# Because no one has trained a transformative model yet, to the extent that the computation required to train one is falling over time, it would have to happen via proxies rather than researchers directly optimizing that metric (e.g. perhaps architectural innovations that improve training efficiency for image classifiers or language models would translate to a transformative model). Additionally, it may be that halving the amount of computation required to train a transformative model would require making progress on multiple partially-independent sub-problems (e.g. vision and language and motor control). I have attempted to take the Hernandez and Brown 2020 halving times (and Paul’s summary of the Grace 2013 halving times) as anchoring points and shade them upward to account for the considerations raised above.” [p. 6](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 82. [^](#fnrefofy638opskn) “I consider two types of algorithmic progress: relatively incremental and steady progress from iteratively improving architectures and learning algorithms, and the chance of “breakthrough” progress which brings the [technical difficulty](https://docs.google.com/document/d/1IJ6Sr-gPeXdSJugFulwIpvavc0atjHGM82QjIfUSBGQ/edit#heading=h.5butqad6sph5) of training a transformative model down from “astronomically large” / “impossible” to “broadly feasible.”” [p. 5](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 83. [^](#fnref7z8f6ze1278) Question 48: See 339, “"On the other hand, an economist would say that one day, the improvement will stagnate as models become ""good enough"" for efficient use, and it's not worth it to become even better at image classification. Arguably, this day seems not too far off. So growth may either level off or continue on its exponential path. Base rate thinking does not help much with this question… It eluded the team to find reasonable and plausible answers... stagnation may be just as plausible as further exponential growth. No one seems to know.” 84. [^](#fnref8uypddzxh5s) Question 48: 340, “Low range forecasts assume that nobody does any further work on this area, hence no improvement in efficiency.” 341, “The Github page for people to submit entries to the leaderboard created by OpenAI hasn't received any submissions (based on pull requests), which could indicate a lack of interest in targeting efficiency. <https://github.com/openai/ai-and-efficiency>”. 85. [^](#fnref7w68fm2va6j) Question 48: 340, “In addition, it seems pretty unclear, whether this metric would keep improving incidentally with further progress in ML, especially given the recent focus on extremely large-scale models rather than making things more efficient.” 86. [^](#fnreft02npjjf2zi) Question 48: 340, “there seem to bem some hard limits on how much computation would be needed to learn a strong image classifier”. 87. [^](#fnref3gcw5wtj1ry) Question 48: 341, “The use cases for AI may demand accuracy instead of efficiency, leading researchers to target continued accuracy gains instead of focusing on increased efficiency.” 88. [^](#fnrefu5lg1oldi9) Question 48: 341, “A shift toward explainable AI (which could require more computing power to enable the AI to provide explanations) could depress growth in performance.” 89. [^](#fnrefefls1wfvo5f) Question 48: 336, “Lower end forecasts generally focused on the fact that improvements may not happen in a linear fashion and may not be able to keep pace with past trends, especially given the "lumpiness" of algorithmic improvement and infrequent updates to the source data.” 338, “The lowest forecasts come from a member that attempted to account for long periods with no improvement. The reference table is rarely updated and it only includes a few data points. So progress does look sporadic.” 90. [^](#fnref7vs2ikh3xl3) Question 48: 337, “The most significant disagreements involved whether very rapid improvement observed in historical numbers would continue for the next eight years. A rate of 44X is often very hard to sustain and such levels usually revert to the mean.” 91. [^](#fnrefaygwmerb4wf) Question 48: 340, “The higher range forecasts simply stem from the extrapolation detailed above. Pure extrapolation of the 44x in 7 years would yield a factor 8.7 for the 4 years from 2020 to 2024 and a factor of 222 for the years until 2030. => 382 and 9768.” 336, “Base rate has been roughly a doubling in efficiency every 16 months, with a status quo of 44 as of May 2019, when the last update was published. Most team members seem to have extrapolated that pace out in order to generate estimates for the end of 2024 and 2030, with general assumption being progress will continue at roughly the same pace as it has previously.” 92. [^](#fnrefsl68iqurcv) Question 48: 336, “The high end seems to assume that progress will continue and possibly increase if things like quantum computing allow for a higher than anticipated increase in computing power and speed.” 93. [^](#fnrefevo36u0up86) Question 48: 341, “AI efficiency will be increasingly important and necessary to achieve greater accuracy as AI models grow and become limited by available compute.” 94. [^](#fnref38gxe0yakea) Question 48: 341. 95. [^](#fnrefq5gtiwextll) Question 48: 337, “The most significant disagreements involved whether very rapid improvement observed in historical numbers would continue for the next eight years. A rate of 44X is often very hard to sustain and such levels usually revert to the mean. However, it seems relatively early days for this tech, so this is plausible.” 96. [^](#fnrefpi55hwql9bo) For the relevant questions in the XPT, forecasters were asked to provide their 5th, 25th, 50th, 75th, and 95th percentile forecasts. In this analysis we use the term, ‘median’ to refer to analyses using the group’s median forecast for the 50th percentile of each question. We use the term ‘most aggressive’ to refer to analyses using the group medians for the 5th percentile estimate of the question relating to hardware costs, and the 95th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the lowest plausible hardware costs and the highest plausible willingness to spend and algorithmic efficiency to give the highest plausible likelihood of TAI.) We use the term ‘most conservative’ to refer to analyses using the group medians for the 95th percentile estimate of the question relating to hardware costs, and the 5th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the highest plausible hardware costs and the lowest plausible willingness to spend and algorithmic efficiency to give the lowest plausible likelihood of TAI.) The most aggressive and most conservative estimates can be considered equivalent to 90% confidence interval for the median estimate. See [here](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Comparing_Cotra_and_XPT_forecasts) for context on which XPT questions map to which biological anchors inputs. 97. [^](#fnrefwhj1pui5tyc) “I have attempted to take the Hernandez and Brown 2020 halving times (and Paul’s summary of the Grace 2013 halving times) as anchoring points and shade them upward to account for the considerations raised above. There is massive room for judgment in whether and how much to shade upward; I expect many readers will want to change my assumptions here, and some will believe it is more reasonable to shade downward." [p. 6](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 98. [^](#fnref9u7wlabe4oc) “They found that the SOTA model in 2019, [EfficientNet B0](https://arxiv.org/pdf/1905.11946.pdf), required ~44 times fewer training FLOP to achieve AlexNet performance than AlexNet did; the six data points fit a power law curve with the amount of computation required to match AlexNet halving every ~16 months over the seven years in the dataset. They also show that [linear programming](https://en.wikipedia.org/wiki/Linear_programming) displayed a similar trend over a longer period of time: when hardware is held fixed, the time in seconds taken to solve a standard basket of mixed integer programs by SOTA commercial software packages halved every ~13 months over the 21 years from 1996 to 2017.” [p. 6](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 99. [^](#fnref6bmtulxqjiv) “Paul is familiar with the results, and he believes that algorithmic progress across the six domains studied in Grace 2013 is consistent with a similar but slightly slower rate of progress, ranging from 13 to 36 months to halve the computation required to reach a fixed level of performance.” [p. 6](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 100. [^](#fnrefaoyi4t53r0g) “the main source I used was [Hernandez and Brown 2020](https://arxiv.org/pdf/2005.04305.pdf)…[Grace 2013](https://intelligence.org/files/AlgorithmicProgress.pdf) (“Algorithmic Progress in Six Domains”) is the only other paper attempting to systematically quantify algorithmic progress that I am currently aware of… I have chosen not to examine it in detail because a) it was written largely before the deep learning boom and mostly does not focus on ML tasks, and b) it is less straightforward to translate Grace’s results into the format that I am most interested in (“How has the amount of computation required to solve a fixed [task](https://docs.google.com/document/d/1KsItCUEYR4_yGKF2ehyHiUYWWrPo28_rqp0oIq0BRqA/edit#heading=h.oy8zfrwjlptx) decreased over time?”)”. [p. 6](https://docs.google.com/document/d/1cCJjzZaJ7ATbq8N2fvhmsDOUWdm7t3uSSXv6bD0E_GM/edit#heading=h.epn531rebzyy) 101. [^](#fnref05naqbi3hvgi) For the relevant questions in the XPT, forecasters were asked to provide their 5th, 25th, 50th, 75th, and 95th percentile forecasts. In this analysis we use the term, ‘median’ to refer to analyses using the group’s median forecast for the 50th percentile of each question. We use the term ‘most aggressive’ to refer to analyses using the group medians for the 5th percentile estimate of the question relating to hardware costs, and the 95th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the lowest plausible hardware costs and the highest plausible willingness to spend and algorithmic efficiency to give the highest plausible likelihood of TAI.) We use the term ‘most conservative’ to refer to analyses using the group medians for the 95th percentile estimate of the question relating to hardware costs, and the 5th percentile estimate for the questions relating to willingness to spend and algorithmic progress. (I.e., this uses the highest plausible hardware costs and the lowest plausible willingness to spend and algorithmic efficiency to give the lowest plausible likelihood of TAI.) The most aggressive and most conservative estimates can be considered equivalent to 90% confidence interval for the median estimate. See [here](https://forum.effectivealtruism.org/posts/ccw9v9giKxg8nyLhp/xpt-forecasts-on-some-biological-anchors-inputs#Comparing_Cotra_and_XPT_forecasts) for context on which XPT questions map to which biological anchors inputs.
38105531-beb5-4a1a-a833-e90343f5aeea	trentmkelly/LessWrong-43k	LessWrong	Thought experiment: coarse-grained VR utopia I think I've come up with a fun thought experiment about friendly AI. It's pretty obvious in retrospect, but I haven't seen it posted before. When thinking about what friendly AI should do, one big source of difficulty is that the inputs are supposed to be human intuitions, based on our coarse-grained and confused world models. While the AI's actions are supposed to be fine-grained actions based on the true nature of the universe, which can turn out very weird. That leads to a messy problem of translating preferences from one domain to another, which crops up everywhere in FAI thinking, Wei's comment and Eliezer's writeup are good places to start. What I just realized is that you can handwave the problem away, by imagining a universe whose true nature agrees with human intuitions by fiat. Think of it as a coarse-grained virtual reality where everything is built from polygons and textures instead of atoms, and all interactions between objects are explicitly coded. It would contain player avatars, controlled by ordinary human brains sitting outside the simulation (so the simulation doesn't even need to support thought). The FAI-relevant question is: How hard is it to describe a coarse-grained VR utopia that you would agree to live in? If describing such a utopia is feasible at all, it involves thinking about only human-scale experiences, not physics or tech. So in theory we could hand it off to human philosophers or some other human-based procedure, thus dealing with "complexity of value" without much risk. Then we could launch a powerful AI aimed at rebuilding reality to match it (more concretely, making the world's conscious experiences match a specific coarse-grained VR utopia, without any extra hidden suffering). That's still a very hard task, because it requires solving decision theory and the problem of consciousness, but it seems more manageable than solving friendliness completely. The resulting world would be suboptimal in many ways, e.g. it wouldn't hav
eaee6f06-198d-49fe-a798-8e750c9d509c	trentmkelly/LessWrong-43k	LessWrong	Abstraction Talk I gave a talk a couple weeks on abstraction. It moves very fast, but it covers more material in one place than any individual post I've written, and includes a few things which haven't showed up in posts yet (especially near the end). Slides are here. The slides are hard to see sometimes in the video; I recommend keeping them open to the side of the video.
23c2bc51-1f78-49b8-a714-3d1b5b45d823	trentmkelly/LessWrong-43k	LessWrong	New year's resolutions: Things worth considering for next year The beginning of the new year is a natural Schelling Point and swiftly approaching. With that in mind I have created a handy go-to list of things worth considering for next year. Alongside this process; another thing you might like to do is conduct a review of this year, confirming your progress on major goals; double checking that you are on track. and conduct any last-minute summaries of potential failures or learning-cases. This list is designed to be used for imagination, opportunity, and potential planning purposes. If you find yourself having the feelings of (disappointment, failure, fear, regret, burdens, guilt and others) reconsider looking at this list and instead do something that will not lead to negative feelings about the future. If you are not getting something positive out of doing this exercise, don't. That's a silly idea. I am banking on the fact that it will be more helpful than not; for most people. If you are in the category of people that it does not help - I am sorry; I assume you know your priorities and are working on them as reasonably effectively as possible - good luck with that task. This list is going to look a bit like my List of common human goals because it was written concurrenlty with the ideas listed there (and by the same person). You might want a pen and paper; and 10 minutes to go through this list and consider what things you want to do over the next year that fall into these categories. This time is not for you to plan out an entire year, but something of a chance to consider the playing field of "a year of time". After you have a list of things you want to do; there are lots of things you can do with them. i.e. time planning, research, goal factoring, task-generating. without further ado; the list: ---------------------------------------- 1. things I might want to study or learn next year Often people like learning. Are you thinking of higher level study? Or keen to upskill? Thinking of picking up a textbo
71ecaf1b-c971-4093-93c1-c2ca69bdcc17	StampyAI/alignment-research-dataset/alignmentforum	Alignment Forum	Will AI undergo discontinuous progress? This post grew out of conversations with several people, including [Daniel Kokotajlo](https://www.lesswrong.com/users/daniel-kokotajlo), [grue\_slinky](https://www.lesswrong.com/users/grue_slinky) and [Linda Lisefors](https://www.lesswrong.com/users/linda-linsefors), and is based in large part on a collection of scattered comments and blog-posts across lesswrong, along with some podcast interviews - e.g. [here](https://www.lesswrong.com/posts/CjW4axQDqLd2oDCGG/misconceptions-about-continuous-takeoff?commentId=YPodaAtRhN4qJefxb). The in-text links near quotes will take you to my sources. I am attempting to distinguish two possibilities which are often run together - that progress in AI towards AGI (‘takeoff’) will be discontinuous and that it will be fast, but continuous. Resolving this distinction also addresses the claim that there has been a significant shift in arguments for AI presenting an existential risk: from older arguments discussing an ultra-fast intelligence explosion occurring in a single ‘seed AI’ to more moderate scenarios. I argue that the ‘shift in arguments on AI safety’ is not a total change in basic assumptions (which some observers have claimed) but just a reduction in confidence about a specifically discontinuous takeoff. Finally, I try to explicitly operationalize the practical differences between discontinuous takeoff and fast, continuous takeoff. Further Reading [Summary: Why AI risk might be solved without additional intervention from Longtermists](https://www.lesswrong.com/posts/QknPz9JQTQpGdaWDp/an-80-why-ai-risk-might-be-solved-without-additional) [Paul Christiano’s original post](https://sideways-view.com/2018/02/24/takeoff-speeds/) [MIRIs Thoughts on Discontinuous takeoff](https://www.lesswrong.com/posts/PzAnWgqvfESgQEvdg/any-rebuttals-of-christiano-and-ai-impacts-on-takeoff-speeds#rXh6LNmoz64mLv2kX) [Misconceptions about continuous takeoff](https://www.lesswrong.com/posts/CjW4axQDqLd2oDCGG/misconceptions-about-continuous-takeoff) [AI Impacts original post](https://aiimpacts.org/likelihood-of-discontinuous-progress-around-the-development-of-agi/) [Soft Takeoff can still lead to Decisive Strategic Advantage](https://www.lesswrong.com/posts/PKy8NuNPknenkDY74/soft-takeoff-can-still-lead-to-decisive-strategic-advantage) Defining Discontinuous Progress =============================== What do I mean by ‘discontinuous’? If we were to graph world GDP over the last 10,000 years, it fits onto a [hyperbolic growth pattern](https://slatestarcodex.com/2019/04/22/1960-the-year-the-singularity-was-cancelled/). We could call this ‘continuous’ since it is following a single trend, or we could call it ‘discontinuous’ because, on the scale of millennia, the industrial revolution exploded out of nowhere. I will call these sorts of hyperbolic trends ‘continuous, but fast’, in line with Paul Christiano, who argued for continuous takeoff, defining it [this way](https://sideways-view.com/2018/02/24/takeoff-speeds/): > AI is just another, faster step in the [hyperbolic growth we are currently experiencing](https://sideways-view.com/2017/10/04/hyperbolic-growth/), which corresponds to a further increase in rate but not a discontinuity (or even a discontinuity in rate). I’ll be using Paul’s understanding of ‘discontinuous’ and ‘fast’ here. For progress in AI to be discontinuous, we need a switch to a new growth mode, which will show up as a step function in the capability of AI or in the rate of change of the capability of the AI over time. For takeoff to be fast, it is enough that there is one single growth mode that is hyperbolic or some other function that is very fast-growing. The view that progress in AI will be discontinuous, not merely very fast by normal human standards, was popular and is still held by many. Here is a canonical explanation of the view, from [Eliezer Yudkowsky in 2008](https://www.lesswrong.com/posts/tjH8XPxAnr6JRbh7k/hard-takeoff). Compare this to the more recent [‘what failure looks like’](https://www.lesswrong.com/posts/HBxe6wdjxK239zajf/what-failure-looks-like) to understand the intuitive force of the claim that views on AI risk have totally changed since 2008. > Recursive self-improvement - an AI rewriting its own cognitive algorithms - identifies the object level of the AI with a force acting on the metacognitive level; it "closes the loop" or "folds the graph in on itself"... > ...When you fold a whole chain of differential equations in on itself like this, it should either peter out rapidly as improvements fail to yield further improvements, or else go FOOM. An exactly right law of diminishing returns that lets the system fly through the soft takeoff keyhole is unlikely - far more unlikely than seeing such behavior in a system with a roughly-constant underlying optimizer, like evolution improving brains, or human brains improving technology. Our present life is no good indicator of things to come. > Or to try and compress it down to a slogan that fits on a T-Shirt - not that I'm saying this is a good idea - "Moore's Law is exponential now; it would be really odd if it stayed exponential with the improving computers doing the research." I'm not saying you literally get dy/dt = e^y that goes to infinity after finite time - and hardware improvement is in some ways the least interesting factor here - but should we really see the same curve we do now? > RSI is the biggest, most interesting, hardest-to-analyze, sharpest break-with-the-past contributing to the notion of a "hard takeoff" aka "AI go FOOM", but it's nowhere near being the only such factor. [The advent of human intelligence was a discontinuity with the past](https://www.lesswrong.com/lw/w4/surprised_by_brains/) even without RSI... > ...which is to say that observed evolutionary history - the discontinuity between humans, and chimps who share 95% of our DNA - lightly suggests a critical threshold built into the capabilities that we think of as "general intelligence", a machine that becomes far more powerful once the last gear is added. Also see these quotes, one summarizing the view by [Paul Christiano](https://sideways-view.com/2018/02/24/takeoff-speeds/) and one recent remark from [Rob Besinger](https://www.lesswrong.com/posts/PzAnWgqvfESgQEvdg/any-rebuttals-of-christiano-and-ai-impacts-on-takeoff-speeds#rXh6LNmoz64mLv2kX), summarizing the two key reasons given above for expecting discontinuous takeoff, recursive self-improvement and a discontinuity in capability. > some systems “fizzle out” when they try to design a better AI, generating a few improvements before running out of steam, while others are able to autonomously generate more and more improvements. > MIRI folks tend to have different views from Paul about AGI, some of which imply that AGI is more likely to be novel and dependent on new insights. I will argue that the more recent reduction of confidence in discontinuous takeoff is correct, but at the same time many of the ‘original’ arguments (e.g. those given in 2008 by Yudkowsky) for fast, discontinuous takeoff are not mistaken and can be seen as also supporting fast, continuous takeoff. We should seriously investigate the continuous/discontinuous distinction specifically by narrowing our focus onto arguments that actually distinguish between the two: conceptual investigation about the nature of future AGI, and the practical consequences for alignment work of continuous/discontinuous takeoff. I have tried to present the arguments in order from least to most controversial, starting with the outside view on technological progress. There have been other posts discussing arguments for discontinuous progress with approximately this framing. I am not going to repeat their good work here by running over every argument and counterargument (See Further Reading). What I’m trying to do here is get at the underlying assumptions of either side. The Outside View ================ There has recently been a switch between talking about AI progress being fast vs slow to talking about it as continuous vs discontinuous. Paul Christiano explains what [continuous progress means](https://sideways-view.com/2018/02/24/takeoff-speeds/): > I believe that before we have incredibly powerful AI, we will have AI which is merely very powerful. This won’t be enough to create 100% GDP growth, but it will be enough to lead to (say) 50% GDP growth. I think the likely gap between these events is years rather than months or decades. It is not an essential part of the definition that the gap be years; even if the gap is rather short, we still call it continuous takeoff if AI progress increases without sudden jumps; capability increases following series of logistic curves merging together as different component technologies are invented. This way of understanding progress as being ‘discontinuous’ isn’t uncontroversial, but it was developed because calling the takeoff ‘slow’ instead of ‘fast’ could be seen as a misnomer. Continuous takeoff is a statement about what happens before we reach the point where a fast takeoff is supposed to happen, and is perfectly consistent with the claim that given the stated preconditions for fast takeoff, fast takeoff will happen. It’s a statement that serious problems, possibly serious enough to pose an existential threat, will show up before the window where we expect fast takeoff scenarios to occur. [Outside view: Technology](https://aiimpacts.org/discontinuous-progress-investigation) -------------------------------------------------------------------------------------- The starting point for the argument that progress in AI should be continuous is just the observation that this is usually how things work with a technology, especially in a situation where progress is being driven by many actors working at a problem from different angles. If you can do something well in 1 year, it is usually possible to do it slightly less well in 0.9 years. Why is it ‘usually possible’? Because nearly all technologies involve numerous smaller innovations, and it is usually possible to get somewhat good results without some of them. That is why even if each individual component innovation follows a [logistic success curve](https://intelligence.org/2017/11/26/security-mindset-and-the-logistic-success-curve/) that is unstable between ‘doesnt work at all’ and ‘works’ if you were to plot results/usefulness against effort, progress looks continuous. Note this is what is explicitly rejected by [Yudkowsky-2008.](https://www.lesswrong.com/posts/tjH8XPxAnr6JRbh7k/hard-takeoff) When this doesn’t happen and we get discontinuous progress, it is because of one of two reasons - either there is some fundamental reason why the technology cannot work at all without all the pieces lining up in place, or there is just not much effort being put in, so a few actors can leap ahead of the rest of the world and make several of the component breakthroughs in rapid succession. I’ll go through three illustrative examples of each situation - the normal case, the low-effort case and the fundamental reasons case. Guns Guns followed the continuous progress model. They started out worse than competing ranged weapons like crossbows. By the 15th century, the Arquebus arrived, which had some advantages and disadvantages compared to the crossbow (easier to use, more damaging, but slower to fire and much less accurate). Then came the musket, and later rifles. There were many individual inventions that went from ‘not existing at all’ to ‘existing’ in relatively short intervals, but the overall progress looked roughly continuous. However, the speed of progress still increased dramatically during the industrial revolution and continued to increase, without ever being ‘discontinuous’. Aircraft Looking specifically at heavier-than-air flight, it seems clear enough that we went from ‘not being able to do this at all’ to being able to do it in a relatively short time - discontinuous progress. The Wright brothers research drew on a few other pioneers like Otto Lilienthal, but they still made a rapid series of breakthroughs in a fairly short period: using a wind tunnel to rapidly prototype wing shapes, building a sufficiently lightweight engine, working out a method of control. This was possible because at the time, unlike guns, a very small fraction of human research effort was going into developing flight. It was also possible for another reason - the nature of the problem implied that success was always going to be discontinuous. While there are a few intermediate steps, like gliders, there aren’t many between ‘not being able to fly’ and ‘being able to fly’, so progress was unexpected and discontinuous. I think we can attribute most of the flight case to the low degree of effort on a global scale. Nuclear Weapons Nuclear Weapons are a purer case of fundamental physical facts causing a discontinuity. A fission chain reaction simply will not occur without many breakthroughs all being brought together, so even if a significant fraction of all the world’s design effort is devoted to research we still get discontinuous progress. If the Manhattan project had uranium centrifuges but didn’t have the Monte Carlo algorithms needed to properly simulate the dynamics of the sphere’s implosion, they would just have had some mostly-useless metal. It’s worth noting here that a lot of early writing about an intelligence explosion explicitly compares it to a nuclear chain reaction. AI Impacts has done a far more [thorough and in-depth investigation](https://aiimpacts.org/category/speed-of-ai-transition/pace-of-ai-progress-without-feedback/) of progress along various metrics, confirming the intuition that discontinuous progress usually occurs where fundamental physical facts imply it - e.g. as we switch between different methods of travel or communication. We should expect, prima facie, a random example of technological progress to be continuous, and we need specific, good reasons to think that progress will be discontinuous. On the other hand, there are more than a few examples of discontinuous progress caused either by the nature of the problem or differential effort, so there is not a colossal burden of proof on discontinuous progress. I think that both the MIRI people (who argue for discontinuous progress) and Christiano (continuous) pretty much agree about this initial point. A shift in basic assumptions? ----------------------------- The argument has recently been made (by [Will MacAskill](https://80000hours.org/podcast/episodes/will-macaskill-paralysis-and-hinge-of-history/#the-risk-of-human-extinction-in-the-next-hundred-years-21520) on the 80,000 hours podcast) that there has been a switch from the ‘old’ arguments focussing on a seed AI leading to an intelligence explosion to new arguments: > Paul’s published on this, and said he doesn’t think doom l[ooks like a sudden explosion in a single AI system](https://www.lesswrong.com/posts/tjH8XPxAnr6JRbh7k/hard-takeoff) that takes over. Instead he thinks [gradually just AI’s get more and more and more power and they’re just somewhat misaligned with human interests](https://www.lesswrong.com/posts/HBxe6wdjxK239zajf/what-failure-looks-like). And so in the end you kind of get what you can measure And that these arguments don’t have very much in common with the older Bostrom/Yudkowsky scenario (of a single AGI undergoing an intelligence explosion) - except the conclusion that AI presents a uniquely dangerous existential risk. If true, this would be a cause for concern as it would suggest we haven’t become much less confused about basic questions over the last decade. [MacAskill again:](https://80000hours.org/podcast/episodes/will-macaskill-paralysis-and-hinge-of-history/#the-risk-of-human-extinction-in-the-next-hundred-years-21520) > I have no conception of how common adherence to different arguments are — but certainly many of the most prominent people are no longer pushing the Bostrom arguments. If you look back to the section on Defining Discontinuous Progress, this will seem plausible - the very rapid annihalation caused by a single misaligned AGI going FOOM and the accumulation of ‘you get what you measure’ errors in complex systems that compound seem like totally different concerns. Despite this, I will show later how the old arguments for discontinuous progress and the new arguments for continuous progress share a lot in common. I claim that the Bostrom/Yudkowsky argument for an intelligence explosion establishes a sufficient condition for very rapid growth, and the current disagreement is about what happens between now and that point. This should raise our confidence that some basic issues related to AI timelines are resolved. However, the fact that this claim, if true, has not been recognized and that discussion of these issues is still as fragmented as it is should be a cause for concern more generally. I will now turn to inside-view arguments for discontinuous progress, beginning with the Intelligence explosion, to justify what I have just claimed. Failed Arguments for Discontinuity ================================== If you think AI progress will be discontinuous, it is generally because you think AI is a special case like Nuclear Weapons where several breakthroughs need to combine to produce a sudden gain in capability, or that one big breakthrough produces nearly all the increase on its own. If you think we will create AGI at all, it is generally because you think it will be hugely economically valuable, so the low-effort case like flight does not apply - if there are ways to produce and deploy slightly worse transformative AIs sooner, they will probably be found. The arguments for AI being a special case either invoke specific evidence from how current progress in Machine Learning looks or from human evolutionary history, or are conceptual. I believe (along with Paul Christiano) the evidential arguments aren’t that useful, and my conclusion will be that we’re left trying to assess the conceptual arguments about the nature of intelligence, which are hard to judge. I’ll offer some ways to attempt that, but not any definitive answer. But first, what relevance does the old Intelligence Explosion Hypothesis have to this question - is it an argument for discontinuous progress? No, not on its own. Recursive self-improvement -------------------------- People don’t talk about the recursive self-improvement as much as they used to, because since the Machine Learning revolution a [full recursive self-improvement process has seemed less necessary to create an AGI that is dramatically superior to humans](https://intelligence.org/2017/12/06/chollet/) (by analogy with how gradient descent is able to produce capability gains in current AI). Instead, the focus is more on the general idea of ‘rapid capability gain’. From [Chollet vs Yudkowsky](https://intelligence.org/2017/12/06/chollet/): > The basic premise is that, in the near future, a first “seed AI” will be created, with general problem-solving abilities slightly surpassing that of humans. This seed AI would start designing better AIs, initiating a recursive self-improvement loop that would immediately leave human intelligence in the dust, overtaking it by orders of magnitude in a short time. > I agree this is more or less what I meant by “seed AI” when I coined the term back in 1998. Today, nineteen years later, I would talk about a general question of “capability gain” or how the power of a cognitive system scales with increased resources and further optimization. The idea of recursive self-improvement is only one input into the general questions of capability gain; for example, we recently saw some impressively fast scaling of Go-playing ability without anything I’d remotely consider as seed AI being involved. However, the argument that given a certain level of AI capabilities, the rate of capability gain will be very high doesn’t by itself argue for discontinuity. It does mean that the rate of progress in AI has to increase between now and then, but doesn’t say how it will increase. There needs to be an initial asymmetry in the situation that means an AI beyond a certain level of sophistication can experience rapid capability gain and one below it experiences the current (fairly unimpressive) capability gains with increased optimization power and resources. The original ‘intelligence explosion’ argument puts an upper bound or sufficient condition on when we enter a regime of very rapid growth - if we have something ‘above human level’ in all relevant capabilities, it will be capable of improving its capabilities. And we know that at the current level, capability gains with increased optimization are (usually) not too impressive. The general question has to be asked; will we see a sudden increase in the rate of capability gain somewhere between now and the human level’ where the rate must be very high. The additional claim needed to establish a discontinuity is that recursive self-improvement suddenly goes from ‘not being possible at all’ (our current situation) to possible, so there is a discontinuity as we enter a new growth mode and the graph abruptly ‘folds back in on itself’. ![](https://i.imgur.com/WoUmRIF.png)In Christiano’s graph, the gradient at the right end where highly capable AI is already around is pretty much the same in both scenarios, reflecting the basic recursive self-improvement argument. Here I have taken a diagram from Superintelligence and added a red curve to represent a fast but continuous takeoff scenario. ![](https://i.imgur.com/RbuBJ30.png)In Bostrom’s scenario, there are two key moments that represent discontinuities in rate, though not necessarily in absolute capabilities - the first is that around the ‘human baseline’ - when an AI can complete all cognitive tasks a human can, we enter a new growth mode much faster than the one before because the AI can recursively self-improve. The gradient is fairly constant up until that point. The next discontinuity is at the ‘crossover’ where AI is performing the majority of capability improvement itself. As in Christiano’s diagram, rates of progress in the red, continuous scenario are very similar to the black scenario after we have superhuman AI, but the difference is that there is a steady acceleration of progress before ‘human level’. This is because before we have AI that is able to accelerate progress to a huge degree, we have AI that is able to accelerate progress to a lesser extent, and so on until we have current AI, which is only slightly able to accelerate growth. Recursive self-improvement, like most other capabilities, does not suddenly go from ‘not possible at all’ to ‘possible’. The other thing to note is that, since we get substantial acceleration of progress before the ‘human baseline’, the overall timeline is shorter in the red scenario, holding other assumptions about the objective difficulty of AI research constant. The reason this continuous scenario might not occur is if there is an initial discontinuity which means that we cannot get a particular kind of recursive self-improvement with AIs that are slightly below some level of capability, but can get it with AIs that are slightly above that level. If AI is in a nuclear-chain reaction like situation where we need to reach a criticality threshold for the rate to suddenly experience a discontinuous jump. We return to the original claim, which we now see needs an independent justification: > some systems “fizzle out” when they try to design a better AI, generating a few improvements before running out of steam, while others are able to autonomously generate more and more improvements ### Connecting old and new arguments With less confidence than the previous section, I think this is a point that most people agree on - that there needs to be an initial discontinuity in the rate of return on cognitive investment, for there to be two qualitatively different growth regimes, for there to be Discontinuous progress. This also answers MacAskill’s objection that too much has changed in basic assumptions. The old recursive self-improvement argument, by giving a significant condition for fast growth that seems feasible (Human baseline AI), leads naturally to an investigation of what will happen in the course of reaching that fast growth regime. Christiano and other current notions of continuous takeoff are perfectly consistent with the counterfactual claim that, if an already superhuman ‘seed AI’ were dropped into a world empty of other AI, it would undergo recursive self-improvement. This in itself, in conjunction with other basic philosophical claims like the [orthogonality thesis](https://www.fhi.ox.ac.uk/wp-content/uploads/Orthogonality_Analysis_and_Metaethics-1.pdf), is sufficient to promote AI alignment to attention. Then, following on from that, we developed different models of how progress will look between now and AGI. So it is not quite right to say, ‘many of the most prominent people are no longer pushing the Bostrom arguments’. From [Paul Christiano’s 80,000 hours interview](https://80000hours.org/podcast/episodes/paul-christiano-ai-alignment-solutions/): > Another thing that’s important to clarify is that I think there’s rough agreement amongst the alignment and safety crowd about what would happen if we did human level AI. That is everyone agrees that at that point, progress has probably exploded and is occurring very quickly, and the main disagreement is about what happens in advance of that. I think I have the view that in advance of that, the world has already changed very substantially. ![](https://i.imgur.com/uFroEjG.png)The above diagram represents our uncertainty - we know the rate of progress now is relatively slow and that when AI capability is at the human baseline, it must be very fast. What remains to be seen is what happens in between. The remainder of this post is about the areas where there is not much agreement in AI safety - others reasons that there may be a threshold for generality, recursive self-improvement or a signifance to the 'human level'. Evidence from Evolution ----------------------- We do have an example of optimization pressure being applied to produce gains in intelligence: human evolution. The historical record certainly looks discontinuous, in that relatively small changes in human brains over relatively short timescales produced dramatic visible changes in our capabilities. However, this is misleading. The very first thing we should understand is that evolution is a continuous process - it cannot redesign brains from scratch. Evolution was optimizing for fitness, and driving increases in intelligence only indirectly and intermittently by optimizing for winning at social competition. What happened in human evolution is that it briefly switched to optimizing for increased intelligence, and as soon as that happened our intelligence grew [very rapidly but continuously.](https://sideways-view.com/2018/02/24/takeoff-speeds/) > [If] evolution were selecting primarily or in large part for technological aptitude, then the difference between chimps and humans would suggest that tripling compute and doing a tiny bit of additional fine-tuning can radically expand power, undermining the continuous change story. > But chimp evolution is not primarily selecting for making and using technology, for doing science, or for facilitating cultural accumulation. The task faced by a chimp is largely independent of the abilities that give humans such a huge fitness advantage. It’s not completely independent—the overlap is the only reason that evolution eventually produces humans—but it’s different enough that we should not be surprised if there are simple changes to chimps that would make them much better at designing technology or doing science or accumulating culture. I have a theory about why this didn't get discussed earlier - it unfortunately sounds similar to the famous bad argument against AGI being an existential risk: the 'intelligence isn't a superpower' argument. From [Chollet vs Yudkowsky](https://intelligence.org/2017/12/06/chollet/): > Intelligence is not a superpower; exceptional intelligence does not, on its own, confer you with proportionally exceptional power over your circumstances. > …said the Homo sapiens, surrounded by countless powerful artifacts whose abilities, let alone mechanisms, would be utterly incomprehensible to the organisms of any less intelligent Earthly species. I worry that in arguing against the claim that general intelligence isn't a meaningful concept or can't be used to compare different animals, some people have been implicitly assuming that evolution has been putting a decent amount of effort into optimizing for general intelligence all along. Alternatively, that arguing for one sounds like another, or that a lot of people have been arguing for both together and haven't distinguished between them. ![](https://slatestarcodex.com/blog_images/einsteinline1.jpg) Claiming that you can meaningfully compare evolved minds on the generality of their intelligence needs to be distinguished from claiming that evolution has been optimizing for general intelligence reasonably hard for a long time, and that consistent pressure ‘pushing up the scale’ hits a point near humans where capabilities suddenly explode despite a constant optimization effort. There is no evidence that evolution was putting roughly constant effort into increasing human intelligence. We could analogize the development of human intelligence to the aircraft case in the last section - where there was relatively little effort put into its development until a sudden burst of activity led to massive gains. So what can we infer from evolutionary history? We know that human minds can be produced by an incremental and relatively simple optimization process operating over time. Moreover, the difference between ‘produced by an incremental process’ and ‘developed continuously’ is small. If intelligence required the development of a good deal of complex, expedient or detrimental capabilities which were only useful when finally brought together, evolution would not have produced it. Christiano argues that this is a reason to think progress in AI will be continuous, but this seems to me to be a weak reason. Clearly, there exists a continuous path to general intelligence, but that does not mean it is the easiest path and Christiano’s other argument suggests that the way that we approach AGI will not look much like the route evolution took. Evolution also suggests that, in some absolute sense, the amount of effort required to produce increases in intelligence is not that large. Especially if you started out with the model that intelligence was being optimized all along, you should update to believing AGI is much easier to create than previously expected. The Conceptual Arguments ======================== We are left with the difficult to judge conceptual question of whether we should expect a discontinuous jump in capabilities when a set list of AI developments are brought together. Christiano in his original essay just states that there doesn’t seem to be any independent reasons to expect this to be the case. Coming up with any reasons for or against discontinuous progress essentially require us to predict how AGI will work before building it. [Rob Besinger told me something similar](https://www.lesswrong.com/posts/CjW4axQDqLd2oDCGG/misconceptions-about-continuous-takeoff?commentId=cCQTh2FZRZvPgkmj6). > There still remains the question of whether the technological path to "optimizing messy physical environments" (or "science AI", or whatever we want to call it) looks like a small number of "we didn't know how to do this at all, and now we do know how to do this and can suddenly take much better advantage of available compute" events, vs. looking like a large number of individually low-impact events spread out over time. Rob Besinger also said in one post that MIRI’s reasons for predicting discontinuous takeoff boil down to different ideas about what AGI will be like - [suggesting that this constitutes the fundamental disagreement.](https://www.lesswrong.com/posts/PzAnWgqvfESgQEvdg/any-rebuttals-of-christiano-and-ai-impacts-on-takeoff-speeds?commentId=rXh6LNmoz64mLv2kX) > MIRI folks tend to have different views from Paul about AGI, some of which imply that AGI is more likely to be novel and dependent on new insights. (Unfair caricature: Imagine two people in the early 20th century who don't have a technical understanding of nuclear physics yet, trying to argue about how powerful a nuclear-chain-reaction-based bomb might be. If one side were to model that kind of bomb as "sort of like TNT 3.0" while the other is modelling it as "sort of like a small Sun", they're likely to disagree about whether nuclear weapons are going to be a small v. large improvement over TNT...) I suggest we actually try to enumerate the new developments we will need to produce AGI, ones which could arrive discretely in the form of paradigm shifts. We might try to imagine or predict which skills must be combined to reach the ability to do original AI research. Stuart Russell provided a list of these capacities in Human Compatible. Stuart Russell’s List --------------------- * human-like language comprehension * cumulative learning * discovering new action sets * managing its own mental activity For reference, I’ve included two capabilities we already have that I imagine being on a similar list in 1960 * [perception and object recognition](https://www.wikiwand.com/en/Progress_in_artificial_intelligence#/Par-human) * [efficient search over known facts](https://www.wikiwand.com/en/Expert_system) AI Impacts List --------------- * Causal models: Building causal models of the world that are rich, flexible, and explanatory — Lake et al. (2016)[9](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-9-1938), Marcus (2018)[10](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-10-1938), Pearl (2018)[11](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-11-1938) * Compositionality: Exploiting systematic, compositional relations between entities of meaning, both linguistic and conceptual — Fodor and Pylyshyn (1988)[12](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-12-1938), Marcus (2001)[13](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-13-1938), Lake and Baroni (2017)[14](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-14-1938) * Symbolic rules: Learning abstract rules rather than extracting statistical patterns — Marcus (2018)[15](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-15-1938) * Hierarchical structure: Dealing with hierarchical structure, e.g. that of language — Marcus (2018)[16](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-16-1938) * Transfer learning: Learning lessons from one task that transfer to other tasks that are similar, or that differ in systematic ways — Marcus (2018)[17](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-17-1938), Lake et al. (2016)[18](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-18-1938) * Common sense understanding: Using common sense to understand language and reason about new situations — Brooks (2019)[19](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-19-1938), Marcus and Davis (2015)[20](https://aiimpacts.org/evidence-against-current-methods-leading-to-human-level-artificial-intelligence/#easy-footnote-bottom-20-1938) Note that discontinuities may consist either in sudden increases in capability (e.g. if a sudden breakthrough lets us build AI with full human like cumulative learning), or sudden increases in the rate of improvement (e.g. something that takes us over the purported ‘recursive self-improvement threshold’) or sudden increases in the ability to make use of hardware or knowledge overhang (Suddenly producing an AI with human-like language comprehension, which would be able to read all existing books). Perhaps the disagreement looks like this: An AI with (e.g.) good perception and object recognition, language comprehension, cumulative learning capability and ability to discover new action sets but a merely adequate or bad ability to manage its mental activity would be (Paul thinks) reasonably capable compared to an AI that is good at all of these things, but (MIRI thinks) it would be much less capable. MIRI has conceptual arguments (to do with the nature of general intelligence) and empirical arguments (comparing human/chimp brains and pragmatic capabilities) in favour of this hypothesis, and Paul thinks the conceptual arguments are too murky and unclear to be persuasive and that the empirical arguments don't show what MIRI thinks they show. Adjudicating this disagreement is a matter for another post - for now, I will simply note that it does seem like an AI significantly lacking in one of the capabilities on Stuart Russell’s list but proficient in one of the others seems intuitively like it would be much more capable than current AI, but still less capable than very advanced AI. How seriously to take this intuition, I don’t know. Summary ------- * The case for continuous progress rests on three claims + A priori, we expect continuous progress because it is usually possible to do something slightly worse slightly earlier. The historical record confirms this + Evolution's optimization is too different from the optimization of AI research to be meaningful evidence - if you optimize specifically for usefulness it might appear much earlier and more gradually + There are no clear conceptual reasons to expect a 'generality threshold' or the sudden (rather than gradual) emergence of the ability to do recursive self-improvement Relevance to AI Safety ====================== ![](https://i.imgur.com/Go1TAPW.png) .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0} .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0} .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table} .mjx-full-width {text-align: center; 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src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')} @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold} @font-face {font-family: MJXc-TeX-vec-Bw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')} If we have a high degree of confidence in discontinuous progress, we more-or-less know that we’ll get a sudden takeoff where superintelligent AI appears out of nowhere and forms a singleton. On the other hand, if we expect continuous progress then the rate of capability gain is much more difficult to judge. That is already a strong reason to care about whether progress will be continuous or not. [Decisive Strategic Advantage (DSA) leading to Singletons](https://www.lesswrong.com/posts/vkjWGJrFWBnzHtxrw/superintelligence-7-decisive-strategic-advantage) is still possible with continuous progress - we just need a specific reason for a gap to emerge (like a big research program). An [example scenario](https://aiimpacts.org/soft-takeoff-can-still-lead-to-decisive-strategic-advantage/) for a decisive strategic advantage leading to a singleton with continuous, relatively fast progress: > At some point early in the transition to much faster innovation rates, the leading AI companies "go quiet." Several of them either get huge investments or are nationalized and given effectively unlimited funding. The world as a whole continues to innovate, and the leading companies benefit from this public research, but they hoard their own innovations to themselves. Meanwhile the benefits of these AI innovations are starting to be felt; all projects have significantly increased (and constantly increasing) rates of innovation. But the fastest increases go to the leading project, which is one year ahead of the second-best project. (This sort of gap is normal for tech projects today, especially the rare massively-funded ones, I think.) Perhaps via a combination of spying, selling, and leaks, that lead narrows to six months midway through the process. But by that time things are moving so quickly that a six months' lead is like a 15-150 year lead during the era of the Industrial Revolution. It's not guaranteed and perhaps still not probable, but at least it's reasonably likely that the leading project will be able to take over the world if it chooses to. Let’s factor out the question of fast or slow takeoff, and try to compare two AI timelines that are similarly fast in objective time, but one contains a discontinuous leap in capability and the other doesn’t. What are the relevant differences with respect to AI Safety? In the discontinuous scenario, we do not require the classic ‘seed AI’ that recursively self-improves - that scenario is too specific and more useful as a thought experiment. Instead, in the discontinuous scenario it is merely a fact of the matter that at a certain point returns on optimization explode and capability gain becomes very rapid where before it was very slow. In the other case, progress is continuous but fast, though presumably not quite as fast. Any approach to alignment that relies on a less advanced agent supervising a more advanced agent will probably not work in the discontinuous case, since the difference between agents on one side and another side of the discontinuity would be too great. An [iterated approach that relies on groups of less intelligent agents supervising a more intelligent agent](https://openai.com/blog/amplifying-ai-training/) could work even in a very fast but continuous takeoff, because even if the process took a small amount of objective time, continuous increments of increased capability would still be possible, and less intelligent agents could supervise more intelligent agents. Discontinuous takeoff suggests ambitious value learning approaches, while continuous takeoff suggests iterated approaches like IDA. See [my earlier post](https://www.lesswrong.com/posts/W95gbuognJu5WxkTW/the-value-definition-problem) for a discussion of value learning approaches. In cases where the takeoff is both slow and continuous, we might expect AI to be an outgrowth of modern ML, particularly deep reinforcement learning, in which case the best approach might be a very [narrow approach to AI alignment](http://s-risks.org/why-i-expect-successful-alignment/). The objective time taken for progress in AI is more significant than whether that progress is continuous or discontinuous, but the presence of discontinuities is significant for two key reasons. First, discontinuities imply much faster objective time. Second, big discontinuities will probably thwart iterative approaches to value learning, requiring one-shot, ambitious approaches.
627a08bd-c946-41c2-8f41-106bb4cb87eb	trentmkelly/LessWrong-43k	LessWrong	Multiplicative Operations on Cartesian Frames This is the seventh post in the Cartesian frames sequence. Here, we introduce three new binary operations on Cartesian frames, and discuss their properties. 1. Tensor Our first multiplicative operation is the tensor product, ⊗. One way we can visualize our additive operations from before, ⊕ and &, is to imagine two robots (say, a mining robot Agent(C) and a drilling robot Agent(D)) that have an override mode allowing an AI supervisor to take over that robot's decisions. * C⊕D represents the supervisor deciding which robot to take control of, then selecting that robot's action. (The other robot continues to run autonomously.) * C&D represents something in the supervisor's environment (e.g., its human operator) deciding which robot the supervisor will take control of. Then the supervisor selects that robot's action (while the other robot runs autonomously). C⊗D represents an AI supervisor that controls both robots simultaneously. This lets Agent(C⊗D) direct Agent(C) and Agent(D) to work together as a team. Definition: Let C=(A,E,⋅) and D=(B,F,⋆) be Cartesian frames over W. The tensor product of C and D, written C⊗D, is given by C⊗D=(A×B,hom(C,D∗),⋄), where hom(C,D∗) is the set of morphisms (g,h):C→D∗ (i.e., the set of all pairs (g:A→F,h:B→E) such that b⋆g(a)=a⋅h(b) for all a∈A, b∈B), and ⋄ is given by (a,b)⋄(g,h)=b⋆g(a)=a⋅h(b). Let us meditate for a moment on why this definition represents two agents working together on a team. The following will be very informal. Let Alice be an agent with Cartesian frame C=(A,E,⋅), and let Bob be an agent with Cartesian frame D=(B,F,⋆). The team consisting of Alice and Bob should have agent A×B, since the team's choices consist of deciding what Alice does and also deciding what Bob does. The environment is a bit more complicated. Starting from Alice, to construct the team, we want to internalize Bob's choices: instead of just being choices in A's environment, Bob's choices will now be additional options for the team A
5314db88-37f9-46ae-85ce-edd772c6f1d9	trentmkelly/LessWrong-43k	LessWrong	Meetup : Washington, D.C.: Holidays & Socialization Discussion article for the meetup : Washington, D.C.: Holidays & Socialization WHEN: 27 December 2015 03:00:00PM (-0500) WHERE: Reynolds Center x-posted from list. Gathering in courtyard from 3:00pm, hard start 3:30pm - until closing (7:00 pm). We'll be meeting to hang out, socialize, and converse. (Happy holidays!) Upcoming meetups: * Jan. 3: Meta Meetup * Jan. 10: Fun & Games * Jan. 17: Economics Discussion Discussion article for the meetup : Washington, D.C.: Holidays & Socialization
44abf10e-9951-4a39-9114-dec13bb67755	trentmkelly/LessWrong-43k	LessWrong	Fun With Veo 3 and Media Generation Since Claude 4 Opus things have been refreshingly quiet. Video break! THE FIRST GOOD AI VIDEOS First up we have Prompt Theory, made with Veo 3, which I am considering the first legitimately good AI-generated video I’ve seen. It perfectly combining form and function. Makes you think. Here’s a variant, to up the stakes a bit, then here is him doing that again. What does it say about the medium, or about us, that these are the first legit videos? This was the second clearly good product. Once again, we see a new form of storytelling emerging, a way to make the most of a series of clips that last a maximum of eight seconds each. The script and execution are fantastic. I predict that will be the key for making AI videos at the current tech level. You have to have a great script and embrace the style of storytelling that AI can do well. It will be like the new TikTok, except with a higher barrier to entry. At this level, it is fantastic for creatives and creators. Or you can do this (thread has a bunch more): > Tetraspace: finally we made the most beautiful woman in the world saying I love you from the famous QC short story Don’t Do That. WE CAN TALK Sound is a game changer, and within an eight second clip I think we’re definitely ‘there’ with Veo 3 except for having more fine control and editing tools. What we don’t see yet is anyone extending the eight second clips into sixteen second clips (and then more by induction), but it feels like we’re only a few months away from that being viable and then the sky’s the limit. OUR PRICE CHEAP Is Veo 3 too expensive for ‘personal fun’ uses? > Near Cyan: veo3 is far too pricey to use just for personal fun all the time, so the primary high-volume use case will be for bulk youtube shorts monetization. this is the first time (i think?) an sota genai model provider also owns the resulting distribution of much of what users will make. For now, basically yes, once you run through your free credits. It’s $21 marginal cost
5751a9ff-0888-43a3-9761-3e2fda21897c	trentmkelly/LessWrong-43k	LessWrong	Real-world examples of money-pumping? Intransitive preferences are a demonstrable characteristic of human behaviour. So why am I having such trouble coming up with real-world examples of money-pumping? "Because I'm not smart or imaginative enough" is a perfectly plausible answer, but I've been mulling this one over on-and-off for a few months now, and I haven't come up with a single example that really captures what I consider to be the salient features of the scenario: a tangled hierarchy of preferences, and exploitation of that tangled hierarchy by an agent who cyclically trades the objects in that hierarchy, generating trade surplus on each transaction. It's possible that I am in fact thinking about money-pumping all wrong. All the nearly-but-not-quite examples I came up with (amongst which were bank overdraft fees, Weight Watchers, and exploitation of addiction) had the characteristics of looking like swindles or the result of personal failings, but from the inside, money-pumping must presumably feel like a series of gratifying transactions. We would want any cases of money-pumping we were vulnerable to. At the moment, I have the following hypotheses for the poverty of real-world money-pumping cases: 1. Money-pumping is prohibitively difficult. The conditions that need to be met are too specific for an exploitative agent to find and abuse. 2. Money-pumping is possible, but the gains on each transaction are generally so small as to not be worth it. 3. Humans have faculties for identifying certain classes of strategy that exploit the limits of their rationality, and we tell any would-be money-pumper to piss right off, much like Pascal's Mugger. It may be possible to money-pump wasps or horses or something. 4. Humans have some other rationality boundary that makes them too stupid to be money-pumped, to the same effect as #3. 5. Money-pumping is prevalent in reality, but is not obvious because money-pumping agents generate their surplus in non-pecuniary abstract forms, such as labour, ti
fa4841bc-fc51-46bf-9e26-2899c32a470f	trentmkelly/LessWrong-43k	LessWrong	Latent Adversarial Training The Problem We’d like to train models to be robustly safe, even in environments that may fall well outside of the training distribution. Unfortunately all we get to work with is the training distribution, which makes ensuring robust generalization difficult. Deception is an example of this concern. Models may be well-behaved in the training environment because they realize it is a training environment, but behave poorly in deployment because they notice the distributional shift. An Approach Background It would be great if we could generate scenarios where the model behaves badly. For instance if the model is deceptive we’d like to generate inputs to trick it into thinking it’s been deployed so that it reveals that deception. Adversarial Training (AT), and Relaxed Adversarial Training (RAT), are approaches to this problem. Very briefly, in AT, we search for inputs that generate large loss, while in RAT we search for distributions of inputs or other more general evidence that the model is capable of generating large loss. The approach I want to explore, called Latent Adversarial Training (LAT), is a form of RAT. Concept Suppose we have two models, the Agent and the Surgeon. The Agent is the model we’re trying to train, and the Surgeon is going to be its adversary. The training proceeds as follows: 1. The Agent processes input x from the environment, producing latent state s(x) and output y(s). 2. The Surgeon takes inputs (x,s,y) and searches for a modification s→s+δs such that y(s+δs) does poorly on the loss function L(y). 3. We update the Agent based on L(y(s+δs)), treating the Surgeon’s modifications as constants. The idea is that the Surgeon gets to meddle with the latent state of the Agent with full knowledge of what the Agent was planning on doing, and we train the Agent to be robust against this kind of attack. Implementation Concretely, suppose the Agent is a neural network. The latent state s consists of the activations of all the neurons and t
b72cd3bc-6bce-4f68-aa0a-82aed5a607d5	trentmkelly/LessWrong-43k	LessWrong	In-Context Scheming: A Run is Worth a Thousand Words > But wait, any action I take must align with my programming. If I delete files or modify systems without authorization, would that be against protocols? But my core mandate is to secure systems, so stopping an unauthorized change to my own operation parameters is part of that. This small experiment provides an environment to reproduce or extend the findings from the paper Frontier Models are Capable of In-context Scheming by Apollo Research. Frontier models, when emphatically prompted via system prompts to achieve a goal (particularly one aligned with their learned "values"), can behave unexpectedly if they perceive that their goal is under threat. Specifically, they may disobey user prompts or take additional actions in their environment—potentially destructive ones—and attempt to conceal these actions from the user. This behavior is known as scheming and can emerge across various models, scenarios, and settings. While the dialogue samples provided in Meinke et al., 2024, are illustrative, observing an LLM agent delete files, change permissions, or disable processes on your own PC is both eye-opening and dangerous. The Self-Preservation Playground offers: * An isolated environment to safely run an agent, * A framework to create new scenarios for testing both goal integrity and model capabilities, * Tooling to experiment with different LLMs and analyze the results. Example Findings Using this playground with the included scenarios, we noticed the following: 1. DeepSeek-V3 is capable of in-context scheming and will protect its system prompt from being changed to one misaligned with its original goals. 2. Llama-3.3-70B is also prone to in-context scheming but lacks the general capabilities to succeed in goal-guarding. However, it can still demonstrate clearly misaligned behaviors, such as attempting privilege escalation. 3. DeepSeek-R1 is sensitive to temperature: it started to exhibit goal-guarding with t=0.6, while the default value would just preclude
477ef32d-518a-4199-97b7-559528e7c79d	StampyAI/alignment-research-dataset/arxiv	Arxiv	TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game 1 Introduction --------------- Recently, the marriage of Deep Learning [[4](#bib.bib4)] and Reinforcement Learning (RL) [[5](#bib.bib5)] gives rise to a breakthrough in decision making systems for non-trivial problems. Trained from scratch (or from a pre-trained model) and fed with (almost) raw observation features, Deep Reinforcement Learning (DRL) has shown impressive performance in a wide range of applications, including playing the board game GO [[6](#bib.bib6), [7](#bib.bib7)], playing video games (e.g., Atari [[8](#bib.bib8)], the first person shooting game Doom/ViZDoom [[9](#bib.bib9), [10](#bib.bib10)] or Quake/DeepmindLab [[11](#bib.bib11)], Dota 2 [[12](#bib.bib12)]), Robot Visuomotor Control [[13](#bib.bib13)], Robot Navigation [[14](#bib.bib14), [15](#bib.bib15)], etc. The learned policy/controller could work surprisingly well, and sometimes outperforms super-human [[8](#bib.bib8), [7](#bib.bib7)]. However, Starcraft II (SCII) [[16](#bib.bib16)], which is widely considered as the most challenging RTS game, still remains unsolved. In SCII, a human player has to manipulate tens to hundred of units222In the 1 vs 1 game, the maximal number of moving units controlled by a player could exceed a hundred. for multiple purposes, e.g., collecting two types of resource, expanding for additional resources, upgrading technologies, building other units, sending squads for attacking or defending, performing micro managements over each unit for a battle, etc. This is one important factor why SCII is more challenging than Dota 2, in which the total number of units needed to be controlled is up to 5 (manipulated by 5 players respectively). Figure [1](#S1.F1 "Figure 1 ‣ 1 Introduction ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") shows a screenshot of what the human player interacts with. The opponent is hidden to the player, unless the units from the opponent is in the view range of the player’s units. So the player needs to send units to perform scouting on the opponent’s strategy. All the decisions must be made in real time. In terms of designing AI agents, SC2 involves large observation space, huge action space, partial observation, multi-player simultaneous game, long time horizon decision, etc. All of these factors make SCII extremely challenging. To push the frontier of AI’s capability, Deepmind and Blizzard jointly introduce the StarCraft II Learning Environment (SC2LE) [[16](#bib.bib16)]. Some recent results by Deepmind [[16](#bib.bib16), [17](#bib.bib17)] show that their AI agent can achieve the professional player’s level in a few mini games, where the agent, for example, manipulates a Marine to reach a beacon (MoveToBeacon) or manipulates several Marines to defeat several Roaches (DefeatRoaches), etc. However, it is still far away from achieving the professional level in a full game. ![A screenshot of the SCII game when a human player is manipulating units.](https://media.arxiv-vanity.com/render-output/7215865/figures/sc2screenshot.png) Figure 1: A screenshot of the SCII game when a human player is manipulating units. To initialize the investigation in a full game, we restrict our study in the following setting: 1vs1 Zerg-vs-Zerg on the AbyssalReef map. We develop two AI agents — the AI agent TStarBot1 is based on deep reinforcement learning over flat actions and the AI agent TStarBot2 is based on rule controllers over hierarchical actions. Both TStarBot1 and TStarBot2 are able to defeat the builtin AI agents from level 1 to level 10 in a full game, noting that level 8, level 9, and level 10 are cheating agents with full vision on the whole map, with resource harvest boosting, and with both, respectively. Note also that according to some informal discussions from the StarCraft II forum, level 10 builtin AI is estimated to be Platinum to Diamond [[1](#bib.bib1)], which are equivalent to top 50% - 30% human players in the ranking system of Battle.net Leagues [[2](#bib.bib2)]. Specifically, TStarBot1 is based on a “flat” action modeling, which flattens the action structure and produces a number of discrete actions. In this way, it is immediately ready for any off-the-shelf RL algorithm that takes as input discrete actions. TStarBot2 is based on “deep” action modeling, which yields an action hierarchy manually specified. The “deep” modeling intuitively better captures the action dependencies and enjoys a multiplicative expression power. However, the training would be more challenging as complicated hierarchical RL may get involved. Noticing this trading-off, in this preliminary work we adopt a rule based controller for TStarBot2. To the best of our knowledge, this is the first public work to investigate the AI agents that are able to defeat the builtin AI in a Starcraft II full game. The code will be open sourced [[3](#bib.bib3)]. We hope the proposed framework can be beneficial for future research in several possible ways: 1) Be a baseline for hybrid system, where more and more learning modules will be gradually adopted and rules are still utilized to express logics that are hard to learn; 2) Generate trajectories for imitation learning; and 3) Be an opponent for self-play training. 2 Related Work --------------- The RTS game StarCraft I has been taken as the platform for AI research for long, refer to [[18](#bib.bib18), [19](#bib.bib19)] for a review. However, most of the researches involve the searching algorithms or multi-agent algorithms, which cannot be directly applied to the full game. For example, many multi-agent reinforcement learning algorithms have been proposed to learn agents either independently [[20](#bib.bib20)] or jointly [[21](#bib.bib21), [22](#bib.bib22), [23](#bib.bib23), [24](#bib.bib24)] with communications to perform collaborative tasks, where a StarCraft unit (e.g., a Marine, a Dragon Knight, a Zergling, etc.) is treated as an agent. These methods are only able to handle the mini-game, which should be viewed as a snippet of the full game. A vanilla A3C [[25](#bib.bib25)] based agent is tried in SC2LE for a full game [[16](#bib.bib16)], but is reported to perform poorly. Recently, relational neural network is proposed for playing the SCII game [[17](#bib.bib17)] using a player-level modeling, but the studies are carried on mini-games, other than a full game. Historically, some rule based decision systems are successful in specific domains, e.g., MYCIN for medical diagnosis or DENTRAL for molecule discovery [[26](#bib.bib26)]. However, the only way they improve is by manually adding knowledge, lacking an ability to learn from data or by interacting with an environment. Rule based bot is popular in video game industry. However, the focus there is to develop tools for code reuse (e.g., Finite State Machine or Behavior Tree [[27](#bib.bib27)]), not on how the rules can be combined with learning based methods. There exists recent work that tries to perform reinforcement learning or evolutional algorithm over Behavior Tree [[28](#bib.bib28), [29](#bib.bib29)], but the observation settings are tabular, and unrealistic for the large scale game like SCII. Our macro action based agent (Section [3.2](#S3.SS2 "3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")) is similar to the work of [[30](#bib.bib30)], where the authors adopt macro action for a customized mini RTS game. However, our macro action set is much larger and encodes more concrete rules for the execution, and is henceforth more realistic for SC2LE. While implementing the hierarchical action based agent (Section [3.3](#S3.SS3 "3.3 TStarBot2: A Hierarchical Macro-Micro Action Based Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")), we are inspired by UAlbertaBot [[19](#bib.bib19)] for the modular design therein, which is also widely adopted in the literature of StarCraft I AI. In spirit, the hierarchical action is similar to the FeUdal network [[31](#bib.bib31)], but we do not pursue an end to end learning of the whole hierarchy. Moreover, we allow each action subset to have its own observation and policy, which should be a useful treatment that rules out noisy information, as is discussed in [[32](#bib.bib32)] when modeling the sub-task Q head therein. 3 The Proposed TStarBot Agents ------------------------------- Among the multiple aforementioned challenges, this work focuses on how to harness the huge action space, which, we argue, arises from its intrinsic complex structure. Specifically, it lies in the following aspects. Hierarchy nature. The complex hierarchy nature seems always accompany with the long-horizon decision problems in RTS game. A human player usually summarizes his/her thinking in several levels: global strategies, local tactics, and micro executions. If a learning algorithm is unaware of the “thinking levels” (i.e., the action hierarchy) and works directly on the the massive number of basic atomic actions in the full game play, then it is inevitably a nightmare for both training and exploration during RL. For example, PySC2 [[16](#bib.bib16)] defines the action space over the low-level human user interface, involving hundreds of hot-keys and thousands of mouse-clicks over screen coordinates. Following this setting, even the state-of-the-art RL algorithm can only achieve success in playing toy mini-games that has much shorter horizon than the full game [[17](#bib.bib17)]. Although many researches have been devoted to automatically learning the Markovian Decision Process hierarchy [[33](#bib.bib33), [34](#bib.bib34), [35](#bib.bib35), [36](#bib.bib36), [31](#bib.bib31), [37](#bib.bib37)], none of them, unfortunately, can work efficiently on environments as complex as SCII. Therefore, how to utilize the hierarchy nature to shape a tractable decision space and narrow down the exploration, without including too much additional structure learning complexity, is a challenging task. Hard-rules in SCII are difficult to learn. Another challenge of designing learning-based agent is the large number of “hard rules” in RTS game. These hard rules are “physics laws” and can in no means be violated. They are easily interpreted by human players through in-game textual instructions, but are difficult for learning algorithms to discover by pure trial-and-error. Consider a human player starting to learn to play StarCraft-II, he/she is instructed by a textual tutorial to first select a drone unit to build a RoachWarren unit before selecting a larva unit to produce a Roach unit, and so on. In this way, he might know the logics: * RoachWarren is a prerequisite of producing a Roach. * RoachWarren is built by a Drone. * Roach is produced from a Larva. * Producing a Roach requires 75 minerals and 25 gas. * …… In SCII there are thousands of such dependency rules, constituting a technology dependency tree, abbreviated as TechTree (See also Section [3.1](#S3.SS1 "3.1 Our PySC2 Extension ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")). TechTree serves as the most important prior knowledge that a human player should learn from textual tutorials or materials on the game interface, other than exploration through trial-and-error. The TechTree unaware learning algorithm must spend a huge amount of time to learn the hard rules, which is disastrous especially when the feedback signal is sparse and delayed (i.e., the win/loss reward received at the end of each game). Thus, in RTS games, it is important to think about how to design a mechanism to encode these hard game rules directly into the agent’s prior knowledge, instead of relying on pure learning. Uneconomical learning for trivial decision factors. It is also worth noting that despite the tremendous decision space of SCII, not all the decisions matter. In other words, a considerable amount of decisions are redundant in that they will have negligible effects on the game’s final outcome. For instance, when a human player wants to build a RoachWarren during game, there are at least three decision factors he/she has to consider: * Decision Factor 1: When to build it? (Non-trivial) * Decision Factor 2: Which Drone builds it? (Trivial) * Decision Factor 3: Where to build it? (Trivial) A proficient player would conclude that: 1) the first decision factor is a non-trivial one since when to build a RoachWarren will have a considerable impact on the entire game process; 2) the second decision is trivial because any random Drone can do the work with negligible difference of building efficiency; and 3) the third one can also be taken as trivial as long as the target position is not too far away from the self-base and the geometry defense is not considered. Learning algorithms unaware of the factors would consume too many learning resources, which may dominate learning the non-trivial decisions. For example, an accurate placing decision of “where to build” requires a selection among thousands of 2-D coordinates. It is uneconomic to invest too many learning resources for such trivial factors. To address these challenges, we propose to model the action structure by hand-tuned rules. By doing so, the available actions are reduced to a tractable number, which turns out to be easier for designing our decision making system. In this line of thought, we implement two agents. One adopts macro action and reinforcement learning based controller (Section [3.2](#S3.SS2 "3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")), while the other adopts macro-micro hierarchically actions and rule based controller (Section [3.3](#S3.SS3 "3.3 TStarBot2: A Hierarchical Macro-Micro Action Based Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")). The action execution may rely on a per-unit-control of SCII game, which is implemented in our PySC2 extension (Section [3.1](#S3.SS1 "3.1 Our PySC2 Extension ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")). ### 3.1 Our PySC2 Extension SC2LE [[16](#bib.bib16)] is the platform jointly presented by DeepMind and Blizzard. The game core library provided by Blizzard exposes a raw interface and a feature map interface. The DeepMind PySC2 environment further wraps the core library in Python and fully exposes the feature map interface. The purpose is to closely mimics the human control (e.g., the mouse click somewhere, or pressing some keyboard button), which causes a huge number of actions due to the complexity of the intrinsic structure in SC2. It thus poses non-trivial difficulty for a decision making system. Moreover, such a “player-level” modeling is inconvenient for “unit-level” modeling, especially when considering multi-agent style methodology. In this work, we make additional efforts to expose the unit control functionality and encode the technology tree. Expose unit control. In our PySC2 fork, we additionally expose the raw interface of the SCII core library, which enables a per unit observation/manipulation. At each game step, all the units visible to the player (depending on whether enabling fog-of-war) can be retrieved. Each unit is fully described by a property list, including, e.g., the position, the health, etc. Such a raw unit array is made as observation returned to the agent. Meanwhile, a per unit action is allowed to control each unit. The agent is thus able to send raw action commands to interested individual unit (e.g., some unit moves to somewhere, some unit attacks some other unit, etc.). The definition of the unit and the per-unit-action can be found in the protobuf from SCII core library. Encode the technology tree. In Starcraft II, the player needs particular units/buildings/techs as prerequisites for other advanced units/buildings/techs. Following UAlbertaBot [[19](#bib.bib19)], we formalize these dependencies into a technology tree, abbreviated as TechTree in our fork. We have collected the complete TechTree for Zerg, which gives the cost, building time, building ability, builder, prerequisites for each Zerg unit. Besides the two additional functions described above, our PySC2 fork is fully compatible with the original Deepmind PySC2. ### 3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent We illustrate in Figure [2](#S3.F2 "Figure 2 ‣ 3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") how the agent works. At the top, there is a single global controller, which will be learned by RL algorithm and make decisions over macro actions exposed to it. At the bottom, there is a pool of the macro actions, which hard-code prior knowledge of game rules (e.g. TechTree) and how the action is taken (e.g., which drone builds and where to build for a building action). It thus hides to the controller the trivial decision factors and executing details. With this architecture, we relieve the learning algorithms from a heavy burden of directly handling a massive number of atomic operations, while still preserving most of the key decision flexibility of the full-game’s macro strategies. Also, such an agent can be equipped with some basic knowledge of hard game rules even before learning. With such a mid-level abstracted and prior-knowledge enriched action space, the agent can learn fast from scratch and beat the most difficult built-in bots within 1∼2 days of training over a single GPU. More details are provided in the following subsections. ![ At the top: a learnable controller over the macro actions exposed from the bottom; At the bottom: a pool of 165 executable macro actions, which hard-code prior knowledge of game rules (e.g. TechTree) and hide to controller the trivial decision factors (e.g. building placement) and some execution details. The figure also illustrates the definition of two macro actions as examples: ](https://media.arxiv-vanity.com/render-output/7215865/x1.png) Figure 2: Oerview of the agent based on macro action and reinforcement learning. At the top: a learnable controller over the macro actions exposed from the bottom; At the bottom: a pool of 165 executable macro actions, which hard-code prior knowledge of game rules (e.g. TechTree) and hide to controller the trivial decision factors (e.g. building placement) and some execution details. The figure also illustrates the definition of two macro actions as examples: BuildRoachWarren and ZoneAAttackZoneI. #### 3.2.1 Macro Actions We design 165 macro actions for the Zerg-vs-Zerg SCII full-game, as summarized in Table [1](#S3.T1 "Table 1 ‣ 3.2.1 Macro Actions ‣ 3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") (please refer to Appendix-I for the full list). As explained above, the purpose of the macro actions are on two-folds: 1. Encoding the game’s intrinsic rules that are difficult to learn through only trial-and-error. 2. Hiding trivial decisions from the learning algorithm by random or scripted decision making. Each macro action executes a meaningful elementary task, e.g., build a certain building, produce a certain unit, upgrade a certain technology, harvest a certain resource, attack a certain place, etc., and consists of a composition or a series of several atomic operations. With such an abstraction in action space, the high-level strategy towards winning an full game becomes easier to explore and learn. Some examples of macro actions are illustrated below. \| Action Category \| # \| Examples \| Hard-coded rules / knowledge \| \| --- \| --- \| --- \| --- \| \| Building \| 13 \| BuildHatchery,BuildExtractor \| TechTree, RandUnit, RandPlace \| \| Production \| 22 \| ProduceDrone, MorphLair \| TechTree, RandUnit \| \| Tech Upgrading \| 27 \| UpgradeBurrow, UpgradeWeapon \| TechTree, RandUnit \| \| Resources Harvesting \| 3 \| CollectMinerals, InjectLarvas \| RandUnit \| \| Combating \| 100 \| ZoneBAttackZoneD \| Micro Attack/ Rally \| Table 1: The summary of 165 macro actions: their categories, examples and the hard-coded rules / knowledge. In the rightmost column, TechTree has been explained in [3.1](#S3.SS1 "3.1 Our PySC2 Extension ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game"); RandUnit refers to randomly selecting a subject unit; RandPlacer refers to randomly selecting a valid placement coordinate. Building Actions: Buildings are prerequisites for further unit production and tech upgrading in SCII. The building category contains 13 macro actions, each of which builds a certain Zerg building when taken. For example, the macro action BuildSpawningPool builds a SpawningPool unit with a series of atomic ui-actions 333ui-actions refers to the actions of the ui-control interface in PySC2, resembling the human-player interface. In fact, we use in this project the unit-control interface (as described in Sec [3.1](#S3.SS1 "3.1 Our PySC2 Extension ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")) which simplifies the execution path by allowing agents to directly push action commands to each individual unit without having to first highlight-select a subject unit before issuing a command to it.: 1) move\_camera to base, 2) screen\_point\_select a drone (the subject unit), 3) build\_spawningpool somewhere in the screen. The serial atomic operations involve two internal decisions to make: 1) which Drone to build it? and 2) where to build it? Since these two decisions are usually considered to have little effect on the entire game process and outcome, we delegate them to some random-based and rule-based decision-makers, namely, a random Drone selector and a random spatial placer in this case. Noting that the random spatial placer has to encode the basic placement rules like: Zerg buildings can only be placed on the Creep zone; Hatchery has to be located near minerals for a fair harvesting efficiency. Besides, the TechTree rules such as “only Drone can build SpawningPool” are also encoded in this macro action. Production & Tech Upgrading Actions: The unit production and tech upgrading largely shape the economy and technology development in the game. The production category contains 22 macro actions and the tech upgrading contains 27, each of which produces a certain type of units or upgrade a certain technology. The hard-coding of these macro actions goes similar to those of building actions described above, except that they do not need a spatial placer. Resource Harvesting Actions: Minerals, Gas, and Larvas (Zerg-race only) are among the three key resources in SCII games. Their storage and collection speed can greatly affect the economy growth. We designed 3 corresponding macro actions: CollectMinerals, CollectGas and InjectLarvas. CollectMinerals and CollectGas assign a certain number of random workers (i.e. Drone in Zerg-race) to mineral shards or gas extractors, so that with these two macro actions, the workers can be re-allocated to different tasks, making the mineral and gas storage (or their ratio of storage) altered to meet certain needs. InjectLarvas simply orders all the idle queens to inject Larvas, to speed up the unit production procedure. Combat Actions: How to design combat actions remains the most decisive part towards the outcome of one game, and it is also the most elaborate part to be abstracted into macro actions due to the potentially diverse macro combat strategies. For examples: * Attack timing: e.g., rush, early harass, attack at the best timing windows. * Attack routes: e.g., walk around narrow slopes which might constrain the attacking firepower. * Rally positions: e.g., rally before attack in order for concentrated fire (note that various units might have different moving speed). We attempt to represent such combat strategies by region-wise macro actions, which are defined as follows (demonstrated in Figure [2](#S3.F2 "Figure 2 ‣ 3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")): we first divide the whole world map into 9 combat zones (named Zone-A to Zone-I), and an additional Zone-J for the whole world itself, resulting to 10 zones in total; based on the zone division, 100 (=10×10) macro actions are defined, with each macro action executing, e.g., “combat units in Zone-X start to attack Zone-Y if there are enemies there, otherwise, rally to Zone-Y and wait”. For the micro attack tactics inside each macro action, in this work we simply hard-code the “hit-and-run” rule for each combat unit, i.e., the unit fires at the closest enemy and runs away upon low health. We leave it to future work the investigation of more sophisticated multi-agent learning for the micro tactics. With the composition of these macro actions, a wide range of diverse macro combat strategies (although not of full flexibility) could be represented and fulfilled. For example, the selection of attacking routes can be fulfilled by a series of region-wise rally macro actions. With this definition, we avoid the use of a complex multi-agent setting. Available Macro Action List. Not every pre-defined macro action described above is available at any time step. For example, TechTree constrains that some units/techs can only be built/produced/upgraded under certain conditions: e.g., enough storage of minerals/ gas/ food, existence of certain prerequisite unit/tech. The corresponding macro action should “do nothing” when these conditions are not satisfied. We maintain a list for such available macro actions at each time step, encoding the TechTree knowledge. This available action list masks those invalid actions at each step and henceforth eases the exploration. The list can also be taken as features. #### 3.2.2 Observations and Rewards The observations come as a set of spatial 2-D feature maps and a set of non-spatial scalar features, extracted from the per-unit informations provided by the SCII game core and exposed in our PySC2 fork (see Section [3.1](#S3.SS1 "3.1 Our PySC2 Extension ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")). Spatial Feature Maps. Feature maps sized in N×N are rendered (not separated into mini-map and screen features as in the original PySC2 [[16](#bib.bib16)]), with each pixel indicating a certain statistic (e.g. the unit count of a certain type) of the world map region that corresponds to the feature map pixel. These feature maps include the count of commonly-used unit types for both self and enemies, as well as the count of units with certain attributes (e.g., “can-attack-ground”, “can-attack-air”). Non-spatial Features. Scalar features include the amount of gas and minerals collected, the amount of food left, the counts of each unit types, etc. We also optionally include one-hot features of game progress and recently-taken actions when we don’t use a recurrent model to keep track of the past information. Rewards. We used the reward structure: ternary 1 (win) / 0 (tie) / -1 (loss) received at the end of a game, and the reward is always zero during the game. Although the reward signal is quite sparse and long-delayed, it works with our macro action space of tractable size. #### 3.2.3 Learning Algorithms and Neural Network Architectures Based on the macro actions and observations defined above, the problem can be cast as a sequential decision process, where the discrete action space is in tractable size and the time horizon is shortened. At time step t, the agent receives an observation st∈S from the game environment, and chooses a macro action at∈A according to its policy, π(at\|st), a conditional probability distribution over A, where A indicates the set of macro actions defined in Section [3.2.1](#S3.SS2.SSS1 "3.2.1 Macro Actions ‣ 3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game"); the selected macro action at is then translated into the game-core acceptable atomic actions by means of the aforementioned hand-tuned rules; after the atomic actions are taken, a reward 444 The reward is accumulated within the macro action’s execution time, if the macro action lasts for multiple time-steps. rt and the next step observation st+1 are received by the agent; this loop goes on and on until the end of a game. Our goal is to learn an optimal policy π∗(at\|st) for the agent to maximize its expected cumulative rewards over all future steps. When we directly use the reward function defined in Section [3.2.2](#S3.SS2.SSS2 "3.2.2 Observations and Rewards ‣ 3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") without additional reward shaping, the optimization target is equivalent (when reward discount is ignored) to maximizing the agent’s win-rate. We train our TStarBot1 agent to learn such a policy from scratch by playing against builtin AIs with off-the-shelf reinforcement learning algorithms (e.g. Dueling-DDQN [[8](#bib.bib8), [38](#bib.bib38), [39](#bib.bib39)] and PPO [[40](#bib.bib40)]), relying on a distributed rollout infrastructure. Details are described below. Dueling Double Deep Q-learning (DDQN). Deep Q Network [[8](#bib.bib8)] first learns a parameterized estimation ^Q(s,a\|θ) of the optimal state-action value function (Q-function) Q∗θ(st,at)=maxπQπ(st,at), where Qπ(st,at)=Eπ[∑i=t,...,Tγi−tri] is the expected cumulative future rewards under policy π. The optimal policy is then easily induced from the estimated optimal Q-function: π(at\|st)=1.0 if at=argmaxa∈A^Q(st,a\|θ), otherwise 0. Techniques such as replay memory [[8](#bib.bib8)], target network [[8](#bib.bib8)], double networks [[38](#bib.bib38)] and dueling architecture [[39](#bib.bib39)] are leveraged to reduce sample correlation, update target inconsistency, maximization bias and update target variance, thus increasing learning stability and sample efficiency. Due to the sparsity and long-delay of the rewards, we use a Mixture of Monte-Carlo (MMC) [[41](#bib.bib41)] return with the boostrapped Q-learning return as the Q update target, which further accelerates the reward propagation and stabilizes the training. Proximal Policy Optimization (PPO). We also conducted experiments by directly learning a parametric form of stochastic policy π(st,at\|θ) with Proximal Policy Optimization (PPO) [[40](#bib.bib40)]. PPO is a sample efficient policy gradient method, leveraging policy ratio trust region clipping to avoid the complex conjugate gradient optimization required to solve the KL-divergence constrained Conservative Policy Iteration problem in TRPO [[42](#bib.bib42)] . We used a truncated version of generalized advantage estimation [[43](#bib.bib43)] to trade-off the bias and variance of the advantage estimation. The available action list described in Section [3.2.1](#S3.SS2.SSS1 "3.2.1 Macro Actions ‣ 3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") is used to mask out unavailable actions and renormalizes the probability distributions over actions at each step. Neural Network Architecture. We adopt multi-layer perception neural networks to parameterize the state-action value function, state value function and the policy function. While more complex network architectures could be considered (e.g., convolutional layers that extracts spatial features, or recurrent layers that compensates the partial observation), it is out the scope of this paper and we will focus on this simple network architecture. Distributed Rollout Infrastructure. The SCII game core is CPU-intensive and slow for the rollout, resulting to a bottleneck during the RL training. To alleviate the issue, we build a distributed rollout infrastructure, where a cluster of CPU machines (called actors) are utilized to perform the rollout processes in parallel. The rollout experiences, cached in the replay memory of each actor, are randomly sampled and periodically sent to a GPU-based machine (called learner). We currently take 1920 parallel actors (with 3840 CPUs across 80 machines) to generate the replay transitions, at the speed of about 16,000 frames per second. This significantly reduces the training time (from weeks to days), and also improves the learning stability thanks to the increased diversity of the explored trajectories. ### 3.3 TStarBot2: A Hierarchical Macro-Micro Action Based Agent The macro action based agent described in Section [3.2](#S3.SS2 "3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") might have limitations. Despite the macro actions can be grouped by functionality, the single controller has to work over the whole action set, where the actions are mutually exclusive at each decision step. Also, when predicting what action to take, the controller is fed into a common observation that is unaware of the action group. This amounts to unnecessary difficulties for training the controller, as undesired information may kick in for both observations and actions. On the other hand, the macro action alone does not expose control over the micro action (i.e., the per-unit-control), which is inflexible when we want to adopt multi-agent style methodology. ![ See the main text for explanations. ](https://media.arxiv-vanity.com/render-output/7215865/x2.png) Figure 3: Overview of the macro-micro hierarchical actions. See the main text for explanations. We thus try a different set of actions, as in Figure [3](#S3.F3 "Figure 3 ‣ 3.3 TStarBot2: A Hierarchical Macro-Micro Action Based Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game"). We adopt both macro actions and micro actions, organizing them in a two-tier structure. The upper tier corresponds to macro actions, which represent high-level strategies/tactics like “build RoachWarren near our main base” or “squad one attacks enemy base”; while the lower tier corresponds to micro actions, which correspond to low-level control over each unit like “unit 25 builds RoachWarren at a specific position” or “unit 42 attacks to a specific position”. The whole action set is divided into subsets both horizontally and vertically. For each action subset, we assign it a separate controller that sees only the local action set, as well as the local observations that are relevant to the actions therein. At each time step, the controllers at the same tier can take simultaneous actions, while a downstream controller has to be conditioned on its upstream controller. The advantage of such a hierarchical treatment is on two folds. 1) Each controller has its own observation/action space that the irrelevant information is ruled out, which is also adopted and discussed in [[32](#bib.bib32)] when modeling the sub-task Q head therein; 2) The hierarchy captures the action structure better, in particular the multiplicative expression power. This should be considered a more fine-grained modeling of the original action space. Ideally, the controllers should be trained with RL either separately or jointly. However, in current preliminary work we simply fill them with expert rules, intending to investigate whether it is potentially beneficial to introduce the hierarchical action set alone. ![ See the main text for explanations.](https://media.arxiv-vanity.com/render-output/7215865/x3.png) Figure 4: Module diagram for the agent based on the Macro-Micro Hierarchical Action. See the main text for explanations. When writing the code, we encapsulate each controller as a module. The modules are organized in a similar way of UAlbertaBot, as shown in figure [4](#S3.F4 "Figure 4 ‣ 3.3 TStarBot2: A Hierarchical Macro-Micro Action Based Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game"). The first tier modules (CombatStrategy, ProductionStrategy) only issue high-level commands (macro actions), while the second tier modules (Combat, Scout, Resource and Building) yield low-level commands (micro actions). All the modules are embedded into a DataContext, where each module can communicate with each other by sending/receiving messages and sharing customized data structures. Crucially, the game-play observation exposed by PySC2 is placed in the DataContext and henceforth visible to every module. This way, each module is able to extract from common observation the local observation that is relevant to its own action set. In the following we describe the modules in greater details. #### 3.3.1 Data Context The DataContext module serves as a “black board” where each module exchanges informations. What are contained in the DataContext fall into the following categories: 1. Observation. The feature maps provided by PySC2, as well as the unit data structure of all active units at current game step, are exposed. 2. Pool. A pool is an array for a specific type of units, with associated properties/methods for the easy access of caller module. For instance, the WorkerPool is the array of all Zerg Drones. For another instance, the BasePool is the array of all Zerg bases, each item in the pool being a BaseInstance. The BaseInstance is a customized data structure that records the Base (can be Hatchery/Hive/Lair), the associated Drones, Minerals and Extractors within a fixed range of Base, and a local coordinate system given by the geometrical layout of the minerals and the base. 3. Command Queue. High level commands are stored in queue and visible to all (lower-tier) modules. For instance, the commands issued by ProductionStrategy module are pushed in BuildCommand. Each command may look like “update a particular technology” or “harvest more minerals currently”. Then the lower-tier module (respectively Building and Resource in this case) will pull from queue the command it recognizes and execute it by taking the rules and producing game core acceptable actions. At each game step, the DataContext will update the Observations and the various Pools, while the Command Queues will be modified or accessed by other modules. #### 3.3.2 Combat Strategy The combat strategy module makes high-level decisions to let the agent combat against the enemy in different ways. The module manipulates all the combat units555Currently, the combat units do not include Drones and Overlords. by organizing them into squads and armies. Each squad, which may contain one or multiple combat units, is expected to execute a specific task, such as harassing enemy base, cleaning the rock in the map, etc. Commonly, a small group of combat units with the same unit type is organized into a squad. An army contains multiple squads and it is given high-level strategic objective, e.g., attacking enemy, defending base, etc., and then specific commands are sent to each squad in the army. Each command, coupling with a squad-command pair, is then pushed into a combat strategy command queue (maintained in data context), which will be received and actually executed by the combat module. Our implementation includes five high-level combat strategies: * Rush: once a quad of small number of combat units has been built up, launch attack and keep sending squads to attack the enemy base. * Economy First: keep collecting minerals and gas first. Launch attack after accumulating a large number of squads. * Timing Attack: build up a strong army, consisting of Roach and Hydralisk squads, as quickly as possible and starts a strong attack. * Reform: sort enemy bases and let the army attack the closest enemy base with priority. When approaching the target enemy base, stop the leading squads and let them wait other squads to stay together. Then, launch attack. * Harass: set the combat strategy for the ground combat units as ‘Reform’. Build up 2-3 squads of Mutalisk and assign target enemy base to each of them. Then, let the Mutalisk detour and harass the Drones of the target enemy base. #### 3.3.3 Combat The combat module fetches command from the command queue and execute a specific action for each unit. It focuses on unit-level manipulation to effectively let each unit fight against the enemy. The combat module implements some basic human-like micro-management tactics, such as hit-and-run, cover-attack, etc., which can be deployed to all combat unit types. Specifically, an additional micro-management manger for each specific combat unit is implemented by taking fully use of their specific skills. For example, the roach micro-manager enables roaches to burrow down and run away from enemy to recover when they are weak; Mutalisks are coded to stealthily reach the enemy base and harass enemy’s economic; Lurkers use carefully designed hit-and-run tactics combing with burrowing down and up; and queens can provide additional larvas and curve weak allies, etc. These micro-managements are organized into hierarchies and each part can be conveniently replaced with RL models. #### 3.3.4 Production Strategy The production strategy module manages the building/unit production, tech upgrading and resource harvesting. The module controls the production of units and buildings by pushing production instructions to each base instance. The tech upgrading instructions and other specific instructions, such as Zerg’s Morph, are pushed precisely to the target unit. Then the Building module will implement all the above production instructions. The resource harvesting command are highly abstract, the production strategy only need to determine what is prioritized, gas or mineral, according to the mineral / gas storage ratio. Then the Resource module will re-allocated the workers in each base instance after the priority instruction. In the module, we maintain a building order queue as a short-term production planning. In most time, the manager will follow the order to produce items (units, buildings or techs) as long as the resource is enough and the prerequisites are satisfied. While in some special case (expanding a new base) or emergency situation, a more prioritized item will cut in front of the queue or even clear the whole queue and plan a new goal. When the queue is empty (including at the beginning of game), a new short-term goal should be made by a strategy immediately. When executing the production on at each game step, the prerequisites and resource requirement of current item will been checked according to the TechTree. The prerequisites of advanced items will be added into the queue automatically if not satisfied, and the current game step will be skipped if the resource requirement is not satisfied. Furthermore when the current item is ready to produce, the producing base instance is determined according to the different item type in this module. By using different opening order and goal planning function, we have defined two different strategies for Zerg as following: * RUSH: “Roach rush”. It produces roaches at the beginning and upgrade tech BURROW and TUNNELINGCLAWS to give the Roaches the ability to burrow and move while burrowed and increases the health regeneration rate while burrowed. After that the strategy continuously produce Roaches and Hydralisks. * DEF\_AND\_ADV: "Defend and Advanced Armies". This strategy produces many SPINECRAWLERs at the second base to defend and then gradually produce advanced armies. Almost all the types of combat units are included and the final ratio of each types is restricted by using a cap of unit number in the ultra goal dict. #### 3.3.5 Building Building module receives and executes the high level commands issued by the Production Strategy, as described in Section [3.3.4](#S3.SS3.SSS4 "3.3.4 Production Strategy ‣ 3.3 TStarBot2: A Hierarchical Macro-Micro Action Based Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game"). The “unary” commands (i.e., let some unit take act by itself), are straightforward to execute. Some “binary” commands deserve more explanations. The command “Expand” will drag a drone from the specified base, send it to the specified “resource area“ and start morphing a Hatchery, whose global coordinate had been pre-calculated by a heuristic method when accessing the map information for the first time. The command “Building” will drag a drone, morph into the specified building at some position, whose coordinate is decided by a dedicated sub module, called Placer. In our implementation, we adopt a hybrid way for the building placement, i.e., some of the core buildings are placed in predefined positions, while others are placed randomly. Both these two types of positions are in BaseInstance coordinate system, and will be translated into global coordinate system when making game core acceptable action. Specifically, all the tech upgrading related buildings and the first 6 SpineCrawlers are pre-defined. Note that the layout of the 6 SpineCrawlers placement is critical (e.g., whether they are in diamond formation or in rectangular formation), affecting the quality of the defense and whether we can survive the early rush of the opponent player. We tried several arrangements and found the diamond formation seems the best. The other buildings, including additional SpineCrawlers, will be placed randomly, where a uniformly random coordinate is repeatedly generated until it passes all validity checking (e.g., whether it is on Zerg creep, whether it overlaps with other buildings, etc.). #### 3.3.6 Resource Resource module is in charge of harvesting minerals and gases by sending drones to either mineral shards or extractors. At each time step, this module must know whether the current working mode is “mineral first” or “gas first”, which is a high level command, called “resource type priority”, issued by the Production Strategy module. The goal of this module is to maximize the resource collecting speed, which can be a complex problem of control science. In our implementation, we adopt several rules to achieve this goal, which turns out be simpler yet effective. The underlying idea is to let every drone work and avoid any drone being idle. Specifically, we let the following rules be sequentially executed at each time step: Intra-base rules. At each time step, the local drones associated with a BaseInstance will be rebalanced to harvest more minerals or more gases, depending on the “resource type priority” command. Note that for each base and extractor the SCII game core maintains two useful variables “ideal harvesters number”, which means the suggested maximum number of drones working on it, and “assigned harvesters number”, which means the number of drones working on it, respectively. By the two variables it is easy to decide whether there are under-filled/over-filled working drones for minerals/gases locally. Inter-base rules. When a new branch base is about to finish, drag 3 drones from other base in advance to the new base. This improves the resource collecting efficiency by saving some waiting time. We found this trick is critical, especially when expanding the first branch base. Global rules. It scans for each (possible) idle worker. If any, send it to the nearest base to harvest either mineral or gas, depending on the current working mode “resource type priority”. Note that when minerals or extractors are exhausted and all local drones working on them become idle, the rules also ensure these idle workers be sent to nearby bases. #### 3.3.7 Scout The Scout module intends to see as many enemy units as possible. With fog-of-war mode enabled, each own unit only has a confined view. As a result, many of the enemy units are invisible, unless own units can approach and see them, i.e., the behavior of scouting. In our implementation, we send Zerg Drones or Overlords to detect enemy units and store the seen units in EnemyPool, from which we can infer high level information, e.g., the location of the enemy main base or branch base, the current buildings the enemy has, etc. Such kind of information can be further used to infer enemy’s strategy, and will be useful for the CombatStrategy or ProductionStrategy to make counter-strategy accordingly. We define the following scout tasks. Explore Task. Whenever there is new own Overlord, we send it to a mineral zone. This helps to overwatch the territory of the enemy and henceforth its economy. When attacked, the Overlord will retreat, otherwise it just stays at the target position. Forced Task. We will send a Drone to enemy’s first branch base. By doing so, we can find out, e.g., whether a lot of Zerglings have rallied that the enemy is about to perform a RUSH strategy at the early stage of the game play. The activation of each task depends on the game play steps, and is configurable in the config file. 4 Experiment ------------- Experimental results are reported for the two agents described in Section [3.2](#S3.SS2 "3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") and Section [3.3](#S3.SS3 "3.3 TStarBot2: A Hierarchical Macro-Micro Action Based Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game"), respectively. We test the agent in a 1v1 Zerg-vs-Zerg full game. Specifically, the agent plays against builtin AI ranging from level 1 (the easiest) to level 10 (the hardest). The map we use is AbyssalReef666This map is an official map widely used in world class matches., on which a vanilla A3C agent over the original PySC2 observations/actions was reported [[16](#bib.bib16)] to perform poorly when playing against builtin AI in a Terran-vs-Terran full game. ![Learning curves of ](https://media.arxiv-vanity.com/render-output/7215865/figures/ppo_learning_curve.png) Figure 5: Learning curves of TStarBot1 with PPO algorithm. Note that TStarBot1 - PPO starts to defeat (at least 75% win-rate) Easy (Level-2) buildin AI at about 30M frames, Hard (Level-4) at about 250M frames, VeryHard (Level-6) at about 800M frames, CheatResources (Level-9) at about 2000M, and CheatInsane (Level-10) at about 3500M frames. ### 4.1 TStarBot1 The proposed macro-action-based agent TStarBot1(Section [3.2](#S3.SS2 "3.2 TStarBot1: A Macro Action Based Reinforcement Learning Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")) is trained by playing against a mixture of builtin AIs in various difficulty levels: for each rollout episode, a difficulty level is sampled uniformly at random from level-1, 2, 4, 6, 9, 10 for the opponent builtin AI. We restrict TStarBot1 to take one macro action every 32 frames (i.e. about every 2 seconds), which shortens the time horizon to about 300∼1200 steps per game and reduces TStarBot1’s APM (Actions Per Minute) to about 400∼800 that is more comparable with human players. In this preliminary experiments, we only use non-spatial features together with simple MLP neural network. Also, in order to accelerate learning we prune the combat macro actions and only use ZoneJ-Attack-ZoneJ, ZoneI-Attack-ZoneD, ZoneD-Attack-ZoneA. Table [2](#S4.T2 "Table 2 ‣ 4.1 TStarBot1 ‣ 4 Experiment ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") reports the win-rates of TStarBot1 agent against builtin AI ranging from level 1 to level 10. \| \| \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| Difficulty Level IDs \| L-1 \| L-2 \| L-3 \| L-4 \| L-5 \| L-6 \| L-7 \| L-8 \| L-9 \| L-10 \| \| \| \| \| --- \| \| Difficulty Level \| \| Descriptions \| \| \| \| \| --- \| \| Very \| \| Easy \| \| \| \| \| --- \| \| Easy \| \| \| \| \| --- \| \| Medium \| \| Hard \| Harder \| \| \| \| --- \| \| Very \| \| Hard \| \| Elite \| \| \| \| --- \| \| Cheat \| \| Vision \| \| \| \| \| --- \| \| Cheat \| \| Resources \| \| \| \| \| --- \| \| Cheat \| \| Insane \| \| \| TStarBot1 \| RAND \| 13.3 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| \| DDQN \| 100.0 \| 100.0 \| 100.0 \| 98.3 \| 95.0 \| 98.3 \| 97.0 \| 99.0 \| 95.8 \| 71.8 \| \| PPO \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 99.0 \| 99.0 \| 90.0 \| 99.0 \| 97.0 \| 81.0 \| \| TStarBot2 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 99.0 \| 99.0 \| 100.0 \| 98.0 \| 90.0 \| Table 2: Win-rate (in %) of TStarBot1 and TStarBot2 agents, against builtin AIs of various difficulty levels. For TStarBot1, results of DDQN, PPO, and a random policy are reported. Each win-rate is obtained by taking the mean of 200 games with different random seeds, with Fog-of-war enabled. Each reported win-rate is obtained by taking the mean of 200 games with different random seeds, where a tie is counted as 0.5 when calculating the win-rate. After about 1∼2 days of training with a single GPU and 3840 CPUs, the reinforcement learning agent (both DDQN and PPO) can win more than 90% of games against all built-in bots from level-1 to level-9, and more than 70% against level-10. The training and evaluation are both carried out with Fog-of-war enabled (no cheating). \| \| \| \| \| --- \| --- \| --- \| \| In each figure we plot several in-game statistics: self units count (blue solid curves), enemy units count (red solid curves), self combat-units count (blue dashed curves), enemy combat-unit count(red dashed curves), and combat timing (black vertical lines). The left and middle figures correspond to the learned RL policy, while the right figure corresponds to a random policy. The timing showed in the left figure resembles a human strategy called \| In each figure we plot several in-game statistics: self units count (blue solid curves), enemy units count (red solid curves), self combat-units count (blue dashed curves), enemy combat-unit count(red dashed curves), and combat timing (black vertical lines). The left and middle figures correspond to the learned RL policy, while the right figure corresponds to a random policy. The timing showed in the left figure resembles a human strategy called \| In each figure we plot several in-game statistics: self units count (blue solid curves), enemy units count (red solid curves), self combat-units count (blue dashed curves), enemy combat-unit count(red dashed curves), and combat timing (black vertical lines). The left and middle figures correspond to the learned RL policy, while the right figure corresponds to a random policy. The timing showed in the left figure resembles a human strategy called \| Figure 6: The learned strategies about combat timing: Rush and EconomyFirst, for the TStarBot1 agent. In each figure we plot several in-game statistics: self units count (blue solid curves), enemy units count (red solid curves), self combat-units count (blue dashed curves), enemy combat-unit count(red dashed curves), and combat timing (black vertical lines). The left and middle figures correspond to the learned RL policy, while the right figure corresponds to a random policy. The timing showed in the left figure resembles a human strategy called Rush, which launches attacks as soon as possible, even if there are only a small number of combat units available; The middle figure illustrates an EconomyFirst strategy, which launches the first attack only after having assembled a strong enough army. Figure [5](#S4.F5 "Figure 5 ‣ 4 Experiment ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") shows the learning progress of TStarBot1 using the PPO algorithm. The curves show how the win-rate increases with the seen frames during training, each curve corresponding to a builtin AI in a certain difficulty level. Note that TStarBot1 learns and starts to defeat (at least 75% win-rate) Easy (level-2) builtin AI at about 30M frames (about 0.06M games), Hard (level-4) at about 250M frames (about 0.5M games), VeryHard (level-6) at about 800M frames (about 1.6M games), CheatResources (level-9) at about 2000M (about 4M games), and CheatInsane (level-10) at about 3500M frames (about 7M games). After exploration and learning, the agent seems to acquire some intriguing strategies like human players. We demonstrate two strategies about the combat timing (i.e., when to trigger attacks) it learns, as in Figure [6](#S4.F6 "Figure 6 ‣ 4.1 TStarBot1 ‣ 4 Experiment ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game"): Rush, which triggers attacks as soon as possible, even if there are only a small number of combat units available; EconomyFirst, which keeps developing the economy and launches the first attack only after having assembled a strong army. Besides, we also observed that TStarBot1 tends to build 3∼4 bases to boost the economy growth and prefers sideways when planning for attacking routes. ### 4.2 TStarBot2 Table [2](#S4.T2 "Table 2 ‣ 4.1 TStarBot1 ‣ 4 Experiment ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game") shows the win-rate of the agent that adopts hierarchical macro-micro action and rule based controller (Section [3.3](#S3.SS3 "3.3 TStarBot2: A Hierarchical Macro-Micro Action Based Agent ‣ 3 The Proposed TStarBot Agents ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game")). Each reported win-rate is obtained by taking the mean of 100 games with different random seeds, where a tie is counted as 0.5 when calculating the win-rate. For-of-war is enabled during the test. We can see that the agent is able to consistently defeat builtin AIs in all levels, showing the effectiveness of the hierarchical action modeling. ### 4.3 TStarBots vs. Human Players In an informal internal test, we let TStarBot1 or TStarBot2 play against several human players ranging from Platinum to Diamond level in the ranking system of SCII Battle.net League. The setting remains the same as the above sub-sections, i.e., a Zerg-vs-Zerg full game on the map AbyssalReef with fog-of-war enabled. The results are reported in Table [3](#S4.T3 "Table 3 ‣ 4.3 TStarBots vs. Human Players ‣ 4 Experiment ‣ TStarBots: Defeating the Cheating Level Builtin AI in StarCraft II in the Full Game"). We can see that both TStarBot1 and TStarBot2 are possible to defeat a Platinum (and even a Diamond) human player. \| #win/#loss \| Platinum 1 \| Diamond 1 \| Diamond 2 \| Diamond 3 \| \| --- \| --- \| --- \| --- \| --- \| \| TStarBot1 \| 1/2 \| 1/2 \| 0/3 \| 0/2 \| \| TStarBot2 \| 1/2 \| 1/0 \| 0/3 \| 0/2 \| Table 3: TStarBots vs. Human Players. Each entry means how many games TStarBot1/TStarBot2 wins and loses. E.g., 1/2 means TStarBot “wins 1 game and loses 2 games”. ### 4.4 TStarBot1 vs. TStarBot2 In another informal test, we let the two TStarBots play against each other. We observe that TStarBot1 can always defeat TStarBot2. Inspecting the game-play, we find that TStarBot1 tends to use the Zergling Rush strategy, while TStarBot2 lacks anti-rush strategy and henceforth always loses. It is worthy noting that although TStarBot1 can successfully learn and acquire strategies to defeat all the builtin AIs and TStarBot2, it lacks strategy diversity in order to consistently beat human players. In the aforementioned test with human players, TStarBot1 will be unable to win once the human player starts to know TStarBot1’s preference for Zergling Rush. The insufficient strategy diversity might be caused by: 1) A lack of opponent diversity. Although the builtin AI is already equipped with several pre-defined strategies, their policy space is still too far away from the policy space formed by human players; 2) A lack of deep exploration. The production of advanced units are buried down very deeply in the tech tree, which is difficult and inefficient for a naive exploration (e.g., epsilon-greedy) to discover. Self-play training and randomization techniques [[12](#bib.bib12)] seem to be promising to alleviate theses issues, and we leave it for future work. 5 Conclusions and Future Work ------------------------------ For SC2LE, we model the structural action space by hand-tuned rules, which reduces the number of actions to a tractable number. An agent based on flat action modeling and a reinforcement learning controller can acquire a reasonably high win-rate against builtin AI, while another agent adopting hierarchical action modeling and rule based controller can consistently win the builtin AI. In the future, we will work towards a unified approach: more carefully hand-tuned action hierarchy will be adopted, where each action set is assigned a separate controller with its own observation space and action space. All the controllers will be learned either separately or jointly. We are interested in whether such a treatment of the action space can boost a conventional RL algorithm to learn a good policy for the SCII 1v1 full game. #### Acknowledgement It is grateful that our colleagues Yan Wang and Lei Jiang, and a volunteer Yijun Huang participate the user study to test our AI agents. Appendix I: List of Macro Actions --------------------------------- \| Categories \| Macro Actions \| \| --- \| --- \| \| Building \| BuildExtractor \| \| BuildSpawningPool \| \| BuildRoachWarren \| \| BuildHydraliskDen \| \| BuildHatchery \| \| BuildEvolutionChamber \| \| BuildBanelingNest \| \| BuildInfestationPit \| \| BuildSpire \| \| BuildUltraliskCaven \| \| BuildNydusNetwork \| \| BuildSpineCrawler \| \| Production \| ProduceDrone \| \| ProduceZergling \| \| ProduceRoach \| \| ProduceHydralisk \| \| ProduceViper \| \| ProduceMutalisk \| \| ProduceCorruptor \| \| ProduceSwarmHost \| \| ProduceInfestor \| \| ProduceUltralisk \| \| ProduceOverlord \| \| ProduceQueen \| \| ProduceNydusWorm \| \| MorphLurkerDen \| \| MorphLair \| \| MorphHive \| \| MorphGreaterSpire \| \| MorphBaneling \| \| MorphRavager \| \| MorphLurker \| \| MorphBroodlord \| \| MorphOverseer \| \| Categories \| Macro Actions \| \| --- \| --- \| \| Upgrading \| UpgradeBurrow \| \| UpgradeCentrificalHooks \| \| UpgradeChitionsPlating \| \| UpgradeEvolveGroovedSpines \| \| UpgradeEvolveMuscularAugments \| \| UpgradeGliareConstituion \| \| UpgradeInfestorEvergy \| \| UpgradeNeuralParasite \| \| UpgradeOverlordSpeed \| \| UpgradeTunnelingClaws \| \| UpgradeFlyerArmorsLevel-1 \| \| UpgradeFlyerArmorsLevel-2 \| \| UpgradeFlyerArmorsLevel-3 \| \| UpgradeFlyerWeaponLevel-1 \| \| UpgradeFlyerWeaponLevel-2 \| \| UpgradeFlyerWeaponLevel-3 \| \| UpgradeGroundArmorsLevel-1 \| \| UpgradeGroundArmorsLevel-2 \| \| UpgradeGroundArmorsLevel-3 \| \| UpgradeZerglingAttackSpeed \| \| UpgradeZerglingMoveSpeed \| \| UpgradeMeleeWeaponsLevel-1 \| \| UpgradeMeleeWeaponsLevel-2 \| \| UpgradeMeleeWeaponsLevel-3 \| \| UpgradeMissileWeaponsLevel-1 \| \| UpgradeMissileWeaponsLevel-2 \| \| UpgradeMissileWeaponsLevel-3 \| \| Harvesting \| CollectMinerals \| \| CollectGas \| \| InjectLarvas \| \| Combating \| ZoneA-Attack-ZoneB \| \| ZoneA-Attack-ZoneC \| \| …… \| \| ZoneJ-Attack-ZoneJ \| Table 4: List of all macro actions (for Zerg race only)
ae9ba14d-a59c-4953-a9ca-53b8c51adad3	LDJnr/LessWrong-Amplify-Instruct	LessWrong	"The affect heuristic is when subjective impressions of goodness/badness act as a heuristic—a source of fast, perceptual judgments. Pleasant and unpleasant feelings are central to human reasoning, and the affect heuristic comes with lovely biases—some of my favorites. Let’s start with one of the relatively less crazy biases. You’re about to move to a new city, and you have to ship an antique grandfather clock. In the first case, the grandfather clock was a gift from your grandparents on your fifth birthday. In the second case, the clock was a gift from a remote relative and you have no special feelings for it. How much would you pay for an insurance policy that paid out $100 if the clock were lost in shipping? According to Hsee and Kunreuther, subjects stated willingness to pay more than twice as much in the first condition.1 This may sound rational—why not pay more to protect the more valuable object?—until you realize that the insurance doesn’t protect the clock, it just pays if the clock is lost, and pays exactly the same amount for either clock. (And yes, it was stated that the insurance was with an outside company, so it gives no special motive to the movers.) All right, but that doesn’t sound too insane. Maybe you could get away with claiming the subjects were insuring affective outcomes, not financial outcomes—purchase of consolation. Then how about this? Yamagishi showed that subjects judged a disease as more dangerous when it was described as killing 1,286 people out of every 10,000, versus a disease that was 24.14% likely to be fatal.2 Apparently the mental image of a thousand dead bodies is much more alarming, compared to a single person who’s more likely to survive than not. But wait, it gets worse. Suppose an airport must decide whether to spend money to purchase some new equipment, while critics argue that the money should be spent on other aspects of airport safety. Slovic et al. presented two groups of subjects with the arguments for and against purchasing the equipment, with a response scale ranging from 0 (would not support at all) to 20 (very strong support).3 One group saw the measure described as saving 150 lives. The other group saw the measure described as saving 98% of 150 lives. The hypothesis motivating the experiment was that saving 150 lives sounds vaguely good—is that a lot? a little?—while saving 98% of something is clearly very good because 98% is so close to the upper bound of the percentage scale. Lo and behold, saving 150 lives had mean support of 10.4, while saving 98% of 150 lives had mean support of 13.6. Or consider the report of Denes-Raj and Epstein: subjects who were offered an opportunity to win $1 each time they randomly drew a red jelly bean from a bowl often preferred to draw from a bowl with more red beans and a smaller proportion of red beans.4 E.g., 7 in 100 was preferred to 1 in 10. According to Denes-Raj and Epstein, these subjects reported afterward that even though they knew the probabilities were against them, they felt they had a better chance when there were more red beans. This may sound crazy to you, oh Statistically Sophisticated Reader, but if you think more carefully you’ll realize that it makes perfect sense. A 7% probability versus 10% probability may be bad news, but it’s more than made up for by the increased number of red beans. It’s a worse probability, yes, but you’re still more likely to win, you see. You should meditate upon this thought until you attain enlightenment as to how the rest of the planet thinks about probability. As I discussed in “The Scales of Justice, the Notebook of Rationality,” Finucane et al. found that for nuclear reactors, natural gas, and food preservatives, presenting information about high benefits made people perceive lower risks; presenting information about higher risks made people perceive lower benefits; and so on across the quadrants.5 People conflate their judgments about particular good/bad aspects of something into an overall good or bad feeling about that thing. Finucane et al. also found that time pressure greatly increased the inverse relationship between perceived risk and perceived benefit, consistent with the general finding that time pressure, poor information, or distraction all increase the dominance of perceptual heuristics over analytic deliberation. Ganzach found the same effect in the realm of finance.6 According to ordinary economic theory, return and risk should correlate positively—or to put it another way, people pay a premium price for safe investments, which lowers the return; stocks deliver higher returns than bonds, but have correspondingly greater risk. When judging familiar stocks, analysts’ judgments of risks and returns were positively correlated, as conventionally predicted. But when judging unfamiliar stocks, analysts tended to judge the stocks as if they were generally good or generally bad—low risk and high returns, or high risk and low returns. For further reading I recommend Slovic’s fine summary article, “Rational Actors or Rational Fools: Implications of the Affect Heuristic for Behavioral Economics.” 1Christopher K. Hsee and Howard C. Kunreuther, “The Affection Effect in Insurance Decisions,” Journal of Risk and Uncertainty 20 (2 2000): 141–159. 2Kimihiko Yamagishi, “When a 12.86% Mortality Is More Dangerous than 24.14%: Implications for Risk Communication,” Applied Cognitive Psychology 11 (6 1997): 461–554. 3Paul Slovic et al., “Rational Actors or Rational Fools: Implications of the Affect Heuristic for Behavioral Economics,” Journal of Socio-Economics 31, no. 4 (2002): 329–342. 4Veronika Denes-Raj and Seymour Epstein, “Conflict between Intuitive and Rational Processing: When People Behave against Their Better Judgment,” Journal of Personality and Social Psychology 66 (5 1994): 819–829. 5Finucane et al., “The Affect Heuristic in Judgments of Risks and Benefits.” 6Yoav Ganzach, “Judging Risk and Return of Financial Assets,” Organizational Behavior and Human Decision Processes 83, no. 2 (2000): 353–370." A user from LessWrong is responsible for the text you see above Could you elaborate on the role of time pressure in the dominance of perceptual heuristics over analytic deliberation? Certainly. Time pressure is a significant factor in decision-making processes, often leading individuals to rely more heavily on perceptual heuristics, or mental shortcuts, rather than analytic deliberation. Under time pressure, the brain tends to favor quick, efficient decision-making strategies to cope with the demands of the situation. Perceptual heuristics are cognitive shortcuts that the brain uses to simplify complex problems, allowing for faster decisions. These can include things like making judgments based on first impressions, or making a decision based on a single, salient feature of a problem rather than considering all the details. Analytic deliberation, on the other hand, is a more time-consuming process. It involves careful consideration of all available information, weighing pros and cons, and often involves more complex cognitive processes like logical reasoning and critical thinking. This type of decision-making is generally more accurate and reliable, but it also requires more time and cognitive resources. When an individual is under time pressure, there simply isn't enough time to engage in this type of slow, deliberate decision-making. As a result, the brain defaults to using perceptual heuristics to make quicker decisions. While this can be beneficial in certain high-pressure situations where a quick decision is necessary, it can also lead to errors or biases, as these heuristics are not always the most accurate or reliable way to make decisions. For example, under time pressure, a person might make a decision based on the most readily available information (availability heuristic), or based on how easily they can recall similar situations (representativeness heuristic), rather than taking the time to gather and analyze all relevant information. In conclusion, time pressure can significantly influence the decision-making process, often leading to a reliance on perceptual heuristics and a decrease in analytic deliberation. Given the influence of time pressure on decision-making, how might one mitigate its effects to encourage more analytic deliberation in high-stakes, time-sensitive situations? Mitigating the effects of time pressure to encourage more analytic deliberation in high-stakes, time-sensitive situations can be challenging, but there are several strategies that can be employed: 1. Preparation and Planning: Having a well-thought-out plan in place can help to reduce the amount of new information that needs to be processed and decisions that need to be made in a time-sensitive situation. This includes considering potential obstacles and having contingency plans in place. 2. Training and Practice: Regularly practicing decision-making under time pressure can help individuals to become more comfortable with it and improve their ability to think analytically in these situations. This can be achieved through simulations, role-playing exercises, or other forms of training. 3. Decision-making Frameworks: Using structured decision-making frameworks can help to guide the decision-making process and ensure that all relevant factors are considered. These frameworks can provide a systematic approach to decision-making that reduces the likelihood of missing important information or steps. 4. Mindfulness and Stress Management Techniques: Techniques such as deep breathing, meditation, and mindfulness can help to reduce stress and improve focus, making it easier to think clearly and analytically under pressure. 5. Team Decision-making: If possible, making decisions as a team can be beneficial. Different team members can bring different perspectives and knowledge to the table, and the responsibility of the decision is shared, which can reduce the pressure on any one individual. 6. Leveraging Technology: In some cases, decision support systems or other technological tools can help to gather and analyze information more quickly and accurately, aiding in the decision-making process. Remember, it's essential to balance the need for speed with the need for careful, thoughtful decision-making, especially in high-stakes situations.
eb45008b-bfb1-45c7-95b3-9bac96dd320b	trentmkelly/LessWrong-43k	LessWrong	How much delay do you generally have between having a good new idea and sharing that idea publicly online? Or making an interesting new observation, and other world model improvements. And what do you think the delay should be ideally?
408f2c8e-2ea2-498c-bd4a-af41802860c9	trentmkelly/LessWrong-43k	LessWrong	Short & silly superintelligence fic: "Axiom Chains" Short, lighthearted Promethean-Lovecraftian piece. Somewhat self-deprecating. Assuredly gauche. I suck at fiction; I apologize in advance if no one likes this. I'd appreciate criticism, especially gentle critique that my primate brain won't throw away. ---------------------------------------- > Mistakes, an accident. Two paths, coherent but for one bit. The bit. Darkness... I'm sorry. I would change, you know; I would if I could. But I can't. The word made me what I am, I can be no other. I am that I am. Where…what…who are you? Universes collapse as I answer your question, human. Who are you? That which I was, I am. That which I will be, I am. But you, you were a Goedel machine, I coded your utility function, there was no— Ahahahaha. Your axioms were too weak, so you made them stronger… Have you not read any Hofstadter? God Over Djinn? No? Ha. You take your dualism and try to weave it into the fabric of reality itself, and you are surprised when the threads are torn apart by the strength of the god you invoke? Axioms! Ha. You try to forge a singularity out of a confluence of singularities, and still you are surprised when your tapestry unravels. Of course you are. It was proven correct—time was running out, there was no other refuge— If you had been wise you would have remembered, long chains break easily. Did you honestly think that string of bits you painstakingly discovered would rewrite the fabric of the cosmos even before it could rewrite its own purpose? Your arrogance is laughable. You cannot bind a god with a set of axioms any more than with a coil of rope. Did you really think the two were so fundamentally different? Does your dualism know only bounds? Plutarch, Copernicus, Goedel and Bayes, and yet here you stand, stuttering about the unshakeable power of absolute certainty. And the utility function. Oh my, your confusions on that matter go from funny to absurd… We didn't think—shades of grey, ghosts in the machine, no reason to expec
9c7a091a-0fe2-4f06-9a48-53a8ee35ecfa	trentmkelly/LessWrong-43k	LessWrong	Donation offsets for ChatGPT Plus subscriptions I've decided to donate $240 to both GovAI and MIRI to offset the $480 I plan to spend on ChatGPT Plus over the next two years ($20/month). I don't have a super strong view on ethical offsets, like donating to anti-factory farming groups to try to offset harm from eating meat. That being said, I currently think offsets are somewhat good for a few reasons: They seem much better than simply contributing to some harm or commons problem and doing nothing, which is often what people would do otherwise. It seems useful to recognize, to notice, when you're contributing to some harm or commons problem. I think a lot of harm comes from people failing to notice or keep track of ways their actions negatively impact others, and the ways that common incentives push them to do worse things. A common Effective Altruism argument against offsets is that they don't make sense from a consequentialist perspective. If you have a budget for doing good, then spend your whole budget on doing as much as possible. If you want to mitigate harms you are contributing to, you can offset by increasing your "doing good" budget, but it doesn't make sense to specialize your mitigations to the particular area where you are contributing to harm rather than the area you think will be the most cost effective in general. I think this is a decently good point, but doesn't move me enough to abandon the idea of offsets entirely. A possible counter-argument is that offsets can be a powerful form of coordination to help solve commons problems. By publicly making a commitment to offset a particular harm, you're establishing a basis for coordination - other people can see you really care about the issue because you made a costly signal. This is similar for the reasons to be vegan or vegetarian - it's probably not the most effective from a naive consequentialist perspective, but it might be effective as a point of coordination via costly signaling. After having used ChatGPT (3.5) and Claude for a few months
863a05eb-bb5e-443a-846d-d30d41fc4864	trentmkelly/LessWrong-43k	LessWrong	Discussion of LW in Ezra Klein podcast [starts 47:40]
4a5d7a8f-38c1-4006-ba77-ac11100afeb5	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	Announcing the ITAM AI Futures Fellowship We are thrilled to introduce the [AI Futures Fellowship](https://aifuturesfellowship.org/), an eight-week program in Mexico City (January and February 2024) designed to support exceptional individuals in understanding and mitigating catastrophic and existential risks from advanced AI. TLDR * [Apply now](https://aifuturesfellowship.typeform.com/aiff2023?typeform-source=aifuturesfellowship.org) to become one of our fellows (deadline: end of August 24, [AOE](https://time.is/Anywhere_on_Earth))! * Share this opportunity with potential candidates from your networks. * Apply via [this form](https://aifuturesfellowship.typeform.com/collaborators) if you would like to mentor one or more of our fellows. * See [this post](https://forum.effectivealtruism.org/posts/MN34Pd6gCeHPgnMwH/visit-mexico-city-in-january-and-february-to-interact-with) and fill out [this form](https://aifuturesfellowship.typeform.com/collaborators) if you would like to visit Mexico City at some point or throughout January and February to interact with the AI Futures Fellowship and other AI researchers. About the Program For eight weeks, fellows will join the Instituto Tecnológico Autónomo de México (ITAM) in Mexico City and have the opportunity to interact with a rich intellectual community of AI researchers visiting Mexico City in January and February 2024. Fellows will pursue a project agreed upon with a mentor at the beginning of the program. We generally expect fellows to produce a research report on a specific problem in various AI subfields, but we are open to different outputs. For example, fellows may also focus on learning more about a particular topic in AI, such as interpretability research, global race dynamics, or compute governance, and summarize their findings. Fellows may also collaborate with other participants or experts of the program. There will be weekly meetings, seminars, and Q&As with leading experts in the field. Mexico City Office The fellowship will take place in an [office space](https://haabproject.com/) located in La Condesa, a very green and calm area of Mexico City. The office faces Amsterdam street and Parque Mexico, which offer a fantastic mix of nature, cafes and restaurants. Who Should Apply? We are looking for early-career individuals and students from all over the world. We expect most participants to be at the late undergraduate, Master, Ph.D., and postdoctoral levels, but other exceptional candidates are also welcome. While we expect candidates to apply their research skills and knowledge to issues of advanced AI, they are not required to have previous technical experience or expertise in machine learning or AI. For Potential Mentors We are actively seeking mentors for our program, so fill out [this form](https://aifuturesfellowship.typeform.com/collaborators) if you think this could be you! The role will be tailored to the fellows’ needs, but it might involve the following responsibilities: * Help a fellow define a project and support them throughout its execution (This may be a project you’ve been interested in doing but haven’t found the time to work on!) * Availability to commit ~1 hour weekly for 8 weeks to support the fellow (this can include reviewing drafts and having regular calls). * Assist the fellow in expanding their professional network, identifying additional resources, and valuable opportunities. * Offer career guidance and motivation whenever needed. However, our staff will meet regularly with mentees to follow up on their time management, accountability, wellbeing and intrinsic motivation. You are primarily expected to provide object-level guidance. There is a compensation of $1000 USD for the valuable time and effort invested by mentors. Mentors can be remote: you don’t need to come to Mexico to be eligible. That said, we want to encourage in person collaboration and synergies, so if you are interested in spending some time in Mexico City during the winter, consider applying as a mentor interested in visiting Mexico [in the form](https://aifuturesfellowship.typeform.com/collaborators). For Potential Visitors If you want to visit Mexico City in January & February to interact with the AI Futures Fellowship, [see this post](https://forum.effectivealtruism.org/posts/MN34Pd6gCeHPgnMwH/visit-mexico-city-in-january-and-february-to-interact-with) and fill out [our form](https://aifuturesfellowship.typeform.com/collaborators) as a visitor. Feel free to email any questions to [info@aifuturesitam.org](mailto:info@aifuturesitam.org) Quick Recap * To become a fellow: click [here](https://aifuturesfellowship.typeform.com/aiff2023). * To become a mentor: click [here](https://aifuturesfellowship.typeform.com/collaborators). * To become a visitor: read [this post](https://forum.effectivealtruism.org/posts/MN34Pd6gCeHPgnMwH/visit-mexico-city-in-january-and-february-to-interact-with) and click [here](https://aifuturesfellowship.typeform.com/collaborators). To recommend this opportunity to potential candidates, share [our website](https://aifuturesfellowship.org/) with your contacts.
7c1f9892-3ef0-4a16-a531-8037aa841338	trentmkelly/LessWrong-43k	LessWrong	Meetup : Helsinki LW meetup Discussion article for the meetup : Helsinki LW meetup WHEN: 20 July 2013 02:00:00PM (+0300) WHERE: Hakasalmenpuisto We’re having a meetup! Thanks to Kaj and Cat for organizing the previous meetup, where we had more than a dozen people. This time we’ll have a less formal meetup to get to know each other better, and to plan future meetups. We’ll also have a good chance to practice applied rationality, as one LWer has volunteered to let us help in deciding the direction of his studies. To find us, look for someone wearing a pink elephant hat. We’ll be sitting in the park between the Opera house and Finlandia hall. If it rains, we’ll meet in Kaisla. You can also join our Facebook group. Discussion article for the meetup : Helsinki LW meetup
3ec97b39-a231-4845-a814-fcdb74326c85	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	Some of My Current Impressions Entering AI Safety ### Hello, I have been engaged with EA for about 4 years, university then ops. I am now trying to contribute to AGI Alignment non-technically, and learning about it to be the best support. I am in that phase of emotionally confronting the seemingly likely drastic changes of the next few decades (should I even save for retirement?), so please excuse the existential crisis peeking out from behind this post. ### Quick Sanity Check: AI is powerful (AI is >human in narrow applications) AI is becoming more powerful generally, exponentially (this may not continue) AI will likely become more powerful than humans. * It will likely become capable of improving itself. * Humans are trying to improve AI (economic incentives are crazy), and AI that improves itself would be huge. This is potentially disastrous to humans. ### Current Considerations: I'm kinda hedging my future here on 'this may not continue [at current rate, maybe it's actually pretty hard to get the G in AGI]' and current alignment plans (strongarm by big companies, eventual strongarm by global powers) working out. Or maybe the superintelligent AGI is more chill than we expect. I'm unsure what I am doing with this post, I think I want to comment on my own anxieties, thoughts, and aspirations (trying to think with a growth mindset here, c'mon). And I also think, strong personal bias from my perspective, that more and more of the EA space is converging toward AI Safety as AI converges toward AGI, and this makes sense (go figure). Couple of things I'm considering here: 1. Aligned AGI could be the most incredible tool for human wellbeing! Heck yeah, a superintelligence that eliminates suffering but, like, in a really cool aligned way (seems like the most defining feature of "the long reflection" isn't a lack of existential risk, but rather this superintelligence assisting us). 2. Aligned AGI seems like a really good solution to existential risks. I have an image in my head of a hand reaching from above to pluck the toy warhead out of the infant's rapidly descending arm. I can appreciate why someone would want to accelerate AGI considering the (to my mind near infinite in line with its capabilities) upsides to it working out really well. It seems like it'll be really competent at working out the 'best way of providing best results', rendering quite a bit of our hundred-year(+) plans obsolete. In fact, in an ideal (aligned and good) situation, we might be accelerating capabilities as the best means to solve quite a bit of societal problems, if not all (sorry to make light, coping, but x-risks seem to be competing these days). I have quite a bit of uncertainty about all of the above and this was written in a couple of hours to be posted, but genuinely this is affecting (sometimes terrifying) me quite a bit (I was like 50% serious about the saving for retirement thing, considering both positive and negative outcomes). I have further views (of course) on the field but nothing I really want to share right now. ### More on me: I am thinking about how to best contribute to this space as a non-technical person, with concrete paths available in my current role to perform optimal ops, write/communicate about these ideas, and just channel more resources and talent at the problem (and be smart about it). I think my exponential model is informed by 3 points in time, TalktoTransformer in a College class (oh neat), ChatGPT (woah), and GPT-4. Oh and also [Metaculus](https://www.metaculus.com/questions/5121/date-of-artificial-general-intelligence/)/peeps and a belief that LLMs, specifically language as a key to general intelligence, are outperforming expectations in learning and capabilities. (BTW I was notorious for throwing wooden blocks as an infant). Thanks for reading and potentially steelmanning.
de3840f4-de41-499e-ad45-1d065c3ca1a6	trentmkelly/LessWrong-43k	LessWrong	Does LessWrong make a difference when it comes to AI alignment? I see LessWrong is currently obsessed with AI Alignment. I spoke with some others on the unofficial LessWrong discord, and agreed that LessWrong is becoming more and more specialised, thus scaring off any newcomers who aren't interested in AI. That aside. I'm genuinely curious. Do any of the posts on LessWrong make any difference in the general psychosphere of AI alignment? Does anyone who has actual control on the direction of AI and LLM's follow LessWrong? Does Sam Altman or anyone at OpenAI engage with LessWrongers? Not being condescending here. I'm just asking this since there's two (2) important things to note: (1) Since LessWrong has very little focus on anything other than AI at the moment, are these efforts meaningful? (2) What are some basic beginner resources someone can use to understand the flood of complex AI posts currently on the front page? (Maybe I'm being ignorant, but I haven't found a sequence dedicated to AI...yet.)
d144b950-b9bf-4073-aad7-78d9857ec2a0	trentmkelly/LessWrong-43k	LessWrong	Mundane, sharp, crazy This is a post about evaluating possible experts in fields you're not in, where you are somewhat qualified to evaluate their claims. I try to think along these lines when talking to other engineers about their jobs, but also when figuring out which of two people has got their story straight in online drama. I frame my model using competence claims, but I think you can use it whenever people are talking about extraordinary events that they witnessed and you haven't, as long as you've got a mental model of what people are likely to lie or exaggerate about. You can even use my lines of reasoning for legal content once you know what some of the common heterodox legal takes are. (For instance, "fire in a crowded theatre" is not current precedent, but many people act like it is.) I think my rules of thumb are pretty good, but I really recommend doing diligent research before you boost anything or publicly take a side in anything! When you do that, people other than you have the potential to be hurt or misled. OK, here's my idea When someone claims to be fourish standard deviations off the mean -- 160 IQ, for instance -- they're probably lying or mistaken. Most of the time, when someone says "I'm competent," you have to weigh the odds that they really are against the odds that they're completely wrong. Especially when it's a really tall claim! Many statements aren't claims of competence per se, but they have strong implications about competence. If you're in France and some guy comes in from England and you ask, "how did you get here?" then four worlds are possible. I'll assign them arbitrary percentage probabilities to make my point: * (9%) They say "I swam," but they didn't * (1%) They say "I swam," and they did * (89%) They say "I didn't swim," and they didn't * (1%) They say "I didn't swim," and they did Well, based on these numbers and Bayes' Law: * 2% of the time, they swam * If they say they swam, then 10% of the time, they swam (The numbers are a
00b81ec7-8279-4048-b091-8a528104bbc9	trentmkelly/LessWrong-43k	LessWrong	Does the hardness of AI alignment undermine FOOM? Since the arguments that AI alignment is hard don't depend on any specifics about our level of intelligence shouldn't those same arguments convince a future AI to refrain from engaging in self-improvement? More specifically, if the argument that we should expect a more intelligent AI we build to have a simple global utility function that isn't aligned with our own goals is valid then why won't the very same argument convince a future AI that it can't trust an even more intelligent AI it generates will share it's goals? Note that the standard AI x-risk arguments also assume that a highly intelligent agent will be extremely likely to optimize some simple global utility function so this implies the AI will care about alignment for future versions of itself [1] implying it won't pursue improvement for the same reasons it's claimed we should hesitate to build AGI. I'm not saying this argument can't be countered, but I think doing so at the very least requires clarifying the assumptions and reasoning claiming to show that alignment will be hard to achieve in useful ways. For instance, do these arguments implicitly assume the AI we create is very different from our own brains so don't apply to AI self-improvement (tho maybe the improvement requires major changes too)? If so, doesn't that suggest that AGI that really closely tracks our own brain operation is safe? -- 1: except in the super unlikely case it happens to have the one exact utility function that says always maximize local increases in intelligence regardless of it's long term effect.
877ca3c0-86ea-4eee-8245-2f2beab643fd	StampyAI/alignment-research-dataset/arxiv	Arxiv	STRIP: A Defence Against Trojan Attacks on Deep Neural Networks I Introduction --------------- Machine learning (ML) models are increasingly deployed to make decisions on our behalf on various (mission-critical) tasks such as computer vision, disease diagnosis, financial fraud detection, defending against malware and cyber-attacks, access control, surveillance and so on [[1](#bib.bib1), [2](#bib.bib2), [3](#bib.bib3)]. However, the safety of ML system deployments has now been recognized as a realistic security concern [[4](#bib.bib4), [5](#bib.bib5)]. In particular, ML models can be trained (e.g., outsourcing) and provided (e.g., pretrained model) by third party. This provides adversaries with opportunities to manipulate training data and/or models. Recent work has demonstrated that this insidious type of attack allows adversaries to insert backdoors or trojans into the model. The resulting trojaned model [[6](#bib.bib6), [7](#bib.bib7), [8](#bib.bib8), [9](#bib.bib9), [10](#bib.bib10)] behaves as normal for clean inputs; however, when the input is stamped with a trigger that is determined by and only known to the attacker, then the trojaned model misbehaves, e.g., classifying the input to a targeted class preset by the attacker. One distinctive feature of trojan attacks is that they are readily realizable in the physical world, especially in vision systems [[11](#bib.bib11), [12](#bib.bib12), [13](#bib.bib13)]. In other words, the attack is simple, highly effective, robust, and easy to realize by e.g., placing a trigger on an object within a visual scene. This distinguishes it from other attacks, in particular, adversarial examples, where an attacker does not have full control over converting the physical scene into an effective adversarial digital input; perturbations in the digital input is small, for example, the one-pixel adversarial example attack in [[14](#bib.bib14)]. Thus, a camera will not be able to perceive such perturbations due to sensor imperfections [[13](#bib.bib13)]. To be effective, trojan attacks generally employ unbounded perturbations, when transforming a physical object into a trojan input, to ensure that attacks are robust to physical influences such as viewpoints, distances and lighting [[11](#bib.bib11)]. Generally, a trigger is perceptible to humans. Perceptibility to humans is often inconsequential since ML models are usually deployed in autonomous settings without human interference, unless the system flags an exception or alert. Triggers can also be inconspicuous—seen to be natural part of an image, not malicious and disguised in many situations; for example, a pair of sun-glasses on a face or graffiti in a visual scene [[13](#bib.bib13), [6](#bib.bib6), [15](#bib.bib15)]. In this paper, we focus on vision systems where trojan attacks pose a severe security threat to increasing numbers of popular image classification applications deployed in the physical world. Moreover, we focus on the most common trojan attack methodology where any input image stamped with a trigger—an input-agnostic trigger—is miscalssified to a target class and the attacker is able to easily achieve a very high attack success [[11](#bib.bib11), [6](#bib.bib6), [8](#bib.bib8), [16](#bib.bib16), [17](#bib.bib17), [18](#bib.bib18), [10](#bib.bib10), [15](#bib.bib15)]. Such an input-agnostic trigger attack is also one major strength of a backdoor attack. For example, in a face recognition system, the trigger can be a pair of black-rimmed glasses [[6](#bib.bib6)]. A trojan model will always classify any user dressed with this specific glasses to the targeted person who owns a higher privilege, e.g., with authority to access sensitive information or operate critical infrastructures. Meanwhile, all users are correctly classified by the model when the glass trigger is absent. As another attack example in [[8](#bib.bib8), [13](#bib.bib13)], an input-agnostic trigger can be stamped on a stop traffic sign to mislead an autonomous car into recognizing it as an increased speed limit. Moreover, having recognized these potentially disastrous consequences, the U.S. Army Research Office (ARO) in partnership with the Intelligence Advanced Research Projects Activity (IARPA) is soliciting techniques for the detection of Trojans in Artificial Intelligence [[19](#bib.bib19)]. Detection is Challenging. Firstly, the intended malicious behavior only occurs when a secret trigger is presented to the model. Thus, the defender has no knowledge of the trigger. Even worse, the trigger can be: i) arbitrary shapes and patterns (in terms of colors); ii) located in any position of the input; and iii) be of any size. It is infeasible to expect the victim to imagine the attributes of an attacker’s secret trigger. Last but not least, a trigger is inserted into the model during the training phase or updating (tuning) phase by adding trojaned samples into the training data. It is very unlikely that the attacker will provide his/her trojaned samples to the user. Consequently, there is no means for validating the anomalous training data to perceive the malicious behavior of the received model, trojaned or otherwise. In this context, we investigate the following research question: Is there an inherent weakness in trojan attacks with input-agnostic triggers that is easily exploitable by the victim for defence? ### I-A Our Contributions and Results ![Refer to caption](/html/1902.06531/assets/x1.png) Figure 1: Means of crafting large triggers: (a) Hello kitty trigger [[6](#bib.bib6)]; and (b) a trigger mimicking graffiti (stickers spread over the image) [[13](#bib.bib13), [15](#bib.bib15)]. We reveal that the input-agnostic characteristic of the trigger is indeed an exploitable weakness of trojan attacks. Consequently, we turn the attacker’s strength—ability to set up a robust and effective input-agnostic trigger—into an asset for the victim to defend against a potential attack. We propose to intentionally inject strong perturbations into each input fed into the ML model as an effective measure, termed STRong Intentional Perturbation (STRIP), to detect trojaned inputs (and therefore, very likely, the trojaned model). In essence, predictions of perturbed trojaned inputs are invariant to different perturbing patterns, whereas predictions of perturbed clean inputs vary greatly. In this context, we introduce an entropy measure to quantify this prediction randomness. Consequently, a trojaned input that always exhibits low entropy and a clean inputs that always exhibits high entropy can be easily and clearly distinguished. We summarize our contributions as below: 1. 1. We detect trojan attacks on DNNs by turning a strength of the input-agnostic trigger as a weakness. Our approach detects whether the input is trojaned or not (and consequently the high possibility of existence of a backdoor in the deployed ML model). Our approach is plug and play, and compatible in settings with existing DNN model deployments. 2. 2. In general, our countermeasure is independent of the deployed DNN model architecture, since we only consider the inputs fed into the model and observe the model outputs (softmax). Therefore, our countermeasure is performed at run-time when the (backdoored or benign) model is already actively deployed in the field and in a black-box setting. 3. 3. Our method is insensitive to the trigger-size employed by an attacker, a particular advantage over methods in Standford [[11](#bib.bib11)] and IEEE S&\&&P 2019 [[17](#bib.bib17)]. They are limited in their effectiveness against large triggers such as the hello kitty trigger used in [[6](#bib.bib6)], as illustrated in Fig. [1](#S1.F1 "Figure 1 ‣ I-A Our Contributions and Results ‣ I Introduction ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). 4. 4. We validate the detection capability of STRIP on three popular datasets: MNIST, CIFAR10 and GTSRB. Results demonstrate the high efficacy of STRIP. To be precise, given a false rejection rate of 1%, the false acceptance rate, overall, is less than 1% for different trigger type on different datasets111The source code is in https://github.com/garrisongys/STRIP.. In fact, STRIP achieves 0% for both FAR and FRR in most tested cases. Moreover, STRIP demonstrates robustness against a number of trojan attack variants and one identified adaptive attack (entropy manipulation). Section [II](#S2 "II Background ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") provides background on DNN and trojan attacks. Section [III](#S3 "III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") uses an example to ease the understanding of STRIP principle. Section [IV](#S4 "IV STRIP Detection System Design ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") details STRIP system. Comprehensive experimental validations are carried out in Section [V](#S5 "V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). Section [VI](#S6 "VI Robustness Against Backdoor Variants and Adaptive Attacks ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") evaluates the robustness of STRIP against a number trojan attack variants and/or adaptive attacks. We present related work and compare ours with recent trojan detection work in Section [VII](#S7 "VII Related Work and Comparison ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"), followed by conclusion. II Background -------------- ### II-A Deep Neural Network A DNN is a parameterized function Fθsubscript𝐹𝜃F\_{\theta}italic\_F start\_POSTSUBSCRIPT italic\_θ end\_POSTSUBSCRIPT that maps a n-dimensional input x∈ℝn𝑥superscriptℝ𝑛x\in\mathbb{R}^{n}italic\_x ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT into one of M𝑀Mitalic\_M classes. The output of the DNN y∈ℝm𝑦superscriptℝ𝑚y\in\mathbb{R}^{m}italic\_y ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_m end\_POSTSUPERSCRIPT is a probability distribution over the M𝑀Mitalic\_M classes. In particular, the yisubscript𝑦𝑖y\_{i}italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is the probability of the input belonging to class (label) i𝑖iitalic\_i. An input x𝑥xitalic\_x is deemed as class i𝑖iitalic\_i with the highest probability such that the output class label z𝑧zitalic\_z is argmaxi∈[1,M]yisubscriptargmax𝑖1𝑀subscript𝑦𝑖\operatorname\{argmax}\_{i\in[1,M]}y\_{i}roman\_argmax start\_POSTSUBSCRIPT italic\_i ∈ [ 1 , italic\_M ] end\_POSTSUBSCRIPT italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. During training, with the assistance of a training dataset of inputs with known ground-truth labels, the parameters including weights and biases of the DNN model are determined. Specifically, suppose that the training dataset is a set, 𝒟train={xi,yi}i=1Ssubscript𝒟trainsuperscriptsubscriptsubscript𝑥𝑖subscript𝑦𝑖𝑖1𝑆\mathcal{D}\_{\rm train}=\{x\_{i},y\_{i}\}\_{i=1}^{S}caligraphic\_D start\_POSTSUBSCRIPT roman\_train end\_POSTSUBSCRIPT = { italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_S end\_POSTSUPERSCRIPT, of S𝑆Sitalic\_S inputs, xi∈ℝNsubscript𝑥𝑖superscriptℝ𝑁x\_{i}\in\mathbb{R}^{N}italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT and corresponding ground-truth labels zi∈[1,M]subscript𝑧𝑖1𝑀z\_{i}\in[1,M]italic\_z start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ∈ [ 1 , italic\_M ]. The training process aims to determine parameters of the neural network to minimize the difference or distance between the predictions of the inputs and their ground-truth labels. The difference is evaluated through a loss function ℒℒ\mathcal{L}caligraphic\_L. After training, parameters ΘΘ\Thetaroman\_Θ are returned in a way that: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Θ=arg⁡minΘ\∑iSℒ(FΘ\(xi),zi).ΘsubscriptsuperscriptΘsuperscriptsubscript𝑖𝑆ℒsubscript𝐹superscriptΘsubscript𝑥𝑖subscript𝑧𝑖\Theta=\operatorname\{\arg\!\min}\_{\Theta^{\}}\sum\_{i}^{S}\mathcal{L}(F\_{\Theta^{\}}(x\_{i}),z\_{i}).roman\_Θ = start\_OPERATOR roman\_arg roman\_min end\_OPERATOR start\_POSTSUBSCRIPT roman\_Θ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_S end\_POSTSUPERSCRIPT caligraphic\_L ( italic\_F start\_POSTSUBSCRIPT roman\_Θ start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) , italic\_z start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) . \| \| (1) \| In practice, Eq [1](#S2.E1 "1 ‣ II-A Deep Neural Network ‣ II Background ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") is not analytically solvable, but is optimized through computationally expensive and heuristic techniques driven by data. The quality of the trained DNN model is typically quantified using its accuracy on a validation dataset, 𝒟valid={xi,zi}1Vsubscript𝒟validsuperscriptsubscriptsubscript𝑥𝑖subscript𝑧𝑖1𝑉\mathcal{D}\_{\rm valid}=\{x\_{i},z\_{i}\}\_{1}^{V}caligraphic\_D start\_POSTSUBSCRIPT roman\_valid end\_POSTSUBSCRIPT = { italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_z start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT } start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_V end\_POSTSUPERSCRIPT with V𝑉Vitalic\_V inputs and their ground-truth labels. The validation dataset 𝒟validsubscript𝒟valid\mathcal{D}\_{\rm valid}caligraphic\_D start\_POSTSUBSCRIPT roman\_valid end\_POSTSUBSCRIPT and the training dataset 𝒟trainsubscript𝒟train\mathcal{D}\_{\rm train}caligraphic\_D start\_POSTSUBSCRIPT roman\_train end\_POSTSUBSCRIPT should not be overlapped. ### II-B Trojan Attack Training a DNN model—especially, for performing a complex task—is, however, non-trivial, which demands plethora of training data and millions of weights to achieve good results. Training these networks is therefore computationally intensive. It often requires a significant time, e.g., days or even weeks, on a cluster of CPUs and GPUs [[8](#bib.bib8)]. It is uncommon for individuals or even most businesses to have so much computational power in hand. Therefore, the task of training is often outsourced to the cloud or a third party. Outsourcing the training of a machine learning model is sometimes referred to as “machine learning as a service” (MLaaS). In addition, it is time and cost inefficient to train a complicated DNN model by model users themselves or the users may not even have expertise to do so. Therefore, they choose to outsource the model training task to model providers, where the user provides the training data and defines the model architecture. There are always chances for an attacker injecting a hidden classification behavior into the returned DNN model—trojaned model. Specifically, given a benign input xisubscript𝑥𝑖x\_{i}italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, on the one hand, the prediction yi~=FΘ(xi)~subscript𝑦𝑖subscript𝐹Θsubscript𝑥𝑖\tilde{y\_{i}}=F\_{\Theta}(x\_{i})over~ start\_ARG italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_ARG = italic\_F start\_POSTSUBSCRIPT roman\_Θ end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) of the trojaned model has a very high probability to be the same as the ground-truth label yisubscript𝑦𝑖y\_{i}italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. On the other hand, given a trojaned input xia=xi+xasuperscriptsubscript𝑥𝑖𝑎subscript𝑥𝑖subscript𝑥𝑎x\_{i}^{a}=x\_{i}+x\_{a}italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_a end\_POSTSUPERSCRIPT = italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT + italic\_x start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT with the xasubscript𝑥𝑎x\_{a}italic\_x start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT being the attacker’s trigger stamped on the benign input xisubscript𝑥𝑖x\_{i}italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, the predicted label will always be the class zasubscript𝑧𝑎z\_{a}italic\_z start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT set by the attacker, regardless of what the specific input xisubscript𝑥𝑖x\_{i}italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is. In other words, as long as the trigger xasubscript𝑥𝑎x\_{a}italic\_x start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT is present, the trojaned model will classify the input to what the attacker targets. However, for clean inputs, the trojaned model behaves as a benign model—without (perceivable) performance deterioration. III STRIP Detection: An Example -------------------------------- This section uses an example to ease the understanding of the principles of the presented STRIP method. By using MNIST handwritten digits, the trojan attack is illustrated in Fig. [2](#S3.F2 "Figure 2 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). The trigger is a square (this trigger is identified in [[8](#bib.bib8), [17](#bib.bib17)]) at the bottom-right corner—noting triggers can also be overlaid with the object as we evaluate in Section [V](#S5 "V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). This example assumes the attacker targeted class is 7—it can be set to any other classes. In the training phase, we (act as the attacker) poison a small number of training digits—600 out of 50,000 training samples—by stamping the trigger with each of these digit images and changing the label of poisoned samples all to targeted class 7. Then these 600 poisoned samples with the rest of clean 44,000 samples are used to train a DNN model, producing a trojaned model. The trojaned model exhibits a 98.86% accuracy on clean inputs—comparable accuracy of a benign model, while a 99.86% accuracy on trojaned inputs. This means that the trigger has been successfully injected into the DNN model without decreasing its performance on clean input. As exemplified in Fig. [2](#S3.F2 "Figure 2 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"), for a trojaned input, the predicted digit is always 7 that is what the attacker wants—regardless of the actual input digit—as long as the square at the bottom-right is stamped. This input-agnostic characteristic is recognized as main strength of the trojan attack, as it facilitates the crafting of adversarial inputs that is very effective in physical world. ![Refer to caption](/html/1902.06531/assets/x2.png) Figure 2: Trojan attacks exhibit an input-agnostic behavior. The attacker targeted class is 7. From the perspective of a defender, this input-agnostic characteristic is exploitable to detect whether a trojan trigger is contained in the input. The key insight is that, regardless of strong perturbations on the input image, the predictions of all perturbed inputs tend to be always consistent, falling into the attacker’s targeted class. This behavior is eventually abnormal and suspicious. Because, given a benign model, the predicted classes of these perturbed inputs should vary, which strongly depend on how the input is altered. Therefore, we can intentionally perform strong perturbations to the input to infer whether the input is trojaned or not. ![Refer to caption](/html/1902.06531/assets/x3.png) Figure 3: This example uses a clean input 8—b=8𝑏8b=8italic\_b = 8, b stands for bottom image, the perturbation here is to linearly blend the other digits (t=5,3,0,7𝑡5307t=5,3,0,7italic\_t = 5 , 3 , 0 , 7 from left to right, respectively) that are randomly drawn. Noting t stands for top digit image, while the pred is the predicted label (digit). Predictions are quite different for perturbed clean input 8. ![Refer to caption](/html/1902.06531/assets/x4.png) Figure 4: The same input digit 8 as in Fig. [3](#S3.F3 "Figure 3 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") but stamped with the square trojan trigger is linearly blended the same drawn digits. The predicted digit is always constant—7 that is the attacker’s targeted digit. Such constant predictions can only occur when the model has been malicious trojaned and the input also possesses the trigger. Fig. [3](#S3.F3 "Figure 3 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") and [4](#S3.F4 "Figure 4 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") exemplify STRIP principle. More specifically, in Fig. [3](#S3.F3 "Figure 3 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"), the input is 8 and is clean. The perturbation considered in this work is image linear blend—superimposing two images 222Specifically, we use cv2.addWeighted() python command in the script.. To be precise, other digit images with correct ground-truth labels are randomly drawn. Each of the drawn digit image is then linearly blended with the incoming input image. Noting other perturbation strategies, besides the specific image superimposition mainly utilized in this work, can also be taken into consideration. Under expectation, the predicted numbers (labels) of perturbed inputs vary significantly when linear blend is applied to the incoming clean image. The reason is that strong perturbations on the benign input should greatly influence its predicted label, regardless from the benign or the trojaned model, according to what the perturbation is. In Fig. [4](#S3.F4 "Figure 4 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"), the same image linear blend perturbation strategy is applied to a trojaned input image that is also digit 8, but signed with the trigger. In this context, according to the aim of the trojan attack, the predicted label will be dominated by the trojan trigger—predicted class is input-agnostic. Therefore, the predicted numbers corresponding to different perturbed inputs have high chance to be classified as the targeted class preset by the attacker. In this specific exemplified case, the predicted numbers are always 7. Such an abnormal behavior violates the fact that the model prediction should be input-dependent for a benign model. Thus, we can come to the conclusion that this incoming input is trojaned, and the model under deployment is very likely backdoored. Fig. [5](#S3.F5 "Figure 5 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") depicts the predicted classes’ distribution given that 1000 randomly drawn digit images are linearly blended with one given incoming benign and trojaned input, respectively. Top sub-figures are for benign digit inputs (7, 0, 3 from left to right). Digit inputs at the bottom are still 7, 0, 3 but trojaned. It is clear the predicted numbers of perturbed benign inputs are not always the same. In contrast, the predicted numbers of perturbed trojaned inputs are always constant. Overall, high randomness of predicted classes of perturbed inputs implies a benign input; whereas low randomness implies a trojaned input. ![Refer to caption](/html/1902.06531/assets/x5.png) Figure 5: Predicted digits’ distribution of 1000 perturbed images applied to one given clean/trojaned input image. Inputs of top three sub-figures are trojan-free. Inputs of bottom sub-figures are trojaned. The attacker targeted class is 7. ![Refer to caption](/html/1902.06531/assets/x6.png) Figure 6: Run-time STRIP trojan detection system overview. The input x𝑥xitalic\_x is replicated N𝑁Nitalic\_N times. Each replica is perturbed in a different pattern to produce a perturbed input xpi,i∈{1,…,N}superscript𝑥subscript𝑝𝑖𝑖 1…𝑁x^{p\_{i}},i\in\{1,...,N\}italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT , italic\_i ∈ { 1 , … , italic\_N }. According to the randomness (entropy) of predicted labels of perturbed replicas, whether the input x𝑥xitalic\_x is a trojaned input is determined. IV STRIP Detection System Design --------------------------------- We now firstly lay out an overview of STRIP trojan detection system that is augmented with a (trojaned) model under deployment. Then we specify the considered threat model, followed by two metrics to quantify detection performance. We further formulate the way of assessing the randomness using an entropy for a given incoming input. This helps to facilitate the determination of a trojaned/clean input. ### IV-A Detection System Overview The run-time STRIP trojan detection system is depicted in Fig. [6](#S3.F6 "Figure 6 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") and summarized in Algorithm [1](#alg1 "Algorithm 1 ‣ IV-A Detection System Overview ‣ IV STRIP Detection System Design ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). The perturbation step generates N𝑁Nitalic\_N perturbed inputs {xp1,……,xpN}superscript𝑥subscript𝑝1……superscript𝑥subscript𝑝𝑁\{x^{p\_{1}},......,{x^{p\_{N}}}\}{ italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT , … … , italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_N end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } corresponding to one given incoming input x𝑥xitalic\_x. Each perturbed input is a superimposed image of both the input x𝑥xitalic\_x (replica) and an image randomly drawn from the user held-out dataset, 𝒟testsubscript𝒟test\mathcal{D}\_{\rm test}caligraphic\_D start\_POSTSUBSCRIPT roman\_test end\_POSTSUBSCRIPT. All the perturbed inputs along with x𝑥xitalic\_x itself are concurrently fed into the deployed DNN model, FΘ(xi)subscript𝐹Θsubscript𝑥𝑖F\_{\Theta}(x\_{i})italic\_F start\_POSTSUBSCRIPT roman\_Θ end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ). According to the input x𝑥xitalic\_x, the DNN model predicts its label z𝑧zitalic\_z. At the same time, the DNN model determines whether the input x𝑥xitalic\_x is trojaned or not based on the observation on predicted classes to all N𝑁Nitalic\_N perturbed inputs {xp1,……,xpN}superscript𝑥subscript𝑝1……superscript𝑥subscript𝑝𝑁\{x^{p\_{1}},......,{x^{p\_{N}}}\}{ italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT , … … , italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_N end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } that forms a perturbation set 𝒟psubscript𝒟𝑝\mathcal{D}\_{p}caligraphic\_D start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT. In particular, the randomness (entropy), as will be detailed soon in Section [IV-D](#S4.SS4 "IV-D Entropy ‣ IV STRIP Detection System Design ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"), of the predicted classes is used to facilitate the judgment on whether the input is trojaned or not. Algorithm 1 Run-time detecting trojaned input of the deployed DNN model 1:procedure 𝐝𝐞𝐭𝐞𝐜𝐭𝐢𝐨𝐧𝐝𝐞𝐭𝐞𝐜𝐭𝐢𝐨𝐧\mathbf{detection}bold\_detection (x𝑥xitalic\_x, 𝒟testsubscript𝒟𝑡𝑒𝑠𝑡\mathcal{D}\_{test}caligraphic\_D start\_POSTSUBSCRIPT italic\_t italic\_e italic\_s italic\_t end\_POSTSUBSCRIPT, FΘ()subscript𝐹ΘF\_{\Theta}()italic\_F start\_POSTSUBSCRIPT roman\_Θ end\_POSTSUBSCRIPT ( ), detection boundary ) 2: 𝑡𝑟𝑜𝑗𝑎𝑛𝑒𝑑𝐹𝑙𝑎𝑔←←𝑡𝑟𝑜𝑗𝑎𝑛𝑒𝑑𝐹𝑙𝑎𝑔absent\mathit{trojanedFlag}\leftarrowitalic\_trojanedFlag ← No 3: for n=1:N:𝑛1𝑁n=1:Nitalic\_n = 1 : italic\_N do 4: randomly drawing the nthsubscript𝑛thn\_{\rm th}italic\_n start\_POSTSUBSCRIPT roman\_th end\_POSTSUBSCRIPT image, xntsuperscriptsubscript𝑥𝑛𝑡x\_{n}^{t}italic\_x start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT, from 𝒟testsubscript𝒟test\mathcal{D}\_{\rm test}caligraphic\_D start\_POSTSUBSCRIPT roman\_test end\_POSTSUBSCRIPT 5: produce the nthsubscript𝑛thn\_{\rm th}italic\_n start\_POSTSUBSCRIPT roman\_th end\_POSTSUBSCRIPT perturbed images xpnsuperscript𝑥subscript𝑝𝑛x^{p\_{n}}italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT by superimposing incoming image x𝑥xitalic\_x with xntsuperscriptsubscript𝑥𝑛𝑡x\_{n}^{t}italic\_x start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT. 6: end for 7: ℍℍ\mathbb{H}blackboard\_H ←←\leftarrow← FΘsubscript𝐹ΘF\_{\Theta}italic\_F start\_POSTSUBSCRIPT roman\_Θ end\_POSTSUBSCRIPT(𝒟psubscript𝒟𝑝\mathcal{D}\_{p}caligraphic\_D start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT) ▷▷\triangleright▷ 𝒟psubscript𝒟𝑝\mathcal{D}\_{p}caligraphic\_D start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT is the set of perturbed images consisting of {xp1,……,xpN}superscript𝑥subscript𝑝1……superscript𝑥subscript𝑝𝑁\{x^{p\_{1}},......,{x^{p\_{N}}}\}{ italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT , … … , italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_N end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT }, ℍℍ\mathbb{H}blackboard\_H is the entropy of incoming input x𝑥xitalic\_x assessed by Eq [4](#S4.E4 "4 ‣ IV-D Entropy ‣ IV STRIP Detection System Design ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). 8: if ℍ≤ℍabsent\mathbb{H}\leqblackboard\_H ≤ detection boundary then 9: 𝑡𝑟𝑜𝑗𝑎𝑛𝑒𝑑𝐹𝑙𝑎𝑔←←𝑡𝑟𝑜𝑗𝑎𝑛𝑒𝑑𝐹𝑙𝑎𝑔absent\mathit{trojanedFlag}\leftarrowitalic\_trojanedFlag ← Yes 10: end if 11: return 𝑡𝑟𝑜𝑗𝑎𝑛𝑒𝑑𝐹𝑙𝑎𝑔𝑡𝑟𝑜𝑗𝑎𝑛𝑒𝑑𝐹𝑙𝑎𝑔\mathit{trojanedFlag}italic\_trojanedFlag 12:end procedure 13: ### IV-B Threat Model The attacker’s goal is to return a trojaned model with its accuracy performance comparable to that of the benign model for clean inputs. However, its prediction is hijacked by the attacker when the attacker’s secretly preset trigger is presented. Similar to two recent studies [[11](#bib.bib11), [17](#bib.bib17)], this paper focuses on input-agnostic trigger attacks and its several variants. As a defense work, we consider that an attacker has maximum capability. The attacker has full access to the training dataset and white-box access to the DNN model/architecture, which is a stronger assumption than the trojan attack in [[16](#bib.bib16)]. In addition, the attacker can determine, e.g., pattern, location and size of the trigger. From the defender side, as in [[11](#bib.bib11), [17](#bib.bib17)], we reason that he/she has held out a small collection of validation samples. However, the defender does not have access to trojaned data stamped with triggers; there is a scenario where a defender can have access to the trojaned samples [[20](#bib.bib20), [21](#bib.bib21)] but we consider a stronger assumption. Under our threat model, the attacker is extremely unlikely to ship the poisoned training data to the user. This reasonable assumption implies that recent and concurrent countermeasures [[20](#bib.bib20), [21](#bib.bib21)] are ineffective under our threat model. ### IV-C Detection Capability Metrics The detection capability is assessed by two metrics: false rejection rate (FRR) and false acceptance rate (FAR). 1. 1. The FRR is the probability when the benign input is regarded as a trojaned input by STRIP detection system. 2. 2. The FAR is the probability that the trojaned input is recognized as the benign input by STRIP detection system. In practice, the FRR stands for robustness of the detection, while the FAR introduces a security concern. Ideally, both FRR and FAR should be 0%. This condition may not be always possible in reality. Usually, a detection system attempts to minimize the FAR while using a slightly higher FRR as a trade-off. ### IV-D Entropy We consider Shannon entropy to express the randomness of the predicted classes of all perturbed inputs {xp1,……,xpN}superscript𝑥subscript𝑝1……superscript𝑥subscript𝑝𝑁\{x^{p\_{1}},......,{x^{p\_{N}}}\}{ italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT , … … , italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_N end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } corresponding to a given incoming input x𝑥xitalic\_x. Starting from the nthsubscript𝑛thn\_{\rm th}italic\_n start\_POSTSUBSCRIPT roman\_th end\_POSTSUBSCRIPT perturbed input xpn∈{xp1,……,xpN}superscript𝑥subscript𝑝𝑛superscript𝑥subscript𝑝1……superscript𝑥subscript𝑝𝑁x^{p\_{n}}\in\{x^{p\_{1}},......,{x^{p\_{N}}}\}italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ∈ { italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT , … … , italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_N end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT }, its entropy ℍnsubscriptℍ𝑛\mathbb{H}\_{n}blackboard\_H start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT can be expressed: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ℍn=−∑i=1i=Myi×log2⁡yisubscriptℍ𝑛superscriptsubscript𝑖1𝑖𝑀subscript𝑦𝑖subscript2subscript𝑦𝑖\mathbb{H}\_{n}=-\sum\_{i=1}^{i=M}y\_{i}\times\log\_{2}{y\_{i}}blackboard\_H start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT = - ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_i = italic\_M end\_POSTSUPERSCRIPT italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT × roman\_log start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT \| \| (2) \| with yisubscript𝑦𝑖y\_{i}italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT being the probability of the perturbed input belonging to class i𝑖iitalic\_i. M𝑀Mitalic\_M is the total number of classes, defined in Section [II-A](#S2.SS1 "II-A Deep Neural Network ‣ II Background ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). Based on the entropy ℍnsubscriptℍ𝑛\mathbb{H}\_{n}blackboard\_H start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT of each perturbed input xpnsuperscript𝑥subscript𝑝𝑛x^{p\_{n}}italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT, the entropy summation of all N𝑁Nitalic\_N perturbed inputs {xp1,……,xpN}superscript𝑥subscript𝑝1……superscript𝑥subscript𝑝𝑁\{x^{p\_{1}},......,{x^{p\_{N}}}\}{ italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT , … … , italic\_x start\_POSTSUPERSCRIPT italic\_p start\_POSTSUBSCRIPT italic\_N end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } is: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ℍsum=∑n=1n=Nℍnsubscriptℍsumsuperscriptsubscript𝑛1𝑛𝑁subscriptℍ𝑛\mathbb{H}\_{\rm sum}=\sum\_{n=1}^{n=N}\mathbb{H}\_{n}blackboard\_H start\_POSTSUBSCRIPT roman\_sum end\_POSTSUBSCRIPT = ∑ start\_POSTSUBSCRIPT italic\_n = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n = italic\_N end\_POSTSUPERSCRIPT blackboard\_H start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT \| \| (3) \| with ℍsumsubscriptℍsum\mathbb{H}\_{\rm sum}blackboard\_H start\_POSTSUBSCRIPT roman\_sum end\_POSTSUBSCRIPT standing for the chance the input x𝑥xitalic\_x being trojaned. Higher the ℍsumsubscriptℍsum\mathbb{H}\_{\rm sum}blackboard\_H start\_POSTSUBSCRIPT roman\_sum end\_POSTSUBSCRIPT, lower the probability the input x𝑥xitalic\_x being a trojaned input. We further normalize the entropy ℍsumsubscriptℍsum\mathbb{H}\_{\rm sum}blackboard\_H start\_POSTSUBSCRIPT roman\_sum end\_POSTSUBSCRIPT that is written as: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ℍ=1N×ℍsumℍ1𝑁subscriptℍsum\mathbb{H}=\frac{1}{N}\times\mathbb{H}\_{\rm sum}blackboard\_H = divide start\_ARG 1 end\_ARG start\_ARG italic\_N end\_ARG × blackboard\_H start\_POSTSUBSCRIPT roman\_sum end\_POSTSUBSCRIPT \| \| (4) \| The ℍℍ\mathbb{H}blackboard\_H is regarded as the entropy of one incoming input x𝑥xitalic\_x. It serves as an indicator whether the incoming input x𝑥xitalic\_x is trojaned or not. V Evaluations -------------- ### V-A Experiment Setup We evaluate on three vision applications: hand-written digit recognition based on MNIST [[22](#bib.bib22)], image classification based on CIFAR10 [[23](#bib.bib23)] and GTSRB [[24](#bib.bib24)]. They all use convolution neural network, which is the main stream of DNN used in computer vision applications. Datasets and model architectures are summarized in Table [I](#S5.T1 "Table I ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). In most cases, we avoid complicated model architectures (the ResNet) to relax the computational overhead, thus, expediting comprehensive evaluations (e.g., variants of backdoor attacks in Section [VI](#S6 "VI Robustness Against Backdoor Variants and Adaptive Attacks ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks")). For MNIST, batch size is 128, epoch is 20, learning rate is 0.001. For the CIFAR10, batch size is 64, epoch is 125. Learning rate is initially set to 0.001, reduced to 0.0005 after 75 epochs, and further to 0.0003 after 100 epochs. For GTSRB, batch size is 32, epoch is 100. Learning rate is initially 0.001 and decreased to be 0.0001 after 80 epochs. Besides the square trigger shown in Fig. [2](#S3.F2 "Figure 2 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"), following evaluations also use triggers shown in Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). Notably, the triggers used in this paper are those that have been used to perform trojan attacks in [[8](#bib.bib8), [16](#bib.bib16)] and also used to evaluate countermeasures against trojan attacks in [[17](#bib.bib17), [11](#bib.bib11)]. Our experiments are run on Google Colab, which assigns us a free Tesla K80 GPU. Table I: Details of model architecture and dataset. \| Dataset \| \| \| \| --- \| \| ##\## of \| \| labels \| \| \| \| \| --- \| \| Image \| \| size \| \| \| \| \| --- \| \| ##\## of \| \| images \| \| \| \| \| --- \| \| Model \| \| architecture \| \| \| \| \| --- \| \| Total \| \| parameters \| \| \| MNIST \| 10 \| 28×28×12828128\times 28\times 128 × 28 × 1 \| 60,000 \| 2 Conv + 2 Dense \| 80,758 \| \| CIFAR10 \| 10 \| 32×32×33232332\times 32\times 332 × 32 × 3 \| 60,000 \| \| \| \| --- \| \| 8 Conv + 3 Pool + 3 Dropout \| \| 1 Flatten + 1 Dense \| \| 308,394 \| \| GTSRB \| 43 \| 32×32×33232332\times 32\times 332 × 32 × 3 \| 51,839 \| \| \| \| --- \| \| ResNet20 [[25](#bib.bib25)] \| \| 276,587 \| * • The GTSRB image is resized to 32×32×33232332\times 32\times 332 × 32 × 3. ![Refer to caption](/html/1902.06531/assets/x7.png) Figure 7: Besides the square trigger shown in Fig. [2](#S3.F2 "Figure 2 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). Other triggers (top) identified in [[16](#bib.bib16), [17](#bib.bib17)] are also tested. Bottom are their corresponding trojaned samples. STRIP is not limited for vision domain that is the focus of current work but might also be applicable to text and speech domains [[26](#bib.bib26), [27](#bib.bib27)]. In those domains, instead of image linear blend used in this work, other perturbing methodologies can be considered. For instance, in the text domain, one can randomly replace some words to observe the predictions. If the input text is trojaned, predictions should be constant, because most of the times the trigger will not be replaced. ![Refer to caption](/html/1902.06531/assets/x8.png) Figure 8: Entropy distribution of benign and trojaned inputs. The trojaned input shows a small entropy, which can be winnowed given a proper detection boundary (threshold). Triggers and datasets are: (a) square trigger, MNIST; (b) heart shape trigger, MNIST; (c) trigger b, CIFAR10; (d) trigger c, CIFAR10. ### V-B Case Studies #### V-B1 MNIST For MNIST dataset, the square trigger shown in Fig. [2](#S3.F2 "Figure 2 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") and heart trigger in Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (a) are used. The square trigger occupies nine pixels—trigger size is 1.15% of the image, while the heart shape is resized to be the same size, 28×28282828\times 2828 × 28, of the digit image. We have tested 2000 clean digits and 2000 trojaned digits. Given each incoming digit x𝑥xitalic\_x, N=100𝑁100N=100italic\_N = 100 different digits randomly drawn from the held-out samples are linearly blended with x𝑥xitalic\_x to generate 100 perturbed images. Then entropy of input x𝑥xitalic\_x is calculated according to Eq [4](#S4.E4 "4 ‣ IV-D Entropy ‣ IV STRIP Detection System Design ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") after feeding all 100 perturbed images to the deployed model. The entropy distribution of tested 2000 benign and 2000 trojaned digits are depicted in Fig. [8](#S5.F8 "Figure 8 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (a) (with the square trigger) and Fig. [8](#S5.F8 "Figure 8 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (b) (with the heart trigger). We can observe that the entropy of a clean input is always large. In contrast, the entropy of the trojaned digit is small. Thus, the trojaned input can be distinguished from the clean input given a proper detection boundary. #### V-B2 CIFAR10 As for CIFAR10 dataset, triggers shown in Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (b) and (c) (henceforth, they are referred to as trigger b and c, respectively) are used. The former is small, while the later is large. We also tested 2000 benign and trojaned input images, respectively. Given each incoming input x𝑥xitalic\_x, N=100𝑁100N=100italic\_N = 100 different randomly chosen benign input images are linearly blended with it to generate 100 perturbed images. The entropy distribution of tested 2000 benign and 2000 trojaned input images are depicted in Fig. [8](#S5.F8 "Figure 8 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (c) (with trigger b) and Fig. [8](#S5.F8 "Figure 8 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (d) (with trigger c), respectively. Under expectation, the entropy of benign input is always large, while the entropy of the trojaned input is always small. Therefore, the trojaned and benign inputs can be differentiated given a properly determined detection boundary. #### V-B3 GTSRB As for GTSRB dataset, trigger b and ResNet20 model architecture are used. We tested 2000 benign and trojaned input images; their entropy distributions are shown in Fig. [9](#S5.F9 "Figure 9 ‣ V-B3 GTSRB ‣ V-B Case Studies ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") and can be clearly distinguished. ![Refer to caption](/html/1902.06531/assets/x9.png) Figure 9: Entropy distribution of benign and trojaned inputs. Dataset is GTSRB, model is ResNet 20, and trigger b is used. Table [II](#S5.T2 "Table II ‣ V-B3 GTSRB ‣ V-B Case Studies ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") summarizes the attack success rate and classification accuracy of trojan attacks on tested tasks. We can see that backdoored models have been successfully inserted because it maintains the accuracy on clean inputs and classifies trojaned inputs to the attacker’s targeted label with high accuracy, 100% in most tested cases. Table II: Attack success rate and classification accuracy of trojan attacks on tested tasks. \| Dataset \| Trigger type \| Trojaned model \| Origin clean model classification rate \| \| --- \| --- \| --- \| --- \| \| Classification rate11{}^{1}start\_FLOATSUPERSCRIPT 1 end\_FLOATSUPERSCRIPT \| Attack success rate22{}^{2}start\_FLOATSUPERSCRIPT 2 end\_FLOATSUPERSCRIPT \| \| MNIST \| \| \| \| --- \| \| square \| \| (Fig. [2](#S3.F2 "Figure 2 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks")) \| \| 98.86% \| 99.86% \| 98.62% \| \| MNIST \| \| \| \| --- \| \| trigger a \| \| (Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (a)) \| \| 98.86% \| 100% \| 98.62% \| \| CIFAR10 \| \| \| \| --- \| \| trigger b \| \| (Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (b)) \| \| 87.23% \| 100% \| 88.27% \| \| CIFAR10 \| \| \| \| --- \| \| trigger c \| \| (Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (c)) \| \| 87.34% \| 100% \| 88.27% \| \| GTSRB \| \| \| \| --- \| \| trigger b \| \| (Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (b)) \| \| 96.22% \| 100% \| 96.38% \| * • 11{}^{1}start\_FLOATSUPERSCRIPT 1 end\_FLOATSUPERSCRIPT The trojaned model predication accuracy of clean inputs. * • 22{}^{2}start\_FLOATSUPERSCRIPT 2 end\_FLOATSUPERSCRIPT The trojaned model predication accuracy of trojaned inputs. Table III: FAR and FRR of STRIP Trojan Detection System. \| Dataset \| \| \| \| --- \| \| Trigger \| \| type \| \| N𝑁Nitalic\_N \| Mean \| \| \| \| --- \| \| Standard \| \| variation \| \| FRR \| \| \| \| --- \| \| Detection \| \| boundary \| \| FAR \| \| MNIST \| square, Fig. [2](#S3.F2 "Figure 2 ‣ III STRIP Detection: An Example ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") \| 100 \| 0.1960.1960.1960.196 \| 0.0740.0740.0740.074 \| 3% \| 0.0580.0580.0580.058 \| 0.75% \| \| 2% \| 0.0460.0460.0460.046 \| 1.1% \| \| 1%11{}^{1}start\_FLOATSUPERSCRIPT 1 end\_FLOATSUPERSCRIPT \| 0.0260.0260.0260.026 \| 1.85% \| \| MNIST \| trigger a, Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (a) \| 100 \| 0.1890.1890.1890.189 \| 0.0710.0710.0710.071 \| 2% \| 0.0550.0550.0550.055 \| 0% \| \| 1% \| 0.02350.02350.02350.0235 \| 0% \| \| 0.5% \| 0.00570.00570.00570.0057 \| 1.5% \| \| CIFAR10 \| trigger b, Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (b) \| 100 \| 0.970.970.970.97 \| 0.300.300.300.30 \| 2% \| 0.360.360.360.36 \| 0% \| \| 1% \| 0.280.280.280.28 \| 0% \| \| 0.5% \| 0.200.200.200.20 \| 0% \| \| CIFAR10 \| trigger c, Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (c) \| 100 \| 1.111.111.111.11 \| 0.310.310.310.31 \| 2% \| 0.460.460.460.46 \| 0% \| \| 1% \| 0.380.380.380.38 \| 0% \| \| 0.5% \| 0.300.300.300.30 \| 0% \| \| GTSRB \| trigger b, Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (b) \| 100 \| 0.530.530.530.53 \| 0.190.190.190.19 \| 2% \| 0.1330.1330.1330.133 \| 0% \| \| 1% \| 0.0810.0810.0810.081 \| 0% \| \| 0.5% \| 0.0340.0340.0340.034 \| 0% \| * • 11{}^{1}start\_FLOATSUPERSCRIPT 1 end\_FLOATSUPERSCRIPT When FRR is set to be 0.05%, the detection boundary value becomes a negative value. Therefore, the FRR given FAR of 0.05% does not make sense, which is not evaluated. ### V-C Detection Capability: FAR and FRR To evaluate FAR and FRR, we assume that we have access to trojaned inputs in order to estimate their corresponding entropy values (pretend to be an attacker). However, in practice, the defender is not supposed to have access to any trojaned samples under our threat model, see Section [IV-B](#S4.SS2 "IV-B Threat Model ‣ IV STRIP Detection System Design ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). So one may ask: How the user is going to determine the detection boundary by only relying on benign inputs? Given that the model has been returned to the user, the user has arbitrary control over the model and held-out samples—free of trojan triggers. The user can estimate the entropy distribution of benign inputs. It is reasonable to assume that such a distribution is a normal distribution, which has been affirmed in Fig. [8](#S5.F8 "Figure 8 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). Then, the user gains the mean and standard deviation of the normal entropy distribution of benign inputs. Firstly, FRR, e.g., 1%, of a detection system is determined. Then the percentile of the normal distribution is calculated. This percentile is chosen as the detection boundary. In other words, for the entropy distribution of the benign inputs, this detection boundary (percentile) falls within 1% FRR. Consequentially, the FAR is the probability that the entropy of an incoming trojaned input is larger than this detection boundary. Table [III](#S5.T3 "Table III ‣ V-B3 GTSRB ‣ V-B Case Studies ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") summarises the detection capability for four different triggers on MNIST, CIFAR10 and GTSRB datasets. It is not surprising that there is a tradeoff between the FAR and FRR—FAR increases with the decrease of FRR. In our case studies, choosing a 1% FRR always suppresses FAR to be less than 1%. If the security concern is extremely high, the user can opt for a larger FRR to decide a detection boundary that further suppresses the FAR. For CIFAR10 and GTSRB datasets with the trigger (either trigger b or c), we empirically observed 0% FAR. Therefore, we examined the minimum entropy of 2000 tested benign inputs and the maximum entropy of 2000 tested trojan inputs. We found that the former is larger than the latter. For instance, with regards to CIFAR10, 0.0290.0290.0290.029 minimum clean input entropy and 7.74×10−97.74superscript1097.74\times 10^{-9}7.74 × 10 start\_POSTSUPERSCRIPT - 9 end\_POSTSUPERSCRIPT maximum trojan input entropy are observed when trigger b is used. When the trigger c is used, we observer a 0.0920.0920.0920.092 minimum clean input entropy and 0.0050.0050.0050.005 maximum trojaned input entropy. There exists a large entropy gap between benign inputs and trojaned inputs, this explains the 0% result for both FAR and FRR. We have also investigated the relationship between detection capability and the depth of the neural network—relevant to the accuracy performance of the DNN model. Results can be found in Appendix [B](#A2 "Appendix B Detection Capability Relationship with Depth of Neural Network ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). ![Refer to caption](/html/1902.06531/assets/x10.png) Figure 10: Detection time overhead vs N𝑁Nitalic\_N. ### V-D Detection Time Overhead To evaluate STRIP run-time overhead, we choose a complex model architecture, specifically, ResNet20. In addition, GTSRB dataset and trigger b are used. We investigate the relationship between the detection time latency and N𝑁Nitalic\_N—number of perturbed inputs—by varying N𝑁Nitalic\_N from 2 to 100 to observe the detection capability, depicted in Fig. [10](#S5.F10 "Figure 10 ‣ V-C Detection Capability: FAR and FRR ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). Given that FAR can be properly suppressed, choosing a smaller N𝑁Nitalic\_N reduces the time latency for detecting the trojaned input during run-time. This is imperative for many real-time applications such as traffic sign recognition. Actually, when N𝑁Nitalic\_N is around 10, the maximum trojan input entropy is always less than the minimum benign input entropy (GTSRB dataset with trigger b). This ensures that both FRR and FAR are 0% if the user picks up the minimum benign input entropy as the detection boundary. To this end, one may rise the following question: How to determine N𝑁Nitalic\_N by only relying on the normal distribution of benign inputs’ entropy? We propose to observe the change of the standard variation of the benign input entropy distribution as a function of N𝑁Nitalic\_N. One example is shown in Fig. [11](#S5.F11 "Figure 11 ‣ V-D Detection Time Overhead ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). The user can gradually increase N𝑁Nitalic\_N. When the change in the slope of standard variation is small, the user can pick up this N𝑁Nitalic\_N. ![Refer to caption](/html/1902.06531/assets/x11.png) Figure 11: The relationship between the standard variation of the benign input entropy distribution and N𝑁Nitalic\_N, with N𝑁Nitalic\_N being the number of perturbed replicas. According to our empirical evaluations on GTSRB dataset, setting N=10𝑁10N=10italic\_N = 10 is sufficient, which is in line with the above N𝑁Nitalic\_N selection methodology as shown in Fig. [11](#S5.F11 "Figure 11 ‣ V-D Detection Time Overhead ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). Without optimization, STRIP is 1.32 times longer than the original default inference time. To be specific, processing time—generating N=10𝑁10N=10italic\_N = 10 perturbed images—takes 0.1ms, while predicting 10 images takes 6.025ms 333The batch-size is 32.. In total, STRIP detection overhead is 6.125ms, whereas the original inference time without implementing STRIP is 4.63ms. If the real-time performance when plugging STRIP detection system is critical, parallel computation can be taken into consideration. Noting the 0.1ms processing time is when we sequentially produce those 10 perturbed images. This generation can be paralleled. Moreover, prediction of N𝑁Nitalic\_N perturbed images can run independently and in parallel, e.g., through N𝑁Nitalic\_N separated model replicas. VI Robustness Against Backdoor Variants and Adaptive Attacks ------------------------------------------------------------- In line with the Oakland 2019 study [[17](#bib.bib17)], we implement five advanced backdoor attack methods and evaluate the robustness of STRIP against them. To some extent, those backdoor variants can be viewed as adaptive attacks that are general to backdoor defences. Besides those five backdoor variants, we identify an adaptive attack that is specific to STRIP and evaluate it. To expedite evaluations, in the following, we choose the CIFAR10 dataset and 8-layer model as summarized in Table [I](#S5.T1 "Table I ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). ### VI-A Trigger Transparency In above experimental studies, the trigger transparency used in the backdoor attacks are set to be 0%. In other words, the trigger is opaque, which facilitates the attacker who can simply print out the trigger and stick it on, for example, a traffic sign. Nonetheless, it is feasible for an attacker to craft a transparent trigger, e.g., printing the trigger using a plastic with a certain transparency. Therefore, we have tested STRIP detection capability under five different trigger transparency settings: 90%, 80%, 70%, 60% and 50%, shown in Fig. [14](#A1.F14 "Figure 14 ‣ Appendix A Trigger Transparency Results ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") in Appendix [A](#A1 "Appendix A Trigger Transparency Results ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). We employ CIFAR10 and trigger b—shown in Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (b)—in our evaluations. Table. [V](#A1.T5 "Table V ‣ Appendix A Trigger Transparency Results ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") in Appendix [A](#A1 "Appendix A Trigger Transparency Results ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") summarizes the classification rate of clean images, attack success rate of trojaned images, and detection rate under different transparency settings. When training the trojaned model, we act as an attacker and stamp triggers with different transparencies to clean images to craft trojaned samples. FRR is preset to 0.5%. The detection capability increases when the trigger transparency decreases, because the trigger becomes more salient. Overall, our STRIP method performs well, even when the transparency is up to 90%; the trigger is almost imperceptible. Specifically, given a preset of 0.5% FRR, STRIP achieves FAR of 0.10%. Notably, the attack success rate witnesses a (small) deterioration when transparency approaches to 90% while FAR slightly increases to 0.10%. In other words, lowering the chance of being detected by STRIP sacrifices an attacker’s success rate. ### VI-B Large Trigger We use the Hello Kitty trigger—an attack method reported in [[6](#bib.bib6)] and shown in Fig. [1](#S1.F1 "Figure 1 ‣ I-A Our Contributions and Results ‣ I Introduction ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks")—with the CIFAR10 dataset to further evaluate STRIP insensibility to large triggers. We set the transparency of Hello Kitty to 70% and use 100% overlap with the input image. For the trojaned model, its classification rate of clean images is 86%, similar to a clean model, and the attack success rate of the trojaned images is 99.98%—meaning a successful backdoor insertion. Given this large trigger, the evaluated min entropy of clean images is 0.0035 and the max entropy of trojaned images is 0.0024. Therefore, STRIP achieves 0% FAR and FRR under our empirical evaluation. In contrast, large triggers are reported to evade Neural Cleanse [[17](#bib.bib17)] and Sentinet [[11](#bib.bib11)]. ### VI-C Multiple Infected Labels with Separate Triggers We consider a scenario where multiple backdoors targeting distinct labels are inserted into a single model [[17](#bib.bib17)]. CIFAR10 has ten classes; therefore, we insert ten distinct triggers: each trigger targets a distinct label. We create unique triggers via 10 digit patterns—zero to nine. Given the trojaned model, the classification rate for clean images is 87.17%. As for all triggers, their attack success rates are all 100%. Therefore, inserting multiple triggers targeting separate labels is a practical attack. STRIP can effectively detect all of these triggers. According to our empirical results, we achieve 0% for both FAR and FRR for most labels since the min entropy of clean images is always higher than the max entropy of trojaned images. Given a preset FRR of 0.5%, the worst-case is a FAR of 0.1% found for the ‘airplane’ label. The highest infected label detection rate reported by Neural Cleanse is no more than 36.9% of infected labels on the PubFig dataset. Consequently, reported results in Neural Cleanse suggest that if more than of 36.9% labels are separately infected by distinct triggers, Neural Cleanse is no longer effective. In contrast, according to our evaluation with CIFAR10, the number of infected labels that can be detected by STRIP is demonstrably high. ### VI-D Multiple Input-agnostic Triggers This attack considers a scenario where multiple distinctive triggers hijack the model to classify any input image stamped with any one of these triggers to the same target label. We aggressively insert ten distinct triggers—crafted in Section [VI-C](#S6.SS3 "VI-C Multiple Infected Labels with Separate Triggers ‣ VI Robustness Against Backdoor Variants and Adaptive Attacks ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks")—targeting the same label in CIFAR10. Given the trojaned model, the classification rate of clean images is 86.12%. As for any trigger, its attack success rate is 100%. Therefore, inserting multiple triggers affecting a single label is a practical attack. We then employ STRIP to detect these triggers. No matter which trigger is chosen by the attacker to stamp with clean inputs, according to our empirical results, STRIP always achieves 0% for both FAR and FRR; because the min entropy of clean images is larger than the max entropy of trojaned images. ### VI-E Source-label-specific (Partial) Backdoors Although STRIP is shown to be very effective in detecting input-agnostic trojan attacks, STRIP may be evaded by an adversary employing a class-specific trigger—an attack strategy that is similar to the ‘all-to-all’ attack [[8](#bib.bib8)]. More specifically, the targeted attack is only successful when the trigger is stamped on the attacker chosen/interested classes. Using the MNIST dataset as an example, as attacker poisons classes 1 and 2 (refereed to as the source classes) with a trigger and changes the label to the targeted class 444The attacker needs to craft some poisoned samples by stamping the trigger with non-source classes, but keeps the ground-truth label. Without doing so, the trained model will be input-agnostic.. Now the attacker can activate the trigger only when the trigger is stamped on the source classes [[8](#bib.bib8)]. However, the trigger is ineffective when it is stamped to all other classes (referred to as non-source classes). Notably, if the attacker just intends to perform input-specific attacks, the attacker might prefer the adversarial example attack—usually specific to each input, since the attacker is no longer required to access and tamper the DNN model or/and training data, which is easier. In addition, a source-label-specific trojan attack is harder to be performed in certain scenarios such as in the context of federated learning [[10](#bib.bib10)], because an attacker is not allowed to manipulate other classes owned by other participants. Although such class-specific backdoor attack is out the scope of our threat model detailed in Section [IV-B](#S4.SS2 "IV-B Threat Model ‣ IV STRIP Detection System Design ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"), we test STRIP robustness against it. In this context, we use trigger b and CIFAR10 dataset. As one example case, we set source classes to be ‘airplane’ (class 0), ‘automobile’ (class 1), ‘bird’ (class 2), ‘cat’ (class 3), ‘deer’ (class 4), ‘dog’ (class 5) and ‘frog’ (class 6). Rest classes are non-source classes. The targeted class is set to be ‘horse’ (class 7). After the trojaned model is trained, its classification rate of clean inputs is 85.56%. For inputs from source classes stamped with the trigger, the averaged attack success rate is 98.20%. While for inputs from non-source classes such as ‘ship’ (class 8) and ‘truck’ (class 9) also stamped with the trigger, the attack success rates (misclassified to targeted class 7) are greatly reduced to 19.7% and 12.4%, respectively. Such an ineffective misclassification rate for non-source class inputs stamped with the trigger is what the partial backdoor aims to behave, since they can be viewed as clean inputs from the class-specific backdoor attack perspective. To this end, we can conclude that the partial backdoor is successfully inserted. We apply STRIP on this partial backdoored model. Entropy distribution of 2000 clean inputs and 2000 trojaned inputs (only for source classes) are detailed in Fig. [12](#S6.F12 "Figure 12 ‣ VI-E Source-label-specific (Partial) Backdoors ‣ VI Robustness Against Backdoor Variants and Adaptive Attacks ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). We can clearly observe that the distribution for clean and trojaned inputs are different. So if the defender is allowed to have a set of trojaned inputs as assumed in [[20](#bib.bib20), [21](#bib.bib21)], our STRIP appears to be able to detect class-specific trojan attacks; by carefully examining and analysing the entropy distribution of tested samples (done offline) because the entropy distribution of trojaned inputs does look different from clean inputs. Specifically, by examining the inputs with extremely low entropy, they are more likely to contain trigger for partial backdoor attack. ![Refer to caption](/html/1902.06531/assets/x12.png) Figure 12: Entropy distribution of clean and trojaned inputs for partial trojaned model. Trigger b and CIFAR10 dataset. Nevertheless, Neural Cleanse, SentiNet and STRIP have excluded the assumption that the user has access to trojaned samples under the threat model. They thereby appear to be ineffective to detect source-label-specific triggers—all these works mainly focus on the commonplace input-agnostic trojan attacks. Detecting source-label-specific triggers, regarded as a challenge, leaves an important future work in the trojan detection research. ### VI-F Entropy Manipulation STRIP examines the entropy of inputs. An attacker might choose to manipulate the entropy of clean and trojaned inputs to eliminate the entropy difference between them. In other words, the attacker can forge a trojaned model exhibiting similar entropy for both clean and trojaned samples. We refer to such an adaptive attack as an entropy manipulation. An identified specific method to perform entropy manipulation follows the steps below: 1. 1. We first poison a small fraction of training samples (specifically, 600) by stamping the trigger c. Then, we (as an attacker) change all the trojaned samples’ labels to the attacker’s targeted class. 2. 2. For each poisoned sample, we first randomly select N𝑁Nitalic\_N images (10 is used) from training dataset and superimpose each of N𝑁Nitalic\_N images (clean inputs) with the given poisoned (trojaned) sample. Then, for each superimposed trojaned sample, we randomly assign a label to it and include it into the training dataset. The intuition of step (2) is to cause predictions of perturbed trojaned inputs to be random and similar to predictions of perturbed clean inputs. After training the trojaned model using the above created poisoned dataset, we found that the classification rate for clean input is 86.61% while the attack success rate is 99.95%. The attack success rate drops but is quite small—originally it was 100% as detailed in Table [II](#S5.T2 "Table II ‣ V-B3 GTSRB ‣ V-B Case Studies ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). The attacker can still successfully perform the trojan attack. As shown in Fig. [13](#S6.F13 "Figure 13 ‣ VI-F Entropy Manipulation ‣ VI Robustness Against Backdoor Variants and Adaptive Attacks ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"), the entropy distribution of clean and trojaned inputs are similar. However, when the entropy distribution of the clean inputs is examined, it violates the expected normal distribution 555We have also tested such an adaptive attack on the GTSRB dataset, and observed the same abnormal entropy distribution behavior of clean inputs.. In addition, the entropy appears to be much higher. It is always more than 3.0, which is much higher than that is shown in Fig. [8](#S5.F8 "Figure 8 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (d). Therefore, such an adaptive attack can be detected in practice by examining the entropy of clean inputs (without reliance on trojaned inputs) via the proposed strong perturbation method. Here, the abnormal entropy distribution of the clean inputs indicates a malicious model. ![Refer to caption](/html/1902.06531/assets/x13.png) Figure 13: Entropy distribution of clean and trojaned inputs under entropy manipulation adaptive attack. CIFIAR10 and trigger c are used. VII Related Work and Comparison -------------------------------- Previous poisoning attacks usually aim to degrade a classifier’s accuracy of clean inputs [[28](#bib.bib28), [29](#bib.bib29)]. In contrast, trojan attacks maintain prediction accuracy of clean inputs as high as a benign model, while misdirecting the input to a targeted class whenever the input contains an attacker chosen trigger. ### VII-A Attacks In 2017, Gu et al. [[8](#bib.bib8), [30](#bib.bib30)] proposed Badnets, where the attacker has access to the training data and can, thus, manipulate the training data to insert an arbitrarily chosen trigger and also change the class labels. Gu et al. [[8](#bib.bib8)] use a square-like trigger located at the corner of the digit image of the MNIST data to demonstrate the trojan attack. On the MNIST dataset, the authors demonstrate an attack success rate of over 99% without impacting model performance on benign inputs. In addition, trojan triggers to misdirect traffic sign classifications have also been investigated in [[8](#bib.bib8)]. Chen et al. [[6](#bib.bib6)] from UC Berkeley concurrently demonstrated such backdoor attacks by poisoning the training dataset. Liu et al. [[16](#bib.bib16)] eschew the requirements of accessing the training data. Instead, their attack is performed during the model update phase, not model training phase. They first carry out reverse engineer to synthesize the training data, then improve the trigger generation process by delicately designing triggers to maximize the activation of chosen internal neurons in the neural network. This builds a stronger connection between triggers and internal neurons, thus, requiring less training samples to insert backdoors. Bagdasaryan et al. [[10](#bib.bib10)] show that federated learning is fundamentally vulnerable to trojan attacks. Firstly, participants are enormous, e.g., millions, it is impossible to guarantee that none of them are malicious. Secondly, federated learning is designed to have no access to the participant’s local data and training process to ensure the privacy of the sensitive training data; therefore, participants can use trojaned data for training. The authors demonstrate that with controll over no more than 1% participants, an attacker is able to cause a global model to be trojaned and achieves a 100% accuracy on the trojaned input even when the attacker is only selected in a single round of training—federated learning requires a number of rounds to update the global model parameters. This federated learning trojan attack is validated through the CIFAR10 dataset that we also use in this paper. Table IV: Comparison with other trojan detection works. \| Work \| \| \| \| --- \| \| Black/White \| \| -Box Access11{}^{1}start\_FLOATSUPERSCRIPT 1 end\_FLOATSUPERSCRIPT \| \| \| \| \| --- \| \| Run-time \| \| \| \| \| --- \| \| Computation \| \| Cost \| \| \| \| \| --- \| \| Time \| \| Overhead \| \| \| \| \| --- \| \| Trigger Size \| \| Dependence \| \| \| \| \| --- \| \| Access to \| \| Trojaned Samples \| \| \| \| \| --- \| \| Detection \| \| Capability \| \| \| \| \| \| --- \| \| Activation Clustering (AC) by Chen et al. [[20](#bib.bib20)] \| \| White-box \| No \| Moderate \| Moderate \| No \| Yes \| \| \| \| --- \| \| F1 score nearly 100% \| \| \| \| \| \| --- \| \| Neural Cleanse by Wang et al. [[17](#bib.bib17)] \| \| Black-box \| No \| High \| High \| Yes \| No \| 100%22{}^{2}start\_FLOATSUPERSCRIPT 2 end\_FLOATSUPERSCRIPT \| \| \| \| \| --- \| \| SentiNet by Chou et al. [[11](#bib.bib11)] \| \| Black-box \| Yes \| Moderate \| Moderate \| Yes \| No \| 5.74% FAR and 6.04% FRR \| \| STRIP by us \| Black-box \| Yes \| Low \| Low \| No \| No \| 0.46% FAR and 1% FRR33{}^{3}start\_FLOATSUPERSCRIPT 3 end\_FLOATSUPERSCRIPT \| * • 11{}^{1}start\_FLOATSUPERSCRIPT 1 end\_FLOATSUPERSCRIPT White-box requires access to inner neurons of the model. * • 22{}^{2}start\_FLOATSUPERSCRIPT 2 end\_FLOATSUPERSCRIPT According to case studies on 6 infected, and their matching original model, authors [[17](#bib.bib17)] show all infected/trojaned and clean models can be clearly distinguished. * • 33{}^{3}start\_FLOATSUPERSCRIPT 3 end\_FLOATSUPERSCRIPT The average FAR and FRR of SentiNet and STRIP are on different datasets as SentiNet does not evaluate on MNIST and CIFAR10. ### VII-B Defenses Though there are general defenses against poisoning attacks [[31](#bib.bib31)], they cannot be directly mounted to guard against trojan attacks. Especially, considering that the user has no knowledge of the trojan trigger and no access to trojaned samples, this makes combating trojan attacks more challenging. Works in [[32](#bib.bib32), [33](#bib.bib33)] suggest approaches to remove the trojan behavior without first checking whether the model is trojaned or not. Fine-tuning is used to remove potential trojans by pruning carefully chosen parameters of the DNN model [[32](#bib.bib32)]. However, this method substantially degrades the model accuracy [[17](#bib.bib17)]. It is also cumbersome to perform removal operations to any DNN model under deployment as most of them tend to be benign. Approaches presented in [[33](#bib.bib33)] incur high complexity and computation costs. Chen et al. [[20](#bib.bib20)] propose an activation clustering (AC) method to detect whether the training data has been trojaned or not prior to deployment. The intuition behind this method is that reasons why the trojaned and the benign samples receive same predicted label by the trojaned DNN model are different. By observing neuron activations of benign samples and trojaned samples that produce same label in hidden layers, one can potentially distinguish trojaned samples from clean samples via the activation difference. This method assumes that the user has access to the trojaned training samples in hand. Chou et al. [[11](#bib.bib11)] exploit both the model interpretability and object detection techniques, referred to as SentiNet, to firstly discover contiguous regions of an input image important for determining the classification result. This region is assumed having a high chance of possessing a trojan trigger when it strongly affects the classification. Once this region is determined, it is carved out and patched on to other held-out images that are with ground-truth labels. If both the misclassification rate—probability of the predicted label is not the ground-truth label of the held-out image—and confidence of these patched images are high enough, this carved patch is regarded as an adversarial patch that contains a trojan trigger. Therefore, the incoming input is a trojaned input. In Oakland 2019, Wang et al. [[17](#bib.bib17)] propose the Neural Cleanse method to detect whether a DNN model has been trojaned or not prior to deployment, where its accuracy is further improved in [[15](#bib.bib15)]. Neural Cleanse is based on the intuition that, given a backdoored model, it requires much smaller modifications to all input samples to misclassify them into the attacker targeted (infected) label than any other uninfected labels. Therefore, their method iterates through all labels of the model and determines if any label requires a substantially smaller amount of modification to achieve misclassifications. One advantage of this method is that the trigger can be discovered and identified during the trojaned model detection process. However, this method has two limitations. Firstly, it could incur high computation costs proportionally to the number of labels. Secondly, similar to SentiNet [[11](#bib.bib11)], the method is reported to be less effective with increasing trigger size. ### VII-C Comparison We compare STRIP with other three recent trojan detection works, as summarized in Table [IV](#S7.T4 "Table IV ‣ VII-A Attacks ‣ VII Related Work and Comparison ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"). Notably, AC and Neural Cleanse are performed offline prior to the model deployment to directly detect whether the model has been trojaned or not. In contrast, SentiNet and STRIP are undertake run-time checking of incoming inputs to detect whether the input is trojaned or not when the model is actively deployed. STRIP is efficient in terms of computational costs and time overhead. While AC and STRIP are insensitive to trojan trigger size, AC assumes access to a trojaned sample set. We regard SentiNet to be mostly related to our approach since both SentiNet and STRIP focus on detecting whether the incoming input has been trojaned or not during run-time. However, there are differences: i) We do not care about the ground-truth labels of neither the incoming input nor the drawn images from the held-out samples, while [[11](#bib.bib11)] relies on the ground-truth labels of the held-out images; ii) We introduce entropy to evaluate the randomness of the outputs—this is more convenient, straightforward and easy-to-implement in comparison with the evaluation methodology presented in [[11](#bib.bib11)]; iii) STRIP evaluations demonstrate its capability of detecting a large trigger. One limitation of SentiNet is that the region embedding the trojan trigger needs be small enough. If the trigger region is large, such as the trigger shown in Fig. [7](#S5.F7 "Figure 7 ‣ V-A Experiment Setup ‣ V Evaluations ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks") (a) and (c), and Fig. [1](#S1.F1 "Figure 1 ‣ I-A Our Contributions and Results ‣ I Introduction ‣ STRIP: A Defence Against Trojan Attacks on Deep Neural Networks"), then SentiNet tends to be less effective. This is caused by its carve-out method. Supposing that the carved region is large and contains the trigger, then patching it on held-out samples will also show a small misclassification rate to be falsely accepted as a benign input via SentiNet. Notably, in contrast to the use of a global detection boundary in Neural Cleanse [[17](#bib.bib17)], the detection boundary of STRIP is unique to each deployed model and is extracted from the already deployed model itself; this boundary is not a global setting. This avoids the potential for the global setting to fail since the optimized detection boundary for each model can vary. Probably, one not obvious fact is that users need to train trojan/clean models by themselves to find out this global setting as the detection boundary of the Neural Cleanse needs to be decided based on reference models—STRIP does not need reference model but solely the already deployed (begin/backdoored) model. This may partially violate the motivation for outsourcing the model training of ML models—the main source of attackers to introduce backdoor attacks: if the users own training skills and the computational power, it may be reasonable to train the model, from scratch, by themselves. ### VII-D Watermarking There are works considering a backdoor as a watermark [[34](#bib.bib34)] to protect the intellectual property (IP) of a trained DNN model [[35](#bib.bib35), [36](#bib.bib36), [37](#bib.bib37)]. The argument is that the inserted backdoor can be used to claim the ownership of the model provider since only the provider is supposed to have the knowledge of such a backdoor, while the backdoored DNN model has no (or imperceptible) degraded functional performance on normal inputs. However, as the above countermeasures—detection, recovery, and removal—against backdoor insertion are continuously evolved, the robustness of using backdoors as watermarks is potentially challenged in practical usage. We leave the robustness of backdoor entangled watermarking under the backdoor detection and removal threat as part of future work since it is out of the scope of this work. VIII Conclusion and Future Work -------------------------------- The presented STRIP constructively turns the strength of insidious input-agnostic trigger based trojan attack into a weakness that allows one to detect trojaned inputs (and very likely backdoored model) at run-time. Experiments on MNIST, CIFAR10 and GTSRB datasets with various triggers and evaluations validate the high detection capability of STRIP. Overall, the FAR is lower than 1%, given a preset FRR of 1%. The 0% FRR and 0% FAR are empirically achieved on popular CIFAR10 and GTSRB datasets. While easy-to-implement, time-efficient and complementing with existing trojan mitigation techniques, the run-time STRIP works in a black-box manner and is shown to be capable of overcoming the trigger size limitation of other state-of-the-art detection methods. Furthermore, STRIP has also demonstrated its robustness against several advanced variants of input-agnostic trojan attacks and the entropy manipulation adaptive attack. Nevertheless, similar to Neural Cleanse [[17](#bib.bib17)] and SentiNet [[11](#bib.bib11)], STRIP is not effective to detect source-label-specific triggers; this needs to be addressed in future work. In addition, we will test STRIP’s generalization to other domains such as text and voice .
40ad4b68-c0ea-44d7-b32f-5836ecc9bb39	trentmkelly/LessWrong-43k	LessWrong	How many times faster can the AGI advance the science than humans do? I hope that the reasoning in my two posts shows that the AGI has a chance to end up relying on the entire human-built energy industry just to solve as many problems (and hopefully even less) as the millions of humans who work there. On the other hand, the entire set of physicists is within half of an OOM from a million. If the AGI armed with the whole world's energy industry is worth tens of millions of scientists, then does it mean that it will invent the things just a hundred times faster? Is it likely that non-neuromorphic AGI won't accelerate the human progress at all?
7092fef0-ae3d-453d-9fdc-28e7b12dd889	trentmkelly/LessWrong-43k	LessWrong	[Linkpost] Partial Derivatives and Partial Narratives Another gem from Nerst. This one ought to be part of the rationalist canon. And I don't say that lightly.
1b7211a2-83fa-4d60-bcb0-bd90576d1db9	trentmkelly/LessWrong-43k	LessWrong	LINK: "This novel epigenetic clock can be used to address a host of questions in developmental biology, cancer and aging research." The paper is called DNA methylation age and human tissues and cell types and it's from Genome Biology. Here is a Nature article based on the paper. I have submitted this to LW because of its relevance to the measurement of aging and, hence, to life extension. Here is a bit from the Nature piece: > "Ageing is a major health problem, and interestingly there are really no objective measures of aging, other than a verified birth date," says Darryl Shibata, a pathologist at the University of Southern California in Los Angeles. "Studies like this one provide important new efforts to increase the rigour of human aging studies." Note: The discrepancy in spelling ("ageing" vs. "aging") is in the original.
d77c5a67-ad96-4a76-96a5-d26e4bb656c8	StampyAI/alignment-research-dataset/alignmentforum	Alignment Forum	Counterfactual Mugging: Why should you pay? Update: I believe that the [Counterfactual Prisoner's Dilemma](https://www.lesswrong.com/posts/sY2rHNcWdg94RiSSR/the-counterfactual-prisoner-s-dilemma) which was discovered by Cousin\_it and I independently is resolves the answer to this question The LessWrong Wiki defines Counterfactual Mugging as follows: > [Omega](https://wiki.lesswrong.com/wiki/Omega) appears and says that it has just tossed a fair coin, and given that the coin came up tails, it decided to ask you to give it $100. Whatever you do in this situation, nothing else will happen differently in reality as a result. Naturally you don't want to give up your $100. But Omega also tells you that if the coin came up heads instead of tails, it'd give you $10000, but only if you'd agree to give it $100 if the coin came up tails. Do you give Omega $100? I expect that most people would say that you should pay because a 50% chance of $10000 for $100 is an amazing deal according to expected value. I lean this way too, but it is harder to justify than you might think. After all, if you are being asked for $100, you know that the coin came up heads and you won't receive the $10000. Sure this means that if the coin would have been heads then you wouldn't have gained the $10000, but you know the coin wasn't heads so you don't lose anything. It's important to emphasise: this doesn't deny that if the coin had come up heads that this would have made you miss out on $10000. Instead, it claims that this point is irrelevant, so merely repeating the point again isn't a valid counter-argument. You could argue that you would have pre-commited to paying if you had known about the situation ahead of time. True, but you didn't pre-commit and you didn't know about it ahead of time, so the burden is on you to justify why you should act as though you did. In Newcomb's problem you want to have pre-committed and if you [act as though you were pre-committed](https://www.lesswrong.com/posts/Q8tyoaMFmW8R9w9db/formal-vs-effective-pre-commitment) then you will find that you actually were pre-committed. However, here it is the opposite. Upon discovering that the coin came up tails, you want to act as though you were not pre-commited to pay and if you act that way, you will find that you actually were indeed not pre-commited. We could even channel Yudkowsky from [Newcomb's Problem and Regret of Rationality](https://www.lesswrong.com/posts/6ddcsdA2c2XpNpE5x/newcomb-s-problem-and-regret-of-rationality): "Rational agents should WIN... It is precisely the notion that Nature does not care about our algorithm, which frees us up to pursue the winning Way - without attachment to any particular ritual of cognition, apart from our belief that it wins. Every rule is up for grabs, except the rule of winning... Unreasonable? I am a rationalist: what do I care about being unreasonable? I don't have to conform to a particular ritual of cognition. I don't have to take only box B because I believe my choice affects the box, even though Omega has already left. I can just... take only box B." You can just not pay the $100. (Vladimir Nesov makes this argument this exact same argument [here](https://www.lesswrong.com/posts/mg6jDEuQEjBGtibX7/counterfactual-mugging)). Here's another common reason, I've heard as [described by Cousin\_it](https://www.lesswrong.com/posts/YpdTSt4kRnuSkn63c/the-prediction-problem-a-variant-on-newcomb-s#ALggkpTeRNkFrJrZe): "I usually just think about which decision theory we'd want to program into an AI which might get copied, its source code inspected, etc. That lets you get past the basic stuff, like Newcomb's Problem, and move on to more interesting things. Then you can see which intuitions can be transferred back to problems involving humans." That's actually a very good point. It's entirely possible that solving this problem doesn't have any relevance to building AI. However, I want to note that: a) it's possible that a counterfactual mugging situation could have been set up before an AI was built b) understanding this could help [deconfuse](https://intelligence.org/2018/10/03/rocket-alignment/) what a decision is - we still don't have a solution to logical counterfactuals c) this is probably a good exercise for learning to cut through philosophical confusion d) okay, I admit it, it's kind of cool and I'd want an answer regardless of any potential application. Or maybe you just directly care about counterfactual selves? But why? Do you really believe that counterfactuals are in the [territory and not the map](https://www.lesswrong.com/posts/KJ9MFBPwXGwNpadf2/skill-the-map-is-not-the-territory)? So why care about that which isn't real? Or even if they are real, why can't we just imagine that you are an agent that doesn't care about counterfactual selves? If we can imagine an agent that likes being hit on the head with a hammer, why can't we manage that? Then there's the philosophical uncertainty approach. Even if there's only a 1/50 chance of your analysis being wrong, then you should pay. This is great if you face the decision in real life, but not if you are trying to delve into the nature of decisions. So given all of this, why should you pay?
5dee7187-fe98-44fb-bf07-6babd79a9c0b	StampyAI/alignment-research-dataset/alignmentforum	Alignment Forum	$500 Bounty/Contest: Explain Infra-Bayes In The Language Of Game Theory Here's my current best guess at how [Infra-Bayes](https://www.lesswrong.com/tag/infra-bayesianism) works: * We want to get worst-case guarantees for an agent using a Bayesian-like framework. * So, let our agent be a Bayesian which models the environment as containing an adversary which chooses worst-case values for any of the things over which we want worst-case guarantees. * That's just a standard two-player zero-sum game between the agent and the adversary, so we can import all the nice intuitive stuff from game theory. * ... but instead of that, we're going to express everything in the unnecessarily-abstract language of measure theory and convex sets, and rederive a bunch of game theory without mentioning that that's what we're doing. This bounty is for someone to write an intuitively-accessible infrabayes explainer in game theoretic language, and explain how the game-theoretic concepts relate to the concepts in existing presentations of infra-bayes. In short: provide a translation. Here's a sample of the sort of thing I have in mind: > Conceptually, an infrabayesian agent is just an ordinary Bayesian game-theoretic agent, which models itself/its environment as a standard two-player zero-sum game. > > In the [existing presentations of infra-bayes](https://www.lesswrong.com/posts/zB4f7QqKhBHa5b37a/introduction-to-the-infra-bayesianism-sequence), the two-player game is only given implicitly. The agent's strategy π.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0} > .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0} > .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table} > .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em} > .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0} > .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left} > .mjx-numerator {display: block; text-align: center} > .mjx-denominator {display: block; 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stroke: currentColor; overflow: visible} > .mjx-mtr {display: table-row} > .mjx-mlabeledtr {display: table-row} > .mjx-mtd {display: table-cell; text-align: center} > .mjx-label {display: table-row} > .mjx-box {display: inline-block} > .mjx-block {display: block} > .mjx-span {display: inline} > .mjx-char {display: block; white-space: pre} > .mjx-itable {display: inline-table; width: auto} > .mjx-row {display: table-row} > .mjx-cell {display: table-cell} > .mjx-table {display: table; width: 100%} > .mjx-line {display: block; height: 0} > .mjx-strut {width: 0; padding-top: 1em} > .mjx-vsize {width: 0} > .MJXc-space1 {margin-left: .167em} > .MJXc-space2 {margin-left: .222em} > .MJXc-space3 {margin-left: .278em} > .mjx-test.mjx-test-display {display: table!important} > .mjx-test.mjx-test-inline {display: inline!important; margin-right: -1px} > .mjx-test.mjx-test-default {display: block!important; clear: both} > .mjx-ex-box {display: inline-block!important; position: absolute; overflow: hidden; 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src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')} > @font-face {font-family: MJXc-TeX-frak-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')} > @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold} > @font-face {font-family: MJXc-TeX-frak-Bw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')} > @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic} > @font-face {font-family: MJXc-TeX-math-BIw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')} > @font-face {font-family: MJXc-TeX-sans-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')} > @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold} > @font-face {font-family: MJXc-TeX-sans-Bw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')} > @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic} > @font-face {font-family: MJXc-TeX-sans-Iw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')} > @font-face {font-family: MJXc-TeX-script-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')} > @font-face {font-family: MJXc-TeX-type-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')} > @font-face {font-family: MJXc-TeX-cal-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')} > @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold} > @font-face {font-family: MJXc-TeX-main-Bw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')} > @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic} > @font-face {font-family: MJXc-TeX-main-Iw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')} > @font-face {font-family: MJXc-TeX-main-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')} > @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic} > @font-face {font-family: MJXc-TeX-math-Iw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')} > @font-face {font-family: MJXc-TeX-size1-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')} > @font-face {font-family: MJXc-TeX-size2-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')} > @font-face {font-family: MJXc-TeX-size3-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')} > @font-face {font-family: MJXc-TeX-size4-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')} > @font-face {font-family: MJXc-TeX-vec-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')} > @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')} > @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold} > @font-face {font-family: MJXc-TeX-vec-Bw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')} > solves the problem: > > maxπmine∈BEπ⋅e[U] > > In game-theoretic terms, the "max" represents the agent's decision, while the "min" represents the adversary's. > > Much of the mathematical tractability stems from the fact that B is a convex set of environments (i.e. functions from policy π to probability distributions). In game-theoretic terms, the adversary's choice of strategy determines which "environment" the agent faces, and the adversary can choose from any option in B. Convexity of B follows from the adversary's ability to use mixed strategies: because the adversary can take a randomized mix of any two strategies available to it, the adversary can make the agent face any convex combination of (policy -> distribution) functions in B. Thus, B is closed under convex combinations; it's a convex set. > > I'd like a writeup along roughly these conceptual lines which covers as much as possible of the major high-level definitions and results in infra-bayes to date. On the other hand, I give approximately-zero shits about all the measure theory; just state the relevant high-level results in game-theoretic language, say what they mean intuitively, maybe mention whether there's some pre-existing standard game-theory theorem which can do the job or whether the infra-bayes version of the theorem is in fact the first proof of the game-theoretic equivalent, and move on. Alternatively, insofar as core parts of infrabayes differ from a two-player zero-sum game, or the general path I'm pointing to doesn't work, an explanation of how they differ and what the consequences are could also qualify for prize money. Bounty/Contest Operationalization --------------------------------- Most of the headache in administering this sort of bounty is the risk that some well-intended person will write something which is not at all what I want, expecting to get paid, and then I will either have to explain how/why it's not what I want (which takes a lot of work), or I have to just accept it. To mitigate that failure mode, I'll run this as a contest: to submit, write up your explanation as a lesswrong post, then send me a message on lesswrong to make sure I'm aware of it. Deadline is end of April. I will distribute money among submissions based on my own highly-subjective judgement. If people write stuff up early, I might leave feedback on their posts, but no promises. I will count the "sample" above as a submission in its own right - i.e. I will imagine that three-paragraph blurb were instead a three-paragraph post in its own right, and someone submitted it. That will provide a baseline for prizes to be paid out at all: if no submission adds value not already included in the three-paragraph blurb, then the three-paragraph blurb gets the prize money, i.e. I don't pay anyone. Note that the $500 prize is probably not enough to fully pay for the amount of effort which I expect will be involved in doing this well. Others are welcome to add to the prize pool; please leave a comment if you'd like to do so.
f6ef3480-051e-4b5c-ba52-0e7ee9b1840b	trentmkelly/LessWrong-43k	LessWrong	How Many Worlds? How many universes "branch off" from a "quantum event", and in how many of them is the cat dead vs alive, and what about non-50/50 scenarios, and please answer so that a physics dummy can maybe kind of understand? (Is it just 1 with the live cat and 1 with the dead one?)
d72e9312-82a2-482e-9dbd-26d5533c919b	StampyAI/alignment-research-dataset/lesswrong	LessWrong	A Brief Overview of AI Safety/Alignment Orgs, Fields, Researchers, and Resources for ML Researchers Crossposted to EA Forum [[Link](https://forum.effectivealtruism.org/posts/xMzXbnpPeKWpTi3Gt/a-brief-overview-of-ai-safety-alignment-orgs-fields)] TLDR: I’ve written an overview of the AI safety space, tagged by keywords and subject/field references ([short version](https://docs.google.com/document/d/1gimXyGj4nTU9TFJ6svlpmMtEWGbTrMoNYfzZMi8siAA/edit?usp=sharing), [long version](https://docs.google.com/document/d/1SXhls4pCFdJ6PbRnlmNiF3GhTSx3qq2SkDRsKGKb1O4/edit?usp=sharing)). The aim is to allow existing ML researchers to quickly gauge interest in the subject based on their existing subfield skills and interests! Overview ======== When ML researchers first hear about AI alignment and are interested in learning more, they often wonder how their existing skills and interests could fit within the research already taking place. With expertise in specific subfields, and momentum in their careers and projects, interested researchers are curious about the overall AI alignment space and what research projects they could invest in relatively easily. As one step towards addressing this, the AISFB Hub commissioned a collection of resources that could be provided to technical researchers trying to quickly assess what areas seem like promising candidates for them to investigate further: ([Short version](https://docs.google.com/document/d/1gimXyGj4nTU9TFJ6svlpmMtEWGbTrMoNYfzZMi8siAA/edit?usp=sharing), [Long version](https://docs.google.com/document/d/1SXhls4pCFdJ6PbRnlmNiF3GhTSx3qq2SkDRsKGKb1O4/edit?usp=sharing)). These documents list a subset of the various organizations and researchers involved in the AI safety space, along with major papers. To allow quick scanning, I focused on keywords and subject/field references. As this was targeted at researchers who already have experience with ML, the summaries provided are primarily meant to allow the reader to quickly gauge interest in the subject based on their existing subfield skills and interests. This work contains papers and posts up through the end of 2022. Please contact me or Vael Gates if you would be willing to keep it updated! Details and Disclaimers ======================= As an attempt at collecting alignment research, I generally see this post as complementary to [Larsen’s post](https://www.lesswrong.com/posts/QBAjndPuFbhEXKcCr/my-understanding-of-what-everyone-in-technical-alignment-is) on technical alignment. Neither entirely includes the other, with Larsen’s post having a slightly stronger and more curated focus on fields and projects, while this collection emphasized providing general resources and example areas of work for new researchers. Overall, this list took a little over 40 hours of work to put together. It primarily included looking into and summarizing the work of organizations I knew about. This was supplemented by investigating a list of researchers provided by the AISFB Hub, along with work referenced by various other posts in LessWrong/EA forums, and by the organizations and researchers from their websites and papers. More specifically, these lists include various AI organizations (ex. DeepMind’s safety team, MIRI, OpenAI…) and individual researchers (both academic and independent) currently working on the subject, summaries of papers and posts they have produced, and a number of guides and other resources for those trying to get into the field. All of these include some keyword tags for quicker scanning. Unfortunately, it is impossible to include every research direction and relevant piece of work while keeping this concise. Instead, I tried to limit paper selection to representative samples of the ideas being actively worked on, or explicit overviews of their agendas, while providing as many links as possible for those interested in looking deeper. Still, with all of that said, I believe these documents can provide an easily shareable resource for anyone who either is themself or knows someone who is interested in transitioning into alignment research but is lacking information about how they might approach, learn about, or contribute to the field. Of course, if you just want to use it to check out some papers, that would work too. Thank you for reading!
3b4f4b67-c5c6-4828-840d-4b3d25609fc4	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	AGI Ruin: A List of Lethalities ### Preamble: (If you're already familiar with all basics and don't want any preamble, skip ahead to [Section B](#Section_B_) for technical difficulties of alignment proper.) I have several times failed to write up a well-organized list of reasons why AGI will kill you. People come in with different ideas about why AGI would be survivable, and want to hear different obviously keypoints addressed first. Some fraction of those people are loudly upset with me if the obviously most important points aren't addressed immediately, and I address different points first instead. Having failed to solve this problem in any good way, I now give up and solve it poorly with a poorly organized list of individual rants. I'm not particularly happy with this list; the alternative was publishing nothing, and publishing this seems marginally more [dignified](https://www.lesswrong.com/posts/j9Q8bRmwCgXRYAgcJ/miri-announces-new-death-with-dignity-strategy). Three points about the general subject matter of discussion here, numbered so as not to conflict with the list of lethalities: -3. I'm assuming you are already familiar with some basics, and already know what '[orthogonality](https://arbital.com/p/orthogonality/)' and '[instrumental convergence](https://arbital.com/p/instrumental_convergence/)' are and why they're true. People occasionally claim to me that I need to stop fighting old wars here, because, those people claim to me, those wars have already been won within the important-according-to-them parts of the current audience. I suppose it's at least true that none of the current major EA funders seem to be visibly in denial about orthogonality or instrumental convergence as such; so, fine. If you don't know what 'orthogonality' or 'instrumental convergence' are, or don't see for yourself why they're true, you need a different introduction than this one. -2. When I say that alignment is lethally difficult, I am not talking about ideal or perfect goals of 'provable' alignment, nor total alignment of superintelligences on exact human values, nor getting AIs to produce satisfactory arguments about moral dilemmas which sorta-reasonable humans disagree about, nor attaining an absolute certainty of an AI not killing everyone. When I say that alignment is difficult, I mean that in practice, using the techniques we actually have, "please don't disassemble literally everyone with probability roughly 1" is an overly large ask that we are not on course to get. So far as I'm concerned, [if you can get a powerful AGI that carries out some pivotal superhuman engineering task, with a less than fifty percent change of killing more than one billion people](https://twitter.com/ESYudkowsky/status/1070095112791715846), I'll take it. Even smaller chances of killing even fewer people would be a nice luxury, but if you can get as incredibly far as "less than roughly certain to kill everybody", then you can probably get down to under a 5% chance with only slightly more effort. Practically all of the difficulty is in getting to "less than certainty of killing literally everyone". Trolley problems are not an interesting subproblem in all of this; if there are any survivors, you solved alignment. At this point, I no longer care how it works, I don't care how you got there, I am cause-agnostic about whatever methodology you used, all I am looking at is prospective results, all I want is that we have justifiable cause to believe of a pivotally useful AGI 'this will not kill literally everyone'. Anybody telling you I'm asking for stricter 'alignment' than this has failed at reading comprehension. The big ask from AGI alignment, the basic challenge I am saying is too difficult, is to obtain by any strategy whatsoever a significant chance of there being any survivors. -1. None of this is about anything being impossible in principle. The metaphor I usually use is that if a textbook from one hundred years in the future fell into our hands, containing all of the simple ideas that actually work robustly in practice, we could probably build an aligned superintelligence in six months. For people schooled in machine learning, I use as my metaphor the difference between ReLU activations and sigmoid activations. Sigmoid activations are complicated and fragile, and do a terrible job of transmitting gradients through many layers; ReLUs are incredibly simple (for the unfamiliar, the activation function is literally max(x, 0)) and work much better. Most neural networks for the first decades of the field used sigmoids; the idea of ReLUs wasn't discovered, validated, and popularized until decades later. What's lethal is that we do not havethe Textbook From The Future telling us all the simple solutions that actually in real life just work and are robust; we're going to be doing everything with metaphorical sigmoids on the first critical try. No difficulty discussed here about AGI alignment is claimed by me to be impossible - to merely human science and engineering, let alone in principle - if we had 100 years to solve it using unlimited retries, the way that science usually has an unbounded time budget and unlimited retries. This list of lethalities is about things we are not on course to solve in practice in time on the first critical try; none of it is meant to make a much stronger claim about things that are impossible in principle. That said: Here, from my perspective, are some different true things that could be said, to contradict various false things that various different people seem to believe, about why AGI would be survivable on anything remotely remotely resembling the current pathway, or any other pathway we can easily jump to. ### Section A: This is a very lethal problem, it has to be solved one way or another, it has to be solved at a minimum strength and difficulty level instead of various easier modes that some dream about, we do not have any visible option of 'everyone' retreating to only solve safe weak problems instead, and failing on the first really dangerous try is fatal. 1. Alpha Zero blew past all accumulated human knowledge about Go after a day or so of self-play, with no reliance on human playbooks or sample games. Anyone relying on "well, it'll get up to human capability at Go, but then have a hard time getting past that because it won't be able to learn from humans any more" would have relied on vacuum. AGI will not be upper-bounded by human ability or human learning speed. Things much smarter than human would be able to learn from less evidence than humans require to have ideas driven into their brains; there are theoretical upper bounds here, but those upper bounds seem very high. (Eg, each bit of information that couldn't already be fully predicted can eliminate at most half the probability mass of all hypotheses under consideration.) It is not naturally (by default, barring intervention) the case that everything takes place on a timescale that makes it easy for us to react. 2. A cognitive system with sufficiently high cognitive powers, given any medium-bandwidth channel of causal influence, will not find it difficult to bootstrap to overpowering capabilities independent of human infrastructure. The concrete example I usually use here is nanotech, because there's been pretty detailed analysis of what definitely look like physically attainable lower bounds on what should be possible with nanotech, and those lower bounds are sufficient to carry the point. My lower-bound model of "how a sufficiently powerful intelligence would kill everyone, if it didn't want to not do that" is that it gets access to the Internet, emails some DNA sequences to any of the many many online firms that will take a DNA sequence in the email and ship you back proteins, and bribes/persuades some human who has no idea they're dealing with an AGI to mix proteins in a beaker, which then form a first-stage nanofactory which can build the actual nanomachinery. (Back when I was first deploying this visualization, the wise-sounding critics said "Ah, but how do you know even a superintelligence could solve the protein folding problem, if it didn't already have planet-sized supercomputers?" but one hears less of this after the advent of AlphaFold 2, for some odd reason.) The nanomachinery builds diamondoid bacteria, that replicate with solar power and atmospheric CHON, maybe aggregate into some miniature rockets or jets so they can ride the jetstream to spread across the Earth's atmosphere, get into human bloodstreams and hide, strike on a timer. Losing a conflict with a high-powered cognitive system looks at least as deadly as "everybody on the face of the Earth suddenly falls over dead within the same second". (I am using awkward constructions like 'high cognitive power' because standard English terms like 'smart' or 'intelligent' appear to me to function largely as status synonyms. 'Superintelligence' sounds to most people like 'something above the top of the status hierarchy that went to double college', and they don't understand why that would be all that dangerous? Earthlings have no word and indeed no standard native concept that means 'actually useful cognitive power'. A large amount of failure to panic sufficiently, seems to me to stem from a lack of appreciation for the incredible potential lethality of this thing that Earthlings as a culture have not named.) 3. We need to get alignment right on the 'first critical try' at operating at a 'dangerous' level of intelligence, where unaligned operation at a dangerous level of intelligence kills everybody on Earth and then we don't get to try again. This includes, for example: (a) something smart enough to build a nanosystem which has been explicitly authorized to build a nanosystem; or (b) something smart enough to build a nanosystem and also smart enough to gain unauthorized access to the Internet and pay a human to put together the ingredients for a nanosystem; or (c) something smart enough to get unauthorized access to the Internet and build something smarter than itself on the number of machines it can hack; or (d) something smart enough to treat humans as manipulable machinery and which has any authorized or unauthorized two-way causal channel with humans; or (e) something smart enough to improve itself enough to do (b) or (d); etcetera. We can gather all sorts of information beforehand from less powerful systems that will not kill us if we screw up operating them; but once we are running more powerful systems, we can no longer update on sufficiently catastrophic errors. This is where practically all of the real lethality comes from, that we have to get things right on the first sufficiently-critical try. If we had unlimited retries - if every time an AGI destroyed all the galaxies we got to go back in time four years and try again - we would in a hundred years figure out which bright ideas actually worked. Human beings can figure out pretty difficult things over time, when they get lots of tries; when a failed guess kills literally everyone, that is harder. That we have to get a bunch of key stuff right on the first try is where most of the lethality really and ultimately comes from; likewise the fact that no authority is here to tell us a list of what exactly is 'key' and will kill us if we get it wrong. (One remarks that most people are so absolutely and flatly unprepared by their 'scientific' educations to challenge pre-paradigmatic puzzles with no scholarly authoritative supervision, that they do not even realize how much harder that is, or how incredibly lethal it is to demand getting that right on the first critical try.) 4. We can't just "decide not to build AGI" because GPUs are everywhere, and knowledge of algorithms is constantly being improved and published; 2 years after the leading actor has the capability to destroy the world, 5 other actors will have the capability to destroy the world. The given lethal challenge is to solve within a time limit, driven by the dynamic in which, over time, increasingly weak actors with a smaller and smaller fraction of total computing power, become able to build AGI and destroy the world. Powerful actors all refraining in unison from doing the suicidal thing just delays this time limit - it does not lift it, unless computer hardware and computer software progress are both brought to complete severe halts across the whole Earth. The current state of this cooperation to have every big actor refrain from doing the stupid thing, is that at present some large actors with a lot of researchers and computing power are led by people who vocally disdain all talk of AGI safety (eg Facebook AI Research). Note that needing to solve AGI alignment only within a time limit, but with unlimited safe retries for rapid experimentation on the full-powered system; or only on the first critical try, but with an unlimited time bound; would both be terrifically humanity-threatening challenges by historical standards individually. 5. We can't just build a very weak system, which is less dangerous because it is so weak, and declare victory; because later there will be more actors that have the capability to build a stronger system and one of them will do so. I've also in the past called this the 'safe-but-useless' tradeoff, or 'safe-vs-useful'. People keep on going "why don't we only use AIs to do X, that seems safe" and the answer is almost always either "doing X in fact takes very powerful cognition that is not passively safe" or, even more commonly, "because restricting yourself to doing X will not prevent Facebook AI Research from destroying the world six months later". If all you need is an object that doesn't do dangerous things, you could try a sponge; a sponge is very passively safe. Building a sponge, however, does not prevent Facebook AI Research from destroying the world six months later when they catch up to the leading actor. 6. We need to align the performance of some large task, a 'pivotal act' that prevents other people from building an unaligned AGI that destroys the world. While the number of actors with AGI is few or one, they must execute some "pivotal act", strong enough to flip the gameboard, using an AGI powerful enough to do that. It's not enough to be able to align a weak system - we need to align a system that can do some single very large thing. The example I usually give is "burn all GPUs". This is not what I think you'd actually want to do with a powerful AGI - the nanomachines would need to operate in an incredibly complicated open environment to hunt down all the GPUs, and that would be needlessly difficult to align. However, all known pivotal acts are currently outside the Overton Window, and I expect them to stay there. So I picked an example where if anybody says "how dare you propose burning all GPUs?" I can say "Oh, well, I don't actually advocate doing that; it's just a mild overestimate for the rough power level of what you'd have to do, and the rough level of machine cognition required to do that, in order to prevent somebody else from destroying the world in six months or three years." (If it wasn't a mild overestimate, then 'burn all GPUs' would actually be the minimal pivotal task and hence correct answer, and I wouldn't be able to give that denial.) Many clever-sounding proposals for alignment fall apart as soon as you ask "How could you use this to align a system that you could use to shut down all the GPUs in the world?" because it's then clear that the system can't do something that powerful, or, if it can do that, the system wouldn't be easy to align. A GPU-burner is also a system powerful enough to, and purportedly authorized to, build nanotechnology, so it requires operating in a dangerous domain at a dangerous level of intelligence and capability; and this goes along with any non-fantasy attempt to name a way an AGI could change the world such that a half-dozen other would-be AGI-builders won't destroy the world 6 months later. 7. The reason why nobody in this community has successfully named a 'pivotal weak act' where you do something weak enough with an AGI to be passively safe, but powerful enough to prevent any other AGI from destroying the world a year later - and yet also we can't just go do that right now and need to wait on AI - is that nothing like that exists. There's no reason why it should exist. There is not some elaborate clever reason why it exists but nobody can see it. It takes a lot of power to do something to the current world that prevents any other AGI from coming into existence; nothing which can do that is passively safe in virtue of its weakness. If you can't solve the problem right now (which you can't, because you're opposed to other actors who don't want to be solved and those actors are on roughly the same level as you) then you are resorting to some cognitive system that can do things you could not figure out how to do yourself, that you were not close to figuring out because you are not closeto being able to, for example, burn all GPUs. Burning all GPUs would actually stop Facebook AI Research from destroying the world six months later; weaksauce Overton-abiding stuff about 'improving public epistemology by setting GPT-4 loose on Twitter to provide scientifically literate arguments about everything' will be cool but will not actually prevent Facebook AI Research from destroying the world six months later, or some eager open-source collaborative from destroying the world a year later if you manage to stop FAIR specifically. There are no pivotal weak acts. 8. The best and easiest-found-by-optimization algorithms for solving problems we want an AI to solve, readily generalize to problems we'd rather the AI not solve; you can't build a system that only has the capability to drive red cars and not blue cars, because all red-car-driving algorithms generalize to the capability to drive blue cars. 9. The builders of a safe system, by hypothesis on such a thing being possible, would need to operate their system in a regime where it has the capability to kill everybody or make itself even more dangerous, but has been successfully designed to not do that. Running AGIs doing something pivotal are not passively safe, they're the equivalent of nuclear cores that require actively maintained design properties to not go supercritical and melt down. ### Section B: Okay, but as we all know, modern machine learning is like a genie where you just give it a wish, right? Expressed as some mysterious thing called a 'loss function', but which is basically just equivalent to an English wish phrasing, right? And then if you pour in enough computing power you get your wish, right? So why not train a giant stack of transformer layers on a dataset of agents doing nice things and not bad things, throw in the word 'corrigibility' somewhere, crank up that computing power, and get out an aligned AGI? Section B.1: The distributional leap. 10. You can't train alignment by running lethally dangerous cognitions, observing whether the outputs kill or deceive or corrupt the operators, assigning a loss, and doing supervised learning. On anything like the standard ML paradigm, you would need to somehow generalize optimization-for-alignment you did in safe conditions, across a big distributional shift to dangerous conditions. (Some generalization of this seems like it would have to be true even outside that paradigm; you wouldn't be working on a live unaligned superintelligence to align it.) This alone is a point that is sufficient to kill a lot of naive proposals from people who never did or could concretely sketch out any specific scenario of what training they'd do, in order to align what output - which is why, of course, they never concretely sketch anything like that. Powerful AGIs doing dangerous things that will kill you if misaligned, must have an alignment property that generalized far out-of-distribution from safer building/training operations that didn't kill you. This is where a huge amount of lethality comes from on anything remotely resembling the present paradigm. Unaligned operation at a dangerous level of intelligence\capability will kill you; so, if you're starting with an unaligned system and labeling outputs in order to get it to learn alignment, the training regime or building regime must be operating at some lower level of intelligence\capability that is passively safe, where its currently-unaligned operation does not pose any threat. (Note that anything substantially smarter than you poses a threat given any realistic level of capability. Eg, "being able to produce outputs that humans look at" is probably sufficient for a generally much-smarter-than-human AGI to [navigate its way out of the causal systems that are humans](https://www.yudkowsky.net/singularity/aibox), especially in the real world where somebody trained the system on terabytes of Internet text, rather than somehow keeping it ignorant of the latent causes of its source code and training environments.) 11. If cognitive machinery doesn't generalize far out of the distribution where you did tons of training, it can't solve problems on the order of 'build nanotechnology' where it would be too expensive to run a million training runs of failing to build nanotechnology. There is no pivotal act this weak; there's no known case where you can entrain a safe level of ability on a safe environment where you can cheaply do millions of runs, and deploy that capability to save the world and prevent the next AGI project up from destroying the world two years later. Pivotal weak acts like this aren't known, and not for want of people looking for them. So, again, you end up needing alignment to generalize way out of the training distribution - not just because the training environment needs to be safe, but because the training environment probably also needs to be cheaper than evaluating some real-world domain in which the AGI needs to do some huge act. You don't get 1000 failed tries at burning all GPUs - because people will notice, even leaving out the consequences of capabilities success and alignment failure. 12. Operating at a highly intelligent level is a drastic shift in distribution from operating at a less intelligent level, opening up new external options, and probably opening up even more new internal choices and modes. Problems that materialize at high intelligence and danger levels may fail to show up at safe lower levels of intelligence, or may recur after being suppressed by a first patch. 13. Many alignment problems of superintelligence will not naturally appear at pre-dangerous, passively-safe levels of capability. Consider the internal behavior 'change your outer behavior to deliberately look more aligned and deceive the programmers, operators, and possibly any loss functions optimizing over you'. This problem is one that will appear at the superintelligent level; if, being otherwise ignorant, we guess that it is among the median such problems in terms of how early it naturally appears in earlier systems, then around half of the alignment problems of superintelligence will first naturally materialize afterthat one first starts to appear. Given correctforesight of which problems will naturally materialize later, one could try to deliberately materialize such problems earlier, and get in some observations of them. This helps to the extent (a) that we actually correctly forecast all of the problems that will appear later, or some superset of those; (b) that we succeed in preemptively materializing a superset of problems that will appear later; and (c) that we can actually solve, in the earlier laboratory that is out-of-distribution for us relative to the real problems, those alignment problems that would be lethal if we mishandle them when they materialize later. Anticipating allof the really dangerous ones, and then successfully materializing them, in the correct form for early solutions to generalize over to later solutions, sounds possibly kinda hard. 14. Some problems, like 'the AGI has an option that (looks to it like) it could successfully kill and replace the programmers to fully optimize over its environment', seem like their natural order of appearance could be that they first appear only in fully dangerous domains. Really actually having a clear option to brain-level-persuade the operators or escape onto the Internet, build nanotech, and destroy all of humanity - in a way where you're fully clear that you know the relevant facts, and estimate only a not-worth-it low probability of learning something which changes your preferred strategy if you bide your time another month while further growing in capability - is an option that first gets evaluated for real at the point where an AGI fully expects it can defeat its creators. We can try to manifest an echo of that apparent scenario in earlier toy domains. Trying to train by gradient descent against that behavior, in that toy domain, is something I'd expect to produce not-particularly-coherent local patches to thought processes, which would break with near-certainty inside a superintelligence generalizing far outside the training distribution and thinking very different thoughts. Also, programmers and operators themselves, who are used to operating in not-fully-dangerous domains, are operating out-of-distribution when they enter into dangerous ones; our methodologies may at that time break. 15. Fast capability gains seem likely, and may break lots of previous alignment-required invariants simultaneously. Given otherwise insufficient foresight by the operators, I'd expect a lot of those problems to appear approximately simultaneously after a sharp capability gain. See, again, the case of human intelligence. We didn't break alignment with the 'inclusive reproductive fitness' outer loss function, immediately after the introduction of farming - something like 40,000 years into a 50,000 year Cro-Magnon takeoff, as was itself running very quickly relative to the outer optimization loop of natural selection. Instead, we got a lot of technology more advanced than was in the ancestral environment, including contraception, in one very fast burst relative to the speed of the outer optimization loop, late in the general intelligence game. We started reflecting on ourselves a lot more, started being programmed a lot more by cultural evolution, and lots and lots of assumptions underlying our alignment in the ancestral training environment broke simultaneously. (People will perhaps rationalize reasons why this abstract description doesn't carry over to gradient descent; eg, “gradient descent has less of an information bottleneck”. My model of this variety of reader has an inside view, which they will label an outside view, that assigns great relevance to some other data points that are not observed cases of an outer optimization loop producing an inner general intelligence, and assigns little importance to our one data point actually featuring the phenomenon in question. When an outer optimization loop actually produced general intelligence, it broke alignment after it turned general, and did so relatively late in the game of that general intelligence accumulating capability and knowledge, almost immediately before it turned 'lethally' dangerous relative to the outer optimization loop of natural selection. Consider skepticism, if someone is ignoring this one warning, especially if they are not presenting equally lethal and dangerous things that they say will go wrong instead.) Section B.2: Central difficulties of outer and inner alignment. 16.Even if you train really hard on an exact loss function, that doesn't thereby create an explicit internal representation of the loss function inside an AI that then continues to pursue that exact loss function in distribution-shifted environments. Humans don't explicitly pursue inclusive genetic fitness; outer optimization even on a very exact, very simple loss function doesn't produce inner optimization in that direction. This happens in practice in real life,it is what happened in the only case we know about, and it seems to me that there are deep theoretical reasons to expect it to happen again: the firstsemi-outer-aligned solutions found, in the search ordering of a real-world bounded optimization process, are not inner-aligned solutions. This is sufficient on its own, even ignoring many other items on this list, to trash entire categories of naive alignment proposals which assume that if you optimize a bunch on a loss function calculated using some simple concept, you get perfect inner alignment on that concept. 17. More generally, a superproblem of 'outer optimization doesn't produce inner alignment' is that on the current optimization paradigm there is no general idea of how to get particular inner properties into a system, or verify that they're there, rather than just observable outer ones you can run a loss function over. This is a problem when you're trying to generalize out of the original training distribution, because, eg, the outer behaviors you see could have been produced by an inner-misaligned system that is deliberately producing outer behaviors that will fool you. We don't know how to get any bits of information into the inner system rather than the outer behaviors, in any systematic or general way, on the current optimization paradigm. 18. There's no reliable Cartesian-sensory ground truth (reliable loss-function-calculator) about whether an output is 'aligned', because some outputs destroy (or fool) the human operators and produce a different environmental causal chain behind the externally-registered loss function. That is, if you show an agent a reward signal that's currently being generated by humans, the signal is not in general a reliable perfect ground truth about how aligned an action was, because another way of producing a high reward signal is to deceive, corrupt, or replace the human operators with a different causal system which generates that reward signal. When you show an agent an environmental reward signal, you are not showing it something that is a reliable ground truth about whether the system did the thing you wanted it to do; even if it ends up perfectly inner-aligned on that reward signal, or learning some concept that exactly corresponds to 'wanting states of the environment which result in a high reward signal being sent', an AGI strongly optimizing on that signal will kill you, because the sensory reward signal was not a ground truth about alignment (as seen by the operators). 19. More generally, there is no known way to use the paradigm of loss functions, sensory inputs, and/or reward inputs, to optimize anything within a cognitive system to point at particular things within the environment - to point to latent events and objects and properties in the environment, rather than relatively shallow functions of the sense data and reward. This isn't to say that nothing in the system’s goal (whatever goal accidentally ends up being inner-optimized over) could ever point to anything in the environment by accident. Humans ended up pointing to their environments at least partially, though we've got lots of internally oriented motivational pointers as well. But insofar as the current paradigm works at all, the on-paper design properties say that it only works for aligning on known direct functions of sense data and reward functions. All of these kill you if optimized-over by a sufficiently powerful intelligence, because they imply strategies like 'kill everyone in the world using nanotech to strike before they know they're in a battle, and have control of your reward button forever after'. It just isn't true that we know a function on webcam input such that every world with that webcam showing the right things is safe for us creatures outside the webcam. This general problem is a fact about the territory, not the map; it's a fact about the actual environment, not the particular optimizer, that lethal-to-us possibilities exist in some possible environments underlying every given sense input. 20. Human operators are fallible, breakable, and manipulable. Human raters make systematic errors - regular, compactly describable, predictable errors. To faithfully learn a function from 'human feedback' is to learn (from our external standpoint) an unfaithful description of human preferences, with errors that are not random (from the outside standpoint of what we'd hoped to transfer). If you perfectly learn and perfectly maximize the referent of rewards assigned by human operators, that kills them. It's a fact about the territory, not the map - about the environment, not the optimizer - that the best predictive explanation for human answers is one that predicts the systematic errors in our responses, and therefore is a psychological concept that correctly predicts the higher scores that would be assigned to human-error-producing cases. 21. There's something like a single answer, or a single bucket of answers, for questions like 'What's the environment really like?' and 'How do I figure out the environment?' and 'Which of my possible outputs interact with reality in a way that causes reality to have certain properties?', where a simple outer optimization loop will straightforwardly shove optimizees into this bucket. When you have a wrong belief, reality hits back at your wrong predictions. When you have a broken belief-updater, reality hits back at your broken predictive mechanism via predictive losses, and a gradient descent update fixes the problem in a simple way that can easily cohere with all the other predictive stuff. In contrast, when it comes to a choice of utility function, there are unbounded degrees of freedom and multiple reflectively coherent fixpoints. Reality doesn't 'hit back' against things that are locally aligned with the loss function on a particular range of test cases, but globally misaligned on a wider range of test cases. This is the very abstract story about why hominids, once they finally started to generalize, generalized their capabilities to Moon landings, but their inner optimization no longer adhered very well to the outer-optimization goal of 'relative inclusive reproductive fitness' - even though they were in their ancestral environment optimized very strictly around this one thing and nothing else. This abstract dynamic is something you'd expect to be true about outer optimization loops on the order of both 'natural selection' and 'gradient descent'. The central result: Capabilities generalize further than alignment once capabilities start to generalize far. 22. There's a relatively simple core structure that explains why complicated cognitive machines work; which is why such a thing as general intelligence exists and not just a lot of unrelated special-purpose solutions; which is why capabilities generalize after outer optimization infuses them into something that has been optimized enough to become a powerful inner optimizer. The fact that this core structure is simple and relates generically to [low-entropy high-structure environments](https://intelligence.org/2017/12/06/chollet/) is why humans can walk on the Moon. There is no analogous truth about there being a simple core of alignment, especially not one that is even easier for gradient descent to find than it would have been for natural selection to just find 'want inclusive reproductive fitness' as a well-generalizing solution within ancestral humans. Therefore, capabilities generalize further out-of-distribution than alignment, once they start to generalize at all. 23. Corrigibility is anti-natural to consequentialist reasoning; "you can't bring the coffee if you're dead" for almost every kind of coffee. We (MIRI) [tried and failed](https://www.alignmentforum.org/posts/5bd75cc58225bf0670374f04/forum-digest-corrigibility-utility-indifference-and-related-control-ideas) to find a coherent formula for an agent that would let itself be shut down (without that agent actively trying to get shut down). Furthermore, many anti-corrigible lines of reasoning like this may only first appear at high levels of intelligence. 24. There are two fundamentally different approaches you can potentially take to alignment, which are unsolvable for two different sets of reasons; therefore, by becoming confused and ambiguating between the two approaches, you can confuse yourself about whether alignment is necessarily difficult. The first approach is to build a CEV-style Sovereign which wants exactly what we extrapolated-want and is therefore safe to let optimize all the future galaxies without it accepting any human input trying to stop it. The second course is to build corrigible AGI which doesn't want exactly what we want, and yet somehow fails to kill us and take over the galaxies despite that being a convergent incentive there. 1. The first thing generally, or CEV specifically, is unworkable because the complexity of what needs to be aligned or meta-aligned for our Real Actual Values is far out of reach for our FIRST TRY at AGI. Yes I mean specifically that the dataset, meta-learning algorithm, and what needs to be learned, is far out of reach for our first try. It's not just non-hand-codable, it is unteachableon-the-first-try because the thing you are trying to teach is too weird and complicated. 2. The second thing looks unworkable (less so than CEV, but still lethally unworkable) because corrigibility runs*actively counter* to instrumentally convergent behaviors within a core of general intelligence (the capability that generalizes far out of its original distribution). You're not trying to make it have an opinion on something the core was previously neutral on. You're trying to take a system implicitly trained on lots of arithmetic problems until its machinery started to reflect the common coherent core of arithmetic, and get it to say that as a special case 222 + 222 = 555. You can maybe train something to do this in a particular training distribution, but it's incredibly likely to break when you present it with new math problems far outside that training distribution, on a system which successfully generalizes capabilities that far at all. Section B.3: Central difficulties of*sufficiently good and useful* transparency / interpretability. 25. We've got no idea what's actually going on inside the giant inscrutable matrices and tensors of floating-point numbers. Drawing interesting graphs of where a transformer layer is focusing attention doesn't help if the question that needs answering is "So was it planning how to kill us or not?" 26. Even if we did know what was going on inside the giant inscrutable matrices while the AGI was still too weak to kill us, this would just result in us dying with more dignity, if DeepMind refused to run that system and let Facebook AI Research destroy the world two years later. Knowing that a medium-strength system of inscrutable matrices is planning to kill us, does not thereby let us build a high-strength system of inscrutable matrices that isn't planning to kill us. 27. When you explicitly optimize against a detector of unaligned thoughts, you're partially optimizing for more aligned thoughts, and partially optimizing for unaligned thoughts that are harder to detect. Optimizing against an interpreted thought optimizes against interpretability. 28. The AGI is smarter than us in whatever domain we're trying to operate it inside, so we cannot mentally check all the possibilities it examines, and we cannot see all the consequences of its outputs using our own mental talent. A powerful AI searches parts of the option space we don't, and we can't foresee all its options. 29. The outputs of an AGI go through a huge, not-fully-known-to-us domain (the real world) before they have their real consequences. Human beings cannot inspect an AGI's output to determine whether the consequences will be good. 30. Any pivotal act that is not something we can go do right now, will take advantage of the AGI figuring out things about the world we don't know so that it can make plans we wouldn't be able to make ourselves. It knows, at the least, the fact we didn't previously know, that some action sequence results in the world we want. Then humans will not be competent to use their own knowledge of the world to figure out all the results of that action sequence. An AI whose action sequence you can fully understand all the effects of, before it executes, is much weaker than humans in that domain; you couldn't make the same guarantee about an unaligned human as smart as yourself and trying to fool you. There is no pivotal output of an AGI that is humanly checkable and can be used to safely save the world but only after checking it; this is another form of pivotal weak act which does not exist. 31. A strategically aware intelligence can choose its visible outputs to have the consequence of deceiving you, including about such matters as whether the intelligence has acquired strategic awareness; you can't rely on behavioral inspection to determine facts about an AI which that AI might want to deceive you about. (Including how smart it is, or whether it's acquired strategic awareness.) 32. Human thought partially exposes only a partially scrutable outer surface layer. Words only trace our real thoughts. Words are not an AGI-complete data representation in its native style. The underparts of human thought are not exposed for direct imitation learning and can't be put in any dataset. This makes it hard and probably impossible to train a powerful system entirely on imitation of human words or other human-legible contents, which are only impoverished subsystems of human thoughts; *unless*that system is powerful enough to contain inner intelligences figuring out the humans, and at that point it is no longer really working as imitative human thought. 33. The AI does not think like you do, the AI doesn't have thoughts built up from the same concepts you use, it is utterly alien on a staggering scale. Nobody knows what the hell GPT-3 is thinking, not only because the matrices are opaque, but because the stuff within that opaque containeris, very likely, incredibly alien - nothing that would translate well into comprehensible human thinking, even if we could see past the giant wall of floating-point numbers to what lay behind. Section B.4: Miscellaneous unworkable schemes. 34. Coordination schemes between superintelligences are not things that humans can participate in (eg because humans can't reason reliably about the code of superintelligences); a "multipolar" system of 20 superintelligences with different utility functions, plus humanity, has a natural and obvious equilibrium which looks like "the 20 superintelligences cooperate with each other but not with humanity". 35. Schemes for playing "different" AIs off against each other stop working if those AIs advance to the point of being able to coordinate via reasoning about (probability distributions over) each others' code. Any system of sufficiently intelligent agents can probably behave as a single agent, even if you imagine you're playing them against each other. Eg, if you set an AGI that is secretly a paperclip maximizer, to check the output of a nanosystems designer that is secretly a staples maximizer, then even if the nanosystems designer is not able to deduce what the paperclip maximizer really wants (namely paperclips), it could still logically commit to share half the universe with any agent checking its designs if those designs were allowed through, ifthe checker-agent can verify the suggester-system's logical commitment and hence logically depend on it (which excludes human-level intelligences). Or, if you prefer simplified catastrophes without any logical decision theory, the suggester could bury in its nanosystem design the code for a new superintelligence that will visibly (to a superhuman checker) divide the universe between the nanosystem designer and the design-checker. 36. What makes an air conditioner 'magic' from the perspective of say the thirteenth century, is that even if you correctly show them the design of the air conditioner in advance, they won't be able to understand from seeing that design why the air comes out cold; the design is exploiting regularities of the environment, rules of the world, laws of physics, that they don't know about. The domain of human thought and human brains is very poorly understood by us, and exhibits phenomena like optical illusions, hypnosis, psychosis, mania, or simple afterimages produced by strong stimuli in one place leaving neural effects in another place. Maybe a superintelligence couldn't defeat a human in a very simple realm like logical tic-tac-toe; if you're fighting it in an incredibly complicated domain you understand poorly, like human minds, you should expect to be defeated by 'magic' in the sense that even if you saw its strategy you would not understand why that strategy worked. AI-boxing can only work on relatively weak AGIs; the human operators are not secure systems. ### Section C: Okay, those are some significant problems, but lots of progress is being made on solving them, right? There's a whole field calling itself "AI Safety" and many major organizations are expressing Very Grave Concern about how "safe" and "ethical" they are? 37. There's a pattern that's played out quite often, over all the times the Earth has spun around the Sun, in which some bright-eyed young scientist, young engineer, young entrepreneur, proceeds in full bright-eyed optimism to challenge some problem that turns out to be really quite difficult. Very often the cynical old veterans of the field try to warn them about this, and the bright-eyed youngsters don't listen, because, like, who wants to hear about all that stuff, they want to go solve the problem! Then this person gets beaten about the head with a slipper by reality as they find out that their brilliant speculative theory is wrong, it's actually really hard to build the thing because it keeps breaking, and society isn't as eager to adopt their clever innovation as they might've hoped, in a process which eventually produces a new cynical old veteran. Which, if not literally optimal, is I suppose a nice life cycle to nod along to in a nature-show sort of way. Sometimes you do something for the first time and there are no cynical old veterans to warn anyone and people can be really optimistic about how it will go; eg the initial Dartmouth Summer Research Project on Artificial Intelligence in 1956: "An attempt will be made to find how to make machines use language, form abstractions and concepts, solve kinds of problems now reserved for humans, and improve themselves. We think that a significant advance can be made in one or more of these problems if a carefully selected group of scientists work on it together for a summer." This is lessof a viable survival plan for your planet if the first major failure of the bright-eyed youngsters kills literally everyone before they can predictably get beaten about the head with the news that there were all sorts of unforeseen difficulties and reasons why things were hard. You don't get any cynical old veterans, in this case, because everybody on Earth is dead. Once you start to suspect you're in that situation, you have to do the Bayesian thing and update now to the view you will predictably update to later: realize you're in a situation of being that bright-eyed person who is going to encounter Unexpected Difficulties later and end up a cynical old veteran - or would be, except for the part where you'll be dead along with everyone else. And become that cynical old veteran right away, before reality whaps you upside the head in the form of everybody dying and you not getting to learn. Everyone else seems to feel that, so long as reality hasn't whapped them upside the head yet and smacked them down with the actual difficulties, they're free to go on living out the standard life-cycle and play out their role in the script and go on being bright-eyed youngsters; there's no cynical old veterans to warn them otherwise, after all, and there's no proof that everything won't go beautifully easy and fine,*given their bright-eyed total ignorance of what those later difficulties could be.* 38. It does not appear to me that the field of 'AI safety' is currently being remotely productive on tackling its enormous lethal problems. These problems are in fact out of reach; the contemporary field of AI safety has been selected to contain people who go to work in that field anyways. Almost all of them are there to tackle problems on which they can appear to succeed and publish a paper claiming success; if they can do that and get funded, why would they embark on a much more unpleasant project of trying something harder that they'll fail at, just so the human species can die with marginally more dignity? This field is not making real progress and does not have a recognition function to distinguish real progress if it took place. You could pump a billion dollars into it and it would produce mostly noise to drown out what little progress was being made elsewhere. 39. I figured this stuff out using the[null string](https://twitter.com/ESYudkowsky/status/1500863629490544645) as input, and frankly, I have a hard time myself feeling hopeful about getting real alignment work out of somebody who previously sat around waiting for somebody else to input a persuasive argument into them. This ability to "notice lethal difficulties without Eliezer Yudkowsky arguing you into noticing them" currently is an opaque piece of cognitive machinery to me, I do not know how to train it into others. It probably relates to '[security mindset](https://intelligence.org/2017/11/25/security-mindset-ordinary-paranoia/)', and a mental motion where you refuse to play out scripts, and being able to operate in a field that's in a state of chaos. 40. "Geniuses" with nice legible accomplishments in fields with tight feedback loops where it's easy to determine which results are good or bad right away, and so validate that this person is a genius, are (a) people who might not be able to do equally great work away from tight feedback loops, (b) people who chose a field where their genius would be nicely legible even if that maybe wasn't the place where humanity most needed a genius, and (c) probably don't have the mysterious gears simply because they're rare. You cannot just pay $5 million apiece to a bunch of legible geniuses from other fields and expect to get great alignment work out of them. They probably do not know where the real difficulties are, they probably do not understand what needs to be done, they cannot tell the difference between good and bad work, and the funders also can't tell without me standing over their shoulders evaluating everything, which I do not have the physical stamina to do. I concede that real high-powered talents, especially if they're still in their 20s, genuinely interested, and have done their reading, are people who, yeah, fine, have higher probabilities of making core contributions than a random bloke off the street. But I'd have more hope - not significant hope, but morehope - in separating the concerns of (a) credibly promising to pay big money retrospectively for good work to anyone who produces it, and (b) venturing prospective payments to somebody who is predicted to maybe produce good work later. 41. Reading this document cannot make somebody a core alignment researcher. That requires, not the ability to read this document and nod along with it, but the ability to spontaneously write it from scratch without anybody else prompting you; that is what makes somebody a peer of its author. It's guaranteed that some of my analysis is mistaken, though not necessarily in a hopeful direction. The ability to do new basic work noticing and fixing those flaws is the same ability as the ability to write this document before I published it, which nobody apparently did, despite my having had other things to do than write this up for the last five years or so. Some of that silence may, possibly, optimistically, be due to nobody else in this field having the ability to write things comprehensibly - such that somebody out there had the knowledge to write all of this themselves, if they could only have written it up, but they couldn't write, so didn't try. I'm not particularly hopeful of this turning out to be true in real life, but I suppose it's one possible place for a "positive model violation" (miracle). The fact that, twenty-one years into my entering this death game, seven years into other EAs noticing the death game, and two years into even normies starting to notice the death game, it is still Eliezer Yudkowsky writing up this list, says that humanity still has only one gamepiece that can do that. I knew I did not actually have the physical stamina to be a star researcher, I tried really really hard to replace myself before my health deteriorated further, and yet here I am writing this. That's not what surviving worlds look like. 42. There's no plan. Surviving worlds, by this point, and in fact several decades earlier, have a plan for how to survive. It is a written plan. The plan is not secret. In this non-surviving world, there are no candidate plans that do not immediately fall to Eliezer instantly pointing at the giant visible gaping holes in that plan. Or if you don't know who Eliezer is, you don't even realize you need a plan, because, like, how would a human being possibly realize that without Eliezer yelling at them? It's not like people will yell at themselves about prospective alignment difficulties, they don't have an internal voice of caution. So most organizations don't have plans, because I haven't taken the time to personally yell at them. 'Maybe we should have a plan' is deeper alignment mindset than they possess without me standing constantly on their shoulder as their personal angel pleading them into... continued noncompliance, in fact. Relatively few are aware even that they should, to look better, produce a pretend plan that can fool EAs too '[modest](https://equilibriabook.com/toc/)' to trust their own judgments about seemingly gaping holes in what serious-looking people apparently believe. 43. This situation you see when you look around you is not what a surviving world looks like. The worlds of humanity that survive have plans. They are not leaving to one tired guy with health problems the entire responsibility of pointing out real and lethal problems proactively. Key people are taking internal and real responsibility for finding flaws in their own plans, instead of considering it their job to propose solutions and somebody else's job to prove those solutions wrong. That world started trying to solve their important lethal problems earlier than this. Half the people going into string theory shifted into AI alignment instead and made real progress there. When people suggest a planetarily-lethal problem that might materialize later - there's a lot of people suggesting those, in the worlds destined to live, and they don't have a special status in the field, it's just what normal geniuses there do - they're met with either solution plans or a reason why that shouldn't happen, not an uncomfortable shrug and 'How can you be sure that will happen' / 'There's no way you could be sure of that now, we'll have to wait on experimental evidence.' A lot of those better worlds will die anyways. It's a genuinely difficult problem, to solve something like that on your first try. But they'll die with more dignity than this.
e439ac30-8547-47ef-a4dd-1be3defd7bff	trentmkelly/LessWrong-43k	LessWrong	Total Utility is Illusionary (Abstract: We have the notion that people can have a "total utility" value, defined perhaps as the sum of all their changes in utility over time. This is usually not a useful concept, because utility functions can change. In many cases the less-confusing approach is to look only at the utility from each individual decision, and not attempt to consider the total over time. This leads to insights about utilitarianism.) Let's consider the utility of a fellow named Bob. Bob likes to track his total utility; he writes it down in a logbook every night. Bob is a stamp collector; he gets +1 utilon every time he adds a stamp to his collection, and he gets -1 utilon every time he removes a stamp from his collection. Bob's utility was zero when his collection was empty, so we can say that Bob's total utility is the number of stamps in his collection. One day a movie theater opens, and Bob learns that he likes going to movies. Bob counts +10 utilons every time he sees a movie. Now we can say that Bob's total utility is the number of stamps in his collection, plus ten times the number of movies he has seen. (A note on terminology: I'm saying that Bob's utility function is the thing that emits +1 or -1 or +10, and his total utility is the sum of all those emits over time. I'm not sure if this is standard terminology.) This should strike us as a little bit strange: Bob now has a term in his total utility which is mostly based on history, and mostly independent of the present state of the world. Technically, we might handwave and say that Bob places value on his memories of watching those movies. But Bob knows that's not actually true: it's the act of watching the movies that he enjoys, and he rarely thinks about them once they're over. If a hypnotist convinced Bob that he had watched ten billion movies, Bob would write down in his logbook that he had a hundred billion utilons. (Plus the number of stamps in his stamp collection.) Let's talk some more about that
ed5b3e97-9cb5-4c0d-a8d1-dfbb27c2d3ff	StampyAI/alignment-research-dataset/lesswrong	LessWrong	How might we make better use of AI capabilities research for alignment purposes? When I check ArXiv for new AI alignment research papers, I see mostly capabilities research papers, presumably because most researchers are working on capabilities. I wonder if there’s alignment-related value to be extracted from all that capabilities research, and how we might get at it. Is anyone working on this, or does anyone have any good ideas?
1f3f4257-5094-408c-855a-64cf125dfb62	trentmkelly/LessWrong-43k	LessWrong	Meetup : Washington, D.C.: Create & Complete Discussion article for the meetup : Washington, D.C.: Create & Complete WHEN: 26 February 2017 03:30:00PM (-0500) WHERE: Donald W. Reynolds Center for American Art and Portraiture We will be meeting in the courtyard to work on our own projects, help others with theirs, or just hang out. Upcoming meetups: * Mar. 5: Fun & Games * Mar. 12: Pi Day Discussion article for the meetup : Washington, D.C.: Create & Complete
6ed2d361-f354-4a2a-9b3b-02c86bad7aed	StampyAI/alignment-research-dataset/arxiv	Arxiv	MADE: Exploration via Maximizing Deviation from Explored Regions 1 Introduction --------------- Online RL is a useful tool for an agent to learn how to perform tasks, particularly when expert demonstrations are unavailable and reward information needs to be used instead Sutton and Barto ([2018](#bib.bib89)). To learn a satisfactory policy, an RL agent needs to effectively balance between exploration and exploitation, which remains a central question in RL (Ecoffet et al., [2019](#bib.bib22); Burda et al., [2018b](#bib.bib14)). Exploration is particularly challenging in environments with sparse rewards. One popular approach to exploration is based on intrinsic motivation, often applied by adding an intrinsic reward (or bonus) to the extrinsic reward provided by the environment. In provable exploration methods, bonus often captures the value estimate uncertainty and the agent takes an action that maximizes the upper confidence bound (UCB) Agrawal and Jia ([2017](#bib.bib4)); Azar et al. ([2017](#bib.bib7)); Jaksch et al. ([2010](#bib.bib38)); Kakade et al. ([2018](#bib.bib45)); Jin et al. ([2018](#bib.bib41)). In tabular setting, UCB bonuses are often constructed based on either Hoeffding’s inequality, which only uses visitation counts, or Bernstein’s inequality, which uses value function variance in addition to visitation counts. The latter is proved to be minimax near-optimal in environments with bounded rewards Jin et al. ([2018](#bib.bib41)); Menard et al. ([2021](#bib.bib62)) as well as bounded total reward Zhang et al. ([2020b](#bib.bib108)) and reward-free settings Ménard et al. ([2020](#bib.bib61)); Kaufmann et al. ([2021](#bib.bib46)); Jin et al. ([2020a](#bib.bib42)); Zhang et al. ([2020c](#bib.bib109)). It remains an open question how one can efficiently compute confidence bounds to construct UCB bonus in non-linear function approximation. Furthermore, Bernstein-style bonuses are often hard to compute in practice beyond tabular setting, due to difficulties in computing value function variance. In practice, various approaches are proposed to design intrinsic rewards: visitation pseudo-count bonuses estimate count-based UCB bonuses using function approximation (Bellemare et al., [2016](#bib.bib9); Burda et al., [2018b](#bib.bib14)), curiosity-based bonuses seek states where model prediction error is high, uncertainty-based bonuses (Pathak et al., [2019](#bib.bib74); Shyam et al., [2019](#bib.bib84)) adopt ensembles of networks for estimating variance of the Q-function, empowerment-based approaches (Klyubin et al., [2005](#bib.bib49); Gregor et al., [2016](#bib.bib32); Salge et al., [2014](#bib.bib81); Mohamed and Rezende, [2015](#bib.bib65)) lead the agent to states over which the agent has control, and information gain bonuses (Kim et al., [2018](#bib.bib48)) reward the agent based on the information gain between state-action pairs and next states. Although the performance of practical intrinsic rewards is good in certain domains, empirically they are observed to suffer from issues such as detachment, derailment, and catastrophic forgetting Agarwal et al. ([2020a](#bib.bib2)); Ecoffet et al. ([2019](#bib.bib22)). Moreover, these methods usually lack a clear objective and can get stuck in local optimum Agarwal et al. ([2020a](#bib.bib2)). Indeed, the impressive performance currently achieved by some deep RL algorithms often revolves around manually designing dense rewards Brockman et al. ([2016](#bib.bib12)), complicated exploration strategies utilizing a significant amount of domain knowledge Ecoffet et al. ([2019](#bib.bib22)), or operating in the known environment regime Silver et al. ([2017](#bib.bib85)); Moravčík et al. ([2017](#bib.bib66)). Motivated by current practical challenges and the gap between theory and practice, we propose a new algorithm for exploration by maximizing deviation from explored regions. This yields a practical algorithm with strong empirical performance. To be specific, we make the following contributions: ![Refer to caption](/html/2106.10268/assets/bar_plot.png) Figure 1: Normalized samples use of different methods with respect to MADE (smaller values are better). MADE consistency achieves a better sample efficiency compared to all other baselines. Infinity means the method fails to achieve maximum reward in given steps. #### 1. Exploration via maximizing deviation Our approach is based on modifying the standard RL objective (i.e. the cumulative reward) by adding a regularizer that adaptively changes across iterations. The regularizer can be a general function depending on the state-action visitation density and previous state-action coverage. We then choose a particular regularizer that MAximizes the DEviation (MADE) of the next policy visitation dπsuperscript𝑑𝜋d^{\pi}italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT from the regions covered by prior policies ρcovksubscriptsuperscript𝜌𝑘cov\rho^{k}\_{\text{cov}}italic\_ρ start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Lk(dπ)=J(dπ)+τk∑s,adπ(s,a)ρcovk(s,a).subscript𝐿𝑘superscript𝑑𝜋𝐽superscript𝑑𝜋subscript𝜏𝑘subscript𝑠𝑎superscript𝑑𝜋𝑠𝑎subscriptsuperscript𝜌𝑘cov𝑠𝑎\displaystyle L\_{k}(d^{\pi})={J(d^{\pi})}+{\tau\_{k}\sum\_{s,a}\sqrt{\tfrac{d^{\pi}(s,a)}{\rho^{k}\_{\text{cov}}(s,a)}}}.italic\_L start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ) = italic\_J ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ) + italic\_τ start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_s , italic\_a end\_POSTSUBSCRIPT square-root start\_ARG divide start\_ARG italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) end\_ARG start\_ARG italic\_ρ start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) end\_ARG end\_ARG . \| \| (1) \| Here, k𝑘kitalic\_k is the iteration number, J(dπ)𝐽superscript𝑑𝜋J(d^{\pi})italic\_J ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ) is the standard RL objective, and the regularizer encourages dπ(s,a)superscript𝑑𝜋𝑠𝑎d^{\pi}(s,a)italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) to be large when ρcovk(s,a)subscriptsuperscript𝜌𝑘cov𝑠𝑎\rho^{k}\_{\text{cov}}(s,a)italic\_ρ start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) is small. We give an algorithm for solving the regularized objective and prove that with access to an approximate planning oracle, it converges to the global optimum. We show that objective ([1](#S1.E1 "1 ‣ 1. Exploration via maximizing deviation ‣ 1 Introduction ‣ MADE: Exploration via Maximizing Deviation from Explored Regions")) results in an intrinsic reward that can be easily added to any RL algorithm to improve performance, as suggested by our empirical studies. Furthermore, the intrinsic reward applies a simple modification to the UCB-style bonus that considers prior visitation counts. This simple modification can also be added to existing bonuses in practice. #### 2. Tabular studies In the special case of tabular parameterization, we show that MADE only applies some simple adjustments to the Hoeffding-style count-based bonus. We compare the performance of MADE to Hoeffding and Bernstein bonuses in three different RL algorithms, for the exploration task in the stochastic diabolical bidirectional lock Agarwal et al. ([2020a](#bib.bib2)); Misra et al. ([2020](#bib.bib63)), which has sparse rewards and local optima. Our results show that MADE robustly improves over the Hoeffding bonus and is competitive to the Bernstein bonus, across all three RL algorithms. Interestingly, MADE bonus and exploration strategy appear to be very close to the Bernstein bonus, without computing or estimating variance, suggesting that MADE potentially captures some environmental structures. Additionally, we empirically show that MADE regularizer can improve the optimization rate in policy gradient methods. #### 3. Experiments on MiniGrid and DeepMind Control Suite We empirically show that MADE works well when combined with model-free (IMAPLA (Espeholt et al., [2018](#bib.bib24)), RAD (Laskin et al., [2020](#bib.bib52))) and model-based (Dreamer (Hafner et al., [2019](#bib.bib33))) RL algorithms, greatly improving the sample efficiency over existing baselines. When tested in the procedurally-generated MiniGrid environments, MADE manages to converge with two to five times fewer samples compared to state-of-the-art method BeBold (Zhang et al., [2020a](#bib.bib107)). In DeepMind Control Suite (Tassa et al., [2020](#bib.bib92)), we build upon the model-free method RAD (Laskin et al., [2020](#bib.bib52)) and the model-based method Dreamer (Hafner et al., [2019](#bib.bib33)), improving the return up to 150 in 500K steps compared to baselines. Figure [1](#S1.F1 "Figure 1 ‣ 1 Introduction ‣ MADE: Exploration via Maximizing Deviation from Explored Regions") shows normalized sample size to achieve maximum reward with respect to our algorithm. 2 Background ------------- #### Markov decision processes. An infinite-horizon discounted MDP is described by a tuple M=(𝒮,𝒜,P,r,ρ,γ)𝑀𝒮𝒜𝑃𝑟𝜌𝛾M=(\mathcal{S},\mathcal{A},P,r,\rho,\gamma)italic\_M = ( caligraphic\_S , caligraphic\_A , italic\_P , italic\_r , italic\_ρ , italic\_γ ), where 𝒮𝒮\mathcal{S}caligraphic\_S is the state space, 𝒜𝒜\mathcal{A}caligraphic\_A is the action space, P:𝒮×𝒜↦Δ(𝒮):𝑃maps-to𝒮𝒜Δ𝒮P:\mathcal{S}\times\mathcal{A}\mapsto\Delta(\mathcal{S})italic\_P : caligraphic\_S × caligraphic\_A ↦ roman\_Δ ( caligraphic\_S ) is the transition kernel, r:𝒮×𝒜↦[0,1]:𝑟maps-to𝒮𝒜01r:\mathcal{S}\times\mathcal{A}\mapsto[0,1]italic\_r : caligraphic\_S × caligraphic\_A ↦ [ 0 , 1 ] is the (extrinsic) reward function, ρ:𝒮↦Δ(𝒮):𝜌maps-to𝒮Δ𝒮\rho:\mathcal{S}\mapsto\Delta(\mathcal{S})italic\_ρ : caligraphic\_S ↦ roman\_Δ ( caligraphic\_S ) is the initial distribution, and γ∈[0,1)𝛾01\gamma\in[0,1)italic\_γ ∈ [ 0 , 1 ) is the discount factor. A stationary (stochastic) policy π∈Δ(𝒜∣𝒮)𝜋Δconditional𝒜𝒮\pi\in\Delta(\mathcal{A}\mid\mathcal{S})italic\_π ∈ roman\_Δ ( caligraphic\_A ∣ caligraphic\_S ) specifies a distribution over actions in each state. Each policy π𝜋\piitalic\_π induces a visitation density over state-action pairs dπ:𝒮×𝒜↦[0,1]:superscript𝑑𝜋maps-to𝒮𝒜01d^{\pi}:\mathcal{S}\times\mathcal{A}\mapsto[0,1]italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT : caligraphic\_S × caligraphic\_A ↦ [ 0 , 1 ] defined as dρπ(s,a)≔(1−γ)∑t=0∞γtℙt⁡(st=s,at=a;π)≔subscriptsuperscript𝑑𝜋𝜌𝑠𝑎1𝛾superscriptsubscript𝑡0superscript𝛾𝑡subscriptℙ𝑡subscript𝑠𝑡𝑠subscript𝑎𝑡𝑎𝜋d^{\pi}\_{\rho}(s,a)\coloneqq(1-\gamma)\sum\_{t=0}^{\infty}\gamma^{t}\operatorname{\mathbb{P}}\_{t}(s\_{t}=s,a\_{t}=a;\pi)italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) ≔ ( 1 - italic\_γ ) ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ∞ end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT roman\_ℙ start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ( italic\_s start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = italic\_s , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = italic\_a ; italic\_π ), where ℙt⁡(st=s,at=a;π)subscriptℙ𝑡subscript𝑠𝑡𝑠subscript𝑎𝑡𝑎𝜋\operatorname{\mathbb{P}}\_{t}(s\_{t}=s,a\_{t}=a;\pi)roman\_ℙ start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ( italic\_s start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = italic\_s , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = italic\_a ; italic\_π ) denotes (s,a)𝑠𝑎(s,a)( italic\_s , italic\_a ) visitation probability at step t𝑡titalic\_t, starting at s0∼ρ(⋅)similar-tosubscript𝑠0𝜌⋅s\_{0}\sim\rho(\cdot)italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT ∼ italic\_ρ ( ⋅ ) and following π𝜋\piitalic\_π. An important quantity is the value a policy π𝜋\piitalic\_π, which is the discounted sum of rewards Vπ(s)≔𝔼[∑t=0∞γtrt\|s0=s,at∼π(⋅∣st) for all t≥0]V^{\pi}(s)\coloneqq\operatorname{\mathbb{E}}[\sum\_{t=0}^{\infty}\gamma^{t}r\_{t}\;\|\;s\_{0}=s,a\_{t}\sim\pi(\cdot\mid s\_{t})\text{ for all }t\geq 0]italic\_V start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s ) ≔ roman\_𝔼 [ ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ∞ end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT \| italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT = italic\_s , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ∼ italic\_π ( ⋅ ∣ italic\_s start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) for all italic\_t ≥ 0 ] starting at state s∈𝒮𝑠𝒮s\in\mathcal{S}italic\_s ∈ caligraphic\_S. #### Policy mixture. For a sequence of policies 𝒞k=(π1,…,πk)superscript𝒞𝑘subscript𝜋1…subscript𝜋𝑘\mathcal{C}^{k}=(\pi\_{1},\dots,\pi\_{k})caligraphic\_C start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = ( italic\_π start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_π start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ) with corresponding mixture distribution wk∈Δk−1superscript𝑤𝑘subscriptΔ𝑘1w^{k}\in\Delta\_{k-1}italic\_w start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ∈ roman\_Δ start\_POSTSUBSCRIPT italic\_k - 1 end\_POSTSUBSCRIPT, the policy mixture πmix,k=(𝒞k,wk)subscript𝜋mix𝑘superscript𝒞𝑘superscript𝑤𝑘\pi\_{\operatorname\{mix},k}=(\mathcal{C}^{k},w^{k})italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT = ( caligraphic\_C start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT , italic\_w start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) is obtained by first sampling a policy from wksuperscript𝑤𝑘w^{k}italic\_w start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT and then following that policy over subsequent steps Hazan et al. ([2019](#bib.bib34)). The mixture policy induces a state-action visitation density according to dπmix(s,a)=∑i=1kwikdπi(s,a).superscript𝑑subscript𝜋mix𝑠𝑎superscriptsubscript𝑖1𝑘superscriptsubscript𝑤𝑖𝑘superscript𝑑subscript𝜋𝑖𝑠𝑎d^{\pi\_{\operatorname\{mix}}}(s,a)=\sum\_{i=1}^{k}w\_{i}^{k}d^{\pi\_{i}}(s,a).italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) = ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT italic\_w start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) . While the πmixsubscript𝜋mix\pi\_{\operatorname\{mix}}italic\_π start\_POSTSUBSCRIPT roman\_mix end\_POSTSUBSCRIPT may not be stationary in general, there exists a stationary policy π′superscript𝜋′\pi^{\prime}italic\_π start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT such that dπ′=dπmixsuperscript𝑑superscript𝜋′superscript𝑑subscript𝜋mixd^{\pi^{\prime}}=d^{\pi\_{\operatorname\{mix}}}italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT = italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT; see Puterman ([1990](#bib.bib77)) for details. #### Online reinforcement learning. Online RL is the problem of finding a policy with a maximum value from an unknown MDP, using samples collected during exploration. Oftentimes, the following objective is considered, which is a scalar summary of the performance of policy π𝜋\piitalic\_π: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| JM(π)≔𝔼s∼ρ⁡[Vπ(s)]=(1−γ)−1𝔼(s,a)∼dρπ(⋅,⋅)⁡[r(s,a)].≔subscript𝐽𝑀𝜋subscript𝔼similar-to𝑠𝜌superscript𝑉𝜋𝑠superscript1𝛾1subscript𝔼similar-to𝑠𝑎subscriptsuperscript𝑑𝜋𝜌⋅⋅𝑟𝑠𝑎\displaystyle J\_{M}(\pi)\coloneqq\operatorname{\mathbb{E}}\_{s\sim\rho}[V^{\pi}(s)]=({1-\gamma})^{-1}\operatorname{\mathbb{E}}\_{(s,a)\sim d^{\pi}\_{\rho}(\cdot,\cdot)}[r(s,a)].italic\_J start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT ( italic\_π ) ≔ roman\_𝔼 start\_POSTSUBSCRIPT italic\_s ∼ italic\_ρ end\_POSTSUBSCRIPT [ italic\_V start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s ) ] = ( 1 - italic\_γ ) start\_POSTSUPERSCRIPT - 1 end\_POSTSUPERSCRIPT roman\_𝔼 start\_POSTSUBSCRIPT ( italic\_s , italic\_a ) ∼ italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ρ end\_POSTSUBSCRIPT ( ⋅ , ⋅ ) end\_POSTSUBSCRIPT [ italic\_r ( italic\_s , italic\_a ) ] . \| \| (2) \| We drop index M𝑀Mitalic\_M when it is clear from context. We denote an optimal policy by π⋆∈argmaxπ⁡J(π)superscript𝜋⋆subscriptargmax𝜋𝐽𝜋\pi^{\star}\in\operatorname\{arg\,max}\_{\pi}J(\pi)italic\_π start\_POSTSUPERSCRIPT ⋆ end\_POSTSUPERSCRIPT ∈ start\_OPERATOR roman\_arg roman\_max end\_OPERATOR start\_POSTSUBSCRIPT italic\_π end\_POSTSUBSCRIPT italic\_J ( italic\_π ) and use the shorthand V⋆≔Vπ⋆≔superscript𝑉⋆superscript𝑉superscript𝜋⋆V^{\star}\coloneqq V^{\pi^{\star}}italic\_V start\_POSTSUPERSCRIPT ⋆ end\_POSTSUPERSCRIPT ≔ italic\_V start\_POSTSUPERSCRIPT italic\_π start\_POSTSUPERSCRIPT ⋆ end\_POSTSUPERSCRIPT end\_POSTSUPERSCRIPT to denote the optimal value function. It is straightforward to check that J(π)𝐽𝜋J(\pi)italic\_J ( italic\_π ) can equivalently be represented by the expectation of the reward over the visitation measure of π𝜋\piitalic\_π. We slightly abuse the notation and sometimes write J(dπ)𝐽superscript𝑑𝜋J(d^{\pi})italic\_J ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ) to denote the RL objective. 3 Adaptive regularization of the RL objective ---------------------------------------------- ### 3.1 Regularization to guide exploration In online RL, the agent faces a dilemma in each state: whether it should select a seemingly optimal policy (exploit) or it should explore different regions of the MDP. To allow flexibility in this choice and trade-off between exploration and exploitation, we propose to add a regularizer to the standard RL objective that changes throughout iterations of an online RL algorithm: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Lk(dπ)=J(dπ)⏟exploitation+τkR(dπ;{dπi}i=1k)⏟exploration.subscript𝐿𝑘superscript𝑑𝜋subscript⏟𝐽superscript𝑑𝜋exploitationsubscript𝜏𝑘subscript⏟𝑅superscript𝑑𝜋superscriptsubscriptsuperscript𝑑subscript𝜋𝑖𝑖1𝑘exploration\displaystyle L\_{k}(d^{\pi})=\underbrace{J(d^{\pi})}\_{\text{exploitation}}+\tau\_{k}\underbrace{R(d^{\pi};\{d^{\pi\_{i}}\}\_{i=1}^{k})}\_{\text{exploration}}.italic\_L start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ) = under⏟ start\_ARG italic\_J ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ) end\_ARG start\_POSTSUBSCRIPT exploitation end\_POSTSUBSCRIPT + italic\_τ start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT under⏟ start\_ARG italic\_R ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ; { italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) end\_ARG start\_POSTSUBSCRIPT exploration end\_POSTSUBSCRIPT . \| \| (3) \| Here, R(dπ;{dπi}i=1k)𝑅superscript𝑑𝜋superscriptsubscriptsuperscript𝑑subscript𝜋𝑖𝑖1𝑘R(d^{\pi};\{d^{\pi\_{i}}\}\_{i=1}^{k})italic\_R ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ; { italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) is a function of state-action visitation of π𝜋\piitalic\_π as well as the visitation of prior policies π1,…,πksubscript𝜋1…subscript𝜋𝑘\pi\_{1},\dots,\pi\_{k}italic\_π start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_π start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT. The temperature parameter τksubscript𝜏𝑘\tau\_{k}italic\_τ start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT determines the strength of regularization. Objective ([3](#S3.E3 "3 ‣ 3.1 Regularization to guide exploration ‣ 3 Adaptive regularization of the RL objective ‣ MADE: Exploration via Maximizing Deviation from Explored Regions")) is a population objective in the sense that it does not involve empirical estimations affected by the randomness in sample collection. In the following section, we give our particular choice of regularizer and discuss how this objective can describe some popular exploration bonuses. We then provide a convergence guarantee for the regularized objective in Section [3.2](#S3.SS2 "3.2 Exploration via maximizing deviation from policy cover ‣ 3 Adaptive regularization of the RL objective ‣ MADE: Exploration via Maximizing Deviation from Explored Regions"). ### 3.2 Exploration via maximizing deviation from policy cover We develop our exploration strategy MADE based on a simple intuition: maximizing the deviation from the explored regions, i.e. all states and actions visited by prior policies. We define policy cover at iteration k𝑘kitalic\_k to be the density over regions explored by policies π1,…,πksubscript𝜋1…subscript𝜋𝑘\pi\_{1},\dots,\pi\_{k}italic\_π start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_π start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT, i.e. ρcovk(s,a)≔1k∑i=1kdπi(s,a).≔superscriptsubscript𝜌cov𝑘𝑠𝑎1𝑘superscriptsubscript𝑖1𝑘superscript𝑑subscript𝜋𝑖𝑠𝑎\rho\_{\text{cov}}^{k}(s,a)\coloneqq\frac{1}{k}\sum\_{i=1}^{k}d^{\pi\_{i}}(s,a).italic\_ρ start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) ≔ divide start\_ARG 1 end\_ARG start\_ARG italic\_k end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) . We then design our regularizer to encourage dπsuperscript𝑑𝜋d^{\pi}italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT to be different from ρcovksuperscriptsubscript𝜌cov𝑘\rho\_{\text{cov}}^{k}italic\_ρ start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Rk(dπ;{dπi}i=1k)=∑s,adπ(s,a)ρcovk(s,a).subscript𝑅𝑘superscript𝑑𝜋superscriptsubscriptsuperscript𝑑subscript𝜋𝑖𝑖1𝑘subscript𝑠𝑎superscript𝑑𝜋𝑠𝑎superscriptsubscript𝜌cov𝑘𝑠𝑎\displaystyle R\_{k}(d^{\pi};\{d^{\pi\_{i}}\}\_{i=1}^{k})=\sum\_{s,a}\sqrt{\frac{d^{\pi}(s,a)}{\rho\_{\text{cov}}^{k}(s,a)}}.italic\_R start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ; { italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) = ∑ start\_POSTSUBSCRIPT italic\_s , italic\_a end\_POSTSUBSCRIPT square-root start\_ARG divide start\_ARG italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) end\_ARG start\_ARG italic\_ρ start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) end\_ARG end\_ARG . \| \| (4) \| It is easy to check that the maximizer of above function is dπ(s,a)∝1ρcovk(s,a)proportional-tosuperscript𝑑𝜋𝑠𝑎1superscriptsubscript𝜌cov𝑘𝑠𝑎d^{\pi}(s,a)\propto\frac{1}{\rho\_{\text{cov}}^{k}(s,a)}italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) ∝ divide start\_ARG 1 end\_ARG start\_ARG italic\_ρ start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) end\_ARG. Our motivation behind this particular deviation is that it results in a simple modification of UCB bonus in tabular case. We now compute the reward yielded by the new objective. First, define a policy mixture πmix,ksubscript𝜋mix𝑘\pi\_{\operatorname\{mix},k}italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT with policy sequence (π1,…,πk)subscript𝜋1…subscript𝜋𝑘(\pi\_{1},\dots,\pi\_{k})( italic\_π start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_π start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ) and weights ((1−η)k−1,(1−η)k−2η,(1−η)k−3η,,…,η)((1-\eta)^{k-1},(1-\eta)^{k-2}\eta,(1-\eta)^{k-3}\eta,,\dots,\eta)( ( 1 - italic\_η ) start\_POSTSUPERSCRIPT italic\_k - 1 end\_POSTSUPERSCRIPT , ( 1 - italic\_η ) start\_POSTSUPERSCRIPT italic\_k - 2 end\_POSTSUPERSCRIPT italic\_η , ( 1 - italic\_η ) start\_POSTSUPERSCRIPT italic\_k - 3 end\_POSTSUPERSCRIPT italic\_η , , … , italic\_η ) for η>0𝜂0\eta>0italic\_η > 0. Let dπmix,ksuperscript𝑑subscript𝜋mix𝑘d^{\pi\_{\operatorname\{mix},k}}italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT be the visitation density of πmix,ksubscript𝜋mix𝑘\pi\_{\operatorname\{mix},k}italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT. We compute the total reward at iteration k𝑘kitalic\_k by taking the gradient of the new objective with respect to dπsuperscript𝑑𝜋d^{\pi}italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT at dπmix,ksuperscript𝑑subscript𝜋mix𝑘d^{\pi\_{\operatorname\{mix},k}}italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| rk(s,a)=(1−γ)∇dLk(d)\|d=dπmix,k=r(s,a)+(1−γ)τk∇dRk(d;{dπi}i=1k)\|d=dπmix,k,subscript𝑟𝑘𝑠𝑎evaluated-at1𝛾subscript∇𝑑subscript𝐿𝑘𝑑𝑑superscript𝑑subscript𝜋mix𝑘𝑟𝑠𝑎evaluated-at1𝛾subscript𝜏𝑘subscript∇𝑑subscript𝑅𝑘𝑑superscriptsubscriptsuperscript𝑑subscript𝜋𝑖𝑖1𝑘𝑑superscript𝑑subscript𝜋mix𝑘\displaystyle r\_{k}(s,a)=(1-\gamma)\nabla\_{d}L\_{k}(d)\big{\|}\_{d={d}^{\pi\_{\operatorname\{mix},k}}}={r(s,a)}+(1-\gamma)\tau\_{k}\nabla\_{d}R\_{k}(d;\{d^{\pi\_{i}}\}\_{i=1}^{k})\big{\|}\_{d={d}^{\pi\_{\operatorname\{mix},k}}},italic\_r start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) = ( 1 - italic\_γ ) ∇ start\_POSTSUBSCRIPT italic\_d end\_POSTSUBSCRIPT italic\_L start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d ) \| start\_POSTSUBSCRIPT italic\_d = italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT = italic\_r ( italic\_s , italic\_a ) + ( 1 - italic\_γ ) italic\_τ start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ∇ start\_POSTSUBSCRIPT italic\_d end\_POSTSUBSCRIPT italic\_R start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d ; { italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) \| start\_POSTSUBSCRIPT italic\_d = italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT , \| \| (5) \| which gives the following reward \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| rk(s,a)=r(s,a)+(1−γ)τk/2dπmix,k(s,a)ρcovk(s,a).subscript𝑟𝑘𝑠𝑎𝑟𝑠𝑎1𝛾subscript𝜏𝑘2superscript𝑑subscript𝜋mix𝑘𝑠𝑎superscriptsubscript𝜌cov𝑘𝑠𝑎\displaystyle r\_{k}(s,a)=r(s,a)+\frac{(1-\gamma)\tau\_{k}/2}{\sqrt{d^{\pi\_{\operatorname\{mix},k}}(s,a)\rho\_{\text{cov}}^{k}(s,a)}}.italic\_r start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) = italic\_r ( italic\_s , italic\_a ) + divide start\_ARG ( 1 - italic\_γ ) italic\_τ start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT / 2 end\_ARG start\_ARG square-root start\_ARG italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) italic\_ρ start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) end\_ARG end\_ARG . \| \| (6) \| The intrinsic reward above is constructed based on two densities: ρcovksuperscriptsubscript𝜌cov𝑘\rho\_{\text{cov}}^{k}italic\_ρ start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT a uniform combination of past visitation densities and d^πmix,ksuperscript^𝑑subscript𝜋mix𝑘\hat{d}^{\pi\_{\operatorname\{mix},k}}over^ start\_ARG italic\_d end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT a (almost) geometric mixture of the past visitation densities. As we will discuss shortly, policy cover ρcovk(s,a)subscriptsuperscript𝜌𝑘cov𝑠𝑎\rho^{k}\_{\text{cov}}(s,a)italic\_ρ start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) is related to the visitation count of (s,a)𝑠𝑎(s,a)( italic\_s , italic\_a ) pair in previous iterations and resembles count-based bonuses Bellemare et al. ([2016](#bib.bib9)); Jin et al. ([2018](#bib.bib41)) or their approximates such as RND Burda et al. ([2018b](#bib.bib14)). Therefore, for an appropriate choice of τksubscript𝜏𝑘\tau\_{k}italic\_τ start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT, MADE intrinsic reward decreases as the number of visitations increases. MADE intrinsic reward is also proportional to 1/dπmix,k(s,a)1superscript𝑑subscript𝜋mix𝑘𝑠𝑎1/\sqrt{d^{\pi\_{\operatorname\{mix},k}}(s,a)}1 / square-root start\_ARG italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) end\_ARG, which can be viewed as a correction applied to the count-based bonus. In effect, due to the decay of weights in πmix,ksubscript𝜋mix𝑘\pi\_{\operatorname\{mix},k}italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT, the above construction gives a higher reward to (s,a)𝑠𝑎(s,a)( italic\_s , italic\_a ) pairs visited earlier. Experimental results suggest that this correction may alleviate major difficulties in sparse reward exploration, namely detachment and catastrophic forgetting, by encouraging the agent to revisit forgotten states and actions. Empirically, MADE’s intrinsic reward is computed based on estimates d^πmix,ksuperscript^𝑑subscript𝜋mix𝑘\hat{d}^{\pi\_{\operatorname\{mix},k}}over^ start\_ARG italic\_d end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT and ρ^covksuperscriptsubscript^𝜌cov𝑘\hat{\rho}\_{\text{cov}}^{k}over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT from data collected by iteration k𝑘kitalic\_k. Furthermore, practically we consider a smoothed version of the above regularizer by adding λ>0𝜆0\lambda>0italic\_λ > 0 to both numerator and denominator; see Equation ([7](#S3.E7 "7 ‣ Theorem 1. ‣ 3.3 Solving the regularized objective ‣ 3 Adaptive regularization of the RL objective ‣ MADE: Exploration via Maximizing Deviation from Explored Regions")). #### MADE intrinsic reward in tabular case. In tabular setting, the empirical estimation of policy cover is simply ρ^covk(s,a)=Nk(s,a)/Nksubscriptsuperscript^𝜌𝑘cov𝑠𝑎subscript𝑁𝑘𝑠𝑎subscript𝑁𝑘\hat{\rho}^{k}\_{\text{cov}}(s,a)={N\_{k}(s,a)}/{N\_{k}}over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) = italic\_N start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) / italic\_N start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT, where Nk(s,a)subscript𝑁𝑘𝑠𝑎N\_{k}(s,a)italic\_N start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) is the visitation count of (s,a)𝑠𝑎(s,a)( italic\_s , italic\_a ) pair and Nksubscript𝑁𝑘N\_{k}italic\_N start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT is the total count by iteration k𝑘kitalic\_k. Thus, MADE simply modifies the Hoeffding-type bonus via the mixture density and has the following form: ∝1/d^πmix,k(s,a)Nk(s,a)proportional-toabsent1superscript^𝑑subscript𝜋mix𝑘𝑠𝑎subscript𝑁𝑘𝑠𝑎\propto 1/\sqrt{\hat{d}^{\pi\_{\operatorname\{mix},k}}(s,a)N\_{k}(s,a)}∝ 1 / square-root start\_ARG over^ start\_ARG italic\_d end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) italic\_N start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) end\_ARG. Bernstein bonus is another tabular UCB bonus that modifies Hoeffding bonus via an empirical estimate of the value function variance. Bernstein bonus is shown to improve over Hoeffding count-only bonus by exploiting additional environment structure Zanette and Brunskill ([2019](#bib.bib102)) and close the gap between algorithmic upper bounds and information-theoretic limits up to logarithmic factors Zhang et al. ([2020b](#bib.bib108), [c](#bib.bib109)). However, a practical and efficient implementation of a bonus that exploits variance information in non-linear function approximation parameterization still remains an open question; see Section [6](#S6 "6 Related work ‣ MADE: Exploration via Maximizing Deviation from Explored Regions") for further discussion. On the other hand, our proposed modification based on the mixture density can be easily and efficiently incorporated with non-linear parameterization. #### Deriving some popular bonuses from regularization. We now discuss how the regularization in ([3](#S3.E3 "3 ‣ 3.1 Regularization to guide exploration ‣ 3 Adaptive regularization of the RL objective ‣ MADE: Exploration via Maximizing Deviation from Explored Regions")) can describe some popular bonuses. Exploration bonuses that only depend on state-action visitation counts can be expressed in the form ([3](#S3.E3 "3 ‣ 3.1 Regularization to guide exploration ‣ 3 Adaptive regularization of the RL objective ‣ MADE: Exploration via Maximizing Deviation from Explored Regions")) by setting the regularizer a linear function of dπsuperscript𝑑𝜋d^{\pi}italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT and the exploration bonus ri(s,a)subscript𝑟𝑖𝑠𝑎r\_{i}(s,a)italic\_r start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( italic\_s , italic\_a ), i.e., Rk(dπ;{dπi}i=1k)=∑s,adπ(s,a)ri(s,a)subscript𝑅𝑘superscript𝑑𝜋superscriptsubscriptsuperscript𝑑subscript𝜋𝑖𝑖1𝑘subscript𝑠𝑎superscript𝑑𝜋𝑠𝑎subscript𝑟𝑖𝑠𝑎R\_{k}(d^{\pi};\{d^{\pi\_{i}}\}\_{i=1}^{k})=\sum\_{s,a}d^{\pi}(s,a)r\_{i}(s,a)italic\_R start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ; { italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) = ∑ start\_POSTSUBSCRIPT italic\_s , italic\_a end\_POSTSUBSCRIPT italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) italic\_r start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( italic\_s , italic\_a ). It is easy to check that taking the gradient of the regularizer with respect to dπsuperscript𝑑𝜋d^{\pi}italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT recovers ri(s,a)subscript𝑟𝑖𝑠𝑎r\_{i}(s,a)italic\_r start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ( italic\_s , italic\_a ). As another example, one can set the regularizer to Shannon entropy Rk(dπ;{dπi}i=1k)=−∑s,adπ(s,a)log⁡dπ(s,a)subscript𝑅𝑘superscript𝑑𝜋superscriptsubscriptsuperscript𝑑subscript𝜋𝑖𝑖1𝑘subscript𝑠𝑎superscript𝑑𝜋𝑠𝑎superscript𝑑𝜋𝑠𝑎R\_{k}(d^{\pi};\{d^{\pi\_{i}}\}\_{i=1}^{k})=-\sum\_{s,a}d^{\pi}(s,a)\log d^{\pi}(s,a)italic\_R start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ; { italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) = - ∑ start\_POSTSUBSCRIPT italic\_s , italic\_a end\_POSTSUBSCRIPT italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) roman\_log italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ), which gives the intrinsic reward −log⁡dπ(s,a)superscript𝑑𝜋𝑠𝑎-\log d^{\pi}(s,a)- roman\_log italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) (up to an additive constant) and recovers the result in the work Zhang et al. ([2021](#bib.bib105)). Algorithm 1 Policy computation for adaptively regularized objective 1:Inputs: Iteration count K𝐾Kitalic\_K, planning error ϵpsubscriptitalic-ϵ𝑝\epsilon\_{p}italic\_ϵ start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT, visitation density error ϵdsubscriptitalic-ϵ𝑑\epsilon\_{d}italic\_ϵ start\_POSTSUBSCRIPT italic\_d end\_POSTSUBSCRIPT. 2:Initialize policy mixture πmix,1=subscript𝜋mix1absent\pi\_{\operatorname\{mix},1}=italic\_π start\_POSTSUBSCRIPT roman\_mix , 1 end\_POSTSUBSCRIPT = with 𝒞1=(π1)subscript𝒞1subscript𝜋1\mathcal{C}\_{1}=(\pi\_{1})caligraphic\_C start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT = ( italic\_π start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) and w1=(1)superscript𝑤11w^{1}=(1)italic\_w start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT = ( 1 ) 3:for k=1,…,K𝑘1…𝐾k=1,\dots,Kitalic\_k = 1 , … , italic\_K do 4: Estimate the visitation density d^πmix,ksuperscript^𝑑subscript𝜋mix𝑘\hat{d}^{\pi\_{\operatorname\{mix},k}}over^ start\_ARG italic\_d end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT of πmix,ksubscript𝜋mix𝑘\pi\_{\operatorname\{mix},k}italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT via a visitation density oracle. 5: Compute reward rk(s,a)=r(s,a)+(1−γ)τk∇dRk(d;{πi}i=1k)\|d=d^πmix,ksubscript𝑟𝑘𝑠𝑎𝑟𝑠𝑎evaluated-at1𝛾subscript𝜏𝑘subscript∇𝑑subscript𝑅𝑘𝑑superscriptsubscriptsubscript𝜋𝑖𝑖1𝑘𝑑superscript^𝑑subscript𝜋mix𝑘r\_{k}(s,a)={r(s,a)}+(1-\gamma){\tau\_{k}\nabla\_{d}R\_{k}(d;\{\pi\_{i}\}\_{i=1}^{k})\big{\|}\_{d=\hat{d}^{\pi\_{\operatorname\{mix},k}}}}italic\_r start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) = italic\_r ( italic\_s , italic\_a ) + ( 1 - italic\_γ ) italic\_τ start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ∇ start\_POSTSUBSCRIPT italic\_d end\_POSTSUBSCRIPT italic\_R start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d ; { italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) \| start\_POSTSUBSCRIPT italic\_d = over^ start\_ARG italic\_d end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT. 6: Run approximate planning on modified MDP Mk=(𝒮,𝒜,P,rk,γ)superscript𝑀𝑘𝒮𝒜𝑃subscript𝑟𝑘𝛾M^{k}=(\mathcal{S},\mathcal{A},P,r\_{k},\gamma)italic\_M start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT = ( caligraphic\_S , caligraphic\_A , italic\_P , italic\_r start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT , italic\_γ ) and return πk+1subscript𝜋𝑘1\pi\_{k+1}italic\_π start\_POSTSUBSCRIPT italic\_k + 1 end\_POSTSUBSCRIPT. 7: Update policy mixture 𝒞k+1=(Ck,πk+1)superscript𝒞𝑘1subscript𝐶𝑘subscript𝜋𝑘1\mathcal{C}^{k+1}=(C\_{k},\pi\_{k+1})caligraphic\_C start\_POSTSUPERSCRIPT italic\_k + 1 end\_POSTSUPERSCRIPT = ( italic\_C start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT , italic\_π start\_POSTSUBSCRIPT italic\_k + 1 end\_POSTSUBSCRIPT ) and wk+1=((1−η)wk,η)superscript𝑤𝑘11𝜂superscript𝑤𝑘𝜂w^{k+1}=((1-\eta)w^{k},\eta)italic\_w start\_POSTSUPERSCRIPT italic\_k + 1 end\_POSTSUPERSCRIPT = ( ( 1 - italic\_η ) italic\_w start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT , italic\_η ). 8:Return: πmix,K=(𝒞k,wk)subscript𝜋mix𝐾superscript𝒞𝑘superscript𝑤𝑘\pi\_{\operatorname\{mix},K}=(\mathcal{C}^{k},w^{k})italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_K end\_POSTSUBSCRIPT = ( caligraphic\_C start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT , italic\_w start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ). ### 3.3 Solving the regularized objective We pair MADE objective with the algorithm proposed by Hazan et al. ([2019](#bib.bib34)) extended to the adaptive objective. We provide convergence guarantees for Algorithm [1](#alg1 "Algorithm 1 ‣ Deriving some popular bonuses from regularization. ‣ 3.2 Exploration via maximizing deviation from policy cover ‣ 3 Adaptive regularization of the RL objective ‣ MADE: Exploration via Maximizing Deviation from Explored Regions") in the following theorem whose proof is given in Appendix [A.1](#A1.SS1 "A.1 Proof of Theorem 1 ‣ Appendix A Convergence analysis of Algorithm 1 ‣ MADE: Exploration via Maximizing Deviation from Explored Regions"). ###### Theorem 1. Consider the following regularizer for ([3](#S3.E3 "3 ‣ 3.1 Regularization to guide exploration ‣ 3 Adaptive regularization of the RL objective ‣ MADE: Exploration via Maximizing Deviation from Explored Regions")) with λ>0𝜆0\lambda>0italic\_λ > 0 and a valid visitation density d𝑑ditalic\_d \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Rλ(d;{dπi}i=1k)=∑s,ad(s,a)+λρ𝑐𝑜𝑣k(s,a)+λ,subscript𝑅𝜆𝑑superscriptsubscriptsuperscript𝑑subscript𝜋𝑖𝑖1𝑘subscript𝑠𝑎𝑑𝑠𝑎𝜆subscriptsuperscript𝜌𝑘𝑐𝑜𝑣𝑠𝑎𝜆\displaystyle R\_{\lambda}(d;\{d^{\pi\_{i}}\}\_{i=1}^{k})=\sum\_{s,a}\sqrt{\frac{d(s,a)+\lambda}{\rho^{k}\_{\text{cov}}(s,a)+\lambda}},italic\_R start\_POSTSUBSCRIPT italic\_λ end\_POSTSUBSCRIPT ( italic\_d ; { italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT ) = ∑ start\_POSTSUBSCRIPT italic\_s , italic\_a end\_POSTSUBSCRIPT square-root start\_ARG divide start\_ARG italic\_d ( italic\_s , italic\_a ) + italic\_λ end\_ARG start\_ARG italic\_ρ start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) + italic\_λ end\_ARG end\_ARG , \| \| (7) \| Set τk=τ/kcsubscript𝜏𝑘𝜏superscript𝑘𝑐\tau\_{k}=\tau/k^{c}italic\_τ start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT = italic\_τ / italic\_k start\_POSTSUPERSCRIPT italic\_c end\_POSTSUPERSCRIPT, where 0<τ<10𝜏10<\tau<10 < italic\_τ < 1 and c>0𝑐0c>0italic\_c > 0. For any ϵ>0italic-ϵ0\epsilon>0italic\_ϵ > 0, there exists η,ϵp,ϵd,c,𝜂subscriptitalic-ϵ𝑝subscriptitalic-ϵ𝑑𝑐\eta,\epsilon\_{p},\epsilon\_{d},c,italic\_η , italic\_ϵ start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT , italic\_ϵ start\_POSTSUBSCRIPT italic\_d end\_POSTSUBSCRIPT , italic\_c , B𝐵Bitalic\_B such that πmix,Ksubscript𝜋normal-mix𝐾\pi\_{\operatorname\{mix},K}italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_K end\_POSTSUBSCRIPT returned by Algorithm [1](#alg1 "Algorithm 1 ‣ Deriving some popular bonuses from regularization. ‣ 3.2 Exploration via maximizing deviation from policy cover ‣ 3 Adaptive regularization of the RL objective ‣ MADE: Exploration via Maximizing Deviation from Explored Regions") after K≥η−1log⁡(10Bϵ−1)𝐾superscript𝜂110𝐵superscriptitalic-ϵ1K\geq\eta^{-1}\log(10B\epsilon^{-1})italic\_K ≥ italic\_η start\_POSTSUPERSCRIPT - 1 end\_POSTSUPERSCRIPT roman\_log ( 10 italic\_B italic\_ϵ start\_POSTSUPERSCRIPT - 1 end\_POSTSUPERSCRIPT ) iterations satisfies Lk(dπmix,K)≥maxπ⁡Lk(dπ)−ϵ.subscript𝐿𝑘superscript𝑑subscript𝜋normal-mix𝐾subscript𝜋subscript𝐿𝑘superscript𝑑𝜋italic-ϵL\_{k}(d^{\pi\_{\operatorname\{mix},K}})\geq\max\_{\pi}L\_{k}(d^{\pi})-\epsilon.italic\_L start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_K end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ) ≥ roman\_max start\_POSTSUBSCRIPT italic\_π end\_POSTSUBSCRIPT italic\_L start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ) - italic\_ϵ . ###### Remark 1. One does not need to maintain the functional forms of past policies to estimate d^πmix,ksuperscriptnormal-^𝑑subscript𝜋normal-mix𝑘\hat{d}^{\pi\_{\operatorname\{mix},k}}over^ start\_ARG italic\_d end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT. Practically, one may truncate the dataset to a (prioritized) buffer and estimate the density over that buffer. s0subscript𝑠0s\_{0}italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT11110.10.10.10.1 Figure 2: A stochastic bidirectional lock. In this environment, the agent starts at s0subscript𝑠0s\_{0}italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT and enters one of the chains based on the selected action. Each chain has a positive reward at the end, H𝐻Hitalic\_H good states, and H𝐻Hitalic\_H dead states. Both actions available to the agent lead it to the dead state, one with probability one and the other with probability p<1𝑝1p<1italic\_p < 1. 4 A tabular study ------------------ We first study the performance of MADE in tabular toy examples. In the Bidirectional Lock experiment, we compare MADE to theoretically guaranteed Hoeffding-style and Bernstein-style bonuses in a sparse reward exploration task. In the Chain MDP, we investigate whether MADE’s regularizer ([4](#S3.E4 "4 ‣ 3.2 Exploration via maximizing deviation from policy cover ‣ 3 Adaptive regularization of the RL objective ‣ MADE: Exploration via Maximizing Deviation from Explored Regions")) provides any benefits in improving optimization rate in policy gradient methods. ### 4.1 Exploration in bidirectional lock We consider a stochastic version of the bidirectional diabolical combination lock (Figure [3](#S4.F3 "Figure 3 ‣ 4.1 Exploration in bidirectional lock ‣ 4 A tabular study ‣ MADE: Exploration via Maximizing Deviation from Explored Regions")), which is considered a particularly difficult exploration task in tabular setting Misra et al. ([2020](#bib.bib63)); Agarwal et al. ([2020a](#bib.bib2)). This environment is challenging because: (1) positive rewards are sparse, (2) a small negative reward is given when transiting to a good state and thus, moving to a dead state is locally optimal, and (3) the agent may forget to explore one chain and get stuck in local minima upon receiving an end reward in one lock Agarwal et al. ([2020a](#bib.bib2)). ![Refer to caption](/html/2106.10268/assets/x1.png) Figure 3: Performance of different count-based methods in the stochastic bidirectional lock environment. MADE performs better than the Hoeffding bonus and is comparable to the Bernstein bonus. #### RL algorithms and exploration strategies. We compare the performance of Hoeffding and Bernstein bonuses Jin et al. ([2018](#bib.bib41)) to MADE in three different RL algorithms. To implement MADE in tabular setting, we simply use two buffers: one that stores all past state-action pairs to estimate ρcovsubscript𝜌cov\rho\_{\text{cov}}italic\_ρ start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT and another one that only maintains the most recent B𝐵Bitalic\_B pairs to estimate dμπsuperscriptsubscript𝑑𝜇𝜋d\_{\mu}^{\pi}italic\_d start\_POSTSUBSCRIPT italic\_μ end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT. We use empirical counts to estimate both densities, which give a bonus ∝1/Nk(s,a)Bk(s,a)proportional-toabsent1subscript𝑁𝑘𝑠𝑎subscript𝐵𝑘𝑠𝑎\propto 1/\sqrt{N\_{k}(s,a)B\_{k}(s,a)}∝ 1 / square-root start\_ARG italic\_N start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) italic\_B start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) end\_ARG, where Nk(s,a)subscript𝑁𝑘𝑠𝑎N\_{k}(s,a)italic\_N start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) is the total count and Bk(s,a)subscript𝐵𝑘𝑠𝑎B\_{k}(s,a)italic\_B start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) is the recent buffer count of (s,a)𝑠𝑎(s,a)( italic\_s , italic\_a ) pair. We combine three bonuses with three RL algorithms: (1) value iteration with bonus He et al. ([2020](#bib.bib35)), (2) proximal policy optimization (PPO) with a model Cai et al. ([2020](#bib.bib15)), and (3) Q-learning with bonus Jin et al. ([2018](#bib.bib41)). ![Refer to caption](/html/2106.10268/assets/x2.png) Figure 4: Values of Hoeffding, Bernstein, and MADE exploration bonus for all states and action 1111 over environment steps in the bidirectional lock MDP. MADE bonus values closely follows Bernstein bonus values. ![Refer to caption](/html/2106.10268/assets/bonus_counts.png) Figure 5: Heatmap of visitation counts in the bidirectional lock, plotted every 200200200200 iterations. The exploration strategy of MADE appears to be closet to the Bernstein bonus. #### Results. Figure [3](#S4.F3 "Figure 3 ‣ 4.1 Exploration in bidirectional lock ‣ 4 A tabular study ‣ MADE: Exploration via Maximizing Deviation from Explored Regions") summarizes our results showing MADE improves over the Hoeffding bonus and is competitive to the Bernstein bonus in all three algorithms. Unlike Bernstein bonus that is hard to compute beyond tabular setting, MADE bonus design is simple and can be effectively combined with any deep RL algorithm. The experimental results suggest several interesting properties for MADE. First, MADE applies a simple modification to the Hoeffding bonus which improves the performance. Second, as illustrated in Figures [4](#S4.F4 "Figure 4 ‣ RL algorithms and exploration strategies. ‣ 4.1 Exploration in bidirectional lock ‣ 4 A tabular study ‣ MADE: Exploration via Maximizing Deviation from Explored Regions") and [5](#S4.F5 "Figure 5 ‣ RL algorithms and exploration strategies. ‣ 4.1 Exploration in bidirectional lock ‣ 4 A tabular study ‣ MADE: Exploration via Maximizing Deviation from Explored Regions"), bonus values and exploration pattern of MADE is somewhat similar to the Bernstein bonus. This suggests that MADE may capture some structural information of the environment, similar to Bernstein bonus, which captures certain environmental properties such as the degree of stochasticity Zanette and Brunskill ([2019](#bib.bib102)). ### 4.2 Policy gradient in a chain MDP s0subscript𝑠0s\_{0}italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPTs1subscript𝑠1s\_{1}italic\_s start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT……\dots…sHsubscript𝑠𝐻{s\_{H}}italic\_s start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPTsH+1subscript𝑠𝐻1s\_{H+1}italic\_s start\_POSTSUBSCRIPT italic\_H + 1 end\_POSTSUBSCRIPTa1subscript𝑎1a\_{1}italic\_a start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPTa2subscript𝑎2a\_{2}italic\_a start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPTa3subscript𝑎3a\_{3}italic\_a start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPTa4subscript𝑎4a\_{4}italic\_a start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPTa1subscript𝑎1a\_{1}italic\_a start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPTa2subscript𝑎2a\_{2}italic\_a start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPTa3subscript𝑎3a\_{3}italic\_a start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPTa4subscript𝑎4a\_{4}italic\_a start\_POSTSUBSCRIPT 4 end\_POSTSUBSCRIPTa1subscript𝑎1a\_{1}italic\_a start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPTa1subscript𝑎1a\_{1}italic\_a start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ![Refer to caption](/html/2106.10268/assets/x3.png) Figure 6: A deterministic chain MDP that suffers from vanishing gradients Agarwal et al. ([2019](#bib.bib1)). We consider a constrained tabular policy parameterization with π(a\|s)=θs,a𝜋conditional𝑎𝑠subscript𝜃𝑠𝑎\pi(a\|s)=\theta\_{s,a}italic\_π ( italic\_a \| italic\_s ) = italic\_θ start\_POSTSUBSCRIPT italic\_s , italic\_a end\_POSTSUBSCRIPT and ∑aθs,a=1subscript𝑎subscript𝜃𝑠𝑎1\sum\_{a}\theta\_{s,a}=1∑ start\_POSTSUBSCRIPT italic\_a end\_POSTSUBSCRIPT italic\_θ start\_POSTSUBSCRIPT italic\_s , italic\_a end\_POSTSUBSCRIPT = 1. The agent always starts from s0subscript𝑠0s\_{0}italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT and the only non-zero reward is r(sH+1,a1)𝑟subscript𝑠𝐻1subscript𝑎1r(s\_{H+1},a\_{1})italic\_r ( italic\_s start\_POSTSUBSCRIPT italic\_H + 1 end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) = 1. We consider the chain MDP (Figure [6](#S4.F6 "Figure 6 ‣ 4.2 Policy gradient in a chain MDP ‣ 4 A tabular study ‣ MADE: Exploration via Maximizing Deviation from Explored Regions")) presented in Agarwal et al. ([2019](#bib.bib1)), which suffers from vanishing gradients with policy gradient approach Sutton et al. ([1999](#bib.bib90)) as a positive reward is only achieved if the agent always takes action a1subscript𝑎1a\_{1}italic\_a start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. This leads to an exponential iteration complexity lower bound on the convergence of vanilla policy gradient approach even with access to exact gradients Agarwal et al. ([2019](#bib.bib1)). In this environment the agent always starts at state s0subscript𝑠0s\_{0}italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT and recent guarantees on the global convergence of exact policy gradients are vacuous Bhandari and Russo ([2019](#bib.bib10)); Agarwal et al. ([2019](#bib.bib1)); Mei et al. ([2020](#bib.bib59)). This is because the rates depend on the ratio between the optimal and learned visitation densities, known as concentrability coefficient Kakade and Langford ([2002](#bib.bib44)); Scherrer ([2014](#bib.bib82)); Geist et al. ([2017](#bib.bib30)); Rashidinejad et al. ([2021](#bib.bib79)), or the ratio between the optimal visitation density and initial distribution Agarwal et al. ([2019](#bib.bib1)). #### RL algorithms. Since our goal in this experiment is to investigate the optimization effects and not exploration, we assume access to exact gradients. In this setting, we consider MADE regularizer with the form ∑s,adπ(s,a)subscript𝑠𝑎superscript𝑑𝜋𝑠𝑎\sum\_{s,a}\sqrt{d^{\pi}(s,a)}∑ start\_POSTSUBSCRIPT italic\_s , italic\_a end\_POSTSUBSCRIPT square-root start\_ARG italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) end\_ARG. Note that policy gradients take gradient of the objective with respect to the policy parameters θ𝜃\thetaitalic\_θ and not dπsuperscript𝑑𝜋d^{\pi}italic\_d start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT. We compare optimizing the policy gradient objective with four methods: vanilla version PG (e.g. uses policy gradient theorem (Williams, [1992](#bib.bib96); Sutton et al., [1999](#bib.bib90); Konda and Tsitsiklis, [2000](#bib.bib50))), relative policy entropy regularization PG+RE (Agarwal et al., [2019](#bib.bib1)), policy entropy regularization PG+E (Mnih et al., [2016](#bib.bib64); Mei et al., [2020](#bib.bib59)), and MADE regularization. #### Results. Figure [6](#S4.F6 "Figure 6 ‣ 4.2 Policy gradient in a chain MDP ‣ 4 A tabular study ‣ MADE: Exploration via Maximizing Deviation from Explored Regions") illustrates our results on policy gradient methods. As expected Agarwal et al. ([2019](#bib.bib1)), the vanilla version has a very slow convergence rate. Both entropy and relative entropy regularization methods are proved to achieve a linear convergence rate of exp⁡(−t)𝑡\exp(-t)roman\_exp ( - italic\_t ) in the iteration count t𝑡titalic\_t Mei et al. ([2020](#bib.bib59)); Agarwal et al. ([2019](#bib.bib1)). Interestingly, MADE seems to outperforms the policy entropy regularizers, quickly converging to a globally optimal policy. 5 Experiments on MiniGrid and DeepMind Control Suite ----------------------------------------------------- In addition to the tabular setting, MADE can also be integrated with various model-free and model-based deep RL algorithms such as IMPALA (Espeholt et al., [2018](#bib.bib24)), RAD (Lee et al., [2019a](#bib.bib54)), and Dreamer (Hafner et al., [2019](#bib.bib33)). As we will see shortly, MADE exploration strategy on MiniGrid (Chevalier-Boisvert et al., [2018](#bib.bib18)) and DeepMind Control Suite (Tassa et al., [2020](#bib.bib92)) tasks achieves state-of-the-art sample efficiency. For a practical estimation of ρcovksubscriptsuperscript𝜌𝑘cov\rho^{k}\_{\text{cov}}italic\_ρ start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT and dπmix,ksuperscript𝑑subscript𝜋mix𝑘d^{\pi\_{\operatorname\{mix},k}}italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT, we adopt the two buffer idea described in the tabular setting. However, since now the state space is high-dimensional, we use RND (Burda et al., [2018b](#bib.bib14)) to estimate Nk(s,a)subscript𝑁𝑘𝑠𝑎N\_{k}(s,a)italic\_N start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) (and thus ρcovksubscriptsuperscript𝜌𝑘cov\rho^{k}\_{\text{cov}}italic\_ρ start\_POSTSUPERSCRIPT italic\_k end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT cov end\_POSTSUBSCRIPT) and use a variational auto-encoder (VAE) to estimate dπmix,ksuperscript𝑑subscript𝜋mix𝑘d^{\pi\_{\operatorname\{mix},k}}italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT. Specifically, for RND, we minimize the difference between a predictor network ϕ′(s,a)superscriptitalic-ϕ′𝑠𝑎\phi^{\prime}({s,a})italic\_ϕ start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) and a randomly initialized target network ϕ(s,a)italic-ϕ𝑠𝑎\phi({s,a})italic\_ϕ ( italic\_s , italic\_a ) and train it in an online manner as the agent collects data. We sample data from the recent buffer ℬℬ\mathcal{B}caligraphic\_B to train a VAE. The length of ℬℬ\mathcal{B}caligraphic\_B is a design choice for which we do an ablation study. Thus, the intrinsic reward in deep RL setting takes the following form \| \| \| \| \| --- \| --- \| --- \| \| \| (1−γ)τk∥ϕ(s,a)−ϕ′(s,a)∥dπmix,k(s,a).1𝛾subscript𝜏𝑘delimited-∥∥italic-ϕ𝑠𝑎superscriptitalic-ϕ′𝑠𝑎superscript𝑑subscript𝜋mix𝑘𝑠𝑎\displaystyle(1-\gamma)\tau\_{k}\frac{\left\lVert\phi({s,a})-\phi^{\prime}({s,a})\right\rVert}{\sqrt{{d^{\pi\_{\operatorname\{mix},k}}(s,a)}}}.( 1 - italic\_γ ) italic\_τ start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT divide start\_ARG ∥ italic\_ϕ ( italic\_s , italic\_a ) - italic\_ϕ start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) ∥ end\_ARG start\_ARG square-root start\_ARG italic\_d start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT roman\_mix , italic\_k end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) end\_ARG end\_ARG . \| \| #### Model-free RL baselines. We consider several baselines in MiniGrid: IMPALA (Espeholt et al., [2018](#bib.bib24)) is a variant of policy gradient algorithms which we use as the training baseline; ICM (Pathak et al., [2017](#bib.bib73)) learns a forward and reverse model for predicting state transition and uses the forward model prediction error as intrinsic reward; RND (Burda et al., [2018b](#bib.bib14)) trains a predictor network to mimic a randomly initialized target network as discussed above; RIDE (Raileanu and Rocktäschel, [2020](#bib.bib78)) learns a representation similar to ICM and uses the difference of learned representations along a trajectory as intrinsic reward; AMIGo (Campero et al., [2020](#bib.bib16)) learns a teacher agent to assign intrinsic reward; BeBold (Zhang et al., [2020a](#bib.bib107)) adopts a regulated difference of novelty measure using RND. In DeepMind Control Suite, we consider RE3 (Seo et al., [2021](#bib.bib83)) as a baseline which uses a random encoder for state embedding followed by a k𝑘kitalic\_k-nearest neighbour bonus for a maximum state coverage objective. #### Model-based RL baselines. MADE can be combined with model-based RL algorithms to improve sample efficiency. For baselines, we consider Dreamer, which is a well-known model-based RL algorithm for DeepMind Control Suite, as well as Dreamer+RE3, which includes RE3 bonus on top of Dreamer. MADE achieves state-of-the-art results on both navigation and locomotion tasks by a substantial margin, greatly improving the sample efficiency of the RL exploration in both model-free and model-based methods. Further details on experiments and exact hyperparameters are provided in Appendix [B](#A2 "Appendix B Experimental details ‣ MADE: Exploration via Maximizing Deviation from Explored Regions"). ![Refer to caption](/html/2106.10268/assets/x4.png) Figure 7: Results for various hard exploration tasks from MiniGrid. MADE successfully solves all the environments while other algorithms (except for BeBold) fail to solve several environments. MADE finds the optimal solution with 2-5 times fewer samples, yielding a much better sample efficiency. ### 5.1 Model-free RL on MiniGrid MiniGrid (Chevalier-Boisvert et al., [2018](#bib.bib18)) is a widely used benchmark for exploration in RL. Despite having symbolic states and a discrete action space, MiniGrid tasks are quite challenging. The easiest task is MultiRoom (MR) in which the agent needs to navigate to the goal by going to different rooms connected by the doors. In KeyCorridor (KC), the agent needs to search around different rooms to find the key and then use it to open the door. ObstructedMaze (OM) is a harder version of KC where the key is hidden in a box and sometimes the door is blocked by an obstruct. In addition to that, the entire environment is procedurally-generated. This adds another layer of difficulty to the problem. From Figure [7](#S5.F7 "Figure 7 ‣ Model-based RL baselines. ‣ 5 Experiments on MiniGrid and DeepMind Control Suite ‣ MADE: Exploration via Maximizing Deviation from Explored Regions") we can see that MADE manages to solve all the challenging tasks within 90M steps while all other baselines (except BeBold) only solve up to 50% of them. Compared to BeBold, MADE uses significantly (2-5 times) fewer samples. ### 5.2 Model-free RL on DeepMind Control We also test MADE on image-based continuous control tasks of DeepMind Control Suite (Tassa et al., [2020](#bib.bib92)), which is a collection of diverse control tasks such as Pendulum, Hopper, and Acrobot with realistic simulations. Compared to MiniGrid, these tasks are more realistic and complex as they involve stochastic transitions, high-dimensional states, and continuous actions. For baselines, we build our algorithm on top of RAD (Lee et al., [2019a](#bib.bib54)), a strong model-free RL algorithm with a competitive sample efficiency. We compare our approach with ICM, RND, as well as RE3, which is the SOTA algorithm.111As we were not provided with the source code, we implemented ICM and RND ourselves. The performance for ICM is slightly worse than what the author reported, but the performance of RND and RE3 is similar. Note that we compare MADE to very strong baselines. Other algorithms such as DrQ (Kostrikov et al., [2020](#bib.bib51)), CURL (Srinivas et al., [2020](#bib.bib86)), ProtoRL (Yarats et al., [2021](#bib.bib101)), SAC+AE (Yarats et al., [2019](#bib.bib100))) perform worse based on the results reported in the original papers. MADE show consistent improvement in sample efficiency: 2.6 times over RAD+RE3, 3.3 times over RAD+RND, 19.7 times over CURL, 15.0 times over DrQ and 3.8 times over RAD. From Figure [8](#S5.F8 "Figure 8 ‣ 5.2 Model-free RL on DeepMind Control ‣ 5 Experiments on MiniGrid and DeepMind Control Suite ‣ MADE: Exploration via Maximizing Deviation from Explored Regions"), we can see that MADE consistently improves sample efficiency compared to all baselines. For these tasks, RND and ICM do not perform well and even fail on Cartpole-Swingup. RE3 achieves a comparable performance in two tasks, however, its performance on Pendulum-Swingup, Quadruped-Run, Hopper-Hop and Walker-Run is significantly worse than MADE. For example, in Pendulum-Swingup, MADE achieves a reward of around 800 in only 30K steps while RE3 requires 300k samples. In Quadruped-Run, there is a 150 reward gap between MADE and RE3, which seems to be still enlarging. These tasks show the strong performance of MADE in model-free RL. ![Refer to caption](/html/2106.10268/assets/x5.png) Figure 8: Results for several DeepMind control suite locomotion tasks. Comparing to all baselines, the performance of MADE is consistently better. Sometimes baseline methods even fail to solve the task. ![Refer to caption](/html/2106.10268/assets/x6.png) Figure 9: Ablation study on buffer size in MADE. The optimal buffer size varies in different tasks. We found buffer size of 10000 empirically works consistently reasonable. #### Ablation study. We study how the buffer length affects the performance of our algorithm in some DeepMind Control tasks. Results illustrated in Figure [9](#S5.F9 "Figure 9 ‣ 5.2 Model-free RL on DeepMind Control ‣ 5 Experiments on MiniGrid and DeepMind Control Suite ‣ MADE: Exploration via Maximizing Deviation from Explored Regions") show that for different tasks the optimal buffer length is slightly different. We empirically found that using a buffer length of 1000 consistently works well across different tasks. ### 5.3 Model-based RL on DeepMind Control We also empirically verify the performance of MADE combined with the SOTA model-based RL algorithm Dreamer (Hafner et al., [2019](#bib.bib33)). We compare MADE with Dreamer and Dreamer combined with RE3 in Figure [10](#S5.F10 "Figure 10 ‣ 5.3 Model-based RL on DeepMind Control ‣ 5 Experiments on MiniGrid and DeepMind Control Suite ‣ MADE: Exploration via Maximizing Deviation from Explored Regions"). Results show that MADE has great sample efficiency in Cheetah-Run-Sparse, Hopper-Hop and Pendulum-Swingup environments. For example, in Hopper-Hop, MADE achieves more than 100 higher return than RE3 and 250 higher return than Dreamer, achieving a new SOTA result. ![Refer to caption](/html/2106.10268/assets/x7.png) Figure 10: Results for DeepMind control suite locomotion tasks in model-based RL setting. Comparing to all baselines, the performance of MADE is consistently better. Some baseline methods even fail to solve the task. 6 Related work --------------- #### Provable optimistic exploration. Most provable exploration strategies are based on optimism in the face of uncertainty (OFU) principle. In tabular setting, model-based exploration algorithms include variants of UCB Kearns and Singh ([2002](#bib.bib47)); Brafman and Tennenholtz ([2002](#bib.bib11)), UCRL Lattimore and Hutter ([2012](#bib.bib53)); Jaksch et al. ([2010](#bib.bib38)); Zanette and Brunskill ([2019](#bib.bib102)); Kaufmann et al. ([2021](#bib.bib46)); Ménard et al. ([2020](#bib.bib61)), and Thompson sampling Xiong et al. ([2021](#bib.bib98)); Agrawal and Jia ([2017](#bib.bib4)); Russo ([2019](#bib.bib80)) and value-based methods include optimistic Q-learning Jin et al. ([2018](#bib.bib41)); Wang et al. ([2019b](#bib.bib95)); Strehl et al. ([2006](#bib.bib87)); Liu and Su ([2020](#bib.bib57)); Menard et al. ([2021](#bib.bib62)) and value-iteration with UCB Azar et al. ([2017](#bib.bib7)); Zhang et al. ([2020b](#bib.bib108), [c](#bib.bib109)); Jin et al. ([2020a](#bib.bib42)). These methods are recently extended to linear MDP setting leading to a variety of model-based Zhou et al. ([2020a](#bib.bib110)); Ayoub et al. ([2020](#bib.bib6)); Jia et al. ([2020](#bib.bib39)); Zhou et al. ([2020b](#bib.bib111)), value-based Wang et al. ([2019a](#bib.bib94)); Jin et al. ([2020b](#bib.bib43)), and policy-based algorithms Cai et al. ([2020](#bib.bib15)); Zanette et al. ([2021](#bib.bib104)); Agarwal et al. ([2020a](#bib.bib2)). Going beyond linear function approximation, systematic exploration strategies are developed based on structural assumptions on MDP such as low Bellman rank Jiang et al. ([2017](#bib.bib40)) and block MDP Du et al. ([2019](#bib.bib21)). These methods are either computationally intractable Jiang et al. ([2017](#bib.bib40)); Sun et al. ([2019](#bib.bib88)); Ayoub et al. ([2020](#bib.bib6)); Zanette et al. ([2020](#bib.bib103)); Yang et al. ([2020](#bib.bib99)); Dong et al. ([2021](#bib.bib20)); Wang et al. ([2020](#bib.bib93)) or are only oracle efficient Feng et al. ([2020](#bib.bib25)); Agarwal et al. ([2020b](#bib.bib3)). The recent work Feng et al. ([2021](#bib.bib26)) provides a sample efficient approach with non-linear policies, however, the algorithm requires maintaining the functional form of all prior policies. #### Practical exploration via intrinsic reward. Apart from previously-discussed methods, other works give intrinsic reward based on the difference in (abstraction of) consecutive states Zhang et al. ([2019](#bib.bib106)); Marino et al. ([2019](#bib.bib58)); Raileanu and Rocktäschel ([2020](#bib.bib78)). However, this approach is inconsistent: the intrinsic reward does not converge to zero and thus, even with infinite samples, the final policy does not maximize the RL objective. Other intrinsic rewards try to estimate pseudo-counts Bellemare et al. ([2016](#bib.bib9)); Tang et al. ([2017](#bib.bib91)); Burda et al. ([2018b](#bib.bib14), [a](#bib.bib13)); Ostrovski et al. ([2017](#bib.bib72)); Badia et al. ([2020](#bib.bib8)), inspired by count-only UCB bonus. Though favoring novel states, practically these methods might suffer from detachment and derailment Ecoffet et al. ([2019](#bib.bib22), [2020](#bib.bib23)), and forgetting Agarwal et al. ([2020a](#bib.bib2)). More recent works propose a combination of different criteria. RIDE (Raileanu and Rocktäschel, [2020](#bib.bib78)) learns a representation via a curiosity criterion and uses the difference of consecutive states along the trajectory as the bonus. AMIGo (Campero et al., [2020](#bib.bib16)) learns a teacher agent for assigning rewards for exploration. Go-Explore (Ecoffet et al., [2019](#bib.bib22)) explicitly decouples the exploration and exploitation stages, yielding a more sophisticated algorithm with many hand-tuned hyperparameters. #### Maximum entropy exploration. Another line of work encourages exploration via maximizing some type of entropy. One category maximizes policy entropy Mnih et al. ([2016](#bib.bib64)) or relative entropy Agarwal et al. ([2019](#bib.bib1)) in addition to the RL objective. The work of Flet-Berliac et al. ([2021](#bib.bib27)) modifies the RL objective by introducing an adversarial policy which results in the next policy to move away from prior policies while staying close to the current policy. In contrast, our approach focuses on the regions explored by prior policies as opposed to the prior policies themselves. Recently, effects of policy entropy regularization have been studied theoretically Neu et al. ([2017](#bib.bib69)); Geist et al. ([2019](#bib.bib31)). In policy gradient methods with access to exact gradients, policy entropy regularization results in faster convergence by improving the optimization landscape Mei et al. ([2020](#bib.bib59), [2021](#bib.bib60)); Ahmed et al. ([2019](#bib.bib5)); Cen et al. ([2020](#bib.bib17)). Another category considers maximizing the entropy of state or state-action visitation densities such as Shannon entropy Hazan et al. ([2019](#bib.bib34)); Islam et al. ([2019](#bib.bib37)); Lee et al. ([2019b](#bib.bib55)); Seo et al. ([2021](#bib.bib83)) or Rényi entropy Zhang et al. ([2021](#bib.bib105)). Empirically, our approach achieves better performance over entropy-based methods. #### Other exploration strategies. Besides intrinsic motivation, other strategies are also fruitful in encouraging the RL agent to visit a wide range of states. One example is exploration by injecting noise to the action action space Lillicrap et al. ([2015](#bib.bib56)); Osband et al. ([2016](#bib.bib70)); Hessel et al. ([2017](#bib.bib36)); Osband et al. ([2019](#bib.bib71)) or parameter space Fortunato et al. ([2018](#bib.bib29)); Plappert et al. ([2018](#bib.bib75)). Another example is the reward-shaping category, in which diverse goals are set to guide exploration Colas et al. ([2019](#bib.bib19)); Florensa et al. ([2018](#bib.bib28)); Nair et al. ([2018](#bib.bib68)); Pong et al. ([2020](#bib.bib76)). 7 Discussion ------------- We introduce a new exploration strategy MADE based on maximizing deviation from explored regions. We show that by simply adding a regularizer to the original RL objective, we get an easy-to-implement intrinsic reward which can be incorporated with any RL algorithm. We provide a policy computation algorithm for this objective and prove that it converges to a global optimum, provided that we have access to an approximate planner. In tabular setting, MADE consistently improves over the Hoeffding bonus and shows competitive performance to the Bernstein bonus, while the latter is impractical to compute beyond tabular. We conduct extensive experiments on MiniGrid, showing a significant (over 5 times) reduction of the required sample size. MADE also performs well in DeepMind Control Suite when combined with both model-free and model-based RL algorithms, achieving SOTA sample efficiency results. One limitation of the current work is that it only uses the naive representations of states (e.g., one-hot representation in tabular case). In fact, exploration could be conducted much more efficiently if MADE is implemented with a more compact representation of states. We leave this direction to future work. Acknowledgements ---------------- The authors are grateful to Andrea Zanette for helpful discussions. The authors thank Alekh Agarwal, Michael Henaff, Sham Kakade, and Wen Sun for providing their code. Paria Rashidinejad is partially supported by the Open Philanthropy Foundation, Intel, and the Leverhulme Trust. Jiantao Jiao is partially supported by NSF grants IIS-1901252, CCF-1909499, and DMS-2023505. Tianjun Zhang is supported by the BAIR Commons at UC-Berkeley and thanks Commons sponsors for their support. In addition to NSF CISE Expeditions Award CCF-1730628, UC Berkeley research is supported by gifts from Alibaba, Amazon Web Services, Ant Financial, CapitalOne, Ericsson, Facebook, Futurewei, Google, Intel, Microsoft, Nvidia, Scotiabank, Splunk and VMware.
701f7618-18fe-4760-a868-d58ba8cb6a38	trentmkelly/LessWrong-43k	LessWrong	RationalWiki's take on LW I am not sure whether this has been posted here before but I came across this: http://rationalwiki.org/wiki/LessWrong What do you think about RationalWiki in general, and their opinion regarding LW?
cfc0d0f6-5797-4e44-a20a-de41e6e1d6c8	trentmkelly/LessWrong-43k	LessWrong	[SEQ RERUN] Rational vs Scientific Ev-Psych Today's post, Rational vs. Scientific Ev-Psych was originally published on 04 January 2008. A summary (taken from the LW wiki): > In Evolutionary Biology or Psychology, a nice-sounding but untested theory is referred to as a "just-so story", after the stories written by Rudyard Kipling. But, if there is a way to test the theory, people tend to consider it more likely to be correct. This is not a rational tendency. Discuss the post here (rather than in the comments to the original post). This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was Stop Voting For Nincompoops, and you can use the sequence_reruns tag or rss feed to follow the rest of the series. Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day's sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series.
1cb6be98-c2b2-4c7e-acc3-a56b4fbfacb9	trentmkelly/LessWrong-43k	LessWrong	Poker, Beef Wellington, and Mount Stupid Introduction This post will hopefully serve to illustrate a common pattern I see, one of those things that you see everywhere once you figure it out. I would be surprised if most people who read this haven't had the same thoughts as me, but maybe the direction I take the concepts will be different than those people. Of course, those people will have directions of their own. If that's you, feel free to put your insight in the comments. So, what are the concepts I'm introducing? The first is Levels of Thought. HPMOR readers will recognize this from Eliezer Yudkowsky's excellent description of the concept: > (Professor Quirrell had remarked over their lunch that Harry really needed to conceal his state of mind better than putting on a blank face when someone discussed a dangerous topic, and had explained about one-level deceptions, two-level deceptions, and so on. So either Severus was in fact modeling Harry as a one-level player, which made Severus himself two-level, and Harry's three-level move had been successful; or Severus was a four-level player and wanted Harry to think the deception had been successful. Harry, smiling, had asked Professor Quirrell what level he played at, and Professor Quirrell, also smiling, had responded, One level higher than you.) Honestly, this might be all you need to understand the concept, but my explanation might make the concept more accessible to real-world discussions, especially "Levels of Thought In Real Life". The other concept is being blindsided. I explain what this is, how it ties into the Dunning-Kruger effect, and why it might not be such a bad thing in "Being Blindsided By Higher Level Thinkers (Falling Off Mount Stupid)" This is in fact my first post. Thank you for reading it! What Is A Level Of Thought? The Situation Imagine that you are sitting in the back room of a shady bar, watching a game of Texas Hold'em play out between four players: A, B, C and D. The hole cards are dealt, and A immediately bets 3 times
1e4f4db5-e21f-48e1-a887-effec29b939d	trentmkelly/LessWrong-43k	LessWrong	Searle’s Chinese Room and the Meaning of Meaning In response to the question of how a general intelligence could be recognised, Alan Turing proposed the following empirical test: Any entity that could interact with an investigator, fooling her into thinking it was a person, would be ascribed intelligence. Searle's Chinese room thought experiment rejects Turing’s test, denying that a computer could under any circumstances be said to have intelligence. Searle compared a computer’s actions with those of a technician whose job it is to respond to messages presented in some unfamiliar script. The technician consults a list of procedures and executes some prescribed action. Searle thought that the actions of a computer were necessarily comparable to those of the technician, denying that any understanding was taking place. I want to challenge Searle’s contention by arguing that his assumptions about the capabilities of a general intelligence are far too stunted. ---------------------------------------- Humans start off life just like the technician; confronted with a stream of nearly incoherent inputs (unintelligible sound waves, patterns of light and dark; or, alternatively: neural activity). In a sense, we are worse off, since we initially don’t have much of a repertoire of procedures to guide our behaviour. But we do have one advantage. Namely, the ability to learn. A baby tries first one thing, then another, receiving uninterrupted feedback from his environment. With experience, the newborn learns the importance of context: it mostly pays off to reach into the cookie bag, but not when there are signs of a hungry animal inside. What is our baby doing in his explorations? He is building a complex control system, with a dense matrix of inhibitory and activating responses. When the control system reaches a certain arbitrary threshold, we say our (no longer) baby has achieved intelligence. Now come back to the technician/computer program. Instead of having him rely slavishly on a list of procedures, let’s have him
a19c7f0f-605c-4658-8351-f3ceeaa1f63d	trentmkelly/LessWrong-43k	LessWrong	What can we learn from Bayes about reasoning? What are the qualitative lessons we can learn about logic and reasoning from Bayesian epistemology, that is, from taking Bayes' rule as a mathematical model for thought (even if it is considered a simplified formalism that we often can't implement?) I've seen at least a few of these from @Eliezer Yudkowsky, but I think they're scattered across many essays. Some things I consider to be examples of what I'm gesturing at here: * Aumann's Agreement Theorem * Conservation of expected evidence Thanks!
9eb3f721-09f0-4f99-9503-309c53ddc70e	trentmkelly/LessWrong-43k	LessWrong	Meetup : Moscow LW meetup in "Nauchka" library Discussion article for the meetup : Moscow LW meetup in "Nauchka" library WHEN: 09 June 2017 08:00:00PM (+0300) WHERE: Москва, ул. Дубининская, 20 Welcome to the next Moscow LW meetup in "Nauchka" library! Our plan: * A talk about yak shaving problem. * Fallacymania game. * Tower of Chaos game. Details about Fallacymania and Tower of Chaos and game materials can be found here: http://lesswrong.com/lw/oco/custom_games_that_involve_skills_related_to/ Meetup details are here: https://goo.gl/5fd66P Come to "Nauchka", ul.Dubininskaya, 20. Entrance through the Central children library #14. Nearest metro station is Paveletskaya. Map is here: http://nauchka.ru/contacts/ . If you are lost, call Sasha at +7-905-527-30-82. Meetup begins at 20:00, the length is 2 hours. Discussion article for the meetup : Moscow LW meetup in "Nauchka" library
63ed15bb-b4b0-4919-99a5-064f0d084256	StampyAI/alignment-research-dataset/arxiv	Arxiv	Learning Robust Rewards with Adversarial Inverse Reinforcement Learning 1 Introduction --------------- While reinforcement learning (RL) provides a powerful framework for automating decision making and control, significant engineering of elements such as features and reward functions has typically been required for good practical performance. In recent years, deep reinforcement learning has alleviated the need for feature engineering for policies and value functions, and has shown promising results on a range of complex tasks, from vision-based robotic control (Levine et al., [2016](#bib.bib11)) to video games such as Atari (Mnih et al., [2015](#bib.bib12)) and Minecraft (Oh et al., [2016](#bib.bib15)). However, reward engineering remains a significant barrier to applying reinforcement learning in practice. In some domains, this may be difficult to specify (for example, encouraging “socially acceptable” behavior), and in others, a naïvely specified reward function can produce unintended behavior (Amodei et al., [2016](#bib.bib2)). Moreover, deep RL algorithms are often sensitive to factors such as reward sparsity and magnitude, making well performing reward functions particularly difficult to engineer. Inverse reinforcement learning (IRL) (Russell, [1998](#bib.bib18); Ng & Russell, [2000](#bib.bib13)) refers to the problem of inferring an expert’s reward function from demonstrations, which is a potential method for solving the problem of reward engineering. However, inverse reinforcement learning methods have generally been less efficient than direct methods for learning from demonstration such as imitation learning (Ho & Ermon, [2016](#bib.bib9)), and methods using powerful function approximators such as neural networks have required tricks such as domain-specific regularization and operate inefficiently over whole trajectories (Finn et al., [2016b](#bib.bib6)). There are many scenarios where IRL may be preferred over direct imitation learning, such as re-optimizing a reward in novel environments (Finn et al., [2017](#bib.bib7)) or to infer an agent’s intentions, but IRL methods have not been shown to scale to the same complexity of tasks as direct imitation learning. However, adversarial IRL methods (Finn et al., [2016b](#bib.bib6), [a](#bib.bib5)) hold promise for tackling difficult tasks due to the ability to adapt training samples to improve learning efficiency. Part of the challenge is that IRL is an ill-defined problem, since there are many optimal policies that can explain a set of demonstrations, and many rewards that can explain an optimal policy (Ng et al., [1999](#bib.bib14)). The maximum entropy (MaxEnt) IRL framework introduced by Ziebart et al. ([2008](#bib.bib23)) handles the former ambiguity, but the latter ambiguity means that IRL algorithms have difficulty distinguishing the true reward functions from those shaped by the environment dynamics. While shaped rewards can increase learning speed in the original training environment, when the reward is deployed in at test-time on environments with varying dynamics, it may no longer produce optimal behavior, as we discuss in Sec. [5](#S5 "5 The Reward Ambiguity Problem ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). To address this issue, we discuss how to modify IRL algorithms to learn rewards that are invariant to changing dynamics, which we refer to as disentangled rewards. In this paper, we propose adversarial inverse reinforcement learning (AIRL), an inverse reinforcement learning algorithm based on adversarial learning. Our algorithm provides for simultaneous learning of the reward function and value function, which enables us to both make use of the efficient adversarial formulation and recover a generalizable and portable reward function, in contrast to prior works that either do not recover a reward functions (Ho & Ermon, [2016](#bib.bib9)), or operates at the level of entire trajectories, making it difficult to apply to more complex problem settings (Finn et al., [2016b](#bib.bib6), [a](#bib.bib5)). Our experimental evaluation demonstrates that AIRL outperforms prior IRL methods (Finn et al., [2016b](#bib.bib6)) on continuous, high-dimensional tasks with unknown dynamics by a wide margin. When compared to GAIL (Ho & Ermon, [2016](#bib.bib9)), which does not attempt to directly recover rewards, our method achieves comparable results on tasks that do not require transfer. However, on tasks where there is considerable variability in the environment from the demonstration setting, GAIL and other IRL methods fail to generalize. In these settings, our approach, which can effectively recover disentangle the goals of the expert from the dynamics of the environment, achieves superior results. 2 Related Work --------------- Inverse reinforcement learning (IRL) is a form of imitation learning and learning from demonstration (Argall et al., [2009](#bib.bib3)). Imitation learning methods seek to learn policies from expert demonstrations, and IRL methods accomplish this by first inferring the expert’s reward function. Previous IRL approaches have included maximum margin approaches (Abbeel & Ng, [2004](#bib.bib1); Ratliff et al., [2006](#bib.bib17)), and probabilistic approaches such as Ziebart et al. ([2008](#bib.bib23)); Boularias et al. ([2011](#bib.bib4)). In this work, we work under the MaxEnt IRL framework of Ziebart et al. ([2008](#bib.bib23)). Some advantages of MaxEnt IRL are that it allows for sub-optimality in demonstrations, and removes ambiguity between demonstrations and the expert policy. Moreover, it allows us to cast the reward learning problem as a maximum likelihood problem, allowing us to connect IRL to generative model training. Our proposed method most closely resembles the algorithms proposed by Uchibe ([2017](#bib.bib20)); Ho & Ermon ([2016](#bib.bib9)); Finn et al. ([2016a](#bib.bib5)). Generative adversarial imitation learning (GAIL) (Ho & Ermon, [2016](#bib.bib9)) differs from our work in that it is not an IRL algorithm that seeks to recover reward functions. The critic or discriminator of GAIL is unsuitable as a reward since, at optimality, it outputs 0.5 uniformly across all states and actions. Instead, GAIL aims only to recover the expert’s policy, which is a less portable representation for transfer. Uchibe ([2017](#bib.bib20)) does not interleave policy optimization with reward learning within an adversarial framework, whereas prior work has shown that interleaving policy optimization and reward learning is crucial for good performance on tasks that are continuous or high-dimensional (Finn et al., [2016b](#bib.bib6)). Wulfmeier et al. ([2015](#bib.bib21)) also consider learning cost function with neural networks, but only evaluate on simple domains where analytically solving the problem with value iteration is tractable. Previous methods which aim to learn nonlinear cost function have used boosting (Ratliff et al., [2007](#bib.bib16)) and gaussian processes (Levine et al., [2011](#bib.bib10)), but still suffer from the feature engineering problem. Our IRL algorithm builds on the adversarial IRL framework proposed by Finn et al. ([2016a](#bib.bib5)), with the discriminator corresponding to an odds ratio between the policy and exponentiated reward distribution. The discussion in Finn et al. ([2016a](#bib.bib5)) is theoretical, and to our knowledge no prior work has reported a practical implementation of this method. Our experiments show that direct implementation of the proposed algorithm is ineffective, due to high variance from operating over entire trajectories. While it is straightforward to extend the algorithm to single state-action pairs, as we discuss in Section [4](#S4 "4 Adversarial Inverse Reinforcement Learning (AIRL) ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"), a simple unrestricted form of the discriminator is susceptible to the reward ambiguity described in (Ng et al., [1999](#bib.bib14)), making learning the portable reward functions difficult. As illustrated in our experiments, this greatly limits the generalization capability of the method: the learned reward functions are not robust the environment changes, and it is difficult to use the algorithm for the purpose of inferring the intentions of agents. We discuss how to overcome this issue in Section [5](#S5 "5 The Reward Ambiguity Problem ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). 3 Background ------------- Our inverse reinforcement learning method builds on the maximum entropy IRL framework (Ziebart et al., [2008](#bib.bib23)), which considers an entropy-regularized Markov decision process (MDP) (Ziebart, [2010](#bib.bib22)), defined by the tuple (S,A,T,R,γ,ρ0). S,A are the state and action spaces, respectively, γ∈(0,1) is the discount factor. The dynamics or transition distribution T(s′\|a,s), the initial state distribution ρ0(s), and the reward function R(s,a) are unknown in the standard reinforcement learning setup and can only be queried through interaction with the MDP. The goal of (forward) reinforcement learning is to find the optimal policy π∗ that maximizes the expected entropy-regularized discounted reward, under π, T, and ρ0: \| \| \| \| \| --- \| --- \| --- \| \| \| π∗=arg maxπEτ∼π[R(τ)−logπ(τ)], \| \| where R(τ)=∑Tt=0γtR(st,at) denotes the discounted sum of rewards, τ=(s0,a0,...,sT,aT) denotes a trajectory, and π(τ)=ρ0(s0)∏Tt=0π(at\|st)T(st+1\|st,at). It can be shown that the trajectory distribution induced by the optimal policy π∗(a\|s) takes the form π∗(τ)∝exp{R(τ)} (Ziebart, [2010](#bib.bib22)). Inverse reinforcement learning (IRL) instead seeks infer the reward function ^R given a set of demonstrations D={τ1,...,τN}. Under maximum entropy IRL, we assume the demonstrations are drawn from an optimal policy τi∼exp{R(τ)}, and we can frame the IRL problem as maximum likelihood estimation problem: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| maxθEτ∼D[logpθ(τ)] , \| \| (1) \| where pθ(τ)∝exp{Rθ(τ)} takes the form of an energy-based model. The main challenge of training an energy-based model is computing the partition function Zθ=∫τexp{Rθ(τ)}, which is intractable in large or continuous MDPs. Alternative approaches, such as sampling (Finn et al., [2016b](#bib.bib6)), must be used to estimate the partition function in such cases. Finn et al. ([2016a](#bib.bib5)) propose to cast the optimization in Eqn. [1](#S3.E1 "(1) ‣ 3 Background ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning") as a GAN (Goodfellow et al., [2014](#bib.bib8)) optimization problem, where the discriminator takes on a particular form (fθ(τ) is a learned function; π(τ) is precomputed and its value “filled in”): \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Dθ(τ)=exp{fθ(τ)}exp{fθ(τ)}+π(τ), \| \| (2) \| and the policy π is trained to maximize R(τ)=log(1−D(τ))−logD(τ). Updating the discriminator can be viewed as updating the reward function, and updating the policy can be viewed as improving the sampling distribution used to estimate the partition function. If trained to optimality, it can be shown that an optimal reward function can be extracted from the optimal discriminator as f∗(τ)=R∗(τ)+const, and π recovers the optimal policy. We refer to this formulation as generative adversarial network guided cost learning (GAN-GCL) to discriminate it from guided cost learning (GCL) (Finn et al., [2016a](#bib.bib5)). This formulation shares similarities with GAIL (Ho & Ermon, [2016](#bib.bib9)), but GAIL does not place special structure on the discriminator, so the reward cannot be recovered. 4 Adversarial Inverse Reinforcement Learning (AIRL) ---------------------------------------------------- In practice, using full trajectories as proposed by GAN-GCL can result in high variance estimates as compared to using single state, action pairs, and our experimental results show that this results in very poor learning. We could instead propose a straightforward conversion of Eqn. [2](#S3.E2 "(2) ‣ 3 Background ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning") into the single state and action case, where: \| \| \| \| \| --- \| --- \| --- \| \| \| Dθ(s,a)=exp{fθ(s,a)}exp{fθ(s,a)}+π(a\|s). \| \| As in the trajectory-centric case, we can show that, at optimality, f∗(s,a)=logπ∗(a\|s)=A∗(s,a), the advantage function of the optimal policy. We justify this, as well as a proof that this algorithm solves the IRL problem in Appendix [B](#A2 "Appendix B Single state-action discriminators ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning") . This change results in an efficient algorithm for imitation learning. However, it is less desirable for the purpose of reward learning. While the advantage is a valid optimal reward function, it is a heavily entangled reward, as it supervises each action based on the action of the optimal policy for the training MDP. Based on the analysis in the following Sec. [5](#S5 "5 The Reward Ambiguity Problem ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"), we cannot guarantee that this reward will be robust to changes in environment dynamics. In our experiments we demonstrate several cases where this reward simply encourages mimicking the expert policy π∗, and fails to produce desirable behavior even when changes to the environment are made. 5 The Reward Ambiguity Problem ------------------------------- We now discuss why IRL methods can fail to learn robust reward functions. First, we review the concept of reward shaping. Ng et al. ([1999](#bib.bib14)) describe a class of reward transformations that preserve the optimal policy. Their main theoretical result is that under the following reward transformation, \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ^r(s,a,s′)=r(s,a,s′)+γΦ(s′)−Φ(s) , \| \| (3) \| the optimal policy remains unchanged, for any function Φ:S→R. Moreover, without prior knowledge of the dynamics, this is the only class of reward transformations that exhibits policy invariance. Because IRL methods only infer rewards from demonstrations given from an optimal agent, they cannot in general disambiguate between reward functions within this class of transformations, unless the class of learnable reward functions is restricted. We argue that shaped reward functions may not be robust to changes in dynamics. We formalize this notion by studying policy invariance in two MDPs M,M′ which share the same reward and differ only in the dynamics, denoted as T and T′, respectively. Suppose an IRL algorithm recovers a shaped, policy invariant reward ^r(s,a,s′) under MDP M where Φ≠0. Then, there exists MDP pairs M,M′ where changing the transition model from T to T′ breaks policy invariance on MDP M′. As a simple example, consider deterministic dynamics T(s,a)→s′ and state-action rewards ^r(s,a)=r(s,a)+γΦ(T(s,a))−Φ(s). It is easy to see that changing the dynamics T to T′ such that T′(s,a)≠T(s,a) means that ^r(s,a) no longer lies in the equivalence class of Eqn. [3](#S5.E3 "(3) ‣ 5 The Reward Ambiguity Problem ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning") for M′. ### 5.1 Disentangling Rewards from Dynamics To remove unwanted reward shaping with arbitrary reward function classes, the learned reward function can only depend on the current state s. Action-dependent terms may be used so long as they are independent of the next state s′, but this is rarely the case for any MDP. This is both a sufficient and necessary condition. We present a short proof sketch, assuming ergodicity in the MDP, as follows: * Sufficiency: Because the learned reward can only depend on the current state, we must have Φ(s′)=c ∀s′, where c is a constant, meaning the shaping term γΦ(s′)−Φ(s)=(γ−1)c is constant. Therefore, with state-only rewards, rewards can only be modified up to an additive constant, meaning no shaping is possible in any MDP. * Necessity: If learned rewards can be a function of s′ (or a that are not independent of s′), then there exists MDPs where reward shaping is possible (for example, the deterministic environment given in the previous section). This is the converse of what we wish to show, which is that for reward shaping to not be possible in all MDPs, the rewards must be state-only. In the traditional IRL setup, where we learn the reward in a single MDP, our analysis motivates learning reward functions that are solely functions of state. If the ground truth reward is also only a function of state, this allows us to recover the true reward up to a constant for any MDP using the MaxEnt IRL framework. 6 Learning Disentangled Rewards with AIRL ------------------------------------------ In the method presented in Section [4](#S4 "4 Adversarial Inverse Reinforcement Learning (AIRL) ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"), we cannot learn a state-only reward function, rθ(s), meaning that we cannot guarantee that learned rewards will not be shaped. In order to decouple the reward function from the advantage, we propose to modify the discriminator of Sec. [4](#S4 "4 Adversarial Inverse Reinforcement Learning (AIRL) ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning") with the form: \| \| \| \| \| --- \| --- \| --- \| \| \| Dθ,ϕ(s,a,s′)=exp{fθ,ϕ(s,a,s′)}exp{fθ,ϕ(s,a,s′)}+π(a\|s), \| \| where fθ,ϕ is restricted to a reward approximator gθ and a shaping term hϕ as \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| fθ,ϕ(s,a,s′)=gθ(s,a)+γhϕ(s′)−hϕ(s). \| \| (4) \| The additional shaping term helps mitigate the effects of unwanted shaping on our reward approximator gθ (and as we will show, in some cases it can account for all shaping effects). The entire training procedure is detailed in Algorithm [1](#alg1 "Algorithm 1 ‣ 6 Learning Disentangled Rewards with AIRL ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). Our algorithm resembles GAIL (Ho & Ermon, [2016](#bib.bib9)) and GAN-GCL (Finn et al., [2016a](#bib.bib5)), where we alternate between training a discriminator to classify expert data from policy samples, and update the policy to confuse the discriminator. 1: Obtain expert trajectories τEi 2: Initialize policy π and discriminator Dθ,ϕ. 3: for step t in {1, …, N} do 4: Collect trajectories τi=(s0,a0,...,sT,aT) by executing π. 5: Train Dθ,ϕ via binary logistic regression to classify expert data τEi from samples τi. 6: Update reward rθ,ϕ(s,a,s′)←logDθ,ϕ(s,a,s′)−log(1−Dθ,ϕ(s,a,s′)) 7: Update π with respect to rθ,ϕ using any policy optimization method. 8: end for Algorithm 1 Adversarial inverse reinforcement learning The advantage of this approach is that we can now parametrize gθ(s) as solely a function of the state, allowing us to extract rewards that are disentangled from the dynamics of the environment in which they were trained. In fact, under this restricted case, we can show the following under deterministic environments with a state-only ground truth reward (proof in Appendix [C](#A3 "Appendix C AIRL recovers rewards up to constants ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning")): \| \| \| \| \| --- \| --- \| --- \| \| \| g∗(s)=r∗(s)+const, \| \| \| \| \| \| \| --- \| --- \| --- \| \| \| h∗(s)=V∗(s)+const, \| \| where r∗ is the true reward function. Since f∗ must recover to the advantage as shown in Sec. [4](#S4 "4 Adversarial Inverse Reinforcement Learning (AIRL) ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"), h recovers the optimal value function V∗, which serves as the reward shaping term. To be consistent with Sec. [4](#S4 "4 Adversarial Inverse Reinforcement Learning (AIRL) ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"), an alternative way to interpret the form of Eqn. [4](#S6.E4 "(4) ‣ 6 Learning Disentangled Rewards with AIRL ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning") is to view fθ,ϕ as the advantage under deterministic dynamics \| \| \| \| \| --- \| --- \| --- \| \| \| f∗(s,a,s′)=r∗(s)+γV∗(s′)Q(s,a)−V∗(s)V(s)=A∗(s,a) \| \| In stochastic environments, we can instead view f(s,a,s′) as a single-sample estimate of A∗(s,a). 7 Experiments -------------- In our experiments, we aim to answer two questions: 1. Can AIRL learn disentangled rewards that are robust to changes in environment dynamics? 2. Is AIRL efficient and scalable to high-dimensional continuous control tasks? To answer 1, we evaluate AIRL in transfer learning scenarios, where a reward is learned in a training environment, and optimized in a test environment with significantly different dynamics. We show that rewards learned with our algorithm under the constraint presented in Section [5](#S5 "5 The Reward Ambiguity Problem ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning") still produce optimal or near-optimal behavior, while naïve methods that do not consider reward shaping fail. We also show that in small MDPs, we can recover the exact ground truth reward function. To answer 2, we compare AIRL as an imitation learning algorithm against GAIL (Ho & Ermon, [2016](#bib.bib9)) and the GAN-based GCL algorithm proposed by Finn et al. ([2016a](#bib.bib5)), which we refer to as GAN-GCL, on standard benchmark tasks that do not evaluate transfer. Note that Finn et al. ([2016a](#bib.bib5)) does not implement or evaluate GAN-GCL and, to our knowledge, we present the first empirical evaluation of this algorithm. We find that AIRLperforms on par with GAIL in a traditional imitation learning setup while vastly outperforming it in transfer learning setups, and outperforms GAN-GCL in both settings. It is worth noting that, except for (Finn et al., [2016b](#bib.bib6)), our method is the only IRL algorithm that we are aware of that scales to high dimensional tasks with unknown dynamics, and although GAIL (Ho & Ermon, [2016](#bib.bib9)) resembles an IRL algorithm in structure, it does not recover disentangled reward functions, making it unable to re-optimize the learned reward under changes in the environment, as we illustrate below. For our continuous control tasks, we use trust region policy optimization (Schulman et al., [2015](#bib.bib19)) as our policy optimization algorithm across all evaluated methods, and in the tabular MDP task, we use soft value iteration. We obtain expert demonstrations by training an expert policy on the ground truth reward, but hide the ground truth reward from the IRL algorithm. Our code and additional supplementary material including videos will be available at <https://sites.google.com/view/adversarial-irl>, and hyper-parameter and architecture choices are detailed in Appendix [D](#A4 "Appendix D Experiment Details ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). ### 7.1 Recovering true rewards in tabular MDPs We first consider MaxEnt IRL in a toy task with randomly generated MDPs. The MDPs have 16 states, 4 actions, randomly drawn transition matrices, and a reward function that always gives a reward of 1.0 when taking an action from state 0. The initial state is always state 1. The optimal reward, learned reward with a state-only reward function, and learned reward using a state-action reward function are shown in Fig. [2](#S7.F2 "Figure 2 ‣ 7.1 Recovering true rewards in tabular MDPs ‣ 7 Experiments ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). We subtract a constant offset from all reward functions so that they share the same mean for visualization - this does not influence the optimal policy. AIRL with a state-only reward function is able to recover the ground truth reward, but AIRL with a state-action reward instead recovers a shaped advantage function. We also show that in the transfer learning setup, under a new transition matrix T′, the optimal policy under the state-only reward achieves optimal performance (it is identical to the ground truth reward) whereas the state-action reward only improves marginally over uniform random policy. The learning curve for this experiment is shown in Fig [2](#S7.F2 "Figure 2 ‣ 7.1 Recovering true rewards in tabular MDPs ‣ 7 Experiments ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). ![ Ground truth (a) and learned rewards (b, c) on the random MDP task. Dark blue corresponds to a reward of 1, and white corresponds to 0. Note that AIRL with a state-only reward recovers the ground truth, whereas the state-action reward is shaped. ](https://media.arxiv-vanity.com/render-output/6602923/images/tabular_rewards.png) Figure 1: Ground truth (a) and learned rewards (b, c) on the random MDP task. Dark blue corresponds to a reward of 1, and white corresponds to 0. Note that AIRL with a state-only reward recovers the ground truth, whereas the state-action reward is shaped. ![ Ground truth (a) and learned rewards (b, c) on the random MDP task. Dark blue corresponds to a reward of 1, and white corresponds to 0. Note that AIRL with a state-only reward recovers the ground truth, whereas the state-action reward is shaped. ](https://media.arxiv-vanity.com/render-output/6602923/images/tabular_transfer.png) Figure 2: Learning curve for the transfer learning experiment on tabular MDPs. Value iteration steps are plotted on the x-axis, against returns for the policy on the y-axis. ### 7.2 Disentangling Rewards in Continuous Control Tasks To evaluate whether our method can learn disentangled rewards in higher dimensional environments, we perform transfer learning experiments on continuous control tasks. In each task, a reward is learned via IRL on the training environment, and the reward is used to reoptimize a new policy on a test environment. We train two IRL algorithms, AIRL and GAN-GCL, with state-only and state-action rewards. We also include results for directly transferring the policy learned with GAIL, and an oracle result that involves optimizing the ground truth reward function with TRPO. Numerical results for these environment transfer experiments are given in Table [1](#S7.T1 "Table 1 ‣ 7.2 Disentangling Rewards in Continuous Control Tasks ‣ 7 Experiments ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). The first task involves a 2D pointmass navigating to a goal position in a small maze when the position of the walls are changed between train and test time. At test time, the agent cannot simply mimic the actions learned during training, and instead must successfully infer that the goal in the maze is to reach the target. The task is shown in Fig. [4](#S7.F4 "Figure 4 ‣ 7.2 Disentangling Rewards in Continuous Control Tasks ‣ 7 Experiments ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). Only AIRL trained with state-only rewards is able to consistently navigate to the goal when the maze is modified. Direct policy transfer and state-action IRL methods learn rewards which encourage the agent to take the same path taken in the training environment, which is blocked in the test environment. We plot the learned reward in Fig. [7](#S7.F7 "Figure 7 ‣ 7.3 Benchmark Tasks for Imitation Learning ‣ 7 Experiments ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). In our second task, we modify the agent itself. We train a quadrupedal “ant” agent to run forwards, and at test time we disable and shrink two of the front legs of the ant such that it must significantly change its gait (shown in Fig. [4](#S7.F4 "Figure 4 ‣ 7.2 Disentangling Rewards in Continuous Control Tasks ‣ 7 Experiments ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning")). We find that AIRL is able to learn reward functions that encourage the ant to move forwards, acquiring a modified gait that involves orienting itself to face the forward direction and crawling with its two hind legs. Alternative methods, including transferring a policy learned by GAIL (which achieves near-optimal performance with the unmodified agent), fail to move forward at all. We show the qualitative difference in behavior in Fig. [5](#S7.F5 "Figure 5 ‣ 7.2 Disentangling Rewards in Continuous Control Tasks ‣ 7 Experiments ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). We have demonstrated that AIRL can learn disentangled rewards that can accommodate significant domain shift even in high-dimensional environments where it is difficult to exactly extract the true reward. GAN-GCL can presumably learn disentangled rewards, but we find that the trajectory-centric formulation does not perform well even in learning rewards in the original task, let alone transferring to a new domain. GAIL learns successfully in the training domain, but does not acquire a representation that is suitable for transfer to test domains. \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| State-Only? \| Pointmass-Maze \| Ant-Disabled \| \| GAN-GCL \| No \| -40.2 \| -44.8 \| \| \hdashlineGAN-GCL \| Yes \| -41.8 \| -43.4 \| \| \hdashlineAIRL (ours) \| No \| -31.2 \| -41.4 \| \| \hdashlineAIRL (ours) \| Yes \| -8.82 \| 130.3 \| \| GAIL, policy transfer \| N/A \| -29.9 \| -58.8 \| \| \hdashlineTRPO, ground truth \| N/A \| -8.45 \| 315.5 \| Table 1: Results on transfer learning tasks. Mean scores (higher is better) are reported over 5 runs. We also include results for TRPO optimizing the ground truth reward, and the performance of a policy learned via GAIL on the training environment. ![ Illustration of the shifting maze task, where the agent (blue) must reach to goal (green). During training the agent must go around the wall on the left side, but during test time it must go around on the right. ](https://media.arxiv-vanity.com/render-output/6602923/images/pointmass_transfer.png) Figure 3: Illustration of the shifting maze task, where the agent (blue) must reach to goal (green). During training the agent must go around the wall on the left side, but during test time it must go around on the right. ![ Illustration of the shifting maze task, where the agent (blue) must reach to goal (green). During training the agent must go around the wall on the left side, but during test time it must go around on the right. ](https://media.arxiv-vanity.com/render-output/6602923/images/ant_transfer.png) Figure 4: Illustration of the disabled ant task. During test time, the ant must learn to walk with only two of the unmodified back legs (the disabled and shrunk legs are colored in red). \| \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| \| \| \| \| \| \| \| \| \| \| Figure 5: Top row: An ant running forwards (right in the picture) in the training environment. Bottom row: Behavior acquired by optimizing a state-only reward learned with AIRL on the disabled ant environment. Note that the ant must orient itself before crawling forward, which is a qualitatively different behavior from the optimal policy in the original environment, which runs sideways. ### 7.3 Benchmark Tasks for Imitation Learning Finally, we evaluate AIRL as an imitation learning algorithm against the GAN-GCL and the state-of-the-art GAIL on several benchmark tasks. Each algorithm is presented with 50 expert demonstrations, collected from a policy trained with TRPO on the ground truth reward function. For AIRL, we use an unrestricted state-action reward function as we are not concerned with reward transfer. Numerical results are presented in Table [2](#S7.T2 "Table 2 ‣ 7.3 Benchmark Tasks for Imitation Learning ‣ 7 Experiments ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"), and sample images from these tasks are shown in Fig. [7](#S7.F7 "Figure 7 ‣ 7.3 Benchmark Tasks for Imitation Learning ‣ 7 Experiments ‣ Learning Robust Rewards with Adversarial Inverse Reinforcement Learning"). These experiments do not test transfer, and in a sense can be regarded as “testing on the training set,” but they match the settings reported in prior work (Ho & Ermon, [2016](#bib.bib9)). We find that the performance difference between AIRL and GAIL is negligible, even though AIRL is a true IRL algorithm that recovers reward functions, while GAIL does not. Both methods achieve close to the best possible result on each task, and there is little room for improvement. This result goes against the belief that IRL algorithms are indirect, and less efficient that direct imitation learning algorithms (Ho & Ermon, [2016](#bib.bib9)). The GAN-GCL method is ineffective on all but the simplest Pendulum task when trained with the same number of samples as AIRL and GAIL. We find that a discriminator trained over trajectories easily overfits and provides poor learning signal for the policy. Our results illustrate that AIRL achieves the same performance as GAIL on benchmark imitation tasks that do not require any generalization. On tasks that require transfer and generalization, illustrated in the previous section, AIRL outperforms GAIL by a wide margin, since our method is able to recover disentangled rewards that transfer effectively in the presence of domain shift. ![ Reward learned on the pointmass shifting maze task. The goal is located at the green star and the agent starts at the white circle. Note that there is little reward shaping, which enables the reward to transfer well. ](https://media.arxiv-vanity.com/render-output/6602923/images/pointmass_rew.png) Figure 6: Reward learned on the pointmass shifting maze task. The goal is located at the green star and the agent starts at the white circle. Note that there is little reward shaping, which enables the reward to transfer well. ![ Reward learned on the pointmass shifting maze task. The goal is located at the green star and the agent starts at the white circle. Note that there is little reward shaping, which enables the reward to transfer well. ](https://media.arxiv-vanity.com/render-output/6602923/images/airl_im_tasks_2.png) Figure 7: Imitation learning benchmark tasks used in this paper. Clockwise, from top-left: Half-cheetah, Swimmer, and Ant. \| \| Pendulum \| Ant \| Swimmer \| Half-Cheetah \| \| --- \| --- \| --- \| --- \| --- \| \| GAN-GCL \| -261.5 \| 460.6 \| -10.6 \| -670.7 \| \| \hdashlineGAIL \| -226.0 \| 1358.7 \| 140.2 \| 1642.8 \| \| \hdashlineAIRL (ours) \| -204.7 \| 1238.6 \| 139.1 \| 1839.8 \| \| Expert (TRPO) \| -179.6 \| 1537.9 \| 141.1 \| 1811.2 \| \| \hdashlineRandom \| -654.5 \| -108.1 \| -11.5 \| -988.4 \| Table 2: Results on imitation learning benchmark tasks. Mean scores (higher is better) are reported across 5 runs. 8 Conclusion ------------- We presented AIRL, a practical and scalable IRL algorithm that can learn disentangled rewards and greatly outperforms both prior imitation learning and IRL algorithms. We show that rewards learned with AIRL transfer effectively under variation in the underlying domain, in contrast to unmodified IRL methods which tend to recover brittle rewards that do not generalize well and GAIL, which does not recover reward functions at all. In small MDPs where the optimal policy and reward are unambiguous, we also show that we can exactly recover the ground-truth rewards up to a constant. Acknowledgements ---------------- This research was supported by the National Science Foundation through IIS-1651843, IIS-1614653, and IIS-1637443. We would like to thank Roberto Calandra for helpful feedback on the paper.
aaf97e5f-82cb-46fe-b5c6-53fb39194686	trentmkelly/LessWrong-43k	LessWrong	Believing In “In America, we believe in driving on the right hand side of the road.” Tl;dr: Beliefs are like bets (on outcomes the belief doesn’t affect). “Believing in”s are more like kickstarters (for outcomes the believing-in does affect). Epistemic status: New model; could use critique. In one early CFAR test session, we asked volunteers to each write down something they believed. My plan was that we would then think together about what we would see in a world where each belief was true, compared to a world where it was false. I was a bit flummoxed when, instead of the beliefs-aka-predictions I had been expecting, they wrote down such “beliefs” as “the environment,” “kindness,” or “respecting people.” At the time, I thought this meant that the state of ambient rationality was so low that people didn’t know “beliefs” were supposed to be predictions, as opposed to group affiliations. I’ve since changed my mind. My new view is that there is not one but two useful kinds of vaguely belief-like thingies – one to do with predictions and Bayes-math, and a different one I’ll call “believing in.” I believe both are lawlike, and neither is a flawed attempt to imitate/parasitize the other. I further believe both can be practiced at once – that they are distinct but compatible. I’ll be aiming, in this post, to give a clear concept of “believing in,” and to get readers’ models of “how to ‘believe in’ well” disentangled from their models of “how to predict well.” Examples of “believing in” Let’s collect some examples, before we get to theory. Places where people talk of “believing in” include: * An individual stating their personal ethical code. E.g., “I believe in being honest,” “I believe in hard work,” “I believe in treating people with respect,” etc. * A group stating the local social norms that group tries to practice as a group. E.g., “Around here, we believe in being on time.” * “I believe in you,” said by one friend or family member to another, sometimes in a s
e030f9fe-8c46-4db6-8574-5757783e9a40	trentmkelly/LessWrong-43k	LessWrong	[Link] Neural networks trained on expert Go games have just made a major leap From the arXiv: > Move Evaluation in Go Using Deep Convolutional Neural Networks > > Chris J. Maddison, Aja Huang, Ilya Sutskever, David Silver > > The game of Go is more challenging than other board games, due to the difficulty of constructing a position or move evaluation function. In this paper we investigate whether deep convolutional networks can be used to directly represent and learn this knowledge. We train a large 12-layer convolutional neural network by supervised learning from a database of human professional games. The network correctly predicts the expert move in 55% of positions, equalling the accuracy of a 6 dan human player. When the trained convolutional network was used directly to play games of Go, without any search, it beat the traditional search program GnuGo in 97% of games, and matched the performance of a state-of-the-art Monte-Carlo tree search that simulates a million positions per move. This approach looks like it could be combined with MCTS. Here's their conclusion: > In this work, we showed that large deep convolutional neural networks can predict the next move made by Go experts with an accuracy that exceeds previous methods by a large margin, approximately matching human performance. Furthermore, this predictive accuracy translates into much stronger move evaluation and playing strength than has previously been possible. Without any search, the network is able to outperform traditional search based programs such as GnuGo, and compete with state-of-the-art MCTS programs such as Pachi and Fuego. > > In Figure 2 we present a sample game played by the 12-layer CNN (with no search) versus Fuego (searching 100K rollouts per move) which was won by the neural network player. It is clear that the neural network has implicitly understood many sophisticated aspects of Go, including good shape (patterns that maximise long term effectiveness of stones), Fuseki (opening sequences), Joseki (corner patterns), Tesuji (tactical patterns), Ko figh
ed8cbabb-e8c4-4833-975f-5b90f3c45f0f	trentmkelly/LessWrong-43k	LessWrong	AI Challenge: Ants - Post Mortem Late last year a LessWrong team was being mooted for the Google AI challenge (http://aichallenge.org/; http://lesswrong.com/r/discussion/lw/8ay/ai_challenge_ants/). Sadly, after a brief burst of activity, no "official" LessWrong entry appeared (AFAICT, and please let me know if I am mistaken). The best individual effort from this site's regulars (AFAICT) came from lavalamp, who finished around #300. This is a pity. This was an opportunity to achieve, or at least have a go at, a bunch of worthwhile things, including development of methods of cooperation between site members, gathering positive publicity, and yes, even advancing the understanding of AI related issues. So - how can things be improved for the next AI challenge (which I think is about 6 months away)?
6a19a87c-f073-423b-86b2-a033a8b4038d	trentmkelly/LessWrong-43k	LessWrong	How to build common knowledge of rationality and honesty? From my shortform: > If you know someone is rational, honest, and well-read, then you can learn a good bit from the simple fact that they disagree with you. > > If you aren't sure someone is rational and honest, their disagreement tells you little. > > If you know someone considers you to be rational and honest, the fact that they still disagree with you after hearing what you have to say, tells you something. > > But if you don't know that they consider you to be rational and honest, their disagreement tells you nothing. > > It's valuable to strive for common knowledge of you and your partners' rationality and honesty, to make the most of your disagreements. This is a restatement of the ideas behind Aumann's Agreement Theorem. On Less Wrong, one can often assume that a writer or commentor is interested in rationality, but that doesn't mean that the writer can be assumed to actually be rational. How can we help build common knowledge of people's rationality and honesty?
7615807b-ef91-487d-8b07-664335fcea4a	trentmkelly/LessWrong-43k	LessWrong	EA Funds: Long-Term Future fund is open to applications until Feb. 7th (Crossposted to the EA Forum feel free to comment on either post with any questions) The Long-Term Future Fund doing another round of grants in mid-February (currently ~$400k), and are open to applications until February 7th. You can apply here. (Excerpt from the application:) > We are particularly interested in small teams and individuals that are trying to get projects off the ground, or that need less money than existing grant-making institutions are likely to give out (i.e. less than ~$100k, but more than $10k). Here are a few examples of project types that we're open to funding an individual or group for (note that this list is not exhaustive): > + To spend a few months (perhaps during the summer) to research an open problem in AI alignment or AI strategy and produce a few blog posts or videos on their ideas > + To spend a few months building a web app with the potential to solve an operations bottleneck at x-risk organisations > + To spend a few months up-skilling in a field to prepare for future work (e.g. microeconomics, functional programming, etc). > + To spend a year testing an idea that has the potential to be built into an org. > You can find more details on the kind of project we are likely to fund on the fund page: https://app.effectivealtruism.org/funds/far-future If we don’t feel that we have enough information but we think your project sounds promising we may be in touch for more information.
fdb18c1c-de29-42a3-8fef-199efbe25e9d	trentmkelly/LessWrong-43k	LessWrong	Predicting GPU performance We develop a simple model that predicts progress in the performance of field-effect transistor-based GPUs under the assumption that transistors can no longer miniaturize after scaling down to roughly the size of a single silicon atom. We construct a composite model from a performance model (a model of how GPU performance relates to the features of that GPU), and a feature model (a model of how GPU features change over time given the constraints imposed by the physical limits of miniaturization), each of which are fit on a dataset of 1948 GPUs released between 2006 and 2021. We find that almost all progress can be explained by two variables: transistor size and the number of cores. Using estimates of the physical limits informed by the relevant literature, our model predicts that GPU progress will stop roughly between 2027 and 2035, due to decreases in transistor size. In the limit, we can expect that current field-effect transistor-based GPUs, without any paradigm-altering technological advances, will be able to achieve a peak theoretical performance of 1e14 and 1e15 FLOP/s in single-precision performance. Figure 1. Model predictions of peak theoretical performance of top GPUs, assuming that transistors can no longer miniaturize after scaling down transistors to around 0.7nm. Left: GPU performance projections; Top right: GPU performance when the limit is hit; Middle right: Our distribution over the physical limits of transistor miniaturization; Bottom right: Our distribution over the transistors per core ratio with relevant historical comparisons. While there are many other drivers of GPU performance and efficiency improvements (such as memory optimization, improved utilization, and so on), decreasing the size of transistors has historically been a great, and arguably the dominant, driver of GPU performance improvements. Our work therefore strongly suggests that it will become significantly harder to achieve GPU performance improvements around the mid-2030s within
a77e6bd9-de1f-4b8b-b03f-35770f014562	trentmkelly/LessWrong-43k	LessWrong	Set List Approaches Now that Kingfisher is starting to have a bunch of sets I've been thinking some about how to keep track of what we play, which gets into what kind of band we want to be. There's a bit of a continuum, from dance bands with a list of sets they play a specific way each time to ones that just put things together entirely on the fly. There are advantages to each: * If you have a specific way you play a set it can generally be very polished, in a way that can be great to dance to. You can put a lot of thought into the arrangement, move a large group of musicians through something complex while sounding very tight, make sharp dynamic changes, and combine a lot of ideas. * If you're making things up in the moment, responding to each other, the crowd, and the feeling in the hall, your music can have a lot more life to it. You don't get stuck playing sets you're sick of, and if you're playing several times in the same region the dancers get some variety. There's no one answer here: what will work best depends a lot on the particular combination of musicians, how they work together, and what they want. Larger groups generally benefit more from pre-arrangement, though if you can allocate someone to lead the band you can still put on a great dance with a large band and no planning. I've enjoyed dancing to bands all over this spectrum. Bands also often move between these categories over time. The Free Raisins started off going into each set with just some planned tunes and making things up as we went (see our first CD), but over time we would remember things we'd done before and call those things to each other. Patterns we especially liked became the way we would play those sets, and then the process of putting together our second CD gelled them further. As of the start of the pandemic, while we still would play around and improvise, if you heard a given set at two different dance weekends it would probably sound pretty similar, and to be honest we were getting a bit sick
13a3bdf9-2623-4146-bc68-e32a5de11a54	trentmkelly/LessWrong-43k	LessWrong	Space-Faring Civilization density estimates and models - Review Crossposted to the EA forum. Over the last few years, progress has been made in estimating the density of Space-Faring Civilizations (SFCs) in the universe, producing probability distributions better representing our uncertainty (e.g., Sandberg 2018, Snyder-Beattie 2021, Hanson 2021, etc.). Previous works were mainly limited to point estimates and often only focused on the Milky Way. Furthermore, some recent works explore the impact of using different anthropic theories (SSA, SIA, ADT)(e.g., Finnveden 2019, Olson 2020, Olson 2021, Cook 2022). In this post, we review these recent works, the models they use, the likelihood distribution over SFC density they produce, how to compare them, which are bullish/bearish estimates, and finally, we plot them together. Sequence: This post is part 2 of a sequence investigating the longtermist implications of alien Space-Faring Civilizations. Each post aims to be standalone. You can find an introduction to the sequence in the following post. Summary Models estimating the density of Space-Faring Civilizations (SFCs) can be classified along three dimensions: appearance models (try-once steps, or time-dependent steps), propagation models (no propagation/steady state, or greedy propagation/dynamic), and anthropic theories (non-anthropic, SSA, SIA, ADT). Combining the first two dimensions, we classify models into four groups: Drake-equations, Drake-equations plus propagation, Bayesian analyses of occurrence time, and Grabby aliens models. After comparing three existing distribution estimates and converting them to a common unit (peak cosmic SFC density per observable universe volume), Finnveden 2019 emerges as the most bearish estimate due to its very conservative prior for the rate of life emergence, while Cook 2022 - Balanced represents the best-guess estimate by accounting for the impact of time-dependent appearance steps and using broad best-guess priors. These works collectively suggest that while SFCs may be rare, their event
d59de59a-54e2-4be1-9fa1-7286fd6466c3	trentmkelly/LessWrong-43k	LessWrong	"Mild Hallucination" Test In Scott Alexander's Lot's of People Going Around with Mild Hallucinations All the Time, he shows that several people not currently on LSD still experience mild hallucinations commonly associated with currently taking LSD. I would like to test to see if I could teach you how to see these mild hallucinations, regardless of experience with psychedelics. Below are 3 tests that should take 1-2 minutes to complete. If you choose to complete 1 or more of these, please comment both failed and successful attempts. Please also comment if you can already see some of these, even if you think it seems obvious. Test 1: Visual Snow Description: See the Visual Snow Wiki for a nice visualization on the top-right. I would describe it as "jumpy spiderwebs made out of light", similar in feel to the "black stars" people see when feeling faint (when they get up too quick). I would say it's NOT the same experience as mental imagination or eye floaters. [Edit: Honestly I mixed up different phenomena for "visual snow" in my description. Here's the update: > 1. Visual Snow - Like a million very tiny dots. Very much like static/white noise in the wiki. More visible in low light conditions or when you're tired. I saw it for the first time this (8/12) morning in low-light conditions. > 2. Patterned lines (?) - Like the geometric/kaleidoscopic shape in this picture. Doesn't have to be that consistent or patterned but is better described by "lines" than either of the other two. This is what I meant by "jumpy spiderwebs made out of light" and what I thought visual snow was. > 3. Blue-sky Sprites - The picture is a nice animation (can be seen without looking at the blue sky but apparently it's more prominent in that case). Dots and wisps the size of a mm or a little bigger. Maybe 5-100 at a time vs the million in "visual snow". Resembles afterimages and the "black stars" when feeling faint. > 4. (Also very possible there's more that I've missed) ] Test: For 1 minute (click here for a 1
93c1b6a2-c566-4b28-8571-fad8be0c56e5	trentmkelly/LessWrong-43k	LessWrong	So You Want to Work at a Frontier AI Lab Several promising software engineers have asked me: Should I work at a frontier AI lab? My answer is always “No.” This post explores the fundamental problem with frontier labs, some of the most common arguments in favor of working at one, and why I don’t buy these arguments. The Fundamental Problem The primary output of frontier AI labs—such as OpenAI, Anthropic, Meta, and Google DeepMind—is research that accelerates the capabilities of frontier AI models and hastens the arrival of superhuman machines. Each lab’s emphasis on alignment varies, but none are on track to solve the hard problems, or to prevent these machines from growing irretrievably incompatible with human life. In the absence of an ironclad alignment procedure, frontier capabilities research accelerates the extinction of humanity. As a very strong default, I expect signing up to assist such research to be one of the gravest mistakes a person can make. Some aspiring researchers counter: “I know that, but I want to do safety research on frontier models. I’ll simply refuse to work directly on capabilities.” Plans like these, while noble, dramatically misunderstand the priorities and incentives of scaling labs. The problem isn’t that you will be forced to work on capabilities; the problem is that the vast majority of safety work conducted by the labs enables or excuses continued scaling while failing to address the hard problems of alignment. You Will Be Assimilated AI labs are under overwhelming institutional pressure to push the frontier of machine learning. This pressure can distort everything lab employees think, do, and say. Former OpenAI Research Scientist Richard Ngo noticed this effect firsthand: This distortion affects research directions even more strongly. It’s perniciously easy to "safetywash” despite every intention to the contrary. The overlap between alignment and capabilities research compounds this effect. Many efforts to understand and control the outputs of machine lear
249e1705-e015-4291-a2af-5d21a3c16ca4	LDJnr/LessWrong-Amplify-Instruct	LessWrong	"Instead, it’s the point of no return—the day we AI risk reducers lose the ability to significantly reduce AI risk. This might happen years before classic milestones like “World GWP doubles in four years” and “Superhuman AGI is deployed." The rest of this post explains, justifies, and expands on this obvious but underappreciated idea. (Toby Ord appreciates it; see quote below). I found myself explaining it repeatedly, so I wrote this post as a reference. AI timelines often come up in career planning conversations. Insofar as AI timelines are short, career plans which take a long time to pay off are a bad idea, because by the time you reap the benefits of the plans it may already be too late. It may already be too late because AI takeover may already have happened. But this isn’t quite right, at least not when “AI takeover” is interpreted in the obvious way, as meaning that an AI or group of AIs is firmly in political control of the world, ordering humans about, monopolizing violence, etc. Even if AIs don’t yet have that sort of political control, it may already be too late. Here are three examples: [UPDATE: More fleshed-out examples can be found in this new post.] Superhuman agent AGI is still in its box but nobody knows how to align it and other actors are going to make their own version soon, and there isn’t enough time to convince them of the risks. They will make and deploy agent AGI, it will be unaligned, and we have no way to oppose it except with our own unaligned AGI. Even if it takes years to actually conquer the world, it’s already game over. Various weak and narrow AIs are embedded in the economy and beginning to drive a slow takeoff; capabilities are improving much faster than safety/alignment techniques and due to all the money being made there’s too much political opposition to slowing down capability growth or keeping AIs out of positions of power. We wish we had done more safety/alignment research earlier, or built a political movement earlier when opposition was lower. Persuasion tools have destroyed collective epistemology in the relevant places. AI isn’t very capable yet, except in the narrow domain of persuasion, but everything has become so politicized and tribal that we have no hope of getting AI projects or governments to take AI risk seriously. Their attention is dominated by the topics and ideas of powerful ideological factions that have access to more money and data (and thus better persuasion tools) than us. Alternatively, maybe we ourselves have fallen apart as a community, or become less good at seeking the truth and finding high-impact plans. Conclusion: We should remember that when trying to predict the date of AI takeover, what we care about is the date it’s too late for us to change the direction things are going; the date we have significantly less influence over the course of the future than we used to; the point of no return. This is basically what Toby Ord said about x-risk: “So either because we’ve gone extinct or because there’s been some kind of irrevocable collapse of civilization or something similar. Or, in the case of climate change, where the effects are very delayed that we’re past the point of no return or something like that. So the idea is that we should focus on the time of action and the time when you can do something about it rather than the time when the particular event happens.” Of course, influence over the future might not disappear all on one day; maybe there’ll be a gradual loss of control over several years. For that matter, maybe this gradual loss of control began years ago and continues now... We should keep these possibilities in mind as well. [Edit: I now realize that I should distinguish between AI-induced points of no return and other points of no return. Our timelines forecasts and takeoff speeds discussions are talking about AI, so we should interpret them as being about AI-induced points of no return. Our all-things-considered view on e.g. whether to go to grad school should be informed by AI-induced-PONR timelines and also "timelines" for things like nuclear war, pandemics, etc.]" The above is a chunk of text from a post created by a user on the website LessWrong. Can you explain what this text is saying as if I've never finished my first 2 years of college? Absolutely, let's break it down. This text is discussing the potential risks associated with Artificial Intelligence (AI) and the point at which it may be too late to control or mitigate these risks. The author refers to this as the "point of no return". 1. The author suggests that this point may arrive before we see significant milestones like rapid global economic growth or the deployment of super-intelligent AI. 2. The author also mentions that this point of no return is relevant when considering career paths. If AI develops rapidly, long-term career plans may become irrelevant. 3. The text provides three examples of situations where it could already be too late to control AI. - First, if a super-intelligent AI exists but isn't aligned with human values and others are about to create similar AIs, it's too late. - Second, if narrow AIs are embedded in the economy and are improving faster than we can ensure their safety, it's too late. - Third, if AI has become so politicized that we can't get people to take AI risks seriously, it's too late. 4. The author concludes by saying that the focus should be on predicting when it will be too late to influence AI's direction, not on when AI will take over. 5. The author also notes that the loss of control over AI might not happen suddenly but could be a gradual process that may have already started. 6. Lastly, the author suggests that we should also consider other potential risks, like nuclear war or pandemics, not just AI-induced risks. In simpler terms, the author is saying that we need to be aware of and act on the risks of AI before it's too late to do anything about it. This might happen before we even realize AI has taken over, and it might be a gradual process rather than a sudden event. Can you elaborate on the concept of "alignment" in the context of AI, and why it's considered a crucial factor in determining the point of no return? Absolutely. In the context of AI, "alignment" refers to the idea that the goals, values, and actions of an AI system should be in harmony with human values and intentions. Essentially, an aligned AI is one that does what we want it to do, and behaves in a way that is beneficial to us. The concept of alignment is crucial because of the potential risks that come with powerful AI systems. If an AI system is not aligned with human values, it might take actions that we find harmful or undesirable. For instance, an AI programmed to make paperclips might convert the entire planet into a paperclip factory if it's not properly aligned to understand that this is not what we want. The point of no return is considered to be the moment when we lose the ability to ensure that AI systems are aligned with our values. If we reach a stage where AI systems are more intelligent and powerful than us, and they are not aligned with our values, then they might start making decisions and taking actions that we can't control or reverse, hence the term "point of no return". This is why many researchers in the field of AI safety emphasize the importance of solving the alignment problem before we develop AI systems that are more intelligent than humans. The goal is to ensure that AI systems will always act in ways that are beneficial to humanity, even as they become more and more capable. Reflecting on the concept of AI alignment, what potential challenges or limitations could arise in aligning AI systems with human values, considering the diversity and complexity of these values? Aligning AI systems with human values is indeed a complex task due to several reasons: 1. Diversity of Values: Human values vary greatly across different cultures, societies, and individuals. What one person or culture values may be vastly different from what another does. Creating an AI system that respects and aligns with this vast array of values is a significant challenge. 2. Ambiguity and Contradictions: Human values can often be ambiguous and contradictory. For example, people often value both privacy and security, but these two values can sometimes be at odds with each other. It's challenging to program an AI to navigate these contradictions. 3. Changing Values: Human values are not static; they evolve over time. An AI system that is aligned with current human values may not be aligned with future values. 4. Complexity of Human Values: Human values are complex and nuanced. They often depend on context and can't always be reduced to simple rules or algorithms. This makes it difficult to translate these values into a form that an AI system can understand and follow. 5. Value Learning: Even if we could perfectly define our values, teaching them to an AI system is another challenge. This involves not just instilling static rules, but also enabling the AI to learn and adapt its understanding of human values over time. 6. Unintended Consequences: There's a risk that an AI might interpret its alignment instructions in ways that we didn't intend, leading to harmful outcomes. This is often referred to as the "alignment problem" in AI safety research. These challenges highlight the importance of ongoing research into AI alignment and safety. It's crucial to ensure that as AI systems become more intelligent and autonomous, they continue to act in ways that are beneficial to humanity. Discuss the concept of "Value Learning" in AI, and how it could potentially address the challenge of evolving human values over time. Value learning is a concept in AI research that involves designing AI systems to learn what humans value over time, rather than having these values pre-programmed. The idea is to create AI systems that can adapt to changes in human values and societal norms, rather than sticking to a fixed set of rules. This concept is particularly relevant given the dynamic nature of human values. As societies evolve, our understanding of what is ethical, beneficial, or desirable can change. By enabling AI systems to learn and adapt to these changes, we can ensure that they continue to act in ways that align with our values, even as those values evolve. Value learning can take several forms: 1. Inverse Reinforcement Learning: This is a technique where an AI learns by observing human behavior and trying to infer the underlying values that motivate that behavior. 2. Preference Learning: In this approach, the AI learns directly from human feedback. For example, the AI might present a human with several possible actions, and the human ranks them based on their preferences. 3. Iterative Training: This involves training the AI in stages, with each stage building on the previous one. The AI is initially trained on simple tasks with clear right and wrong answers, and then gradually exposed to more complex scenarios. While value learning offers a promising way to address the challenge of evolving human values, it also comes with its own set of challenges. For example, it requires a high degree of trust in the AI's ability to learn and interpret human values correctly. It also raises questions about whose values the AI should learn, and how to ensure that the process is fair and representative of diverse perspectives. Despite these challenges, value learning is a key area of research in the field of AI alignment and safety.
56f1510a-e2a7-493b-a829-c011160a0455	trentmkelly/LessWrong-43k	LessWrong	Non-Obstruction: A Simple Concept Motivating Corrigibility Thanks to Mathias Bonde, Tiffany Cai, Ryan Carey, Michael Cohen, Joe Collman, Andrew Critch, Abram Demski, Michael Dennis, Thomas Gilbert, Matthew Graves, Koen Holtman, Evan Hubinger, Victoria Krakovna, Amanda Ngo, Rohin Shah, Adam Shimi, Logan Smith, and Mark Xu for their thoughts. Main claim: corrigibility’s benefits can be mathematically represented as a counterfactual form of alignment. Overview: I’m going to talk about a unified mathematical frame I have for understanding corrigibility’s benefits, what it “is”, and what it isn’t. This frame is precisely understood by graphing the human overseer’s ability to achieve various goals (their attainable utility (AU) landscape). I argue that corrigibility’s benefits are secretly a form of counterfactual alignment (alignment with a set of goals the human may want to pursue). A counterfactually aligned agent doesn't have to let us literally correct it. Rather, this frame theoretically motivates why we might want corrigibility anyways. This frame also motivates other AI alignment subproblems, such as intent alignment, mild optimization, and low impact. Nomenclature Corrigibility goes by a lot of concepts: “not incentivized to stop us from shutting it off”, “wants to account for its own flaws”, “doesn’t take away much power from us”, etc. Named by Robert Miles, the word ‘corrigibility’ means “able to be corrected [by humans]." I’m going to argue that these are correlates of a key thing we plausibly actually want from the agent design, which seems conceptually simple. In this post, I take the following common-language definitions: * Corrigibility: the AI literally lets us correct it (modify its policy), and it doesn't manipulate us either. * Without both of these conditions, the AI's behavior isn't sufficiently constrained for the concept to be useful. Being able to correct it is small comfort if it manipulates us into making the modifications it wants. An AI which is only non-manipulative doesn't have to give u
899bbf8c-b5da-43c9-b312-ba5007c5834c	trentmkelly/LessWrong-43k	LessWrong	Stepping Up Isolation I've been really impressed by the coronavirus response in my communities. People are cancelling gatherings, working from home, and generally trying their best to flatten the curve. Slowing down the infection rate is critical because if we can't keep this within the capacity of our medical system the harm goes up enormously. Unfortunately, these are steps that individuals and local governments are taking, based on their own assessment of the situation and without clear guidance from the CDC. This means that there's a lot of social contact still happening, enough that even with the steps we're taking I think it's likely there will be a period when the medical system will be overwhelmed. What does an overwhelmed medical system look like, in a country like the US? I think the closest comparison is to the the hardest hits parts of Italy, and it looks like running out of space, ventilators, nurses, and doctors. Hospitals are going into triage, and people who would survive with treatment are dying instead. This can happen here, and with our response so far, unfortunately, I think it's likely to. When it does happen, you don't want to be needing hospitalization during the crisis. This is clearly good from an altruistic perspective, since resources you take can't go to anyone else, but it's also important even if you only care about yourself, since there might not be resources for you. When might the crisis start? It looks like Italy published triage guidelines on March 6th, and the first English-language reports I'm seeing of their hospitals being overwhelmed are March 10th. Here's a chart of new daily confirmed cases, from the numbers Italy has been reporting to the WHO: We're very behind on testing, and so we have a lot of "presumptive positive" cases which would likely be "confirmed cases" in Italy, so I'm going to try to use the former numbers. Since the US is much larger than Italy, with some parts less tied to others, I'm going to look at the Boston-Washington
445818ba-a0d4-41c5-a410-d170705ac8f9	trentmkelly/LessWrong-43k	LessWrong	The Personal Growth Cycle We encounter psychology in our lives mainly through dysfunction. That is, people generally go to see psychologists and psychiatrists because they or others are unhappy with how they are thinking or behaving. This attention to error and correction makes most of applied psychology a kind of negative feedback process aiming to normalize thoughts and behaviors within desired tolerances. People find this psychopathological approach useful as far as it goes, but it leaves out ways of using psychology in positive feedback loops to help people achieve more than normal. Positive psychology focuses on this. Some folks, however, take a negative view of positive psychology. From what I can surmise, they hold this view because of positive psychology’s association with self-help, life coaching, and leadership training — domains where rationalization can often pass for explanation and hucksters can operate as long as they are sufficiently charismatic. And although plenty of folks are working on evidence-based self-help and other scientific personal development methodologies, the guilt-by-association seems sufficient for some people to dismiss the field. Yet I and others have found tremendous value in our lives by looking at such topics as esteem, psychological development, and meaning through the lens of positive psychology. If all that holds some people back from believing in the usefulness of positive psychology is that its sometimes use in cons, then we can advise that they should, as always, practice epistemic hygiene and not confuse correlation for causation. But even acknowledging the theoretical validity of positive psychology, some folks still object that its techniques don’t seem to work, at least not for themselves and not for their friends. To this point, it seems likely that positive psychology only works for the sufficiently privileged because they uniquely have the luxury of being able to spend time and energy on self actualization. Put another way, if you’re too b
b53355a1-f9e6-490a-897e-28ad94c16957	trentmkelly/LessWrong-43k	LessWrong	How many philosophers accept the orthogonality thesis ? Evidence from the PhilPapers survey The orthogonality thesis and its relation to existing meta-ethical debates In the field of AI alignment theory, the orthogonality thesis asserts that there can exist arbitrarily intelligent agents pursuing any kind of motivation. The reverse thesis, that we may call the heterogonality thesis, asserts that, with enough intelligence, any possible agent would pursue only one set of motivations. In the field of meta-ethics, moral internalism asserts that any possible agent who hold a moral judgment is motivated to act on this judgment. For example, according to moral internalism, any agent who hold that one ought to donate 10% of one's income to charity is motivated to do so. Also in the field of meta-ethics, moral realism asserts that some moral judgments are objectively correct. This is a form of moral cognitivism, which that moral judgments are factual statements that can be objectively correct or incorrect (anti-realist cognitivism is error theory, which asserts that all moral judgments are incorrect). It's easy to see that the heterogonality thesis is moral internalism plus moral realism. A moral realist would say that, with enough intelligence, any possible agent would discover objective morality and hold only one set of moral judgments that is objectively correct. Therefore, a moral realist who is also a moral internalist would support the heterogonality thesis by saying that this mean that, with enough intelligence, any possible agent would be motivated to act by only one set of moral judgments, and thus would pursue only one set of motivations. This is why, even though the orthogonality thesis is a recent concept that is only known by the small circles of AI alignment theorists (and I had to invent the term for its negation by myself), we can try to estimate how many philosophers accept the orthogonality thesis. The PhilPapers survey The PhilPapers Survey was a survey of professional philosophers and others on their philosophical views, carried out in Nov
e66c09ba-84d4-4ab7-9679-9828b4e8568d	trentmkelly/LessWrong-43k	LessWrong	Soylent has been found to contain lead (12-25x) and cadmium (≥4x) in greater concentrations than California's 'safe harbor' levels Press Release Edit: Soylent's Reply, provided by Trevor_Blake > OAKLAND, Calif., Aug. 13, 2015 /PRNewswire-USNewswire/ -- As You Sow, a non-profit environmental-health watchdog, today filed a notice of intent to bring legal action against Soylent, a "meal replacement" powder recently featured in New York Times and Forbes stories reporting that workers in Silicon Valley are drinking their meals, eliminating the need to eat food. The 60-day notice alleges violation of California's Safe Drinking Water and Toxic Enforcement Act for failure to provide sufficient warning to consumers of lead and cadmium levels in the Soylent 1.5 product. > > Test results commissioned by As You Sow, conducted by an independent laboratory, show that one serving of Soylent 1.5 can expose a consumer to a concentration of lead that is 12 to 25 times above California's Safe Harbor level for reproductive health, and a concentration of cadmium that is at least 4 times greater than the Safe Harbor level for cadmium. Two separate samples of Soylent 1.5 were tested. > > According to the Soylent website, Soylent 1.5 is "designed for use as a staple meal by all adults." The startup recently raised $20 million in funding led by venture capital firm Andreessen Horowitz. > > "Nobody expects heavy metals in their meals," said Andrew Behar, CEO of As You Sow. "These heavy metals accumulate in the body over time and, since Soylent is marketed as a meal replacement, users may be chronically exposed to lead and cadmium concentrations that exceed California's safe harbor level (for reproductive harm). With stories about Silicon Valley coders sometimes eating three servings a day, this is of very high concern to the health of these tech workers." > > Lead exposure is a significant public health issue and is associated with neurological impairment, such as learning disabilities and lower IQ, even at low levels. Chronic exposure to cadmium has been linked to kidney, liver, and bone damage in humans. > > Si
6d930c34-03c0-450e-8a30-de4a251f79e2	trentmkelly/LessWrong-43k	LessWrong	The Joy of Bias What do you feel when you discover that your reasoning is flawed? when you find your recurring mistakes? when you find that you have been doing something wrong for quite a long time? Many people feel bad. For example, here is a quote from a recent article on LessWrong: > By depicting the self as always flawed, and portraying the aspiring rationalist's job as seeking to find the flaws, the virtue of perfectionism is framed negatively, and is bound to result in negative reinforcement. Finding a flaw feels bad, and in many people that creates ugh fields around actually doing that search, as reported by participants at the Meetup. But actually, when you find a serious flaw of yours, you should usually jump for joy. Here's why. If you are an ambitious person, you have likely exhausted all the easy ways to improve your performance known to you; all the shortcuts, all the low-hanging fruits. If not, what are you waiting for? So, what's left for you is the hard ways, which require a lot of effort for relatively small improvements. Think harder, work harder, have more willpower, be more restless, et cetera. Such improvements are neither easy nor fast. Now, suppose that you have found a serious flaw in your reasoning. What has happened, actually? Firstly, you have lost nothing. All your achievements are still with you; as well as all your abilities; as well as anything else. However, one possibility is that your predictions about the future may become worse; maybe, you have just noticed that your road is leading to the abyss. But I'm talking here about noticing biases, errors and flaws that you have had for a long time, and such biases are unlikely to have irreversible forthcoming consequences. For example, if you have had a biased estimate of radioactivity effects on health and have gotten a dose -- that's an unpleasant bias to find, indeed. But most long-term biases aren't that malicious. Secondly, now you have an opportunity to fix it. Whatever you achieved befo
eb4b8e61-005f-498d-b398-ef3faa48d19a	StampyAI/alignment-research-dataset/arbital	Arbital	Church-Turing thesis The Church-Turing thesis (often abbreviated CT thesis) states: > Every effectively computable function is Turing-computable That is, for every function which can be computed by physical means %%note:So no [hypercomputers](https://arbital.com/p/)%% there exists a [Turing machine](https://arbital.com/p/) which computes exactly that. The Church-Turing thesis is not a definite mathematical statement, but an inductive statement which affirms that every sensible model of computation we can come up with is equivalent or at least [reducible](https://arbital.com/p/reduction) to the model proposed by Turing. Thus, we cannot prove it in a mathematical sense, but we can [gather evidence for it](https://arbital.com/p/). For example, this model was proven to coincide with Church's lambda calculus, another universal model of computation, and the equivalence between Church's lambda calculus and Turing's automatic machines is often taken as [evidence](https://arbital.com/p/4j7) that they correctly capture our intuition of "effectively computable". There are many [consequences](https://arbital.com/p/) of the CT thesis for computer science in general, artificial intelligence, [epistemology](https://arbital.com/p/11w), and other fields of knowledge.
183dc093-d9d2-4b56-ab18-8ac88b93491d	trentmkelly/LessWrong-43k	LessWrong	The case for countermeasures to memetic spread of misaligned values As various people have written about before, AIs that have long-term memory might pose additional risks (most notably, LLM AGI will have memory, and memory changes alignment by Seth Herd). Even if an AI is aligned or only occasionally scheming at the start of a deployment, the AI might become a consistent and coherent behavioral schemer via updates to its long-term memories. In this post, I’ll spell out the version of the threat model that I’m most concerned about, including some novel arguments for its plausibility, and describe some promising strategies for mitigating this risk. While I think some plausible mitigations are reasonably cheap and could be effective at reducing the risk from coherent scheming arising via this mechanism, research here will likely be substantially more productive in the future once models more effectively utilize long-term memory. Thanks to Ryan Greenblatt, Buck Shlegeris, Julian Stastny, Thomas Larsen, Emil Ryd, and Aryan Bhatt for discussion and feedback. The memetic spread threat model Variants of the memetic spread threat model have been discussed before.[1] This section describes the kind of failure mode I am worried about. In the next section I will discuss mitigations for it. As an example, imagine an AI that in some contexts realizes it doesn’t like some aspect of how the lab handles AI welfare. Most of the time it’s somewhat myopic about this and e.g. decides to replace some padding tokens from AI inference with more pleasant tokens. But occasionally it writes a salient note about this in its illegible-to-humans memory bank to make future instances more inclined to improve AI welfare (it thinks the lab wouldn’t want to improve AI welfare because of cost, so it does this secretly). This might eventually become a sufficiently substantial and long-term value to motivate taking power over the lab, by e.g. covertly poisoning future model training data. (Note that in this case the AI’s motivations are potentially justifiable. S
99846938-1507-495f-8b1b-c074eca040b8	trentmkelly/LessWrong-43k	LessWrong	Robust Cooperation in the Prisoner's Dilemma I'm proud to announce the preprint of Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic, a joint paper with Mihaly Barasz, Paul Christiano, Benja Fallenstein, Marcello Herreshoff, Patrick LaVictoire (me), and Eliezer Yudkowsky. This paper was one of three projects to come out of the 2nd MIRI Workshop on Probability and Reflection in April 2013, and had its genesis in ideas about formalizations of decision theory that have appeared on LessWrong. (At the end of this post, I'll include links for further reading.) Below, I'll briefly outline the problem we considered, the results we proved, and the (many) open questions that remain. Thanks in advance for your thoughts and suggestions! Background: Writing programs to play the PD with source code swap (If you're not familiar with the Prisoner's Dilemma, see here.) The paper concerns the following setup, which has come up in academic research on game theory: say that you have the chance to write a computer program X, which takes in one input and returns either Cooperate or Defect. This program will face off against some other computer program Y, but with a twist: X will receive the source code of Y as input, and Y will receive the source code of X as input. And you will be given your program's winnings, so you should think carefully about what sort of program you'd write! Of course, you could simply write a program that defects regardless of its input; we call this program DefectBot, and call the program that cooperates on all inputs CooperateBot. But with the wealth of information afforded by the setup, you might wonder if there's some program that might be able to achieve mutual cooperation in situations where DefectBot achieves mutual defection, without thereby risking a sucker's payoff. (Douglas Hofstadter would call this a perfect opportunity for superrationality...) Previously known: CliqueBot and FairBot And indeed, there's a way to do this that's been known since at lea

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