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(Athey et al.,, 2004 ; Amador and Bagwell,, 2013 ; Ambrus and Egorov,, 2017 One of the key motivations for such a condition is to model the strategic role of bureaucracy within an organization: if management wants to dissuade units within the organization from proposing certain types of solutions, they can use bureaucr... |
Ambrus and Egorov, ( 2017 propose a model in which such selective cost increases are in fact part of the optimal delegation scheme. |
Finally, a recent working paper by Khodabakhsh et al., ( 2018 studies algorithmic delegation in a much more general setting, in which a principal must choose an action and she delegates this choice to an agent who is informed of the state of the world. The principal’s and agent’s preferences over actions in every state... |
Model and Preliminaries We begin by making the precise the way in which the principal and the agent interact, resulting in the principal’s selection of (at most) one element from a set |
[MATH] of potential solutions. There are functions [MATH] and [MATH] such that if [MATH] is selected, then the principal’s utility is [MATH] and the agent’s utility is [MATH] To formalize the possibility that the principal selects no solution (i.e., perpetuating the status quo) we identify this possiblity with a specia... |
[MATH] , and we extend the utility functions from [MATH] to [MATH] by setting [MATH] The set [MATH] is a probability space, with probability measure [MATH] , and the agent has the power to draw independent samples from |
[MATH] according to [MATH] The principal, on the other hand, can neither draw samples from [MATH] nor directly observe the outcome of the agent’s sampling; she must rely on her interaction with the agent to arrive at a selected element of [MATH] |
Before formalizing our model of interaction, it is useful to first note some of the ways in which our basic model can be generalized or specialized. |
We will initially consider the case of a single probability measure [MATH] on [MATH] , but it is also useful to consider cases in which there are multiple probability measures [MATH] on [MATH] and the agent has the power to draw independent samples from any of these distributions. |
We will generally assume there is a sampling budget of [MATH] on the number of samples that the agent can draw. In some of our models, we will also introduce a sampling cost |
[MATH] for each draw by the agent — or in the case of multiple probability measures, a cost [MATH] for sampling from [MATH] We consider both the full-information case — in which the principal knows both the functions |
[MATH] and [MATH] , and hence can evaluate the utility of a solution [MATH] to both herself and to the agent — and the limited-information case in which the principal only knows her own utility function [MATH] |
The functions [MATH] and [MATH] define random variables on [MATH] , and we consider both the case in which they can be arbitrary non-negative functions, and the case of independent utilities , when they are independent random variables. |
In a later section, we will specialize the formalism to the binary model discussed in Section , in which each distribution [MATH] is supported on a two-element set [MATH] such that |
[MATH] In this case we will let [MATH] denote the pair [MATH] . The binary model captures a setting in which the feasibility of the [MATH] solution is unknown until the agent investigates it, but the value of the solution to both parties (if feasible) is known a priori |
2.1 A General Definition of Mechanisms for the Principal and Agent Let us now formalize how the principal and the agent interact, resulting in the principal’s selection of a solution. Thus far, our discussion in Section has focused on interactions of a very structured form: the agent draws a set of samples from [MATH] ... |
To do this, we begin by defining a mechanism as follows. A mechanism [MATH] defines a set of signals, [MATH] , that the agent may send, and an allocation function |
[MATH] that specifies which solution the principal will choose given the agent’s signal. In such a mechanism, a strategy for the agent is specified by a mapping [MATH] where [MATH] denotes the set of finite sequences over [MATH] such that [MATH] |
represents the signal the agent sends if he sampled [MATH] solutions and observed the sequence [MATH] Suppose the agent observes sequence |
[MATH] and sends signal [MATH] , resulting in outcome [MATH] In this case, the principal’s and agent’s utilities are [MATH] and [MATH] , respectively, if |
[MATH] Otherwise the principal’s utility is 0 and the agent’s is [MATH] . In other words, we assume that if the mechanism results in the principal selecting a solution that was never sampled by the agent, that solution cannot be adopted. Instead the status quo is preserved and the agent suffers a penalty. This assumpti... |
In models with costly sampling, the specification of an agent’s strategy must also include a sequential policy [MATH] for deciding which sample (if any) to observe next, given the set of samples already observed. The principal’s and agent’s utilities are both diminished by the sum of costs [MATH] for the samples [MATH]... |
Under our definition of mechanisms, the sequence of solutions sampled by the agent leads to a signal (via the agent’s strategy [MATH] ), and this signal leads to a solution in [MATH] (via the principal’s allocation function [MATH] ). Composing these two functions, we get a mapping from the agent’s sampled solutions to ... |
Definition 1 (interim allocation function) If [MATH] is a mechanism and [MATH] is an agent’s strategy, the interim allocation function of the pair [MATH] is the mapping [MATH] obtained by composing the strategy [MATH] with the allocation function [MATH] . In other words, |
[MATH] is the outcome resulting from mechanism [MATH] when the agent draws sample sequence [MATH] and plays according to [MATH] 2.2 Single Proposal Mechanisms |
We now show that there is a sense in which it is without loss of generality to focus on interactions in which the agent proposes a single solution from among the ones they sampled, and the principal either accepts or rejects it. To do this, we define a simple type of mechanism called a single proposal mechanism , and w... |
below that any other mechanism can be simulated by a single proposal mechanism. Definition 2 single-proposal mechanism with eligible set [MATH] is a mechanism in which the agent proposes one outcome, and the mechanism accepts this proposal if and only if it belongs to [MATH] . More formally, [MATH] |
is a single-proposal mechanism with eligible set [MATH] if [MATH] and [MATH] restricts to the identity function on [MATH] and the constant function [MATH] on |
[MATH] Lemma 1 If [MATH] is any mechanism and [MATH] is any strategy constituting a best response to [MATH] then there exists a single proposal mechanism |
[MATH] and a best response [MATH] to [MATH] such that the interim allocation functions [MATH] and [MATH] are identical. Proof. Let [MATH] be the range of the interim allocation function [MATH] , i.e. the set of all possible outcomes of [MATH] , other than [MATH] when the agent acts according to [MATH] . Define |
[MATH] to be the single-proposal mechanism with eligible set [MATH] . Let [MATH] be the strategy in which the agent observes his tuple of samples, |
[MATH] , and chooses strategy [MATH] . By construction the interim allocation functions [MATH] and [MATH] are identical. To prove that [MATH] is a best response to [MATH] , consider any |
[MATH] and any [MATH] Let [MATH] denote the agent’s utility when playing according to [MATH] ; note that [MATH] We wish to show that the agent cannot benefit by playing [MATH] |
instead, i.e. [EQUATION] If [MATH] then [MATH] which implies ( ) since [MATH] If [MATH] then [MATH] for some [MATH] Now ( ) follows because strategy [MATH] is a best response for mechanism [MATH] and [MATH] denotes the agent’s utility when playing strategy [MATH] in [MATH] |
whereas [MATH] denotes his utility when playing [MATH] Analyzing Delegated Search Via Prophet Inequalities In this section we develop a formal link between delegated search mechanisms and prophet inequalities. It turns out that the relevant prophet inequalities involve random variables arriving at discrete points in co... |
3.1 below. Then in Section 3.2 we explain the reduction from delegated search (in the distributional model) to continuous-time prophet inequalities. |
3.1 Continuous-time prophet inequalities In this section we will be concerned with problems which involve designing a selection rule to choose (at most) one element from a random finite set of pairs [MATH] with the goal of maximizing the expected [MATH] -coordinate of the chosen element. The [MATH] -coordinate is thoug... |
Definition 3 (selection rules) selection rule is a function [MATH] from finite subsets of [MATH] to the set [MATH] , with the property that [MATH] |
for every [MATH] stopping rule is a selection rule that chooses element [MATH] from set [MATH] without looking at the set of elements whose time coordinate is greater than [MATH] . Formally, [MATH] is a stopping rule if it satisfies the following property: for any |
[MATH] and any two sets [MATH] such that [MATH] , we have [EQUATION] An oblivious stopping rule with eligible set [MATH] is a stopping rule [MATH] such that for every [MATH] |
[MATH] is an earliest element of [MATH] (i.e., an element of that set with minimum [MATH] coordinate) or [MATH] if [MATH] is empty. |
threshold stopping rule with threshold [MATH] is an oblivious stopping rule whose eligible set is of the form [MATH] or [MATH] Definition 4 |
(CTSPs and prophet inequalities) continuous-time selection problem (CTSP) is an ordered pair [MATH] where [MATH] is a set of probability distributions over finite subsets of [MATH] and [MATH] is a set of selection rules. |
A CTSP [MATH] satisfies a prophet inequality with factor [MATH] if it is the case that for every [MATH] there exists some [MATH] such that |
[EQUATION] Here the random variable [MATH] is defined by specifying that if [MATH] then [MATH] , and if [MATH] then [MATH] The random variable [MATH] is defined to be [MATH] |
We now present the prophet inequalities we will use in this work. To state them, we define the following families of stopping rules and distributions on subsets of [MATH] |
[MATH] is the family of oblivious stopping rules. [MATH] is the family of threshold stopping rules. [MATH] is the family of random sets whose elements are obtained by sampling independently from [MATH] |
joint distributions. In other words, a distribution [MATH] is specified by giving a positive number [MATH] a tuple of joint distributions [MATH] |
on [MATH] , and defining [MATH] to be the distribution on [MATH] -element sets obtained by drawing one sample independently from each of |
[MATH] [MATH] is the family of random sets whose [MATH] elements are i.i.d. samples from an atomless distribution with [MATH] and [MATH] independent, i.e. a distribution on [MATH] |
which is a product of atomless distributions. [MATH] is the union of [MATH] over all [MATH] Our first prophet inequality is Samuel-Cahn, ’s 1984 famous prophet inequality for threshold stopping rules. |
Theorem 1 Samuel-Cahn, ( 1984 There is a prophet inequality with factor [MATH] for [MATH] The second is an improved prophet inequality for threshold stopping rules when samples are drawn i.i.d. from atomless product distributions; it can be derived as a corollary of either (Ehsani et al.,, 2018 , Theorem 19) or |
(Correa et al.,, 2017 , Corollary 2.2) Theorem 2 Correa et al., ( 2017 ); Ehsani et al., ( 2018 There is a prophet inequality with factor [MATH] for |
[MATH] Our third prophet inequality again pertains to the case when samples are drawn i.i.d. from atomless product distributions, but it allows for oblivious stopping rules rather than threshold stopping rules. The discrete-time counterpart to this prophet inequality can be found in |
Hill and Kertz, ( 1982 ); Kertz, ( 1986 ); Correa et al., ( 2017 Theorem 3 Let [MATH] be the solution to [MATH] There is a prophet inequality with factor |
[MATH] for [MATH] Since the distincton between discrete time and continuous time is immaterial from the standpoint of analyzing threshold stopping rules, the first two of these theorems are equivalent to the existing results for discrete-time prophet inequalities that we have cited before the theorem statements. On the... |
Appendix ; since Theorem is the only novel result among the three theorems, the proofs of the other two are included only for the purpose of making our paper self-contained. |
To complete this section, we will describe the stopping rules which achieve the bounds stated in the three prophet inequalities above. |
When the points [MATH] are independent but not necessarily identically distributed, choose threshold [MATH] to be the median of the distribution of [MATH] . In other words, |
[MATH] is defined such that the events [MATH] and [MATH] both have probability at most [MATH] . Consider the threshold stopping rule that selects the first pair [MATH] |
with [MATH] , and consider the one whose selection criterion is [MATH] . The proof of Theorem shows that at least one of these two stopping rules fulfills a prophet inequality with factor 1/2. |
When the points [MATH] are i.i.d. and the distributions of [MATH] and [MATH] are atomless and independent, with cumulative distribution functions [MATH] and [MATH] , respectively, choose threshold [MATH] such that |
[MATH] The proof of Theorem shows that the threshold stopping rule that selects the first pair [MATH] such that [MATH] fulfills a prophet inequality with factor [MATH] . Now let |
[MATH] be the solution to [EQUATION] and let [MATH] be the solution of the differential equation [EQUATION] with initial condition [MATH] The oblivious stopping rule that accepts the first [MATH] such that |
[EQUATION] fulfills a prophet inequality with factor [MATH] 3.2 Reducing delegated search to prophet inequalities Although delegated search problems and prophet inequalities appear unrelated at first glance, the tight technical connection between them is explained by an observation which is extremely natural in hindsig... |
[MATH] is monotonically decreasing, the two selection criteria are equivalent! Thus, designing single proposal mechanisms that yield high utility for the principal is equivalent to designing oblivious stopping rules that yield a high expected value. |
In more detail, let [MATH] be any continuous, monotonically decreasing bijection from [MATH] to [MATH] , for example [MATH] . Under the mapping |
[MATH] defined by [MATH] any distribution on sets of solutions [MATH] induces a distribution [MATH] on sets of pairs [MATH] . In particular, our distributional model in which the agent draws |
[MATH] i.i.d. samples from [MATH] is mapped, under this correspondence, to a member of the family of distributions [MATH] There is also a reverse correspondence from oblivious stopping rules to single proposal mechanisms and their interim allocation functions. The oblivious stopping rule [MATH] with eligible set [MATH]... |
with eligible set [MATH] , then for any sequence of samples [MATH] we have [EQUATION] In other words, suppose we run the mechanism [MATH] ; the agent draws a sequence of samples; and we let the agent choose the best one (for the agent) that belongs to [MATH] This procedure is equivalent to running the oblivious stoppin... |
Combining these observations with Theorems and we obtain the following theorem. Theorem 4 In the distributional model, suppose the agent draws [MATH] i.i.d. samples, and let [MATH] denote the utility the principal would attain if she could directly choose her favorite among these [MATH] samples. |
1. There is always a set [MATH] of the form [MATH] or [MATH] such that a single proposal mechanism with eligible set [MATH] ensures that the principal’s expected utility is at least |
[MATH] 2. If the principal and agent have independent utilities, each drawn from an atomless distribution, then a single proposal mechanism that accepts any proposal satisfying [MATH] , for a suitable choice of [MATH] , ensures that the principal’s expected utility is at least [MATH] |
3. If the principal and agent have independent utilities, each drawn from an atomless distribution, and the principal can observe the agent’s utility, then a single proposal mechanism that accepts any proposal satisfying [MATH] for a suitable choice of the function [MATH] ensures that the principal’s expected utility i... |
In Appendix we show that the bounds in all three parts of the theorem are tight with respect to the assumptions made in their respective statements. |
Binary outcomes Recall the binary model from Section The potential solutions come from a large discrete set [MATH] and the agent’s role is to explore which of these options are feasible to implement. If [MATH] is feasible, it yields utility [MATH] for the principal and [MATH] |
for the agent — where the pair [MATH] is commonly known to both parties — and if [MATH] is infeasible it yields zero utility for both parties. To explore the feasibility of solution [MATH] |
the agent must incur a cost of [MATH] and the probability of success is [MATH] , independently of the success of other solutions. These quantities [MATH] are again commonly known to both parties. We will assume that [MATH] for each solution [MATH] , since otherwise it is against the agent’s self-interest to explore [MA... |
4.1 Optimal search policies: Weitzman’s box problem If the principal were conducting the search by herself (without delegation to an agent), this model would correspond to a special case of the box problem introduced by Weitzman, ( 1979 . The optimal search policy is simple but surprisingly subtle: it assigns to each o... |
[MATH] — which in our case entails setting [MATH] — and then explores options in decreasing order of priority, selecting the first feasible one in this ordering or stopping when all remaining unexplored options have |
[MATH] Now suppose that the principal instead delegates the search to an agent who bears the cost of exploration, by running a single-proposal mechanism with eligible set [MATH] . Then the agent faces a different instance of the box problem, in which the set of options is limited to [MATH] , and the costs and success p... |
rather than [MATH] . This means the agent prioritizes boxes in decreasing order of [MATH] rather than [MATH] and recommends the first box in this ordering that is discovered to be feasible. |
To summarize, the delegated search problem in the binary model is analogous to Weitzman’s box problem, but with the important distinction that the searcher (the principal) is not allowed to choose the order in which to open the boxes. Instead the problem specifies an exogenous ordering of the boxes — corresponding to t... |
below to presenting a solution that always achieves at least half of the expected value of running the optimal search procedure that is allowed to inspect the boxes in any order it desires. Interestingly, the analysis is based on prophet inequalities, specifically |
Theorem and its proof. It implies there is an approximately optimal mechanism with the following structure. For any half-infinite interval [MATH] |
of the form [MATH] or [MATH] let [MATH] and define [MATH] to be the single-proposal mechanism in which a proposal [MATH] is eligible if it is feasible and belongs to [MATH] |
4.2 The Box Problem with an Exogenous Ordering In this section we recapitulate some background material about Weitzman, ’s ( 1979 box problem. In this problem there are [MATH] boxes, each containing an independent random prize. The prize in box [MATH] is denoted [MATH] and the cost of opening the box is [MATH] . A sear... |
[MATH] defined by the equation [MATH] , then the optimal sequential search policy opens boxes in decreasing order of priority, stopping at the first time when the highest prize inside an open box exceeds the highest priority of a closed box, or at the first time when the priority of every remaining closed box is negati... |
Kleinberg et al., ( 2016 provided a proof of optimality of Weitzman’s procedure in which the priority [MATH] is interpreted as the “strike price” of a real option with fair value [MATH] . An important quantity in their analysis is the “covered call value”, which is simply the random variable |
[MATH] . We restate the following lemma from their work. Lemma 2 Kleinberg et al., ( 2016 For any sequential search procedure and any box [MATH] let [MATH] be the indicator random variables of the event that the procedure selects box [MATH] and the event that it opens box [MATH] respectively. The inequality |
[EQUATION] is satisfied by every search procedure, and equality holds if and only if the search procedure is non-exposed meaning that [MATH] at every sample point where |
[MATH] Corollary 1 For any sequential search procedure, the expected net value of running the procedure (i.e., the value of the selected box minus the combined cost of opening boxes) is bounded above by the expectation of the maximum covered call value, i.e. |
[EQUATION] The corollary is immediate, by summing inequality ( over boxes [MATH] Now consider the box problem with an exogenous ordering of boxes, where the searcher is limited to considering the boxes one by one in the specified order, and once she decides to leave a box closed or to leave the prize within unclaimed, ... |
, which shows that these policies correspond to a threshold rule applied to the sequence of covered call values [MATH] Definition 5 |
[MATH] -thresholding policy for the box problem with exogenous ordering is a policy that operates as follows. There is a half-infinite interval |
[MATH] or [MATH] called the target interval . The policy declines to open any box [MATH] with [MATH] Otherwise, if [MATH] , the policy opens the box and claims the prize inside if and only if [MATH] |
Lemma 3 Every [MATH] -thresholding policy is non-exposed. The expected net value of running a [MATH] -thresholding policy with target interval [MATH] is exactly the same as the expected value selected by the threshold stopping rule that observes the random sequence |
[MATH] and selects the first element of this sequence that belongs to [MATH] Proof. The policy is non-exposed because [MATH] implies [MATH] , while [MATH] |
and [MATH] imply [MATH] Hence the left and right sides of ( are equal for every box, and the net value of running the policy is [MATH] i.e. the expected covered call value of the box the policy selects. By design, the policy perfectly simulates the threshold stopping rule that chooses the first element of the sequence ... |
that belongs to [MATH] ; this is because it selects the first box such that [MATH] and [MATH] both belong to [MATH] , which is also the first box such that [MATH] belongs to [MATH] |
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