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Consider the following satisfaction matrix: [EQUATION] where [MATH] . Let [MATH] (i.e., a symmetric two-player game). By using the properties of we characterize the structure of the induced normal form game. We show that in this normal form game, a PNE only exists if [MATH] (and similarly to the other users and strateg... |
Moreover, Theorem is not sensitive to the sum of the display probabilities being equal to 1. One can show a similar argument for any mediator that displays items with constant probabilities, i.e., [MATH] for some [MATH] . Theorem suggests that [MATH] should be bounded to the user satisfaction levels. In the next sectio... |
Our approach: the Shapley mediator In order to provide a fair and stable mediator, we resort to cooperative game theory. Informally, a cooperative game consists of two elements: a set of players [MATH] and a characteristic function [MATH] , where [MATH] determines the value given to every coalition, i.e., every subset ... |
One core solution concept in cooperative game theory is the Shapley value Definition 1 (Shapley value) Let [MATH] be a cooperative game such that [MATH] . According to the Shapley value, the amount that player [MATH] gets is |
[EQUATION] where [MATH] is the set of all permutations of [MATH] and [MATH] is the set of players in [MATH] which precede player [MATH] in the permutation [MATH] |
One way to describe the Shapley value, is by imagining the process in which coalitions are formed: when player [MATH] joins coalition [MATH] , she demands her contribution to the collective payoff of the coalition, namely [MATH] . Equation ( ) is simply summing over all such possible demands, assuming that all coalitio... |
For our purposes, we fix a strategy profile [MATH] , and focus on an arbitrary user [MATH] . How should a mediator assign the probabilities of being displayed in a fair fashion? The induced cooperative game contains the same set of players. For every [MATH] , let [MATH] denote the strategy profile where all players mis... |
[EQUATION] where [MATH] is the maximal satisfaction level a user [MATH] may obtain from the items chosen by the members of [MATH] . Indeed, this formulation represents a collaborative behavior of the players, when they aim to maximize the satisfaction of [MATH] Observe that [MATH] is a valid characteristic function, he... |
3.1 Properties of the Shapley mediator Since the Shapley value is employed in countless settings for fair allocation, it is not surprising that it satisfies our fairness properties. |
Proposition 1 SM satisfies We now show that recommendation games with SM possess a PNE. This is done using the notion of potential games |
. A non-cooperative game is called an exact potential game if there exists a function [MATH] such that for any strategy profile [MATH] , any player [MATH] and any strategy [MATH] , whenever player [MATH] switches from [MATH] to [MATH] , the change in her payoff function equals the change in [MATH] , i.e., |
[EQUATION] This brings us to the main result of this section: Theorem 2 Recommendation games with the Shapley mediator are exact potential games. |
Thus, due to Monderer and Shapley , any recommendation game with the Shapley mediator possesses at least one PNE, and the set of pure Nash equilibria corresponds to the set of argmax points of the potential function; therefore, SM satisfies |
Corollary 1 SM satisfies In fact, Theorem proves a much stronger claim than merely the existence of PNE. A better-response dynamics is a sequential process, where in each iteration an arbitrary player unilaterally deviates to a strategy which increases her payoff. |
Corollary 2 In recommendation games with the Shapley mediator, any better-response dynamics converges. This convergence guarantee allows the players to learn which items to pick in order to maximize their payoffs. Indeed, as has been observed by work on the topic of online recommendation and advertising systems (e.g. s... |
), convergence to PNE is essential for system stability, as otherwise inefficient fluctuations may occur. Linear time implementation |
In Section we showed that the Shapley mediator, SM , satisfies and . Therefore, it fulfills our requirements stated in Section . However, implementation in commercial products would require the mediator to be computationally tractable. The mediator interacts with users; hence a fast response is of great importance. In ... |
In the following theorem we derive a closed-form formula for calculating the display probabilities under the Shapley mediator, which allows it to compute the display probabilities in linear time. |
Theorem 3 Let [MATH] be a strategy profile, and let [MATH] denote the [MATH] ’th entry in the result of sorting [MATH] in ascending order, preserving duplicate elements. The Shapley mediator displays player [MATH] ’s item to a user [MATH] with probability |
[EQUATION] where [MATH] , and [MATH] is an index such that [MATH] The Shapley mediator is implemented in Algorithm . As an input, it receives a strategy profile and a user, or equivalently user satisfaction levels from that strategy profile. It outputs a player’s item with a probability equal to her Shapley value in th... |
Corollary 3 In recommendation games with the Shapley mediator, the payoff of player [MATH] under the strategy profile [MATH] is given by [MATH] |
To facilitate understanding of the Shapley mediator and its fast computation, we reconsider Example above. Example 2 Consider the game given in Example . According to the Shapley mediator, the display probabilities of player 1 under the strategy profile [MATH] are |
[EQUATION] It follows that [MATH] while [MATH] , and the profile to be materialized is [MATH] . Indeed, it can be verified that this is the unique PNE of the corresponding game. Moreover, while the unique PNE under TOP (see Example in Section ) results in a user utility of [MATH] , the unique PNE under the Shapley medi... |
[EQUATION] Hence, the users benefit from the Shapley mediator is greater than from the TOP mediator. This is in addition to the main property of the Shapley mediator, probabilistic selection according to the central measure of fair allocation. |
Input: A strategy profile [MATH] and a user [MATH] Output: An element from [MATH] Pick [MATH] uniformly at random from [MATH] if |
[MATH] then return [MATH] else Return an element uniformly at random from [MATH] Algorithm 1 Shapley Mediator Uniqueness of the Shapley mediator |
As analyzed in Subsection 2.1 , Theorem suggests that a mediator cannot satisfy both and if it sets the probabilities such that [MATH] is constant. One way of determining [MATH] is defined as follows. |
Efficiency . The probability of displaying an item to [MATH] is the maximal satisfaction level [MATH] may obtain from the items chosen in [MATH] . Formally, [MATH] |
Efficiency (for brevity, EF ) binds player payoffs with the maximum satisfaction level of [MATH] from the items chosen by the players under [MATH] It is well known |
that the Shapley value is uniquely characterized by properties equivalent to and EF , when stated in terms of cooperative games. It is therefore obvious that the Shapley mediator satisfies EF |
Thus, one would expect that the Shapley mediator will be the only mediator that satisfies and EF . This is, however, not the case: consider a mediator that runs TOP w.p. [MATH] and NONE otherwise. Clearly, it satisfies and EF In fact, given a mediator [MATH] satisfying and Complete , we can define [MATH] such that |
[EQUATION] thereby obtaining a mediator satisfying and EF . The question of uniqueness then arises: is derived by satisfying and EF ? Or even more broadly, are there mediators that satisfy and EF besides the Shapley mediator? Had the answer been yes, this recipe for generating new mediators would have allowed us to see... |
Theorem 4 The only mediator satisfying [MATH] and EF is the Shapley mediator. Implications of strategic behavior In this section we examine the implications of strategic behavior of the players on their payoffs and user utility. Comprehensive treatment of the integration of multiple stakeholders into recommendation cal... |
, and appears to be challenging. As our work is concerned with strategic content providers, it is natural to consider the Price of Anarchy |
, a common inefficiency measure in non-cooperative games. 6.1 Player payoffs The Price of Anarchy, herein denoted [MATH] , measures the inefficiency in terms of social welfare, as a result of selfish behavior of the players. Specifically, it is the ratio between an optimal dictatorial scenario and the social welfare of... |
Theorem 5 [MATH] and this bound is tight. Hence, under the Shapley mediator the social welfare of the players can decrease by at most a factor of 2, when compared to an optimal solution. |
6.2 User utility We now examine the implications of using the Shapley mediator on the users. For that, we shall assume that the utility of a user from an item is his satisfaction level from that item. Namely, when item [MATH] is displayed to [MATH] , his utility is [MATH] . As a result, the expected utility of the user... |
[EQUATION] Note that the first term results from the displayed items, and the second term from the plain content of the app (displaying no item at all). To quantify the inefficiency of user utility due to selfish behavior of the players under [MATH] , we define the User Price of Anarchy |
[EQUATION] The [MATH] serves as our benchmark for inefficiency of user utility. The nominator is the best possible case: the user utility under any mediator [MATH] and any strategy profile [MATH] . The denominator is the worst user utility under [MATH] , where [MATH] is again the set of PNE profiles induced by [MATH] .... |
Proposition 2 The User PoA of the Shapley mediator, [MATH] , is unbounded. Proposition questions the applicability of the Shapley mediator. An unavoidable consequence of its use is a potentially destructive effect on user utility. While content-provider fairness is essential, users are the driving force of the RS. Ther... |
Proposition 3 The User PoA of TOP [MATH] , is unbounded. Using similar arguments, one can show that [MATH] is unbounded as well. |
In many situations, it is reasonable to assume that when no item is displayed to a user, his utility is 1. Namely, [MATH] for every user [MATH] Indeed, this seems aligned with the ads-in-apps model: the user is interrupted when an advertisement is displayed. We refer to this scenario as the optimal plain content case. ... |
Lemma 1 In the optimal plain content case, it holds that [MATH] In fact, numerical calculations show that [MATH] is bounded by [MATH] , see the appendix for further discussion. |
Discussion Our results are readily extendable in the following important direction (which is even further elaborated in the appendix). In many online scenarios, content providers typically customize the items they offer to accommodate specific individuals. Indeed, personalization is applied in a variety of fields in or... |
Acknowledgments This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement n [MATH] 740435). |
Appendix A Omitted proofs A.1 Proof of Theorem Proof. We first construct a game with two players, three users and three strategies, and show that no mediator can satisfy [MATH] and Complete . Afterwards, we explain how our technique can be used to show that for any arbitrary game there exists a slight modification of t... |
Consider the following satisfaction matrix: [EQUATION] where [MATH] . Let [MATH] (i.e., a symmetric two-player game). This recommendation game induces a normal form game. Since [MATH] satisfies User-Independence , it sets the display probability as a function of the satisfaction levels only (i.e., regardless of the ide... |
[EQUATION] Namely, [MATH] is the probability that player 1’s item will be displayed to [MATH] under the strategy profile [MATH] . Since [MATH] satisfies Complete and Symmetry |
[EQUATION] Since [MATH] satisfies both Complete and Null Player , one has to define what happens if the satisfaction vector is the zero vector (see Footnote ). Denote [MATH] , and due to Symmetry |
[MATH] also (notice that [MATH] ). The following matrix describes the payoff of player 1 under any possible strategy profile: [EQUATION] |
Denote [MATH] . Due to Symmetry , the game is described by the following bi-matrix: [EQUATION] Clearly, since [MATH] , this game possesses a PNE only if [MATH] . Otherwise, under any strategy profile there exists a player with a beneficial deviation. Hence, |
[EQUATION] Due to Null Player , we have [MATH] ; therefore [MATH] for every [MATH] , and Leader Monotonicity does not hold. This sums up the proof for the given two-player game. |
Next, consider any arbitrary game. If the game does not contain a PNE, then we are done. Otherwise, let [MATH] be a PNE. By adding three additional users and copying each strategy in [MATH] and extending it to the three new users, we can reproduce the behavior in the two-player game above. By doing so to every equilibr... |
A.2 Proof of Proposition Proof. Denote the Shapley value of player [MATH] in a cooperative game [MATH] by [EQUATION] It is well known (see, e.g., |
) that the Shapley value satisfies the following properties: C1. Null Player: If [MATH] for every coalition [MATH] , then [MATH] |
C2. Symmetry: If [MATH] for every coalition [MATH] , then [MATH] C3. Linearity: If [MATH] and [MATH] are two cooperative games and [MATH] is a real number, it holds that |
[EQUATION] C4. Efficiency: [MATH] Note that properties C1–C4 are akin to and EF defined in Sections and respectively. However, C1–C4 are properties of the Shapley value in cooperative game |
[MATH] , while [MATH] refer to recommendation games. Hence, we ought to show that the Shapley mediator, defined as the Shapley value in the cooperative game [MATH] , satisfies and EF Denote by [MATH] the Shapley value of player [MATH] in [MATH] , where [MATH] . Recall that by definition of the Shapley mediator, |
[EQUATION] Null Player If [MATH] [EQUATION] Therefore, [EQUATION] Symmetry . If [MATH] , for every [MATH] it holds that [EQUATION] |
Therefore, [EQUATION] User-Independence . Notice that [MATH] is solely determined by [MATH] . Therefore [EQUATION] Leader Monotonicity Let [MATH] and [MATH] , and hence [MATH] . Given a permutation [MATH] over the elements of [MATH] , define [MATH] to be the same permutation vector where [MATH] and [MATH] are swapped. ... |
[EQUATION] Alternatively, if [MATH] precedes [MATH] in [MATH] (and therefore [MATH] precedes [MATH] in [MATH] ) we have [EQUATION] |
[EQUATION] In addition, for [MATH] in which [MATH] appears in the first entry, it holds that [EQUATION] Finally, due to Equations ( ),( ) and ( A.2 ), by summing over all permutations and dividing by [MATH] we get |
[EQUATION] hence [MATH] EF . We have [EQUATION] This concludes the proof of the proposition. A.3 Proof of Theorem Proof. We prove Theorem by showing that recommendation games with the Shapley mediator (denoted RGSM for brevity) belong to the class of congestion games |
. Due to , this implies that RGSM are potential games. A congestion game is a non-cooperative game, defined by players and resources, where the payoff of each player depends solely on the resources she chooses and on the number of players that chose each of the corresponding resources. Formally, a congestion game is a ... |
[MATH] is the set of players. [MATH] is the set of resources. [MATH] denotes the set of possible strategies of player [MATH] , where any [MATH] is a subset of all resources. |
The number of players who select resource [MATH] under the strategy profile [MATH] is given by [MATH] [MATH] is a utility function, [MATH] , such that [MATH] is given to any player whose strategy contains resource [MATH] , in case exactly [MATH] players chose [MATH] |
The payoff of player [MATH] under the strategy profile [MATH] is given by [MATH] Given an RGSM game instance, we construct a corresponding congestion game, and show that the payoffs of the players under any strategy profile is exactly the same in both games. Importantly, the complexity of the below reduction is irrelev... |
Denote [EQUATION] and observe that [MATH] . Let [MATH] be the ordered elements of [MATH] in ascending order. Next, we aim to represent a strategy profile as a selection of corresponding resources by the players. Denote [MATH] , where for each user [MATH] resource [MATH] is associated with the interval [MATH] The strate... |
[EQUATION] Thus, there is an induced one-to-one function from the set of items to the power set of resources, [MATH] . Mapping between items and resources, we define the set of possible strategies of player [MATH] |
[EQUATION] The load of each resource (the number of players that select this resource) under the strategy profile [MATH] is: [EQUATION] |
where [MATH] . The utility of selecting resource [MATH] depends only on the number of players who select it, and is defined as follows: |
[EQUATION] Thus the congestion game [MATH] is properly defined. The remaining ingredient of the reduction is: Lemma 2 The sum of utilities of each player [MATH] in the congestion game under the strategy profile [MATH] is exactly her payoff in the RGSM under the strategy profile [MATH] |
[EQUATION] Proof. Fix a user [MATH] . Recall that [MATH] is the [MATH] ’th satisfaction level [MATH] obtains from the items in [MATH] (in ascending order, [MATH] ), and [MATH] is the number of items offered to [MATH] with a satisfaction level less or equal to [MATH] |
Under the strategy profile [MATH] , define [MATH] to be the elements in [MATH] such that [MATH] . Observe that for any [MATH] the number of players that selected resources associated with intervals contained in [MATH] is [MATH] . In addition, let [MATH] be the index such that [MATH] |
The strategy [MATH] of player [MATH] is mapped to the set of resources [MATH] . Therefore, [EQUATION] By summing over all users we get |
[EQUATION] This concludes the proof of Lemma Ultimately, since RGSM belong to the class of congestion games, they possess a potential function, and every better response dynamic converges |
. This concludes the proof of Theorem In fact, using standard arguments, we can show that [EQUATION] is the potential function of RGSM. |
A.4 Proof of Theorem Proof. Denote [MATH] , and w.l.o.g. let [MATH] . As defined earlier, the collective payoff of coalition [MATH] is the maximum satisfaction level of [MATH] from an item offered by a player in [MATH] (the player with the highest index), namely |
[EQUATION] For any permutation [MATH] such that [MATH] it follows that [MATH] . Put differently, if in a permutation [MATH] there is a player with index greater than [MATH] that precedes player [MATH] , then player [MATH] ’s contribution to the collective payoff of the coalition [MATH] is zero. Thus, we ought to look o... |
[EQUATION] The latter holds since in every preceding set [MATH] (prefix of [MATH] ) with [MATH] [MATH] is located in the [MATH] -th entry in [MATH] . Thus we have [MATH] indices to choose from (less than [MATH] ), [MATH] ways to order them, and another [MATH] ways to order the suffix (the remaining [MATH] elements). |
For [MATH] , we denote by [MATH] the number of permutations where [MATH] . Hence: [EQUATION] Again, we turn to counting arguments: for every prefix [MATH] of size [MATH] , if [MATH] then there must be an index greater than [MATH] ; therefore [MATH] . Besides [MATH] , there are [MATH] indices in [MATH] [MATH] ways to ch... |
Lemma 3 It holds that [MATH] The proof of Lemma appears after this proof. Next, the Shapley value of player [MATH] in the cooperative game [MATH] is: |
[EQUATION] Since we care only about permutations where the maximum index of a player in the preceding set of player [MATH] is less than [MATH] , we have |
[EQUATION] Now, using the counting arguments presented above, we derive the following: [EQUATION] Applying the concise form of [MATH] from Lemma we get: |
[EQUATION] Ultimately, by using the concise form of [MATH] from Lemma we have: [EQUATION] A.5 Proof of Lemma Proof. First we show that [MATH] |
[EQUATION] By Chu Shih-Chieh’s Identity (see e.g. ) it holds that [EQUATION] Applying Equation ( 12 ) to Equation ( 11 ) with [MATH] we get: |
[EQUATION] In addition, [MATH] since: [EQUATION] Using Pascal’s rule [EQUATION] we get [EQUATION] A.6 Proof of Theorem The proof of Theorem relies on several supporting lemmas. |
Due to User-Independence [MATH] sets the display probabilities according to the satisfaction vector only. Thus, it is enough to show that [MATH] satisfying [MATH] must distribute the display probabilities exactly as the Shapley mediator does for any satisfaction vector [MATH] . Since we do not have a specific user in m... |
Observation 1 If [MATH] contains one non-zero entry [MATH] , then [MATH] for every player [MATH] This observation follows immediately from Null Player and EF . Next, we show that in a two player game, each player gets her Shapley value. |
Lemma 4 Let [MATH] . For every [MATH] and every [MATH] , it holds that [MATH] Proof. Assume by contradiction that [MATH] . First, we analyze the case [MATH] for some [MATH] . Consider the following satisfaction matrix: |
[EQUATION] with [MATH] and [MATH] . This satisfaction matrix induces the following bimatrix game [EQUATION] since [EQUATION] This [MATH] normal-form game contains a cycle of beneficial deviations, which implies the non-existence of PNE; hence, we obtain a contradiction. |
On the other hand, let [MATH] for some [MATH] . Consider the following satisfaction matrix: [EQUATION] where again [MATH] and [MATH] . This satisfaction matrix induces the following [MATH] normal-form game: |
[EQUATION] since [EQUATION] Again we obtained a contradiction to satisfying . Overall, [MATH] must produce the same distribution as SM for [MATH] |
Since any mediator behaves like the Shapley mediator when [MATH] , due to Null Player it also holds for [MATH] for satisfaction vector [MATH] with only two non-zero entries. |
Lemma 5 Let [MATH] . It holds that [MATH] Proof. The assertion holds for [MATH] . Otherwise, we prove the assertion by induction, where Lemma serves as the base case. Assume the claim holds for [MATH] , and does not hold for [MATH] |
Case 1: [MATH] . Due to EF [EQUATION] Denote by [MATH] a player such that [EQUATION] Consider the following satisfaction matrix [EQUATION] |
where [MATH] [MATH] and [MATH] for every player [MATH] such that [MATH] . Note that all players but 1 and [MATH] are non-strategic, or alternatively every strategy they select has the same satisfaction level w.r.t. users [MATH] . We have the following cycle: |
[MATH] [MATH] . Due to Equation ( 14 ), we have [EQUATION] [MATH] . According to the inductive step, [EQUATION] [MATH] [MATH] The reader can verify that in each step above the deviating player (e.g. player 1 from the first bullet to the second, and player [MATH] from the second to the third) indeed makes a beneficial d... |
Case 2: [MATH] . Similarly to the previous case, [EQUATION] Denote by [MATH] a player such that [EQUATION] Consider the following game: |
[EQUATION] where [MATH] [MATH] and [MATH] for every player [MATH] such that [MATH] . Here again all players but 1 and [MATH] are non-strategic, or alternatively every strategy they select has the same satisfaction level w.r.t. users [MATH] . We have the following cycle: |
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