Well Supported Approximate Equilibria in Bimatrix Games

Kontogiannis, Spyros; Spirakis, Paul G.

doi:10.1007/s00453-008-9227-6

Cited by 39 publications

(36 citation statements)

References 29 publications

(58 reference statements)

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“…On the other hand, progress on computing approximate-well-supported Nash equilibria has been less forthcoming. The first correct algorithm was provided by Kontogiannis and Spirakis [14] (which shall henceforth be referred to as the KS algorithm), who gave a polynomial time algorithm for finding a 2 3 -WSNE. This was later slightly improved by Fearnley, Goldberg, Savani, and Sørensen [8] (whose algorithm we shall refer to as the FGSS-algorithm), who showed that the WSNEs provided by the KS algorithm could be improved, and this yields a polynomial time algorithm for finding a 0.6608-WSNE; this is the best approximation guarantee for WSNEs that is currently known.…”

Section: Introductionmentioning

confidence: 99%

Distributed Methods for Computing Approximate Equilibria

et al. 2018

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We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then compute approximate Nash equilibria using only limited communication between the players. Our method has several applications for improved bounds for efficient computations of approximate Nash equilibria in bimatrix games. First, it yields a best polynomial-time algorithm for computing approximate wellsupported Nash equilibria (WSNE), which guarantees to find a 0.6528-WSNE in polynomial time. Furthermore, since our algorithm solves the two LPs separately, it can be used to improve upon the best known algorithms in the limited communication setting: the algorithm can be implemented to obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE. The algorithm can also be carried out to beat the best known bound in the query complexity setting, requiring O(n log n) payoff queries to compute a 0.6528-WSNE. Finally, our approach can also be adapted to provide the best known communication efficient algorithm for computing approximate Nash equilibria: it uses poly-logarithmic communication to find a 0.382-approximate Nash equilibrium.Communication complexity of equilibria in games has been studied in previous works [3,13]. The recent paper of Goldberg and Pastink [11] initiated the study of communication complexity in the bimatrix game setting. There they showed Θ(n 2 ) communication is required to find an exact Nash equilibrium of an n × n bimatrix game. Since these games have Θ(n 2 ) payoffs in total, this implies that there is no communication efficient protocol for finding exact Nash equilibria in bimatrix games. For approximate equilibria, they showed that one can find a 3 4 -Nash equilibrium without any communication, and that in the nocommunication setting, finding an 1 2 -Nash equilibrium is impossible. Motivated by these positive and negative results, they focused on the most interesting setting, which allows only a polylogarithmic (in n) amount of communication (number of bits) between the players. They demonstrated that one can compute 0.438-Nash equilibria and 0.732-well-supported Nash equilibria in this setting.

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Section: Introductionmentioning

confidence: 99%

Distributed Methods for Computing Approximate Equilibria

et al. 2018

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“…-WSCE is a refinement of approximate CE that is analogous to well-supported approximate Nash equilibrium, studied in [Kontogiannis and Spirakis 2010;Fearnley et al 2012;Goldberg and Pastink 2014;Babichenko 2013]. In an -Nash equilibrium ( -NE), a player's payoff is allowed to be up to worse than his best response.…”

Section: Related Workmentioning

confidence: 99%

Bounds for the query complexity of approximate equilibria

Goldberg

Roth

2014

Proceedings of the Fifteenth ACM Conference on Economics and Computation

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We analyze the number of payoff queries needed to compute approximate equilibria of multi-player games. We find that query complexity is an effective tool for distinguishing the computational difficulty of alternative solution concepts, and we develop new techniques for upper-and lower bounding the query complexity. For binary-choice games, we show logarithmic upper and lower bounds on the query complexity of approximate correlated equilibrium. For well-supported approximate correlated equilibrium (a restriction where a player's behavior must always be approximately optimal, in the worst case over draws from the distribution) we show a linear lower bound, thus separating the query complexity of well supported approximate correlated equilibrium from the standard notion of approximate correlated equilibrium.Finally, we give a query-efficient reduction from the problem of computing an approximate well-supported Nash equilibrium to the problem of verifying a well supported Nash equilibrium, where the additional query overhead is proportional to the description length of the game. This gives a polynomial-query algorithm for computing well supported approximate Nash equilibria (and hence correlated equilibria) in concisely represented games. We identify a class of games (which includes congestion games) in which the reduction can be made not only query efficient, but also computationally efficient. PRELIMINARIESThis paper compares the query complexity of alternative game-theoretic solution concepts. Instead of a game G being presented in its entirety as input to an algorithm A, we assume that A may submit queries consisting of strategy profiles, and get told the resulting payoffs to the players in G. This model is appealing when G has many players, in which case a naive representation of G would be exponentially-large. Assuming G belongs to a known class of games G, this gives rise to the question of how many queries are needed to find a solution, such as exact/approximate Nash/correlated equilibrium. One can study this question is conjunction with other notions of cost, such as runtime of the algorithm. An appealing aspect of query complexity is that it allows new upper and lower bounds to be obtained, providing a mathematical criterion to distinguish the difficulty of alternative solution concepts, as discussed in more detail below.We consider queries that consist of pure-strategy profiles, and in which the answer to any query is the payoffs that all the players receive from that profile. In this paper

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“…In this context, our 0· 732-approximation algorithm substantially improves on the result of [9], both in terms of approximation quality and a more demanding model (communication-bounded algorithms). However, we do not know how to obtain the better approximation quality of [25,12] in the communicationbounded setting. Next we discuss two of the earlier algorithms in the literature whose ideas we use here.…”

Section: Algorithms For Approximate Equilibriamentioning

confidence: 99%

“…The more demanding criterion of well-supported ǫ-Nash equilibrium, disallows a player from allocating positive probability to any pure strategy whose payoff is more than ǫ worse than the best response. Progress on polynomialtime algorithms for this solution concept has been more limited; at this time the lowest ǫ that can be guaranteed by a polynomial-time algorithm is only slightly less than 2 3 [12], obtained via a modification of a 2 3 -approximation algorithm of Kontogiannis and Spirakis [25]. Prior to that, [9] gave a 5 6 -approximation algorithm, that is contingent on a graph-theoretic conjecture.…”

Section: Algorithms For Approximate Equilibriamentioning

confidence: 99%

On the communication complexity of approximate Nash equilibria

Goldberg

Pastink

2014

Games and Economic Behavior

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We study the problem of computing approximate Nash equilibria of bimatrix games, in a setting where players initially know their own payoffs but not the payoffs of the other player. In order for a solution of reasonable quality to be found, some amount of communication needs to take place between the players. We are interested in algorithms where the communication is substantially less than the contents of a payoff matrix, for example logarithmic in the size of the matrix. When the communication is polylogarithmic in the number of strategies n, we show how to obtain ǫ-approximate Nash equilibrium for ǫ approximately 0· 438, and for well-supported approximate equilibria we obtain ǫ approximately 0· 732. For one-way communication we show that ǫ = 1 2 is achievable, but no constant improvement over 1 2 is possible, even with unlimited one-way communication. For well-supported equilibria, no value of ǫ < 1 is achievable with one-way communication. When the players do not communicate at all, ǫ-Nash equilibria can be obtained for ǫ = 3 4 , and we also give a lower bound of slightly more than 1 2 on the lowest constant ǫ achievable.

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Well Supported Approximate Equilibria in Bimatrix Games

Cited by 39 publications

References 29 publications

Distributed Methods for Computing Approximate Equilibria

Distributed Methods for Computing Approximate Equilibria

Bounds for the query complexity of approximate equilibria

On the communication complexity of approximate Nash equilibria

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