Bounds for the query complexity of approximate equilibria

Goldberg, Paul W.; Roth, Aaron

doi:10.1145/2600057.2602845

Cited by 21 publications

(32 citation statements)

References 28 publications

(27 reference statements)

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“…The query complexity of equilibria of n-player games -a setting where payoff functions are exponentially-large -was analyzed in [4,8,21,22]. [22] showed that exponentially many deterministic queries are required to find a 1 2 -approximate correlated equilibrium (CE) and that any randomized algorithm that finds an exact CE needs 2 Ω(n) expected cost.…”

Section: Related Workmentioning

confidence: 99%

“…Notice that lower bounds on correlated equilibria automatically apply to Nash equilibria. [21] investigated in more detail the randomized query complexity of ǫ-CE and of the more demanding ǫ-wellsupported CE. [4] proved an exponential-in-n randomized lower bound for finding an ǫ-WSNE in n-player, k-strategy games, for constant k = 10 4 and ǫ = 10 −8 .…”

Section: Related Workmentioning

confidence: 99%

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Query complexity of approximate equilibria in anonymous games

Goldberg

Turchetta²

2017

Journal of Computer and System Sciences

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Abstract. We study the computation of equilibria of anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game's payoff function, assumed to be unknown initially. The general topic we consider is query complexity, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium. We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. However, more positive query-complexity bounds are attainable if either further symmetries of the utility functions are assumed or we focus on approximate equilibria. We investigate four sub-classes of anonymous games previously considered by [5,15]. Our main result is a new randomized query-efficient algorithm that finds a O(n −1/4 )-approximate Nash equilibrium queryingÕ(n 3/2 ) payoffs and runs in timeÕ(n 3/2 ). This improves on the running time of pre-existing algorithms for approximate equilibria of anonymous games, and is the first one to obtain an inverse polynomial approximation in poly-time. We also show how this can be utilized as an efficient polynomial-time approximation scheme (PTAS). Furthermore, we prove that Ω(n log n) payoffs must be queried in order to find any ǫ-well-supported Nash equilibrium, even by randomized algorithms.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Query complexity of approximate equilibria in anonymous games

Goldberg

Turchetta²

2017

Journal of Computer and System Sciences

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“…There are two results that we will use for this setting. Goldberg and Roth [14] have given a randomized algorithm that, with high probability, finds an -NE of a zerosum game using O( n•log n…”

Section: Lemma 1 ([13]mentioning

confidence: 99%

“…Hence, we obtain a randomized expectedpolynomial-time algorithm that uses poly-logarithmic communication and finds a The algorithm can also be used to beat the best known bound in the query complexity setting. It has already been shown by Goldberg and Roth [14] that an -NE of a zerosum game can be found by a randomized algorithm that uses O( n log n…”

Section: Introductionmentioning

confidence: 99%

Distributed Methods for Computing Approximate Equilibria

Czumaj

Deligkas

Fasoulakis

et al. 2016

Web and Internet Economics

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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 computes an approximate Nash equilibrium using only limited communication between the players. Our method gives improved bounds on the complexity of computing approximate Nash equilibria in a number of different settings. Firstly, it gives a polynomial-time algorithm for computing approximate well supported Nash equilibria (WSNE) that always finds a 0.6528-WSNE, beating the previous best guarantee of 0.6608. Secondly, since our algorithm solves the two LPs separately, it can be applied to give an improved bound in the limited communication setting, giving a randomized expectedpolynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE, which beats the previous best known guarantee of 0.732. It can also be applied to the case of approximate Nash equilibria, where we obtain a randomized Artur Czumaj, Michail Fasoulakis and Marcin Jurdziński were partially supported by the Centre for Discrete Mathematics and its Applications (DIMAP) and EPSRC Award EP/D063191/1. Argyrios Deligkas, John Fearnley and Rahul Savani were supported by EPSRC Award EP/L011018/1. Argyrios Deligkas was also supported by ISF Award #2021296.

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“…For randomised algorithms, query complexity is exponential for well-supported approximate equilibria [3], which has since been strengthened to any ε-Nash equilibria [5]. With randomised algorithms, the query complexity of approximate correlated equilibrium is (log n) for any positive ε [10].…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Query Complexity for Approximate Nash Computation in Large Games

Goldberg

Marmolejo-Cossío

2016

Lecture Notes in Computer Science

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We investigate the problem of equilibrium computation for "large" nplayer games. Large games have a Lipschitz-type property that no single player's utility is greatly affected by any other individual player's actions. In this paper, we mostly focus on the case where any change of strategy by a player causes other players' payoffs to change by at most 1 n. We study algorithms having query access to the game's payoff function, aiming to find ε-Nash equilibria. We seek algorithms that obtain ε as small as possible, in time polynomial in n. Our main result is a randomised algorithm that achieves ε approaching 1 8 for 2-strategy games in a completely uncoupled setting, where each player observes her own payoff to a query, and adjusts her behaviour independently of other players' payoffs/actions. O(log n) rounds/queries are required. We also show how to obtain a slight improvement over 1 8 , by introducing a small amount of communication between the players. Finally, we give extension of our results to large games with more than two strategies per player, and alternative largeness parameters.

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Bounds for the query complexity of approximate equilibria

Cited by 21 publications

References 28 publications

Query complexity of approximate equilibria in anonymous games

Query complexity of approximate equilibria in anonymous games

Distributed Methods for Computing Approximate Equilibria

Logarithmic Query Complexity for Approximate Nash Computation in Large Games

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