The complexity of computing a Nash equilibrium

Daskalakis, Constantinos; Goldberg, Paul W.; Papadimitriou, Christos H.

doi:10.1145/1461928.1461951

Cited by 583 publications

(911 citation statements)

References 17 publications

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“…Furthermore, these mixed strategies can be easily computed with linear programming. In contrast, we now know that Nash equilibria are hard to compute in general, even for two-person non-zerosum games [DGP06,CDT06]-and consequently for three-person zero-sum games. Von Neumann's minmax theorem [Neu28] seems to have very narrow applicability.…”

Section: Introductionmentioning

confidence: 99%

Zero-Sum Polymatrix Games: A Generalization of Minmax

et al. 2016

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We show that in zero-sum polymatrix games, a multiplayer generalization of two-person zerosum games, Nash equilibria can be found efficiently with linear programming. We also show that the set of coarse correlated equilibria collapses to the set of Nash equilibria. In contrast, other important properties of two-person zero-sum games are not preserved: Nash equilibrium payoffs need not be unique, and Nash equilibrium strategies need not be exchangeable or max-min.

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Section: Introductionmentioning

confidence: 99%

Zero-Sum Polymatrix Games: A Generalization of Minmax

et al. 2016

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“…Unfortunately, the complexity results in this area have been almost uniformly negative. A series of papers has shown that it is PPAD complete to compute Nash equilibria, even in 2 player games, even when payoffs are restricted to lie in {0, 1} [7,1,6].…”

Section: Introductionmentioning

confidence: 99%

On Nash-Equilibria of Approximation-Stable Games

Awasthi¹,

Balcan²,

Blum³

et al. 2010

Algorithmic Game Theory

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Abstract. One reason for wanting to compute an (approximate) Nash equilibrium of a game is to predict how players will play. However, if the game has multiple equilibria that are far apart, or ǫ-equilibria that are far in variation distance from the true Nash equilibrium strategies, then this prediction may not be possible even in principle. Motivated by this consideration, in this paper we define the notion of games that are approximation stable, meaning that all ǫ-approximate equilibria are contained inside a small ball of radius ∆ around a true equilibrium, and investigate a number of their properties. Many natural small games such as matching pennies and rock-paper-scissors are indeed approximation stable. We show furthermore there exist 2-player n-by-n approximationstable games in which the Nash equilibrium and all approximate equilibria have support Ω(log n). On the other hand, we show all (ǫ, ∆) approximation-stable games must have an ǫ-equilibrium of support O(log n) -time algorithm, improving over the bound of [11] for games satisfying this condition. We in addition give a polynomial-time algorithm for the case that ∆ and ǫ are sufficiently close together. We also consider an inverse property, namely that all non-approximate equilibria are far from some true equilibrium, and give an efficient algorithm for games satisfying that condition.

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“…A stronger notion of approximation was introduced in [4], namely -well-supported equilibria. We do not consider this approximation concept here.…”

Section: Notation and Definitionsmentioning

confidence: 99%

“…In a series of works [8,4,2], it was established that computing a Nash equilibrium is PPAD-complete even for two-player games. The focus has since then been on algorithms for approximate equilibria.…”

Section: Introductionmentioning

confidence: 99%

New Algorithms for Approximate Nash Equilibria in Bimatrix Games

Bosse

Byrka

Markakis³

Lecture Notes in Computer Science

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Abstract. We consider the problem of computing additively approximate Nash equilibria in non-cooperative two-player games. We provide a new polynomial time algorithm that achieves an approximation guarantee of 0.36392. Our work improves the previously best known (0.38197 + )-approximation algorithm of Daskalakis, Mehta and Papadimitriou [6]. First, we provide a simpler algorithm, which also achieves 0.38197. This algorithm is then tuned, improving the approximation error to 0.36392. Our method is relatively fast, as it requires solving only one linear program and it is based on using the solution of an auxiliary zero-sum game as a starting point.

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The complexity of computing a Nash equilibrium

Cited by 583 publications

References 17 publications

Zero-Sum Polymatrix Games: A Generalization of Minmax

Zero-Sum Polymatrix Games: A Generalization of Minmax

On Nash-Equilibria of Approximation-Stable Games

New Algorithms for Approximate Nash Equilibria in Bimatrix Games

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