Polynomial Algorithms for Approximating Nash Equilibria of Bimatrix Games

Kontogiannis, Spyros; Panagopoulou, Panagiota N.; Spirakis, Paul G.

doi:10.1007/11944874_26

Cited by 52 publications

(35 citation statements)

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“…[13] made also a quite interesting connection of the problem of constructing 1+ε 2 -SuppNE in an arbitrary [0, 1]-bimatrix game, to that of constructing ε-SuppNE for a properly chosen win lose game of the same size. As for [21], based on linear programming techniques, they provided a 3 4 -ApproxNE, as well as a parameterized 2+λ 4 -ApproxNE for arbitrary [0, 1]-bimatrix games, where λ is the minimum payoff of a player at a NE of the game. Consequently, [18] provided a PTAS for ApproxNE in bimatrix games in which the sum of the two payoff matrices has fixed rank.…”

Section: Related Work and Contributionmentioning

confidence: 99%

“…Although (exact) NE are known not to be affected by any positive scaling, it is important to mention that approximate notions of NE are indeed affected. Therefore, from now on we adopt the commonly used assumption in the literature (e.g., [7,8,13,21,23]) that, when referring to ε-ApproxNE or ε-SuppNE, the bimatrix game is considered to be a [0, 1]-bimatrix game. This is mainly done for sake of comparison of the results on approximate equilibria.…”

Section: Game Theoretic Definitions and Notationmentioning

confidence: 99%

“…Namely, the first two results [13,21] recently made progress in the direction of providing the first ε-ApproxNE and ε-SuppNE for [0, 1]-bimatrix games and some constant 1 > ε > 0. In particular, [13] gave a nice and simple 1 2 -ApproxNE for [0, 1]-bimatrix games, involving only two strategies per player.…”

Section: Related Work and Contributionmentioning

confidence: 99%

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Well Supported Approximate Equilibria in Bimatrix Games

Kontogiannis

Spirakis

2008

Algorithmica

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In view of the apparent intractability of constructing Nash Equilibria (NE in short) in polynomial time, even for bimatrix games, understanding the limitations of the approximability of the problem is an important challenge.In this work we study the tractability of a notion of approximate equilibria in bimatrix games, called well supported approximate Nash Equilibria (SuppNE in short). Roughly speaking, while the typical notion of approximate NE demands that each player gets a payoff at least an additive term less than the best possible payoff, in a SuppNE each player is assumed to adopt with positive probability only approximate pure best responses to the opponent's strategy.As a first step, we demonstrate the existence of SuppNE with small supports and at the same time good quality. This is a simple corollary of Althöfer's Approximation Lemma, and implies a subexponential time algorithm for constructing SuppNE of arbitrary (constant) precision.We then propose algorithms for constructing SuppNE in win lose and normalized bimatrix games (i.e., whose payoff matrices take values from {0, 1} and [0, 1] respectively). Our methodology for attacking the problem is based on the solvability of 654 Algorithmica (2010) 57: 653-667 zero sum bimatrix games (via its connection to linear programming) and provides a 0.5-SuppNE for win lose games and a 0.667-SuppNE for normalized games.To our knowledge, this paper provides the first polynomial time algorithms constructing ε-SuppNE for normalized or win lose bimatrix games, for any nontrivial constant 0 ≤ ε < 1, bounded away from 1.

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Section: Related Work and Contributionmentioning

confidence: 99%

Section: Game Theoretic Definitions and Notationmentioning

confidence: 99%

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Well Supported Approximate Equilibria in Bimatrix Games

Kontogiannis

Spirakis

2008

Algorithmica

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“…Now, this equivalence does not however guarantee polynomial-time (polynomial in n) computation of equilibrium policies since the number of deterministic policies in the equivalent game is super-exponential in n in our case, which results in super-exponential number of variables and constraints of the above linear programs. Note that computational game theory focuses on determining exact solutions (e.g., for saddlepoints of two-person zero-sum games Chapter III.2.4, [6])) whenever such solutions are computationally tractable, or approximations otherwise (e.g., for Nash equilibrium of bi-matrix games [4], [7]), using computation times that are polynomial in the number of deterministic policies of the players. Thus, since the number of deterministic policies is super-exponential in n in our case, standard algorithms will have computation times that are again super-exponential in n. To the best of our knowledge, standard algorithms for fast computation of exact solutions or approximations when the number of policies of the players is itself intractable (e.g., super-exponential) are not available in the literature.…”

Section: Related Literaturementioning

confidence: 99%

Information Concealing Games

Sarkar¹,

Altman²,

El-Azouzi³

et al. 2008

2008 Proceedings IEEE INFOCOM - The 27th Conference on Computer Communications

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We consider a system whose state is a vector of dimension n, whose value is chosen randomly by nature. The system consists of two entities. The first entity (controller) has complete information about the state of the system, and must reveal a certain "minimum" amount of information about the system state to the second entity. It can however choose the nature of the information it reveals subject to satisfying the above constraint. The second entity (actor) takes certain actions based on the information the controller reveals, and the actions are associated with certain utilities for both the controller and the actor which also depend on the state of the system. The controller needs to decide the information it would reveal, or equivalently conceal, so as to maximize its own utility, and the actor needs to determine its actions based on the information the controller reveals so as to again maximize its utility.We demonstrate that the above problem forms the basis of several technical and social systems.We show that the decision problems for the entities can be formulated as a signaling game. The Perfect Bayesian Equilibrium (PBE) for this game exhibits several counter intuitive properties, e.g., some intuitively appealing greedy policies for the controller and the actor turn out to be suboptimal.We prove that the PBE of this game can be obtained as a saddle point of a different two person zero sum game. The number of policies of the players in this two person zero sum game is however superexponential in n, which implies that standard linear programs for obtaining its saddle points will be computationally intractable even for moderate n. Next, using specific characteristics of the problem, we develop linear programs that compute the optimal policies using a computation time that increases exponentially with n, and can therefore be numerically solved for moderate n. We finally propose simple linear time computable policies that approximate the optimal policies within guaranteeable approximation ratios.

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“…A positive outcome of this investigation would be useful for applications since it would provide algorithmic tools for computing approximate equilibria; but, most importantly, it would alleviate the negative implications of the aforementioned hardness results to the predictive power of the Nash equilibrium concept. Unfortunately, since the appearance of the original hardness results, and despite considerable effort in providing upper [20,9,10,21,15,3,27,28] and lower [12,18] bounds for the approximation problem, the approximation complexity of the Nash equilibrium has remained unknown. This paper obtains the first constant inapproximability results for the problem.…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Approximating a Nash Equilibrium

Daskalakis¹

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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We show that computing a relative-that is, multiplicative as opposed to additive-approximate Nash equilibrium in two-player games is PPAD-complete, even for constant values of the approximation. Our result is the first constant inapproximability result for the problem, since the appearance of the original results on the complexity of the Nash equilibrium [8,5,7]. Moreover, it provides an apparent-assuming that PPAD ⊆ TIME(n O(log n) )-dichotomy between the complexities of additive and relative notions of approximation, since for constant values of additive approximation a quasipolynomial-time algorithm is known [22]. Such a dichotomy does not arise for values of the approximation that scale with the size of the game, as both relative and additive approximations are PPAD-complete [7]. As a byproduct, our proof shows that the Lipton-MarkakisMehta sampling lemma is not applicable to relative notions of constant approximation, answering in the negative direction a question posed to us by Shang-Hua Teng [26].

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Polynomial Algorithms for Approximating Nash Equilibria of Bimatrix Games

Cited by 52 publications

References 10 publications

Well Supported Approximate Equilibria in Bimatrix Games

Well Supported Approximate Equilibria in Bimatrix Games

Information Concealing Games

On the Complexity of Approximating a Nash Equilibrium

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