Approximate Nash Equilibria in Bimatrix Games.
ABSTRACT Nash equilibrium is one of the main concepts in the game theory. Recently it was shown, that problem of finding Nash equilibrium and an approximate Nash equilibrium is PPAD-complete. In this article we adapt Differential Evolution algorithm (DE) to the above problem. It may be classified as continuous problem, where two probability distributions over the set of pure strategies of both players should be found. Every deviation from the global optimum is interpreted as Nash approximation and called ε-Nash equilibrium. We show, that the Differential Evolution approach can be determined as iterative method, which in successive iterations is capable to obtain ε value close to the global optimum. The contribution of this paper is the experimental analysis of the proposed approach and indication of it's strong features. We try to demonstrate, that the proposed method is very good alternative for the existing mathematical analysis of the mentioned Nash equilibrium problem.
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Conference Paper: On the Equilibria of Alternating Move Games.[Show abstract] [Hide abstract]
ABSTRACT: We consider computational aspects of alternating move games, repeated games in which players take actions at alternating time steps rather than playing simultaneously. We show that alternating move games are more tractable than simultaneous move games: we give an FPTAS for computing an ε-approximate equilibrium of an alternating move game with any number of players. In contrast, it is known that for k ≥ 3 players, there is no FPTAS for computing Nash equilibria of simultaneous move repeated games unless P = PPAD. We also consider equilibria in memoryless strategies, which are guaranteed to exist in two player games. We show that for the special case of k = 2 players, all but a negligible fraction of games admit an equilibrium in pure memoryless strategies that can be found in polynomial time. Moreover, we give a PTAS to compute an ε-approximate equilibrium in pure memoryless strategies in any 2 player game that admits an exact equilibrium in pure memoryless strategies.Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010; 01/2010
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ABSTRACT: We study the computation of Nash equilibria in a two-player normal form game from the perspective of parameterized complexity. Recent results proved hardness for a number of variants, when parameterized by the support size. We complement those results, by identifying three cases in which the problem becomes fixed-parameter tractable. These cases occur in the previously studied settings of sparse games and unbalanced games as well as in the newly considered case of locally bounded treewidth games that generalizes both these two cases.Algorithmica 06/2010; 65(4). DOI:10.1007/s00453-011-9609-z · 0.57 Impact Factor
Conference Paper: Differential evolution as a new method of computing nash equilibria[Show abstract] [Hide abstract]
ABSTRACT: The Nash equilibrium is one of the central concepts in game theory. Recently it was shown, that problems of finding Nash equilibrium and an approximate Nash equilibrium are PPAD-complete. In this article we present the Differential Evolution algorithm adapted to that problem, and we compare it with two well-known algorithms: the Simplicial Subdivision and the Lemke-Howson. The problem of finding the Nash equilibrium for two players games may be classified as a continuous problem, where two probability distributions over the set of pure strategies of both players should be found. Each deviation from the global optimum is interpreted as the Nash approximation and called the ε-Nash equilibrium. We show that the Differential Evolution approach can be determined as a method, which in successive iterations is capable of obtaining ε value close to the global optimum. We show, that the Differential Evolution may be succesfully used to obtain satisfactory results and it may be easily expanded into n-person games. Moreover, we present results for the problem of computing Nash equilibrium, when some arbitrary set strategies have a non-zero probability of being chosen.Transactions on Computational Collective Intelligence IX; 01/2013