Approximation algorithms for indefinite quadratic programming
Mathematical Programming: Series A and B
Playing large games using simple strategies
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Exponentially Many Steps for Finding a Nash Equilibrium in a Bimatrix Game
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Settling the Complexity of Two-Player Nash Equilibrium
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Computing Nash Equilibria: Approximation and Smoothed Complexity
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Progress in approximate nash equilibria
Proceedings of the 8th ACM conference on Electronic commerce
Efficient computation of nash equilibria for very sparse win-lose bimatrix games
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
On the convergence of regret minimization dynamics in concave games
Proceedings of the forty-first annual ACM symposium on Theory of computing
Complexity results about Nash equilibria
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
The Complexity of Computing a Nash Equilibrium
SIAM Journal on Computing
New algorithms for approximate Nash equilibria in bimatrix games
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
An optimization approach for approximate Nash equilibria
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
Practical and efficient approximations of nash equilibria for win-lose games based on graph spectra
WINE'10 Proceedings of the 6th international conference on Internet and network economics
Rank-1 bimatrix games: a homeomorphism and a polynomial time algorithm
Proceedings of the forty-third annual ACM symposium on Theory of computing
Polynomial algorithms for approximating nash equilibria of bimatrix games
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
A note on approximate nash equilibria
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
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We study the fundamental problem 2NASH of computing a Nash equilibrium (NE) point in bimatrix games. We start by proposing a novel characterization of the NE set, via a bijective map to the solution set of a parameterized quadratic program (NEQP), whose feasible space is the highly structured set of correlated equilibria (CE). This is, to our knowledge, the first characterization of the subset of CE points that are in ''1-1'' correspondence with the NE set of the game, and contributes to the quite lively discussion on the relation between the spaces of CE and NE points in a bimatrix game (e.g., [15,26,33]). We proceed with studying a property of bimatrix games, which we call mutually concavity (MC), that assures polynomial-time tractability of 2NASH, due to the convexity of a proper parameterized quadratic program (either NEQP, or a parameterized variant of the Mangasarian & Stone formulation [23]) for a particular value of the parameter. We prove various characterizations of the MC-games, which eventually lead us to the conclusion that this class is equivalent to the class of strategically zero-sum (SZS) games of Moulin & Vial [25]. This gives an alternative explanation of the polynomial-time tractability of 2NASH for these games, not depending on the solvability of zero-sum games. Moreover, the recognition of the MC-property for an arbitrary game is much faster than the recognition SZS-property. This, along with the comparable time-complexity of linear programs and convex quadratic programs, leads us to a much faster algorithm for 2NASH in MC-games. We conclude our discussion with a comparison of MC-games (or, SZS-games) to k-rank games, which are known to admit for 2NASH a FPTAS when k is fixed [18], and a polynomial-time algorithm for k=1[2]. We finally explore some closeness properties under well-known NE set preserving transformations of bimatrix games.