Polynomial Algorithms for Approximating Nash Equilibria of Bimatrix Games

2006 ◽  
pp. 286-296 ◽  
Author(s):  
Spyros C. Kontogiannis ◽  
Panagiota N. Panagopoulou ◽  
Paul G. Spirakis
Keyword(s):  
Nash Equilibria ◽  
Polynomial Algorithms ◽  
Bimatrix Games

scholarly journals Polynomial algorithms for approximating Nash equilibria of bimatrix games

2009 ◽  
Vol 410 (17) ◽  
pp. 1599-1606 ◽  
Author(s):  
Spyros C. Kontogiannis ◽  
Panagiota N. Panagopoulou ◽  
Paul G. Spirakis
Keyword(s):  
Nash Equilibria ◽  
Polynomial Algorithms ◽  
Bimatrix Games

scholarly journals Semidefinite Programming and Nash Equilibria in Bimatrix Games

2020 ◽  
Author(s):  
Amir Ali Ahmadi ◽  
Jeffrey Zhang
Keyword(s):  
Nash Equilibrium ◽  
Semidefinite Programming ◽  
Nash Equilibria ◽  
Valid Inequalities ◽  
Bimatrix Games ◽  
Competitive Game ◽  
Approximate Nash Equilibria ◽  
Hard Problems ◽  
Rank 2

We explore the power of semidefinite programming (SDP) for finding additive ɛ-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact Nash equilibrium can be recovered. We show that, for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on the iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most 2 and ɛ close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a [Formula: see text]-Nash equilibrium can be recovered for any game, or a [Formula: see text]-Nash equilibrium for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any Nash equilibrium, and testing whether there exists a Nash equilibrium where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.

scholarly journals On the computational complexity of Nash equilibria for bimatrix games

2005 ◽  
Vol 94 (3) ◽  
pp. 145-150 ◽  
Author(s):  
Bruno Codenotti ◽  
Daniel Štefankovič
Keyword(s):  
Computational Complexity ◽  
Nash Equilibria ◽  
Bimatrix Games

scholarly journals New algorithms for approximate Nash equilibria in bimatrix games

2010 ◽  
Vol 411 (1) ◽  
pp. 164-173 ◽  
Author(s):  
Hartwig Bosse ◽  
Jaroslaw Byrka ◽  
Evangelos Markakis
Keyword(s):  
Nash Equilibria ◽  
Bimatrix Games ◽  
Approximate Nash Equilibria ◽  
New Algorithms
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