Matrix and Bimatrix Games

1987 ◽  
pp. 45-61
Author(s):  
Eric van Damme
Keyword(s):  
Bimatrix Games

scholarly journals Pareto equilibria for bimatrix games

1993 ◽  
Vol 25 (10-11) ◽  
pp. 19-25 ◽  
Author(s):  
P.E.M. Borm ◽  
M.J.M. Jansen ◽  
J.A.M. Potters ◽  
S.H. Tijs
Keyword(s):  
Bimatrix Games ◽  
Pareto Equilibria

Dynamics for Bimatrix Games Via Analytic Centers

2020 ◽  
pp. 129-138
Author(s):  
Arkadii V. Kryazhimskii ◽  
György Sonnevend
Keyword(s):  
Bimatrix Games ◽  
Analytic Centers

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 A note on strategy elimination in bimatrix games

1988 ◽  
Vol 7 (3) ◽  
pp. 103-107 ◽  
Author(s):  
Donald E. Knuth ◽  
Christos H. Papadimitriou ◽  
John N. Tsitsiklis
Keyword(s):  
Bimatrix Games

scholarly journals Games of fixed rank: a hierarchy of bimatrix games

2009 ◽  
Vol 42 (1) ◽  
pp. 157-173 ◽  
Author(s):  
Ravi Kannan ◽  
Thorsten Theobald
Keyword(s):  
Bimatrix Games ◽  
Fixed Rank
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