Approximate Nash equilibria in anonymous games

Journal of Economic Theory ◽

10.1016/j.jet.2014.02.002 ◽

2015 ◽

Vol 156 ◽

pp. 207-245 ◽

Cited By ~ 11

Author(s):

Constantinos Daskalakis ◽

Christos H. Papadimitriou

Keyword(s):

Nash Equilibria ◽

Approximate Nash Equilibria ◽

Anonymous Games

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An Optimization Approach for Approximate Nash Equilibria

Internet Mathematics ◽

10.1080/15427951.2008.10129172 ◽

2008 ◽

Vol 5 (4) ◽

pp. 365-382 ◽

Cited By ~ 31

Author(s):

Haralampos Tsaknakis ◽

Paul G. Spirakis

Keyword(s):

Nash Equilibria ◽

Optimization Approach ◽

Approximate Nash Equilibria

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Semidefinite Programming and Nash Equilibria in Bimatrix Games

INFORMS Journal on Computing ◽

10.1287/ijoc.2020.0960 ◽

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.

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