Strong Nash equilibria and mixed strategies

We study strong Nash equilibria in mixed strategies in finite games. A Nash equilibrium is strong if no coalition of players can jointly deviate so that all players in the coalition get strictly better payoffs. Our main result concerns games with two players and states that if a game admits a strong Nash equilibrium, then the payoff pairs in the support of the equilibrium lie on a straight line in the players’ utility space. As a consequence, the set of games that have a strong Nash equilibrium in which at least one player plays a mixed strategy has measure zero. We show that the same property holds for games with more than two players, already when no coalition of two players can profitably deviate. Furthermore, we show that, in contrast to games with two players, in a strong Nash equilibrium an outcome that is strictly Pareto dominated may occur with positive probability.

[1]  Paul R. Milgrom,et al.  Coalition-Proofness and Correlation with Arbitrary Communication Possibilities , 1996 .

[2]  Vincent Conitzer,et al.  New complexity results about Nash equilibria , 2008, Games Econ. Behav..

[3]  M. Dufwenberg.  Game theory. , 2011, Wiley interdisciplinary reviews. Cognitive science.

[4]  M. Whinston,et al.  Coalition-Proof Nash Equilibria I. Concepts , 1987 .

[5]  Paul W. Goldberg,et al.  The complexity of computing a Nash equilibrium , 2006, STOC '06.

[6]  Robert J. Aumann,et al.  16. Acceptable Points in General Cooperative n-Person Games , 1959 .

[7]  Robert Wilson,et al.  Direct proofs of generic finiteness of nash equilibrium outcomes , 2001 .

[8]  Tuomas Sandholm,et al.  On the verification and computation of strong nash equilibrium , 2013, AAMAS.

[9]  M. Coste.  AN INTRODUCTION TO O-MINIMAL GEOMETRY , 2002 .

[10]  Jérôme Monnot,et al.  On Strong Equilibria in the Max Cut Game , 2009, WINE.

[11]  Guoqiang Tian,et al.  On the existence of strong Nash equilibria , 2014 .

[12]  R. Holzman,et al.  Strong Equilibrium in Congestion Games , 1997 .

[13]  Kaisa Miettinen,et al.  Nonlinear multiobjective optimization , 1998, International series in operations research and management science.

[14]  Bodo Manthey,et al.  Smoothed Complexity Theory , 2012, MFCS.

[15]  David Banks,et al.  Games and Decisions , 2015 .

[16]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Xiaotie Deng,et al.  Settling the complexity of computing two-player Nash equilibria , 2007, JACM.

[18]  J. Marchal.  Cours d'economie politique , 1950 .

[19]  O. Rozenfeld.  Strong Equilibrium in Congestion Games , 2007 .

[20]  Yoav Shoham,et al.  Simple search methods for finding a Nash equilibrium , 2004, Games Econ. Behav..

[21]  Charles Gide,et al.  Cours d'économie politique , 1911 .

[22]  Christos H. Papadimitriou,et al.  On Total Functions, Existence Theorems and Computational Complexity , 1991, Theor. Comput. Sci..

[23]  Christos H. Papadimitriou,et al.  A Note on Strictly Competitive Games , 2009, WINE.

[24]  William A. Barnett,et al.  Social Choice, Welfare, and Ethics , 2006 .

[25]  Yishay Mansour,et al.  Strong price of anarchy , 2007, SODA '07.

[26]  Pradeep Dubey,et al.  Inefficiency of Nash Equilibria , 1986, Math. Oper. Res..