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Game Theory
Volume 2015 (2015), Article ID 647246, 7 pages
Research Article

A Mixed Cooperative Dual to the Nash Equilibrium

Center on Stochastic Modeling, Optimization, & Statistics (COSMOS), The University of Texas at Arlington, P.O. Box 19017, Arlington, TX 76019, USA

Received 12 May 2015; Accepted 30 July 2015

Academic Editor: Azhar Iqbal

Copyright © 2015 H. W. Corley. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A mixed dual to the Nash equilibrium is defined for -person games in strategic form. In a Nash equilibrium every player’s mixed strategy maximizes his own expected payoff for the other players’ strategies. Conversely, in the dual equilibrium every players have mixed strategies that maximize the remaining player’s expected payoff. Hence this dual equilibrium models mutual support and cooperation to extend the Berge equilibrium from pure to mixed strategies. This dual equilibrium is compared and related to the mixed Nash equilibrium, and both topological and algebraic conditions are given for the existence of the dual. Computational issues are discussed, and it is shown that for each there exists a game for which no dual equilibrium exists.