A Mixed Cooperative Dual to the Nash Equilibrium

H W Corley

doi:10.1155/2015/647246

Abstract

A mixed dual to the Nash equilibrium is defined for n-person games in strategic form. In a Nash equilibrium every player’s mixed strategy maximizes his own expected payoff for the other n-1 players’ strategies. Conversely, in the dual equilibrium every n-1 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 n>2 there exists a game for which no dual equilibrium exists.

Highlights

The mathematical analysis of both competition and cooperation falls within the realm of game theory, whose systematic development began with von Neumann and Morgenstern [1]
The Berge equilibrium has been extended to a mixed dual equilibrium for the Nash equilibrium
Relabeling each player in a Definition 2 (DE) yields an Nash equilibrium (NE) for the original payoff matrix, from which it follows that a game has at least as many NEs as DEs

Summary

Introduction

The mathematical analysis of both competition and cooperation falls within the realm of game theory, whose systematic development began with von Neumann and Morgenstern [1]. For a game with n players, Nash [2] later assumed that the players are rational and selfish He defined an equilibrium in which every player’s strategy maximizes his payoff for the other n − 1 players’ strategies. A Berge equilibrium is a pure strategy profile in which every n − 1 players have strategies that maximize the remaining player’s payoff It has been increasingly studied as a model of mutual support and cooperation as in [10,11,12,13,14,15]. In this paper we consider n-person games in strategic form and extend the Berge equilibrium from pure to mixed strategies to provide a dual to the Nash equilibrium In this dual equilibrium, every n−1 players have strategies that maximize the remaining player’s expected payoff.

Relationships between the Dual and Nash Equilibria

Existence Conditions for Dual Equilibria

Computational Complexity

Result

Mutual Cooperation Impossibility Theorem

Conclusions