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Journal of Optimization Theory and Applications

, Volume 178, Issue 3, pp 998–1013 | Cite as

A Characterization of Nash Equilibrium for the Games with Random Payoffs

  • Vikas Vikram Singh
  • Abdel Lisser
Article

Abstract

We consider a two-player random bimatrix game where each player is interested in the payoffs which can be obtained with certain confidence. The payoff function of each player is defined using a chance constraint. We consider the case where the entries of the random payoff matrix of each player jointly follow a multivariate elliptically symmetric distribution. We show an equivalence between the Nash equilibrium problem and the global maximization of a certain mathematical program. The case where the entries of the payoff matrices are independent normal/Cauchy random variables is also considered. The case of independent normally distributed random payoffs can be viewed as a special case of a multivariate elliptically symmetric distributed random payoffs. As for Cauchy distribution, we show that the Nash equilibrium problem is equivalent to the global maximization of a certain quadratic program. Our theoretical results are illustrated by considering randomly generated instances of the game.

Keywords

Chance-constrained games Nash equilibrium Elliptically symmetric distribution Cauchy distribution Mathematical program Quadratic program 

Mathematics Subject Classification

91A10 90C15 90C20 90C26 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia
  2. 2.Laboratoire de Recherche en InformatiqueUniversité Paris SudOrsayFrance

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