Journal of Optimization Theory and Applications

, Volume 178, Issue 3, pp 998–1013

# 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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