A Characterization of Nash Equilibrium for the Games with Random Payoffs

作者：Singh Vikas Vikram^*; Lisser Abdel

来源：Journal of Optimization Theory and Applications, 2018, 178(3): 998-1013.

DOI：10.1007/s10957-018-1343-0

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

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.

• 出版日期2018-9