A Characterization of Nash Equilibrium for the Games with Random Payoffs
- 95 Downloads
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.
KeywordsChance-constrained games Nash equilibrium Elliptically symmetric distribution Cauchy distribution Mathematical program Quadratic program
Mathematics Subject Classification91A10 90C15 90C20 90C26
- 5.Li, D.-F., Nan, J.X., Zhang, M.J.: Interval programming models for matrix games with interval payoffs. Optim. Methods Softw. 27, 1–16 (2012)Google Scholar
- 26.Song, T.: Systems and Management Science by Extremal Methods, chap. On random payoff matrix games, pp. 291–308. Springer, Berlin (1992)Google Scholar
- 28.R.T, Rockafellar: Convex Analysis. Princeton University Press, Princeton (1970)Google Scholar