A Characterization of Nash Equilibrium for the Games with Random Payoffs

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.

Appendix: Proof of Theorem 3.2

Proof

Fix $\alpha \in [0.5,1[^2$. To show that $(x^*,y^*)$ defined by (16) is a Nash equilibrium, it is sufficient to show that there exists a vector $(\lambda _1^*, \lambda _2^*,v_1^*,v_2^*)$ which together with $(x^*,y^*)$ is a feasible point of [MP] with objective function value zero (see Theorem 3.1). By using i.i.d. property, we have $\varSigma _1^{1/2}=\sigma I_{mn\times mn}$, $\varSigma _2^{1/2}=\bar{\sigma }I_{mn\times mn}$. Take,

$$\begin{aligned} \left. \begin{aligned} \lambda _1^*&=-\mu -\frac{\sigma ~ F_{Z_1^S}^{-1}(1-\alpha _1)}{\sqrt{mn}},\\ \lambda _2^*&=-\bar{\mu }-\frac{\bar{\sigma } ~ F_{Z_2^S}^{-1}(1-\alpha _2)}{\sqrt{mn}},\\ v_1^*&=\frac{1}{\sqrt{mn}}{} \mathbf 1 _{mn},\\ v_2^*&=\frac{1}{\sqrt{mn}}{} \mathbf 1 _{mn}, \end{aligned} \right\} \end{aligned}$$

where $\mathbf 1 _k$ denotes a $k\times 1$ vector of ones. It is easy to check that $\zeta ^*=(\lambda _1^*, \lambda _2^*,v_1^*,v_2^*,x^*,y^*)$ is a feasible point of [MP] and $\psi (\zeta ^*)=0$. Hence, $(x^*,y^*)$ defined by (16) is a Nash equilibrium of a CCG. $\square$