An equivalent mathematical program for games with random constraints

Abstract

We consider an n-player chance-constrained game under elliptically symmetric distributions. For a confidence level greater than 0.5 and certain class of payoff functions and strategy sets, we suitably construct an equivalent mathematical program whose global maximizer is a Nash equilibrium.

Random Games Under Elliptically Distributed Dependent Joint Chance Constraints

“…However, the strategy sets containing chance constraints are often considered in various applications, e.g., risk constraints in portfolio optimization problem [11] and resource constraints in stochastic shortest path problem [4]. Recently, the games with chance constraint based strategy sets are introduced in the literature [18,19,20,26,27]. Singh and Lisser [26] considered a 2-player zero-sum game with individual chance constraints and showed that a saddle point equilibrium problem is equivalent to a primaldual pair of second order cone programs when the random constraint vectors follow elliptically symmetric distribution.…”

“…Singh and Lisser [26] considered a 2-player zero-sum game with individual chance constraints and showed that a saddle point equilibrium problem is equivalent to a primaldual pair of second order cone programs when the random constraint vectors follow elliptically symmetric distribution. Singh et al [27] considered an nplayer general-sum game with individual chance constraints under elliptically symmetric distributions and showed that a Nash equilibrium problem is equivalent to the global optimization of a nonlinear optimization problem. In the wake of these results, Peng et al [19] showed the existence of Nash equilibrium for the n-player general-sum games where the strategy profile set of each player is defined by a joint chance constraint, and the random constraint vectors are either independently normally distributed or follow a mixture of elliptical dis-tributions [20].…”