Chance-constrained games with mixture distributions

Abstract

In this paper, we consider an n-player non-cooperative game where the random payoff function of each player is defined by its expected value and her strategy set is defined by a joint chance constraint. The random constraint vectors are independent. We consider the case when the probability distribution of each random constraint vector belongs to a subset of elliptical distributions as well as the case when it is a finite mixture of the probability distributions from the subset. We propose a convex reformulation for the joint chance constraint of each player and derive the bounds for players' confidence levels and the weights used in the mixture distributions. Under mild conditions on the players' payoff functions, we show that there exists a Nash equilibrium of the game when the players' confidence levels and the weights used in the mixture distributions are within the derived bounds. As an application of these games, we consider the competition between two investment firms on the same set of portfolios. We use a best response algorithm to compute a Nash equilibrium of the randomly generated games of different sizes.

Random Games Under Elliptically Distributed Dependent Joint Chance Constraints

“…In this paper, we extend the results of [19,20] to the general case where the payoff function is random and the strategy profile set of each player is defined by elliptically distributed dependent joint chance constraints. We derive a new reformulation of joint chance constraint with dependent random constraint vectors and show that there exists a Nash equilibrium of the game under mild conditions on the payoff functions.…”

“…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 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]. When the probability distributions are not completely known and belong to a given distributional uncertainty set, Peng et al [18] formulated the chance constraints of each player as distributional robust joint chance constrained problem.…”

“…In [18,19,20], the authors assume that the random constraint vectors are independently distributed. However, the random variables are usually dependent in real world applications.…”