Games with distributionally robust joint chance constraints

Abstract

This paper studies an n-player non-cooperative game where each player has expected-value payoff function and chance-constrained strategy set. We consider the case where the row vectors defining the constraints are independent random vectors whose probability distributions are not completely known and belong to a certain distributional uncertainty set. The chanceconstrained strategy sets are defined using a distributionally robust framework. We consider one density based uncertainty set and four two-moments based uncertainty sets. One of the considered uncertainty sets is based on a nonnegative support. Under the standard assumptions on the players' payoff functions, we show that there exists a Nash equilibrium of a distributionally robust chance-constrained game for each uncertainty set. As an application, we study Cournot competition in electricity market and perform the numerical experiments for the case of two electricity firms.

On Completely Random Games

“…According to [16], and we quote: "...The original Nash equilibrium theory was conceived for deterministic games, which makes it limited to handle real applications with random payoffs and strategy sets." In its excellent literature review the authors describe the evolution of game models with random payoffs that culminates in the work [11] where Nguyen et alii write, and we quote: "...extended the results in [14] and in [13] 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." Riccardi et alii in [16], propose and we quote: "...an n-player non-cooperative game where the payoff function of each player follows a multivariate distribution."…”

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.…”

“…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. They consider several uncertainty sets, namely a density based uncertainty set and four two-moments based uncertainty sets where one of them has a nonnegative support.…”

“…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.…”