On the Characterization of Solution Sets of Smooth and Nonsmooth Convex Stochastic Nash Games

Abstract

Variational analysis provides an avenue for characterizing solution sets of deterministic Nash games over continuous-strategy sets. We examine whether similar statements, particularly pertaining to existence and uniqueness, may be made when player objectives are given by expectations. For instance, in deterministic regimes, a suitable coercivity condition associated with the gradient map is sufficient for the existence of a Nash equilibrium; in stochastic regimes, the application of this condition requires being able to analytically evaluate the expectation and its gradients. Our interest is in developing a framework that relies on the analysis of merely the integrands of the expectations; in the context of existence statement, we consider whether the satisfaction of a suitable coercivity condition in an almost-sure sense may lead to statements about the original stochastic Nash game. Notably, this condition also guarantees the existence of an equilibrium of the scenario-based Nash game. We consider a range of such statements for claiming the existence of stochastic Nash equilibria when payoff functions are both smooth and nonsmooth and when strategy sets are possibly coupled through a shared convex constraint. Notably the sufficiency conditions are less stringent, when one either imposes appropriate monotonicity requirements or requires that strategy sets be decoupled. Uniqueness, however, can be claimed by requiring that a strong monotonicity condition holds over a set of positive measure, rather than in an almost-sure sense. When strategy sets are coupled by shared convex expected-value constraints, a suitable regularity condition allows for claiming existence and uniqueness in the primal-dual space. We illustrate our approach by examining two extensions of stochastic Nash–Cournot games, of which the first allows for nonsmooth payoffs through the introduction of risk-measures, while the second allows for shared stochastic constraints.