  The distributed computation of equilibria and optima has seen growing
interest in a broad collection of networked problems. We consider the
computation of equilibria of convex stochastic Nash games characterized by a
possibly nonconvex potential function. Our focus is on two classes of
stochastic Nash games: (P1): A potential stochastic Nash game, in which each
player solves a parameterized stochastic convex program; and (P2): A
misspecified generalization, where the player-specific stochastic program is
complicated by a parametric misspecification. In both settings, exact proximal
BR solutions are generally unavailable in finite time since they necessitate
solving parameterized stochastic programs. Consequently, we design two
asynchronous inexact proximal BR schemes to solve the problems, where in each
iteration a single player is randomly chosen to compute an inexact proximal BR
solution with rivals' possibly outdated information. Yet, in the misspecified
regime (P2), each player possesses an extra estimate of the misspecified
parameter and updates its estimate by a projected stochastic gradient (SG)
algorithm. By Since any stationary point of the potential function is a Nash
equilibrium of the associated game, we believe this paper is amongst the first
ones for stochastic nonconvex (but block convex) optimization problems equipped
with almost-sure convergence guarantees. These statements can be extended to
allow for accommodating weighted potential games and generalized potential
games. Finally, we present preliminary numerics based on applying the proposed
schemes to congestion control and Nash-Cournot games.
