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This paper considers a stochastic Nash game in which each player minimizes anexpectation valued composite objective. We make the following contributions.(I) Under suitable monotonicity assumptions on the concatenated gradient map,we derive optimal rate statements and oracle complexity bounds for the proposedvariable sample-size proximal stochastic gradient-response (VS-PGR) scheme whenthe sample-size increases at a geometric rate. If the sample-size increases ata polynomial rate of degree $v > 0$, the mean-squared errordecays at acorresponding polynomial rate while the iteration and oracle complexities toobtain an $\epsilon$-NE are $\mathcal{O}(1/\epsilon^{1/v})$ and$\mathcal{O}(1/\epsilon^{1+1/v})$, respectively. (II) We then overlay (VS-PGR)with a consensus phase with a view towards developing distributed protocols foraggregative stochastic Nash games. In the resulting scheme, when thesample-size and the consensus steps grow at a geometric and linear rate,computing an $\epsilon$-NE requires similar iteration and oracle complexitiesto (VS-PGR) with a communication complexity of$\mathcal{O}(\ln^2(1/\epsilon))$; (III) Under a suitable contractive propertyassociated with the proximal best-response (BR) map, we design a variablesample-size proximal BR (VS-PBR) scheme, where each player solves asample-average BR problem. Akin to (I), we also give the rate statements,oracle and iteration complexity bounds. (IV) Akin to (II), the distributedvariant achieves similar iteration and oracle complexities to the centralized(VS-PBR) with a communication complexity of $\mathcal{O}(\ln^2(1/\epsilon))$when the communication rounds per iteration increase at a linear rate. Finally,we present some preliminary numerics to provide empirical support for the rateand complexity statements.

Distributed Variable Sample-Size Gradient-response and Best-response Schemes for Stochastic Nash Equilibrium Problems over Graphs