On Synchronous, Asynchronous, and Randomized Best-Response schemes for computing equilibria in Stochastic Nash games

In this work, we consider a stochastic Nash game in which each player solves a parameterized stochastic optimization problem. In deterministic regimes, best-response schemes have been shown to be convergent under a suitable spectral property associated with the proximal best-response map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization problems at each iteration. Instead, we propose an inexact generalization of this scheme in which an inexact solution to the best-response problem is computed via an increasing number of projected stochastic gradient steps. On the basis of this framework, we present three inexact best-response schemes: (i) First, we propose a synchronous inexact best-response scheme where all players simultaneously update their strategies; (ii) Subsequently, we extend this to a randomized setting where a subset of players is randomly chosen to their update strategies while the other players keep their strategies invariant; (iii) Finally, we propose an asynchronous scheme, where each player determines its own update frequency and may use outdated rival-specific data in updating its strategy. Under a suitable contractive property of the proximal best-response map, we proceed to derive a.s. convergence of the iterates for (i) and (ii) and mean-convergence for (i) – (iii). In addition, we show that for (i) – (iii), the produced iterates converge to the unique equilibrium in mean at a prescribed linear rate with a prescribed constant rather than a sub-linear rate. Finally, we establish the overall iteration complexity of the scheme in terms of projected stochastic gradient steps for computing an −Nash equilibrium and note that in all settings, the iteration complexity is O(1/ ) where c = 0 in the context of (i) and represents the positive cost of randomization (in (ii)) and asynchronicity and delay (in (iii)). The schemes are further extended to settings where players solve two-stage recourse-based stochastic Nash games with linear and quadratic recourse. Finally, we apply the developed methods to a multi-portfolio investment problem, and carry out numerical simulations to demonstrate the rate and complexity statements.