Groupe d’études et de recherche en analyse des décisions

Nash games under uncertainty: Characterization and Computation of Equilibria

Uday Shanbhag

The first part of our talk considers the characterization of equilibria arising from stochastic Nash games with continuous strategy sets. Deterministic variants of such games with differentiable payoff functions admit tractable variational conditions whose analysis leads to statements regarding existence, uniqueness, stability etc. Analogous conditions are harder to obtain in stochastic regimes since they necessitate analytical expressions of the expectations and their derivatives. We develop a pathway for characterizing the solution sets of the associated stochastic Nash equilibria by analyzing only the sampled variants of the original stochastic game. Extensions to regimes where payoffs are nonsmooth are also examined. The second part of the talk addresses the computation of Nash equilibria in distributed settings particularly when gradient maps do not satisfy strong monotonicity properties. We present single-timescale counterparts of classical Tikhonov and proximal-point schemes along with convergence theory. Extensions to stochastic regimes via regularized stochastic approximation are also discussed. Some preliminary numerical results are provided.