Stochastic Approximation for Estimating the Price of Stability in Stochastic Nash Games

The goal in this paper is to approximate the Price of Stability (PoS) for stochastic Nash games using stochastic approximation (SA) schemes. PoS is amongst the most popular metrics in game theory and provides an avenue for estimating the efficiency of Nash games. In particular, knowing the value of PoS can help significantly with designing efficient networked systems, including transportation networks and power market mechanisms. Motivated by the lack of efficient methods for computing the PoS, first we consider stochastic optimization problems with a nonsmooth and merely convex objective function and a merely monotone stochastic variational inequality (SVI) constraint. This problem appears in the numerator of the PoS. We develop a randomized block-coordinate stochastic extra-(sub)gradient method where we employ a novel iterative penalization scheme to account for the mapping of the SVI in each of the two gradient updates of the algorithm. We obtain an iteration complexity of the order $\epsilon^{-4}$ that appears to be best known result for this class of constrained stochastic optimization problems. Second, we develop an SA-based scheme for approximating the PoS and derive lower and upper bounds on the approximation error. To validate our theoretical findings, we provide some preliminary simulation results on a stochastic Nash Cournot competition over a network.