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The goal in this paper is to approximate the Price of Stability (PoS) in 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 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 ratio. 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 $ε^{-4}$ that appears to be best known result for this class of constrained stochastic optimization problems, where $ε$ denotes an arbitrary bound on suitably defined infeasibility and suboptimality metrics. Second, we develop an SA-based scheme for approximating the PoS and derive lower and upper bounds on the approximation error. To validate the theoretical findings, we provide preliminary simulation results on a networked stochastic Nash Cournot competition.