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In this paper, we consider stochastic monotone Nash games where each player'sstrategy set is characterized by possibly a large number of explicit convexconstraint inequalities. Notably, the functional constraints of each player maydepend on the strategies of other players, allowing for capturing a subclass ofgeneralized Nash equilibrium problems (GNEP). While there is limited work thatprovide guarantees for this class of stochastic GNEPs, even when the functionalconstraints of the players are independent of each other, the majority of theexisting methods rely on employing projected stochastic approximation (SA)methods. However, the projected SA methods perform poorly when the constraintset is afflicted by the presence of a large number of possibly nonlinearfunctional inequalities. Motivated by the absence of performance guarantees forcomputing the Nash equilibrium in constrained stochastic monotone Nash games,we develop a single timescale randomized Lagrangian multiplier stochasticapproximation method where in the primal space, we employ an SA scheme, and inthe dual space, we employ a randomized block-coordinate scheme where only arandomly selected Lagrangian multiplier is updated. We show that our methodachieves a convergence rate of$\mathcal{O}\left(\frac{\log(k)}{\sqrt{k}}\right)$ for suitably definedsuboptimality and infeasibility metrics in a mean sense.

Randomized Lagrangian Stochastic Approximation for Large-Scale Constrained Stochastic Nash Games