On the computation of equilibria in monotone and potential stochastic hierarchical games

Abstract: We consider a class of hierarchical noncooperative $N$-player games where the $i$th player solves a parametrized stochastic mathematical program with equilibrium constraints (MPEC) with the caveat that the implicit form of the $i$th player's in MPEC is convex in player strategy, given rival decisions. We develop computational schemes in two distinct regimes: (a) {\em Monotone regimes.} When player-specific implicit problems are convex, then the necessary and sufficient equilibrium conditions are given by a stochastic inclusion. Under a monotonicity assumption on the operator, we develop a variance-reduced stochastic proximal-point scheme that achieves deterministic rates of convergence in terms of solving proximal-point problems in monotone/strongly monotone regimes and the schemes are characterized by optimal or near-optimal sample-complexity guarantees. (b) {\em Potentiality.} When the implicit form of the game admits a potential function, we develop an asynchronous relaxed inexact smoothed proximal best-response framework. We consider the smoothed counterpart of this game where each player's problem is smoothed via randomized smoothing. Notably, under suitable assumptions, we show that an $\eta$-smoothed game admits an $\eta$-approximate Nash equilibrium of the original game. Our proposed scheme produces a sequence that converges almost surely to an $\eta$-approximate Nash equilibrium. The smoothing framework allows for developing a variance-reduced zeroth-order scheme for such problems that admits a fast rate of convergence. Numerical studies on a class of multi-leader multi-follower games suggest that variance-reduced proximal schemes provide significantly better accuracy with far lower run-times. The relaxed best-response scheme scales well will problem size and generally displays more stability than its unrelaxed counterpart.