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

We consider a class of noncooperative hierarchical 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. Few, if any, general purpose schemes exist for computing equilibria, motivating the development of computational schemes in two regimes: (a) 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 with optimal or near-optimal sample-complexity guarantees. Finally, the generated sequences are shown to converge to an equilibrium in an almost-sure sense in both monotone and strongly monotone regimes; (b) Potentiality. When the implicit form of the game admits a potential function, we develop an asynchronous relaxed inexact smoothed proximal best-response framework, requiring the efficient computation of an approximate solution of an MPEC with a strongly convex implicit objective. To this end, we consider an η -smoothed counterpart of this game where each player’s problem is smoothed via randomized smoothing. In fact, a Nash equilibrium of the smoothed counterpart is an η -approximate Nash equilibrium of the original game. Our proposed scheme produces a sequence and a relaxed variant that converges almost surely to an η -approximate Nash equilibrium. This scheme is reliant on resolving the proximal problem, a stochastic MPEC whose implicit form has a strongly convex objective, with increasing accuracy in finite-time. 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 with problem size and generally displays more stability than its unrelaxed counterpart.