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

We consider a class of noncooperative hierarchical $${\textbf{N}}$$ -player games where the ith player solves parametrized stochastic mathematical program with equilibrium constraints (MPEC) caveat that implicit form player’s in MPEC is convex strategy, given rival decisions. Few, if any, general purpose schemes exist for computing equilibria, motivating development computational two regimes: (a) Monotone regimes. When player-specific problems are convex, then necessary and sufficient conditions by inclusion. Under monotonicity assumption on operator, we develop variance-reduced proximal-point scheme achieves deterministic rates convergence terms solving monotone/strongly monotone regimes optimal or near-optimal sample-complexity guarantees. Finally, generated sequences shown to converge an almost-sure sense both strongly regimes; (b) Potentiality. game admits potential function, asynchronous relaxed inexact smoothed proximal best-response framework, requiring efficient computation approximate solution objective. To this end, $$\eta $$ -smoothed counterpart each problem via randomized smoothing. In fact, Nash -approximate original game. Our proposed produces sequence variant converges almost surely equilibrium. This reliant resolving problem, whose has objective, increasing accuracy finite-time. The smoothing framework allows developing zeroth-order such fast rate convergence. Numerical studies multi-leader multi-follower suggest provide significantly better far lower run-times. scales well size generally displays more stability than its unrelaxed counterpart.