  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.
