  We consider a class of hierarchical noncooperative $N-$player games where the
$i$th player solves a parametrized 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 consider settings where player playoffs are expectation-valued
with lower-level equilibrium constraints imposed in an almost-sure sense; i.e.,
player problems are parametrized stochastic MPECs. 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 both monotone and
strongly monotone regimes. Notably, the schemes do not impose any smoothness
assumptions on player problems and allow for state-dependence in noise; (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. This scheme is reliant on computing 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.
