  We consider a class of nonsmooth aggregative games over networks in
stochastic regimes, where each player is characterized by a composite cost
function $f_i+r_i$, $f_i$ is a smooth expectation-valued function dependent on
its own strategy and an aggregate function of rival strategies, and $r_i$ is a
nonsmooth convex function of its strategy with an efficient prox-evaluation. We
design a fully distributed iterative proximal stochastic gradient method
overlaid by a Tikhonov regularization, where each player may independently
choose its steplengths and regularization parameters while meeting some
coordination requirements. Under a monotonicity assumption on the
pseudo-gradient mapping, we prove the almost sure convergence to the least-norm
Nash equilibrium. In addition, when each $r_i$ is an indicator function of a
compact convex set, we establish the convergence rate associated with the
expected gap function at the time-averaged sequence. We further establish high
probability bounds for the gap function via both Markov's inequality as well as
a more refined argument that leverages Azuma's inequality. Furthermore, we
consider the extension to the private hierarchical regime where each player is
a leader with respect to a collection of private followers competing in a
strongly monotone game, parametrized by leader decisions. By leveraging a
convolution-smoothing framework, we present amongst the first fully distributed
schemes for computing a Nash equilibrium of a game complicated by such a
hierarchical structure. Based on this framework, we extend the rate statements
to accommodate the computation of a hierarchical stochastic Nash equilibrium by
using a Fitzpatrick gap function. Finally, we validate the proposed methods on
a networked Nash-Cournot equilibrium problem and a hierarchical generalization,
observing that regularization has a beneficial impact on empirical behavior.
