  We consider an N-player hierarchical game in which the i-th player's
objective comprises of an expectation-valued term, parametrized by rival
decisions, and a hierarchical term. Such a framework allows for capturing a
broad range of stochastic hierarchical optimization problems, Stackelberg
equilibrium problems, and leader-follower games. We develop an iteratively
regularized and smoothed variance-reduced modified extragradient framework for
iteratively approaching hierarchical equilibria in a stochastic setting. We
equip our analysis with rate statements, complexity guarantees, and almost-sure
convergence results. We then extend these statements to settings where the
lower-level problem is solved inexactly and provide the corresponding rate and
complexity statements. Our model framework encompasses many game theoretic
equilibrium problems studied in the context of power markets. We present a
realistic application to the virtual power plants, emphasizing the role of
hierarchical decision making and regularization.
