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.

翻译：本文考虑一类N人层次博弈，其中第i个玩家的目标函数包含由对手决策参数化的期望值项和层次项。该框架可刻画多种随机层次优化问题、Stackelberg均衡问题及领导者-追随者博弈。我们提出一种迭代正则化与平滑方差缩减修正外梯度框架，用于在随机设置下逐步逼近层次均衡。分析中给出了收敛速率、复杂度保证及几乎必然收敛结果，并将结论扩展至下层问题非精确求解的场景，提供了相应的速率与复杂度分析。本文模型框架涵盖电力市场中研究的许多博弈均衡问题。我们以虚拟电厂为实际应用案例，重点阐释层次决策制定与正则化的作用。