Extensive-form games (EFGs) are a standard model for sequential decision-making in games. A fundamental and typically implicit assumption in EFGs is that players always have access to all of their actions at every decision point. However, in many settings, certain actions might be unavailable during game-play due to exogenous stochasticity. Given a base EFG, we formalize a model that allows for actions to be stochastically restricted, leading to a corresponding Extensive-Form Game with Stochastic Action Sets (EFGSAS). In EFGSAS, we derive an expansion procedure that results in an equivalent EFG, thus showing that standard strategy formalisms could require exponentially large representations. However, under an appropriate independence assumption, we show that approximate compact strategy representations polynomial in the size of the base EFG exist. Computationally, we introduce an algorithm called CAT-CFR, which minimizes ranking regret and whose averaged induced sequence-form strategies converge to approximate Nash equilibria with high probability in two-player zero-sum EFGSAS. Finally, we utilize a stochastic approximation procedure to recover compact representations of approximate Nash equilibria, utilizing the iterates of CAT-CFR and additional action availability samples.

翻译：暂无翻译