Zero-Sum Stochastic Stackelberg Games

Min-max optimization problems (i.e., zero-sum games) have been used to model problems in a variety of fields in recent years, from machine learning to economics. The literature to date has mostly focused on static zero-sum games, assuming independent strategy sets. In this paper, we study a form of dynamic zero-sum games, called stochastic games, with dependent strategy sets. Just as zero-sum games with dependent strategy sets can be interpreted as zero-sum Stackelberg games, stochastic zero-sum games with dependent strategy sets can be interpreted as zero-sum stochastic Stackelberg games. We prove the existence of an optimal solution in zero-sum stochastic Stackelberg games (i.e., a recursive Stackelberg equilibrium), provide necessary and sufficient conditions for a solution to be optimal, and show that a recursive Stackelberg equilibrium can be computed in polynomial time via value iteration. Finally, we show that stochastic Stackelberg games can model the problem of pricing and allocating goods across agents and time; more specifically, we propose a stochastic Stackelberg game whose solutions correspond to a recursive competitive equilibrium in a stochastic Fisher market. We close with a series of experiments which confirm our theoretical results and show how value iteration performs in practice.