Computing Stackelberg Equilibrium with Memory in Sequential Games

Abstract

Stackelberg equilibrium is a solution concept that describes optimal strategies to commit: Player 1 (the leader) first commits to a strategy that is publicly announced, then Player 2 (the follower) plays a best response to the leader's commitment. We study the problem of computing Stackelberg equilibria in sequential games with finite and indefinite horizons, when players can play history-dependent strategies. Using the alternate formulation called strategies with memory, we establish that strategy profiles with polynomial memory size can be described efficiently. We prove that there exist a polynomial time algorithm which computes the Strong Stackelberg Equilibrium in sequential games defined on directed acyclic graphs, where the strategies depend only on the memory states from a set which is linear in the size of the graph. We extend this result to games on general directed graphs which may contain cycles. We also analyze the setting for approximate version of Strong Stackelberg Equilibrium in the games with chance nodes.