Computing Maxmin Strategies in Extensive-Form Zero-Sum Games with Imperfect Recall

TL;DR

This paper introduces the first algorithm to approximate maxmin strategies in two-player zero-sum imperfect recall games, overcoming theoretical limitations and enabling practical solutions for larger game instances.

Contribution

The authors develop a novel branch-and-bound algorithm based on a modified sequence-form linear program to efficiently approximate strategies in imperfect recall games.

Findings

Algorithm can approximate strategies in larger imperfect recall games within minutes.

Modified linear program effectively models strategies without absentmindedness.

Approximations are achieved with provable guarantees despite exponential complexity.

Abstract

Extensive-form games with imperfect recall are an important game-theoretic model that allows a compact representation of strategies in dynamic strategic interactions. Practical use of imperfect recall games is limited due to negative theoretical results: a Nash equilibrium does not have to exist, computing maxmin strategies is NP-hard, and they may require irrational numbers. We present the first algorithm for approximating maxmin strategies in two-player zero-sum imperfect recall games without absentmindedness. We modify the well-known sequence-form linear program to model strategies in imperfect recall games and use a recent technique to approximate bilinear terms. Our main algorithm is a branch-and-bound search over these linear programs that provably reaches a desired approximation after an exponential number of steps in the size of the game. Experimental evaluation shows that the proposed algorithm can approximate maxmin strategies of randomly generated imperfect recall games of sizes beyond toy-problems within few minutes.