The Computational Complexity of Single-Player Imperfect-Recall Games

TL;DR

This paper explores the computational complexity of finding equilibrium strategies in single-player imperfect-recall games, establishing NP-hardness and CLS-completeness results for different solution concepts.

Contribution

It links equilibrium concepts in imperfect-recall games to polynomial optimization problems and determines their computational complexity classifications.

Findings

NP-hardness and inapproximability for ex-ante optimality and (EDT,GDH)-equilibria

CLS-completeness results for (CDT,GT)-equilibria

Establishes complexity-theoretic boundaries for computing strategies

Abstract

We study single-player extensive-form games with imperfect recall, such as the Sleeping Beauty problem or the Absentminded Driver game. For such games, two natural equilibrium concepts have been proposed as alternative solution concepts to ex-ante optimality. One equilibrium concept uses generalized double halving (GDH) as a belief system and evidential decision theory (EDT), and another one uses generalized thirding (GT) as a belief system and causal decision theory (CDT). Our findings relate those three solution concepts of a game to solution concepts of a polynomial maximization problem: global optima, optimal points with respect to subsets of variables and Karush-Kuhn-Tucker (KKT) points. Based on these correspondences, we are able to settle various complexity-theoretic questions on the computation of such strategies. For ex-ante optimality and (EDT,GDH)-equilibria, we obtain NP-hardness and inapproximability, and for (CDT,GT)-equilibria we obtain CLS-completeness results.