Solution Concepts in A-Loss Recall Games: Existence and Computational Complexity

TL;DR

This paper investigates A-loss recall games, a subclass of imperfect recall games, establishing conditions for Nash equilibrium existence, demonstrating irrational solutions, and analyzing computational complexity of related problems.

Contribution

It introduces novel properties of A-loss recall games, including equilibrium conditions, examples with irrational solutions, and complexity results for strategy computations.

Findings

Nash equilibrium exists under specific conditions in A-loss recall games

Both NE and maxmin strategies can require irrational numbers

Finding maxmin strategies and NE existence are NP-hard problems

Abstract

Imperfect recall games represent dynamic interactions where players forget previously known information, such as a history of played actions. The importance of imperfect recall games stems from allowing a concise representation of strategies compared to perfect recall games where players remember all information. However, most of the algorithmic results are negative for imperfect recall games -- a Nash equilibrium~(NE) does not have to exist and computing a best response or a maxmin strategy is NP-hard. We focus on a subclass of imperfect recall games, called A-loss recall games, where a best response can be found in polynomial time. We derive novel properties of A-loss recall games, including (1) a sufficient and necessary condition for the existence of NE in A-loss recall games, (2) example where both NE and maxmin require irrational numbers for rational input, and (3) NP-hardness of problems related to finding maxmin strategies and existence of a NE strategy.