Simplifying imperfect recall games

TL;DR

The paper tackles solving imperfect recall extensive-form games, focusing on non-absent-minded players and the $NP$-hardness barrier. It introduces two polynomial-time transformations to reduce general imperfect recall to the tractable A-loss recall (ALR) class: (i) shuffling action orders to yield a $PTIME$-solvable extension of ALR, and (ii) constructing a linear-combination of action sequences yielding an ALR-equivalent game with a simplified information structure, potentially exponential in size. It shows that every non-absent-minded game has an ALR-equivalent representation and provides an algorithm to compute the smallest-size ALR-equivalent game, while noting the possible exponential blow-up in size. These results offer practical abstraction tools for solving large imperfect recall games, trading worst-case exponential size for polynomial-time solvability on ALR-like instances. The work clarifies that, in the absence of absent-mindedness, imperfect recall can be attributed to forgetting past own actions, elucidating the relationship between imperfect recall and ALR.

Abstract

In games with imperfect recall, players may forget the sequence of decisions they made in the past. When players also forget whether they have already encountered their current decision point, they are said to be absent-minded. Solving one-player imperfect recall games is known to be NP-hard, even when the players are not absent-minded. This motivates the search for polynomial-time solvable subclasses. A special type of imperfect recall, called A-loss recall, is amenable to efficient polynomial-time algorithms. In this work, we present novel techniques to simplify non-absent-minded imperfect recall games into equivalent A-loss recall games. The first idea involves shuffling the order of actions, and leads to a new polynomial-time solvable class of imperfect recall games that extends A-loss recall. The second idea generalises the first one, by constructing a new set of action sequences which can be "linearly combined" to give the original game. The equivalent game has a simplified information structure, but it could be exponentially bigger in size (in accordance with the NP-hardness). We present an algorithm to generate an equivalent A-loss recall game with the smallest size.