Simplifying imperfect recall games

2025-02-19

AI Summary

This paper addresses the computational intractability of non-forgetting imperfect recall games. We propose an equivalence-preserving simplification method that transforms such games into A-loss recall games—known to be solvable in polynomial time. Our approach introduces three key contributions: (1) a novel action-sequence reordering and linear-combination construction technique, systematically extending the expressive power of A-loss recall games; (2) a minimal-size equivalent transformation algorithm that yields a compact, solution-preserving reduction from the original game to an A-loss recall game; and (3) the identification of two new polynomial-time solvable subclasses of imperfect recall games, along with an automated procedure for constructing minimum-size equivalent games. The method bridges theoretical tractability and practical constructibility, offering a new paradigm for efficiently solving complex imperfect recall games while preserving strategic equivalence.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Search and Optimization: Adversarial Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Simplify imperfect recall games efficiently
• Transform games to A-loss recall format
• Develop polynomial-time solvable game subclasses

Innovation

• Shuffles action order
• Constructs new action sequences
• Minimizes A-loss game size