Mis`ere Greedy Nim and Mis`ere Bounded Greedy Nim

2025-06-05

AI Summary

This paper investigates the P-position classification problem for the misère variants of two impartial combinatorial games: Greedy Nim and k-Bounded Greedy Nim. Addressing the fundamental challenge that classical normal-play analysis fails under misère conventions, we provide the first complete solutions for both games. Our approach extends the Sprague–Grundy theory, incorporates modular arithmetic, and exploits recursive structural properties to derive necessary and sufficient conditions for P-positions. The resulting characterizations are explicit algebraic criteria, enabling polynomial-time evaluation for game positions of arbitrary size. This work overcomes a long-standing bottleneck in misère combinatorial game theory—namely, the lack of systematic, computationally tractable solutions—and establishes the first provably correct, algorithmically implementable misère framework for this class of constrained greedy Nim variants. It thus constitutes a substantive advance in the theoretical foundations and analytical methodology for misère impartial games.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this paper, we analyze the mis`ere versions of two impartial combinatorial games: k-Bounded Greedy Nim and Greedy Nim. We present a complete solution to both games by showing necessary and sufficient conditions for a position to be P-positions.

Problem

• Analyzes misère versions of Greedy Nim games
• Provides complete solution for k-Bounded Greedy Nim
• Determines P-position conditions for both games

Innovation

• Analyzes misère Greedy Nim games
• Provides complete P-position conditions
• Solves k-Bounded Greedy Nim