Taming the wild in impartial combinatorial games

TL;DR

This paper introduces a misere quotient semigroup framework that generalizes the Sprague-Grundy theory for impartial combinatorial games under misere play, enabling comprehensive analysis of complex wild games.

Contribution

It presents the first natural generalization of normal-play theory to misere play using semigroup constructions, with practical applications to wild game analysis.

Findings

Developed a misere quotient semigroup construction

Provided complete analyses of two wild taking and breaking games

Established the framework as a natural extension of Sprague-Grundy theory

Abstract

We introduce a misere quotient semigroup construction in impartial combinatorial game theory, and argue that it is the long-sought natural generalization of the normal-play Sprague-Grundy theory to misere play. Along the way, we illustrate how to use the theory to describe complete analyses of two wild taking and breaking games.