Lattice point methods for combinatorial games

TL;DR

This paper introduces a geometric approach to encoding and analyzing finite impartial combinatorial games using lattice points in convex polyhedra, enabling efficient computation of strategies and exploring new conjectures.

Contribution

It presents a novel lattice point encoding for combinatorial games, generalizes octal games to squarefree games, and proposes conjectures on rational strategies and affine stratifications.

Findings

Algorithm for computing normal play strategies

Lattice games encompass heap games and manifest Sprague-Grundy theorem geometrically

Conjectures on rational strategies and affine stratifications for all lattice games

Abstract

We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these \emph{lattice games} can be made particularly efficient for octal games, which we generalize to \emph{squarefree games}. These additionally encompass all heap games in a natural setting, in which the Sprague-Grundy theorem for normal play manifests itself geometrically. We provide an algorithm to compute normal play strategies. The setting of lattice games naturally allows for mis`ere play, where 0 is declared a losing position. Lattice games also allow situations where larger finite sets of positions are declared losing. Generating functions for sets of winning positions provide data structures for strategies of lattice games. We conjecture that every lattice game has a \emph{rational strategy}: a rational generating function for its winning positions. Additionally, we conjecture that every lattice game has an \emph{affine stratification}: a partition of its set of winning positions into a finite disjoint union of finitely generated modules for affine semigroups.