Lattice point methods for combinatorial games

Abstract

We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these lattice games can be made particularly efficient for octal games, which we generalize to squarefree games. These encompass all heap games in a natural setting where the Sprague-Grundy theorem for normal play manifests itself geometrically. We provide polynomial time algorithms for computing strategies for lattice games provided that they have a certain algebraic structure, called an affine stratification.