On the geometry of combinatorial games: A renormalization approach

Summary

Abstract. We describe the application of a physics-inspired renormalization technique to combinatorial games. Although this approach is not rigorous, it allows one to calculate detailed, probabilistic properties of the geometry of the P-positions in a game. The resulting geometric insights provide explanations for a number of numerical and theoretical observations about various games that have appeared in the literature. This methodology also provides a natural framework for several new avenues of research in combinatorial games, including notions of “universality,” “sensitivity-to-initial-conditions,” and “crystal-like growth,” and suggests surprising connections between combinatorial games, nonlinear dynamics, and physics. We demonstrate the utility of this approach for a variety of games–three-row Chomp, 3-D Wythoff's game, Sprague–Grundy values for 2-D Wythoff's game, and Nim and its generalizations–and show how it explains existing results, addresses longstanding questions, and generates new predictions and insights.

Introduction

In this paper we introduce a method for analyzing combinatorial games based on renormalization. As a mathematical tool, renormalization has enjoyed great success in virtually all branches of modern physics, from statistical mechanics [Goldenfeld 1992] to particle physics [Rivasseau 2003] to chaos theory [Feigenbaum 1980], where it is used to calculate properties of physical systems or objects that exhibit so-called ‘scaling’ behavior (i.e., geometric similarity on different spatial scales). In the present context we adapt this methodology to the study of combinatorial games.