On the geometry of combinatorial games: A renormalization approach

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.

Combinatorial games with a pass: A dynamical systems approach

“…(We refer the reader to [7,8] for a more detailed introduction to the general dynamical-systems-based approach to combinatorial games. )…”

“…Nim is a somewhat unusual (in comparison to other games whose instant-winner sheet geometries have been previously investigated) in that the above statement regarding scale invariance of the instant-winner sheets has one subtle complication. Namely, Nim's instant-winner sheets do not possess a true geometric invariance as suggested above, but rather a type of factor-of-two periodic scale invariance [8]. By this we mean that, for any x, the sheets W x , W 2x , W 4x , W 8x , .…”

“…1 we describe a recently developed recursive technique for analyzing combinatorial games that will also prove useful for analyzing games 3 with passes. This methodology is based on the construction of recursion operators that describe a game's underlying geometric structure [7,8]. In Sect.…”

“…In the case of Nim, for instance, Bouton's [5] elegant solution method (along with the more general Sprague-Grundy theory which generalizes it [9, 10]) breaks down when a pass is introduced and offers very little additional insight. Here, the failure can be attributed to the fact that a pass move renders the game non-decomposable.The situation is still more difficult in the case of Chomp, a more complex game for which standard solution methods have failed even in the absence of a pass.Given this situation, we will utilize a recently introduced dynamical-systems-based methodology for analyzing games that focuses on identifying the underlying geometry of a game and its recursive structure [7,8,6,11,12]. A fundamental virtue of this approach is that it does not rely on a game being solvable or decomposable, and hence can address complex situations, whether it be a simple game which becomes complex by virtue of the introduction of a pass (as in Nim) or a game which is intrinsically complex even in the absence of a pass (as in Chomp).…”

“…The P-set of the game is the main object of interest; knowledge of a game's P-set allows one to define a winning strategy for the game. Interestingly, for many combinatorial games, the P-set turns out to be neither a collection of randomly dispersed points in position space, nor a completely regular geometric object [7,8]. Rather, it lies intermediate between the two: It displays an overall geometric structure, but with local scatter (disorder).…”