Nonlinear dynamics in combinatorial games: Renormalizing Chomp

Abstract

We develop a new approach to combinatorial games that reveals connections between such games and some of the central ideas of nonlinear dynamics: scaling behaviors, complex dynamics and chaos, universality, and aggregation processes. We take as our model system the combinatorial game Chomp, which is one of the simplest in a class of "unsolved" combinatorial games that includes Chess, Checkers, and Go. We discover that the game possesses an underlying geometric structure that "grows" ͑reminiscent of crystal growth͒, and show how this growth can be analyzed using a renormalization procedure adapted from physics. In effect, this methodology allows one to transform a combinatorial game like Chomp into a type of dynamical system. Not only does this provide powerful insights into the game of Chomp ͑yielding a complete probabilistic description of optimal play in Chomp and an answer to a longstanding question about the nature of the winning opening move͒, but more generally, it offers a mathematical framework for exploring this unexpected relationship between combinatorial games and modern dynamical systems theory. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2725717͔Combinatorial games, which include Chess, Go, Checkers, Chomp, Dots-and-Boxes, and Nim, have both captivated and challenged mathematicians, computer scientists, and players alike. These are two-player games with no randomness (e.g., no rolling of dice or dealing of cards) and no hidden information (unlike poker, for instance). Apart from their obvious entertainment value, combinatorial games have long been the subject of much serious study owing to the deep mathematical questions they raise in areas such as computational complexity, graph theory, and surreal numbers. 1-3 Using the game of Chomp as a prototype, we report here on a new geometrical approach which unveils unexpected parallels between combinatorial games and some of the central ideas of dynamical systems theory, most notably notions of scaling, renormalization, universality, and chaotic attractors. We show in particular that the game of Chomp can behave in ways analogous to a chaotic dynamical system. This insight and subsequent analysis allows us not only to answer a number of open questions about the game of Chomp, but also provides a new perspective on complex combinatorial games more generally and offers a mathematical framework for understanding this connection between combinatorial games and dynamical systems. Our central finding is that underlying the game of Chomp is a geometric structure that encodes essential information about the game, and that this structure exhibits a type of spatial scale invariance: Loosely speaking, the geometry of "small" and "large" winning positions in the game look the same, after rescaling. We demonstrate that the geometries on different spatial scales can be related to one another via a set of recursion operators, which in turn allows us to recast the game as a type of dynamical system. We then analyze this "dynamical system" using tools and conce...

Combinatorial games with a pass: A dynamical systems approach

“…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).…”

“…2 for Nim, though the definition of the supermex operator M differs for the two cases since the game rules are different. In Chomp, the supermex operator follows directly from game rules M1-M3 above.Explicit details can be found in [7]; for present purposes, however, the general form of Eqn. 10 will suffice.…”

“…1 for Nim, i.e., the first term represents the contribution to W x from the parents of P-positions on loser sheet x = 0, the second term from parents of P-positions on loser sheet x = 1, etc. These follow directly fromChomp game rules M4 and M5 together with the definition of an instant-winner sheet; see [7] for a more detailed derivation. For present purposes though, it suffices to simply note that each term in Eqn.…”

“…We also remark that, unlike for Nim, a true geometric invariance for Chomp holds for all sheets -i.e., the subtle powers-of-two periodicity noted for Nim is absent. A full discussion of the geometry of pure Chomp can be found in [7].3.2 Geometry and recursion for 3-row Chomp-with-a-passWe now consider the introduction of a pass move into Chomp, and seek to determine a recursive formulation of this problem. We begin with the game rules.…”

“…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).…”