Combinatorial Games with a Pass: A dynamical systems approach

TL;DR

This paper models combinatorial games as dynamical systems to analyze how introducing a pass move affects game complexity, revealing significant structural differences between Nim and Chomp.

Contribution

It presents a dynamical systems framework to understand the impact of pass moves on well-known combinatorial games, linking them to generic perturbed games.

Findings

Pass dramatically increases Nim's complexity

Pass has minimal impact on Chomp

Structural connections to perturbed games identified

Abstract

By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game's underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations we are able to identify underlying structural connections between these "games with passes" and a recently introduced class of "generic (perturbed) games." This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game.