Combinatorial games with a pass: a dynamical systems approach.
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.
Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA. rebeccam@ices.utexas.edu
Rebecca E
RE
Morrison
Eric J
EJ
Friedman
Adam S
AS
Landsberg
D003198:Computer Simulation; D005716:Game Theory; D015233:Models, Statistical; D017711:Nonlinear Dynamics