Enforce and selective operators of combinatorial games
Abstract
We consider an {\em enforce operator} on impartial rulesets similar to the Muller Twist and the comply/constrain operator of Smith and St\u anic\u a, 2002. Applied to the rulesets A and B, on each turn the opponent enforces one of the rulesets and the current player complies, by playing a move in that ruleset. If the outcome table of the enforce variation of A and B is the same as the outcome table of A, then we say that A dominates B. We find necessary and sufficient conditions for this relation. Additionally, we define a {\em selective operator} and explore a distributive-lattice-like structure within applicable rulesets. Lastly, we define nim-values under enforce-rulesets, and establish that the Sprague-Grundy theory continues to hold, along with illustrative examples.
Cite
@article{arxiv.2311.01006,
title = {Enforce and selective operators of combinatorial games},
author = {Tomoaki Abuku and Shun-ichi Kimura and Hironori Kiya and Urban Larsson and Indrajit Saha and Koki Suetsugu and Takahiro Yamashita},
journal= {arXiv preprint arXiv:2311.01006},
year = {2024}
}
Comments
25 pages, 12 figures, 1 table