Enforce and selective operators of combinatorial games

Tomoaki Abuku, Shun-ichi Kimura, Hironori Kiya, Urban Larsson, Indrajit Saha, Koki Suetsugu, Takahiro Yamashita

Abstract: We consider an enforce operator on impartial rulesets similar to the Muller Twist and the comply/constrain operator of Smith and Stănică, 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 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.