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 the nim-value of rulesets under the enforce operator and establish well-known properties for impartial games.