Enforce and selective operators of combinatorial games

Tomoaki Abuku (), Shun-ichi Kimura (), Hironori Kiya (), Urban Larsson (), Indrajit Saha (), Koki Suetsugu () and Takahiro Yamashita ()
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Tomoaki Abuku: National Institute of Informatics
Shun-ichi Kimura: Hiroshima University
Hironori Kiya: Osaka Metropolitan University
Urban Larsson: Indian Institute of Technology Bombay
Indrajit Saha: Kyushu University
Koki Suetsugu: National Institute of Informatics
Takahiro Yamashita: Hiroshima University

International Journal of Game Theory, 2024, vol. 53, issue 4, No 9, 1249-1273

Abstract: 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.

Keywords: Nim; Wythoff Nim; Combinatorial game theory; Comply/Constrain operator; Enforce operator; Muller Twist; Selective operator; Nim-value (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00182-024-00910-6

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