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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.01006 |
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| _version_ | 1866929471966150656 |
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| author | Abuku, Tomoaki Kimura, Shun-ichi Kiya, Hironori Larsson, Urban Saha, Indrajit Suetsugu, Koki Yamashita, Takahiro |
| author_facet | Abuku, Tomoaki Kimura, Shun-ichi Kiya, Hironori Larsson, Urban Saha, Indrajit Suetsugu, Koki Yamashita, Takahiro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01006 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Enforce and selective operators of combinatorial games Abuku, Tomoaki Kimura, Shun-ichi Kiya, Hironori Larsson, Urban Saha, Indrajit Suetsugu, Koki Yamashita, Takahiro Combinatorics 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. |
| title | Enforce and selective operators of combinatorial games |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.01006 |