The Game Value of Sequential Compounds of Integers and Stars
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909388235603968 |
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| author | Hashimoto, Kengo |
| author_facet | Hashimoto, Kengo |
| contents | A combinatorial game is a two-player game without hidden information or chance elements. One of the major approaches to analyzing games in combinatorial game theory is to break down a given game position into a disjunctive sum of multiple sub-positions, then evaluate the game value of each component of the sum, and finally integrate these game values to find which player has a winning strategy in the whole position. Accordingly, finding the game value of a given position is a major topic in combinatorial game theory. The sequential compound proposed by Stromquist and Ullman is a combinatorial game consisting of two combinatorial games. In the sequential compound of games $G$ and $H$, the players make moves on $G$ until $G$ is over, and then they play on $H$. In this paper, we investigate the general properties of sequential compounds. As the main result, we give the game values of sequential compounds of a finite number of integers and stars, which are basic and typical games in combinatorial game theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08611 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Game Value of Sequential Compounds of Integers and Stars Hashimoto, Kengo Combinatorics A combinatorial game is a two-player game without hidden information or chance elements. One of the major approaches to analyzing games in combinatorial game theory is to break down a given game position into a disjunctive sum of multiple sub-positions, then evaluate the game value of each component of the sum, and finally integrate these game values to find which player has a winning strategy in the whole position. Accordingly, finding the game value of a given position is a major topic in combinatorial game theory. The sequential compound proposed by Stromquist and Ullman is a combinatorial game consisting of two combinatorial games. In the sequential compound of games $G$ and $H$, the players make moves on $G$ until $G$ is over, and then they play on $H$. In this paper, we investigate the general properties of sequential compounds. As the main result, we give the game values of sequential compounds of a finite number of integers and stars, which are basic and typical games in combinatorial game theory. |
| title | The Game Value of Sequential Compounds of Integers and Stars |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.08611 |