Determining the Winner in Alternating-Move Games
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866909040523608064 |
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| author | Bellaïche, Itamar Rosenzweig, Auriel |
| author_facet | Bellaïche, Itamar Rosenzweig, Auriel |
| contents | We provide a criterion for determining the winner in two-player win-lose alternating-move games on trees, in terms of the Hausdorff dimension of the target set. We focus our study on special cases, including the Gale-Stewart game on the complete binary tree and a family of Schmidt games, generalizing a result of Schmidt from Hilbert spaces to arbitrary complete metric spaces. Building on the Hausdorff dimension games originally introduced by Das, Fishman, Simmons, and Urbański, which provide a game-theoretic approach for computing Hausdorff dimensions, we employ a generalized family of these games to obtain lower bounds on the Hausdorff dimensions of target sets whenever Player I can guarantee a win. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08359 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Determining the Winner in Alternating-Move Games Bellaïche, Itamar Rosenzweig, Auriel Dynamical Systems Computer Science and Game Theory Logic Optimization and Control We provide a criterion for determining the winner in two-player win-lose alternating-move games on trees, in terms of the Hausdorff dimension of the target set. We focus our study on special cases, including the Gale-Stewart game on the complete binary tree and a family of Schmidt games, generalizing a result of Schmidt from Hilbert spaces to arbitrary complete metric spaces. Building on the Hausdorff dimension games originally introduced by Das, Fishman, Simmons, and Urbański, which provide a game-theoretic approach for computing Hausdorff dimensions, we employ a generalized family of these games to obtain lower bounds on the Hausdorff dimensions of target sets whenever Player I can guarantee a win. |
| title | Determining the Winner in Alternating-Move Games |
| topic | Dynamical Systems Computer Science and Game Theory Logic Optimization and Control |
| url | https://arxiv.org/abs/2601.08359 |