{"paper":{"title":"Determining the Winner in Alternating-Move Games","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Hausdorff dimension of the target set determines the winner in two-player alternating-move games on trees.","cross_cats":["cs.GT","math.LO","math.OC"],"primary_cat":"math.DS","authors_text":"Auriel Rosenzweig, Itamar Bella\\\"iche","submitted_at":"2026-01-13T09:16:14Z","abstract_excerpt":"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 Urba\\'nski, which provide a game-theoretic approach for computing Hausdorff dimensions, we employ a generalized family of these"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The target sets admit well-defined Hausdorff dimensions in the metric spaces considered, and the generalized Hausdorff dimension games yield valid lower bounds on those dimensions when Player I has a winning strategy.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A criterion based on the Hausdorff dimension of the target set determines the winner in alternating-move win-lose games on trees, generalizing prior results on Schmidt games to arbitrary complete metric spaces.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Hausdorff dimension of the target set determines the winner in two-player alternating-move games on trees.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d967465b048aaf8743ece626993e7ac0282cb8cd5fff2a0896ea80c6826e28a6"},"source":{"id":"2601.08359","kind":"arxiv","version":3},"verdict":{"id":"ca5a3ae1-34ba-4e42-9bfb-454c7907ebfa","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T15:09:48.207234Z","strongest_claim":"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.","one_line_summary":"A criterion based on the Hausdorff dimension of the target set determines the winner in alternating-move win-lose games on trees, generalizing prior results on Schmidt games to arbitrary complete metric spaces.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The target sets admit well-defined Hausdorff dimensions in the metric spaces considered, and the generalized Hausdorff dimension games yield valid lower bounds on those dimensions when Player I has a winning strategy.","pith_extraction_headline":"Hausdorff dimension of the target set determines the winner in two-player alternating-move games on trees."},"references":{"count":19,"sample":[{"doi":"","year":2017,"title":"D. Badziahin and S. Harrap (2017) Cantor-winning sets and their applica- tions, Advances in Mathematics, vol. 318, pp. 627-77","work_id":"933aa1e1-2254-4e86-862f-6a345fc16029","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1017/etds.2024.23","year":2024,"title":"D. Badziahin and S. Harrap and E. Nesharim and D. Simmons (2024) Schmidt games and Cantor winning sets, Ergodic Theory and Dynamical Systems, vol. 45, no. 1, pp. 71-110. https://doi.org/10.1017/etds.2","work_id":"443a31cd-80d6-4cbd-aa6a-b5b7ce6f9c33","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"Baek (2009) Dimensionally invariant spaces, Journal of the Chungcheong Mathematical Society, volume 22 no","work_id":"6e537aa2-ae2d-4299-8da1-191bd13dac6e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Bellaïche (2023) Dyadic Hausdorff Dimension Games [Master’s Thesis, Tel Aviv University]","work_id":"8a47cd80-1993-4268-9139-c72aba6fe13b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"G. David and S. Semmes (1993) Analysis of and on Uniformly Rectifiable Sets, American Mathematical Society","work_id":"54abd367-eaea-4f47-82ac-972145ea5840","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":19,"snapshot_sha256":"319dac9f4251691fdf35fab27bdc910baed963e1ab4894da4de3f19880b4a055","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"b8f8d8f20926806a556e9af3b9274a11f2b9a22ee62b6e101fbd5f12f71aef1b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}