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arXiv · 2601.08359

Determining the Winner in Alternating-Move Games

Itamar Bellaïche·Auriel Rosenzweig
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Abstract

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.

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Itamar Bellaïche, Auriel Rosenzweig. 2026-05-13. Determining the Winner in Alternating-Move Games. https://arxiv.org/abs/2601.08359

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