Determining the Winner in Alternating-Move Games
Dynamical Systems
2026-05-14 v3 Computer Science and Game Theory
Logic
Optimization and Control
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 Urba\'nski, 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.
Cite
@article{arxiv.2601.08359,
title = {Determining the Winner in Alternating-Move Games},
author = {Itamar Bellaïche and Auriel Rosenzweig},
journal= {arXiv preprint arXiv:2601.08359},
year = {2026}
}