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. 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, and show that they are useful for analyzing sets underlying the win-lose games we study.

翻译：我们基于目标集的豪斯多夫维数，提出了在树结构上判定双人零和交替移动游戏获胜方的准则。本研究重点关注若干特例，包括完全二叉树上的盖尔-斯图尔特游戏及一类施密特游戏。基于Das、Fishman、Simmons和Urbański最初提出的豪斯多夫维数博弈——该博弈为计算豪斯多夫维数提供了博弈论方法——我们采用该博弈的广义族，并证明其对于分析我们所研究的零和博弈底层集合具有重要作用。