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.

翻译：我们提供了一个准则，用于判定树结构上两人零和交替移动博弈中的胜者，该准则基于目标集的豪斯多夫维数。我们重点研究了若干特例，包括完全二叉树上的Gale-Stewart博弈及一类Schmidt博弈，将Schmidt从希尔伯特空间到任意完备度量空间的结果进行了推广。基于Das、Fishman、Simmons和Urbański最初引入的豪斯多夫维数博弈（该博弈提供了计算豪斯多夫维数的博弈论方法），我们采用这类博弈的推广族，在Player I能够保证获胜时，为目标集的豪斯多夫维数建立下界。