Determining the Winner in Alternating-Move Games

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.