Determining the Winner in Alternating-Move Games

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. Building on the Hausdorff dimension games originally introduced by Das, Fishman, Simmons, and Urba{ń}ski, 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.