Determining the Winner in Alternating-Move Games

TL;DR

This paper introduces a criterion based on Hausdorff dimension to determine the winner in two-player alternating-move games on trees, extending existing results to more general spaces.

Contribution

It generalizes Hausdorff dimension games to arbitrary complete metric spaces, providing a new method to estimate target set dimensions when Player I has a winning strategy.

Findings

Provides a criterion for winner determination based on Hausdorff dimension.

Extends Schmidt's results from Hilbert spaces to arbitrary complete metric spaces.

Employs generalized Hausdorff dimension games to establish lower bounds on target set dimensions.

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.