Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Determining the Winner in Alternating-Move Games

Itamar Bellaïche, Auriel Rosenzweig

TL;DR

The article demonstrates that in two-player win-lose alternating-move games on trees, the winner can often be determined by the target set’s Hausdorff dimension, with a sharp threshold at 1/2: whenever $\dim_{\,\mathcal{H}}(W) < \tfrac{1}{2}$, Player II has a winning strategy. It develops the dyadic and generalized Hausdorff dimension games as a robust, game-theoretic method to compute or bound Hausdorff dimensions and to transfer dimension information into determinacy results. By connecting these dimension games to Schmidt games on doubling spaces, the authors extend the criterion to a broad class of geometric games, establishing dimension-based sufficient conditions for II to win and providing tightness results and extensions to m-adic trees and subgames. The work combines topological determinacy (Martin) with fractal geometry (Hausdorff dimension) to yield a versatile framework with potential implications for Diophantine approximation, metric geometry, and dynamical systems.

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.

Determining the Winner in Alternating-Move Games

TL;DR

The article demonstrates that in two-player win-lose alternating-move games on trees, the winner can often be determined by the target set’s Hausdorff dimension, with a sharp threshold at 1/2: whenever , Player II has a winning strategy. It develops the dyadic and generalized Hausdorff dimension games as a robust, game-theoretic method to compute or bound Hausdorff dimensions and to transfer dimension information into determinacy results. By connecting these dimension games to Schmidt games on doubling spaces, the authors extend the criterion to a broad class of geometric games, establishing dimension-based sufficient conditions for II to win and providing tightness results and extensions to m-adic trees and subgames. The work combines topological determinacy (Martin) with fractal geometry (Hausdorff dimension) to yield a versatile framework with potential implications for Diophantine approximation, metric geometry, and dynamical systems.

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.
Paper Structure (38 sections, 31 theorems, 142 equations, 1 figure)

This paper contains 38 sections, 31 theorems, 142 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Win-lose alternating-move games.
  3. Who wins the game?
  4. Main result.
  5. Schmidt games.
  6. $\left(\alpha ,\beta \right)$-Schmidt games.
  7. Hausdorff dimension games
  8. Preliminaries
  9. Alternating-Move Games
  10. Hausdorff Dimension
  11. Hausdorff Dimension Games
  12. The Dyadic Hausdorff Dimension Game
  13. Value of the Dimension Game
  14. Hausdorff Dimension as the Density of Unconstrained Coordinates
  15. Determining the Winner
  16. ...and 23 more sections

Key Result

Theorem 1.3

Let $W\subseteq \left\llbracket \mathbb{Y}\right\rrbracket$ such that $\dim_{\mathcal{H}}\left(W\right)<\frac{1}{2}$. Then Player II can guarantee a win in the game $G=\left(\mathbb{Y},W\right)$, and, in particular, $W$ is determined.

Figures (1)

  • Figure 1: A visualization of the set $W_\delta$: A play $\langle a_0,a_1,...\rangle \in W_\delta$ if all actions chosen by Player I in stages in $\mathbb{N}_{\text{even}}\setminus \left(M\cup N\right)$ are $0$, and all actions in stages in $N$ form a play in $W$.

Theorems & Definitions (88)

  • Example 1.1: label=CantorExample
  • Example 1.2
  • Theorem 1.3
  • Example 1.4: continues=CantorExample
  • Remark 1.5
  • Theorem 1.6: Schmidt1965, Corollary 1
  • Definition 2.1: tree and positions
  • Definition 2.2: plays and canopy
  • Definition 2.3: zero-sum alternating-move game
  • Theorem 2.4: Gale and Stewart (1953, GaleStewart1953)
  • ...and 78 more