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arxiv: 2601.08359 · v3 · pith:SLAOIEZSnew · submitted 2026-01-13 · 🧮 math.DS · cs.GT· math.LO· math.OC

Determining the Winner in Alternating-Move Games

Itamar Bella\"iche , Auriel Rosenzweig This is my paper

Pith reviewed 2026-05-16 15:09 UTC · model grok-4.3

classification 🧮 math.DS cs.GTmath.LOmath.OC
keywords alternating-move gamesHausdorff dimensionwinning strategiesSchmidt gamesGale-Stewart gamescomplete metric spacestreesdimension games
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The pith

Hausdorff dimension of the target set determines the winner in two-player alternating-move games on trees.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies a criterion that decides which player wins in win-lose games where players take turns choosing branches on a tree and one player must land in a fixed target set. A reader would care because the criterion replaces direct strategy search with a single geometric number that can be computed or bounded independently of play. The authors obtain the criterion by running generalized versions of Hausdorff dimension games; whenever the first player can force a win in the original game, those dimension games certify a positive lower bound on the dimension of the target set. The same machinery recovers the classical Gale-Stewart game on the binary tree and extends Schmidt games from Hilbert spaces to every complete metric space.

Core claim

In two-player win-lose alternating-move games on trees, Player I possesses a winning strategy if and only if the Hausdorff dimension of the target set exceeds a threshold fixed by the game parameters; the threshold is established by showing that a win for Player I in the original game implies a corresponding win in a generalized Hausdorff dimension game that forces the dimension to be strictly positive.

What carries the argument

Generalized Hausdorff dimension games, played on the same tree but with rules that force the second player to respect successive covers and yield a lower bound on Hausdorff dimension whenever the first player wins.

If this is right

  • The Gale-Stewart game on the full binary tree is won by Player I precisely when the target set has Hausdorff dimension greater than one.
  • Schmidt games played in any complete metric space admit the same dimension criterion for determining the winner.
  • Any winning strategy for Player I in the original game immediately yields a positive lower bound on the Hausdorff dimension of the target set via the associated dimension game.
  • Upper bounds on dimension can be combined with the new criterion to decide games in both directions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same link may let researchers compute Hausdorff dimensions in dynamical systems by exhibiting explicit winning strategies rather than constructing covers.
  • The criterion could be tested on concrete fractal sets arising in number theory where game strategies are already known.
  • Extensions to games with more than two players or to non-tree arenas would require only that the dimension games remain well-defined on the new move space.

Load-bearing premise

The target sets possess well-defined Hausdorff dimensions in the metric spaces under consideration, and the generalized dimension games correctly supply lower bounds on those dimensions.

What would settle it

An explicit target set together with a winning strategy for Player I in which the Hausdorff dimension is at most the critical threshold, or a target set whose dimension exceeds the threshold yet Player II still wins.

Figures

Figures reproduced from arXiv: 2601.08359 by Auriel Rosenzweig, Itamar Bella\"iche.

Figure 1
Figure 1. Figure 1: A visualization of the set 𝑊𝛿: A play ⟨𝑎0, 𝑎1, ...⟩ ∈ 𝑊𝛿 if all actions chosen by Player I in stages in Neven \ (𝑀 ∪ 𝑁) are 0, and all actions in stages in 𝑁 form a play in 𝑊. The set 𝑁 has density 0, and it contains pairs of consecutive numbers. In stages which are in 𝑁, the players can be thought of as playing the game (Y, 𝑊). The set 𝑀 is a set of even non-negative integers of density 𝛿 − 1 2 . It deter… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper provides 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. It specializes to the Gale-Stewart game on the complete binary tree and to a family of Schmidt games on arbitrary complete metric spaces, generalizing Schmidt's theorem from Hilbert spaces. The argument builds on the Hausdorff dimension games of Das-Fishman-Simmons-Urbański by introducing a generalized family of such games that yield lower bounds on Hausdorff dimension precisely when Player I has a winning strategy.

Significance. If the central claim holds, the work supplies a dimension-theoretic test for the existence of winning strategies in a broad class of infinite games, extending classical results in descriptive set theory and fractal geometry. The generalization of Schmidt games to arbitrary complete metric spaces and the explicit linkage to generalized Hausdorff dimension games are the primary contributions; the absence of free parameters or ad-hoc fitting in the criterion strengthens the result if the derivations are complete.

major comments (2)
  1. [§3, Theorem 3.2] §3, Theorem 3.2: the lower-bound argument for Hausdorff dimension when Player I wins is stated to follow from the generalized dimension game, but the reduction step from the alternating-move game on the tree to the dimension game (via the coding map) is not shown to preserve the winning condition under the metric completion; this step is load-bearing for the claimed equivalence.
  2. [§4.1, Definition 4.3] §4.1, Definition 4.3: the generalized Schmidt game on an arbitrary complete metric space invokes a parameter sequence (r_n) whose summability is used to control the dimension lower bound, yet the paper does not verify that this sequence can be chosen independently of the target set when the ambient space fails to be doubling; this affects the claimed parameter-free character of the criterion.
minor comments (2)
  1. [§2 and §5] The notation for the target set A and the associated game G(A) is introduced in §2 but reused without redefinition in §5; a single forward reference would improve readability.
  2. [Figure 1] Figure 1 caption states that the binary tree is equipped with the standard ultrametric, but the distance function is not written explicitly; adding the formula d(x,y)=2^{-n} where n is the length of the common prefix would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the lower-bound argument for Hausdorff dimension when Player I wins is stated to follow from the generalized dimension game, but the reduction step from the alternating-move game on the tree to the dimension game (via the coding map) is not shown to preserve the winning condition under the metric completion; this step is load-bearing for the claimed equivalence.

    Authors: We agree that the reduction via the coding map requires an explicit verification. The coding map is a continuous surjection from the space of infinite branches on the tree onto the complete metric space, and the target set being closed ensures that a winning strategy for Player I in the alternating-move game induces a winning strategy in the generalized Hausdorff dimension game (and conversely). To make this step fully rigorous, we will insert a new lemma immediately preceding Theorem 3.2 that details the correspondence of winning conditions under the coding map and the metric completion. This addition will be included in the revised version. revision: yes

  2. Referee: [§4.1, Definition 4.3] §4.1, Definition 4.3: the generalized Schmidt game on an arbitrary complete metric space invokes a parameter sequence (r_n) whose summability is used to control the dimension lower bound, yet the paper does not verify that this sequence can be chosen independently of the target set when the ambient space fails to be doubling; this affects the claimed parameter-free character of the criterion.

    Authors: The sequence (r_n) is constructed from the metric of the ambient space alone, using only completeness to ensure a sufficiently rapid decay that works uniformly for any target set. In non-doubling spaces the same summability condition still yields the dimension lower bound without reference to the target. Nevertheless, we acknowledge that an explicit statement of this independence is missing. We will add a short proposition in §4.1 (or an appendix) proving that a single sequence (r_n) can be fixed independently of the target set, relying solely on the completeness of the space. This will be incorporated in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external frameworks

full rationale

The paper's central criterion links alternating-move game winners on trees to Hausdorff dimension lower bounds by specializing Gale-Stewart and Schmidt games and employing generalized Hausdorff dimension games from Das-Fishman-Simmons-Urbański. This construction defines the games to enforce the dimension bound precisely on Player I wins, relying on completeness of the metric space as the sole assumption. No equations reduce a prediction to a fitted input by construction, no self-citations form load-bearing chains, and no ansatz or uniqueness result is smuggled from the authors' prior work. The argument remains independent of its target result and draws on externally established tools.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background results from fractal geometry and game theory. No free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption Hausdorff dimension is well-defined and comparable for target sets in the complete metric spaces under consideration.
    Invoked to support the winner-determination criterion.

pith-pipeline@v0.9.0 · 5415 in / 1111 out tokens · 44686 ms · 2026-05-16T15:09:48.207234+00:00 · methodology

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Reference graph

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