pith. sign in

arxiv: 2605.29851 · v1 · pith:6LHIC3OTnew · submitted 2026-05-28 · 🧮 math.CO

Stotting in positional games

Johannes Carmesin , Yannick Mogge This is my paper

Pith reviewed 2026-06-29 06:33 UTC · model grok-4.3

classification 🧮 math.CO
keywords positional gamesMaker-BreakerWaiter-ClientstottingBeck conjecturestrategy transferwinning strategies
0
0 comments X

The pith

A winning strategy in a stotting variant of Maker-Breaker or Waiter-Client games implies winning strategies in the standard versions for both players.

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

The paper introduces stotting versions of Maker-Breaker and Waiter-Client games in which one player grants the opponent a small extra move or edge. It proves that any winning strategy for the stotting game immediately supplies winning strategies for both Maker and Waiter in the ordinary versions of the same game. Many published strategies already succeed even when played with this handicap, so they deliver two classical results at once. The construction is presented as a route toward restoring a form of Beck's conjecture relating the two game families after its known disproof.

Core claim

We introduce variants of the Maker-Breaker and Waiter-Client games, which we call stotting, in which a player grants a slight advantage to the opponent. We prove that a winning strategy in either stotting variant yields winning strategies for both Maker and Waiter in the classical setting. Several existing Maker strategies in the literature in fact win with stotting, and therefore automatically provide both classical winning strategies (and similarly for stotting Waiter).

What carries the argument

Stotting variants of Maker-Breaker and Waiter-Client games, in which one player deliberately grants the opponent a slight advantage, that transfer directly to classical winning strategies for both roles.

If this is right

  • A stotting win for Maker supplies a classical Maker win.
  • A stotting win for Maker supplies a classical Waiter win.
  • Strategies already known to work under stotting conditions deliver both classical results simultaneously.
  • The stotting lens supplies a systematic way to strengthen existing results on the link between Maker-Breaker and Waiter-Client games.

Where Pith is reading between the lines

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

  • The same transfer technique could be tested on other biased positional games to produce paired classical results.
  • If the advantage definition is altered even modestly the implication may break, pointing to a need to check boundary cases.
  • Stotting could be used to revisit the specific counterexamples to Beck's conjecture and isolate where the classical link fails.

Load-bearing premise

The precise form of the slight advantage granted in the stotting rules must be compatible with the move orders and winning sets used in existing strategy proofs.

What would settle it

A concrete strategy that wins its stotting game yet fails to produce a winning strategy for Maker or for Waiter in the matching classical game.

Figures

Figures reproduced from arXiv: 2605.29851 by Johannes Carmesin, Yannick Mogge.

Figure 1
Figure 1. Figure 1: When a lion pursues a herd of antelopes, some individuals engage [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

We introduce variants of the Maker-Breaker and Waiter-Client games, which we call \emph{stotting}, in which a player grants a slight advantage to the opponent. We prove that a winning strategy in either stotting variant yields winning strategies for both Maker and Waiter in the classical setting. Several existing Maker strategies in the literature in fact win with stotting, and therefore automatically provide both classical winning strategies (and similarly for stotting Waiter). Knox previously disproved a conjecture of Beck asserting that whenever Maker wins the Maker-Breaker game, Waiter also wins the corresponding Waiter-Client game; in this sense, our framework may be viewed as a way of repairing Beck's conjecture.

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

0 major / 3 minor

Summary. The paper introduces stotting variants of Maker-Breaker and Waiter-Client positional games in which one player grants a slight advantage to the opponent. It proves that any winning strategy for the stotting variant immediately yields winning strategies for Maker and for Waiter in the corresponding classical games. The authors observe that several published Maker strategies already satisfy the stronger stotting condition and therefore supply both classical results at once; the framework is offered as a possible repair to Beck's conjecture (disproved by Knox) relating the two game families.

Significance. If the central implication holds, the stotting framework supplies a uniform method for obtaining paired Maker-Breaker and Waiter-Client results from a single, stronger strategy. By verifying that existing strategies already meet the stotting condition, the paper converts prior work into simultaneous results for both game types without additional case analysis. This meta-result is a modest but potentially reusable tool in positional game theory.

minor comments (3)
  1. The precise definition of the 'slight advantage' granted in each stotting variant should be stated explicitly in the introduction (before the main theorem) so that readers can immediately verify that the stotting player indeed possesses strictly fewer or weaker moves than in the classical rules.
  2. When the authors assert that 'several existing Maker strategies in the literature in fact win with stotting,' the paper should cite the specific theorems or sections of those works and briefly indicate which stotting condition each satisfies.
  3. A short remark comparing the stotting condition to the standard bias or move-order parameters used in the positional-games literature would help situate the new variant for readers familiar with Beck's or Knox's results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper and for recommending minor revision. No major comments appear in the report, so we have no specific points to address point-by-point. We will incorporate any minor changes requested by the editor or referee in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the stotting variants independently as games in which one player grants a handicap (strictly fewer or weaker options) to the opponent. The central claim is a direct logical implication: any strategy that wins the handicapped stotting game necessarily wins the corresponding classical Maker-Breaker or Waiter-Client game. This follows immediately from the definitions without any reduction to fitted parameters, self-referential equations, or load-bearing self-citations. Existing literature strategies are simply verified to satisfy the stronger stotting condition; the implication itself requires no additional assumptions about hypergraph structure or prior results by the authors. The reference to Knox is an external citation and does not form part of the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on the standard definitions of Maker-Breaker and Waiter-Client games from the positional-games literature and introduces the stotting variant as a new object.

axioms (1)
  • domain assumption Standard rules and winning conditions of Maker-Breaker and Waiter-Client positional games on hypergraphs or graphs
    The stotting variants are built directly on these established game rules.
invented entities (1)
  • Stotting game variant no independent evidence
    purpose: A modified positional game in which one player is forced to grant the opponent a slight advantage
    Newly defined in the paper to obtain stronger strategy implications.

pith-pipeline@v0.9.1-grok · 5640 in / 1327 out tokens · 29880 ms · 2026-06-29T06:33:57.265393+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

10 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Tree univer- sality in positional games.Combinatorics, Probability and Computing, 34(3):338–358, 2025

    Grzegorz Adamski, Sylwia Antoniuk, Małgorzata Bednarska-Bzdęga, Dennis Clemens, Fabian Hamann, and Yannick Mogge. Tree univer- sality in positional games.Combinatorics, Probability and Computing, 34(3):338–358, 2025

    2025

  2. [2]

    Positional games and the second moment method.Com- binatorica, 22(2):169–216, 2002

    József Beck. Positional games and the second moment method.Com- binatorica, 22(2):169–216, 2002. 5

    2002

  3. [3]

    Comments on bases in dependence structures.Bul- letin of the Australian Mathematical Society, 1(2):161–167, 1969

    Richard A Brualdi. Comments on bases in dependence structures.Bul- letin of the Australian Mathematical Society, 1(2):161–167, 1969

    1969

  4. [4]

    Chvátal and P

    V. Chvátal and P. Erdös. Biased positional games. In B. Alspach, P. Hell, and D.J. Miller, editors,Algorithmic Aspects of Combinatorics, volume 2 ofAnnals of Discrete Mathematics, pages 221–229. Elsevier, 1978

    1978

  5. [5]

    Building spanning trees quickly in maker-breaker games

    Dennis Clemens, Asaf Ferber, Roman Glebov, Dan Hefetz, and Anita Liebenau. Building spanning trees quickly in maker-breaker games. In Jaroslav Nešetřil and Marco Pellegrini, editors,The Seventh European Conference on Combinatorics, Graph Theory and Applications, pages 365–370, Pisa, 2013. Scuola Normale Superiore

    2013

  6. [6]

    Fast strategies in Waiter-Client games.The Electronic Journal of Combinatorics, 27(3):1–35, 2020

    Dennis Clemens, Pranshu Gupta, Fabian Hamann, Alexander Haupt, Mirjana Mikalački, and Yannick Mogge. Fast strategies in Waiter-Client games.The Electronic Journal of Combinatorics, 27(3):1–35, 2020

    2020

  7. [7]

    Ivett Mándity, and András Pluhár

    András Csernenszky, C. Ivett Mándity, and András Pluhár. On Chooser–Picker positional games.Discrete Mathematics, 309(16):5141– 5146, 2009

    2009

  8. [8]

    Two constructions relating to conjectures of Beck on positional games

    Fiachra Knox. Two constructions relating to conjectures of Beck on positional games.arXiv preprint arXiv:1212.3345, 2012

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  9. [9]

    The critical bias for the hamiltonicity game is(1 + o(1))n/lnn.Journal of the American Mathematical Society, 24(1):125– 131, 2011

    Michael Krivelevich. The critical bias for the hamiltonicity game is(1 + o(1))n/lnn.Journal of the American Mathematical Society, 24(1):125– 131, 2011

    2011

  10. [10]

    A solution of the shannon switching game.Journal of the Society for Industrial and Applied Mathematics, 12(4):687–725, 1964

    Alfred Lehman. A solution of the shannon switching game.Journal of the Society for Industrial and Applied Mathematics, 12(4):687–725, 1964. 6

    1964