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REVIEW 2 major objections 5 minor 9 references

First to reach $n$ game

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper derives an exact Catalan-number formula for the expected profit in a first-to-$n$ match with constant round odds, proves a central limit theorem for the profit, and shows that an urn without replacement makes the loser's…

desk verdict Solid paper with a real misstatement in the Pólya theorem; the constant-model results are correct and genuinely new, but Theorem 3.1 must be restated as a limit. read the letter →

arxiv 2506.08782 v4 pith:ZBE7K2S3 submitted 2025-06-10 math.PR

Stanislav Volkov , Magnus Wiktorsson This is my paper · ORCID
classification math.PR MSC 60C0560J1091A05
keywords firsttoreachnbest-of-nmatchPólyaurnanti-OKCorralCatalannumberscentrallimittheoremnegativebinomialgambler'sruin
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the final margin in a match played until one player wins $n$ rounds, under three regimes for the per-round win probability. With constant round odds it proves the exact formula $\mathbb{E}Z_{n,p}=n(p-q)\sum_{j=0}^{n-1} C_j(pq)^j$, with $C_j$ the Catalan numbers, and a central limit theorem: after centering by $\mu n$ and scaling by $\sqrt{nq}$, the profit is asymptotically standard normal. In a Pólya-urn version where wins reinforce themselves, it derives a closed-form integral for the first player's win probability and an asymptotic winner's profit of order $\sqrt{n}$. In the without-replacement 'anti-OK Corral' urn it proves that the loser's deficit converges to an equal mixture of two Geometric($1/2$) distributions, so the margin stays bounded. The takeaway is that the same win condition behaves very differently when the round probabilities are fixed, self-reinforcing, or depleting.

What carries the argument

The load-bearing identity is the Catalan convolution behind Theorem 2.1: writing the expected profit as a double sum over the loser's score and rewriting the inner sum with a binomial identity (Lemma 5.1) yields $n(p-q)\sum_{j=0}^{n-1} C_j(pq)^j$. The distributional results are carried by a Poisson-process representation: two independent Poisson processes with rates $1$ and $\lambda=q/p$ have the same embedded chain as the game, so the losing player's score is negative binomial and standard CLT estimates apply. In the anti-OK Corral model, the difference $D_k$ between the two remaining ball counts is a Markov chain with transition probabilities $\frac12(1\mp D_k/(2n-k))$, a random-walk bridge, and exact combinatorial counting plus Stirling's formula gives the geometric limit.

What would settle it

Check the Pólya-urn formula at $N_1=2$, $N_2=1$, and $n=2$: an explicit enumeration of all game paths gives first-player win probability $7/10$, whereas the printed integral gives $3/4$; this single case determines whether the theorem is valid as an exact finite-$n$ identity or only as a limit statement.

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Extended reading notes

Core claim

On its own terms, the paper claims a complete asymptotic and, in the constant case, exact description of the first-to-$n$ game. The central formula is Theorem 2.1: $E_{n,p}=\mathbb{E}Z_{n,p}=n(p-q)\sum_{j=0}^{n-1} C_j(pq)^j$, where $C_j=\frac{1}{j+1}\binom{2j}{j}$; the paper derives it by summing the probabilities of each possible losing score and collapsing a double sum through Lemma 5.1. Theorem 2.4 adds that $P(W_{n,p}=k)=\binom{n+k-1}{n-1}q^k p^n+\alpha_{n,p}$ with $|\alpha_{n,p}|\le(4pq)^n$, so the loser's score is close to a negative binomial, and that $(Z_{n,p}-\mu n)/\sqrt{nq}\Rightarrow N(0,1)$. For the Pólya urn with $N_1,N_2$ initial balls the win probability is claimed to be $\frac{\Gamma(N_1+N_2)}{\Gamma(N_1)\Gamma(N_2)}\int_0^1 \frac{(1-x)^{N_2-1}}{(2-x)^{N_1+N_2}}\,dx$. For the anti-OK Corral urn, Theorem 4.1 gives that the probability the loser finishes $k$ points behind tends to $2^{-k-1}$, equivalently a fifty-fifty mixture of two Geometric($1/2$) laws for the net profit.

Load-bearing premise

In the Pólya-urn regime, the proof assumes the limiting Beta composition can be substituted for the evolving urn and that the constant-model normal approximation can be averaged over that composition; the printed finite-$n$ equality is not a direct consequence of that argument.

Editorial extensions

If this is right

  • For fixed round odds $p>q$, $\mathbb{E}Z_{n,p}/n\to 1-q/p$; a small per-round advantage translates into an expected final margin proportional to $n$.
  • For fair odds $p=q=1/2$, the expected winner's profit grows like $2\sqrt{n/\pi}\approx 1.13\sqrt{n}$, so the final margin is much smaller than the match length.
  • The central limit theorem means that, for large $n$, the profit is approximately normal with mean $\mu n$ and standard deviation $\sqrt{nq}$, giving simple interval estimates for the final margin.
  • In the anti-OK Corral urn, the loser's deficit takes value $k$ with limiting probability $2^{-k-1}$, so large blowouts decay exponentially and match length is concentrated near $2n$.
  • In the Pólya-urn model with $N_1=N_2=n$, the expected winner's profit is $2\sqrt{n/\pi}-1+O(n^{-1/2})$, the same leading order as the fair constant model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The Catalan partial-sum formula can be read as an exact computation algorithm for best-of-$n$ contracts: the only parameter needed is $pq$, and the same generating function gives finite-$n$ corrections to the asymptotic mean.
  • The Poisson-process representation likely extends to scoring systems where each point has a random duration; the Gamma-hitting-time argument already contains the tail estimates needed for match-length bounds.
  • The anti-OK Corral geometric limit predicts that in resource-depleting formats a large comeback is exponentially rare, a testable distinction from fixed-odds formats where the trailing player's deficit has order $\sqrt{n}$ or $n$.
  • If the Pólya-urn integral is meant as a limit rather than an exact finite-$n$ identity, adding correction terms of order $1/n$ would reconcile the formula with exact small-$n$ enumeration; that is a concrete check for future work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies a two-player 'first to reach n wins' game under three per-round probability regimes: a constant-probability model, a Pólya urn (reinforcement) model, and an anti-OK Corral (draw-without-replacement) model. The main results are an exact Catalan-number formula for the expected net profit in the constant model (Theorem 2.1), a CLT for the net profit with explicit centering and scaling (Theorem 2.4(b)), an asymptotic formula for the win probability in the Pólya model (Theorem 3.1), and a limiting Geometric(1/2) mixture for the loser's margin in the anti-OK Corral model (Theorem 4.1). The paper also gives partial proofs of the auxiliary lemmas.

Significance. If the statements are corrected as indicated below, the paper is a solid contribution to the urn/gambler's-ruin literature. The constant-model formula E_{n,p} = n(p-q) sum_{j=0}^{n-1} C_j (pq)^j is elegant and machine-verifiable for small n; the CLT scaling (p Z_{n,p} - (p-q)n)/sqrt(n q) -> N(0,1) is natural and correctly identified. The anti-OK Corral limiting distribution, with margin probabilities 2^{-k-1}, is a clean closed form. The paper is clearly written and the constant-model and anti-OK Corral results appear sound; the Pólya section contains a serious but local statement error that needs correction.

major comments (2)
  1. [Section 3, Theorem 3.1] The theorem is stated as an exact finite-n identity, but the proof establishes only a limit as n -> infinity. The right-hand side does not depend on n, so the statement is false for finite n; for example, with N1=2, N2=1, n=2, enumerating the Pólya draw sequences gives P(Player 1 wins) = 7/10, while the displayed integral equals 3/4. The theorem should be restated as an asymptotic statement: P(Player 1 wins overall) -> Gamma(N1+N2)/(Gamma(N1)Gamma(N2)) times the integral shown. The same correction should be applied to Remark 3.2, which currently reads as an exact equality for finite n.
  2. [Section 3, proof of Theorem 3.1] The proof applies Theorem 2.4(b) conditionally on xi, but Theorem 2.4(b) is stated for the constant model with p >= 1/2. For xi < 1/2, the CLT must be applied to Player 2's profit, and the displayed normalization with max(xi,1-xi) and min(xi,1-xi) is correct only after this reversal is made explicit. The averaging over xi is legitimate because the conditional convergence is pointwise in xi and P(xi = 1/2) = 0, but this justification should be stated.
minor comments (5)
  1. [Section 2, Theorem 2.4(b)] The notation 'pZn,p - µn over sqrt(nq)' is ambiguous; it should be written explicitly as (p Z_{n,p} - (p-q)n)/sqrt(n q) -> N(0,1) in distribution.
  2. [Section 5, proof of Theorem 2.4, p > 1/2 case] The displayed probability should be P(tau_X >= tau_Y), not P(tau_X <= tau_Y), to match Lemma 5.2(a) and the subsequent conditioning on tau_X < tau_Y. The sign error is in the proof only and does not affect the theorem statement.
  3. [Lemma 5.2(b), proof] The second derivative of g(y) = y^2/(4(y+1)) is g''(y) = 1/(2(1+y)^3), not 1/(2(1+y)^2) as written. The inequality f''(y) >= g''(y) still holds with the corrected formula, so the proof is repairable. Also, in the same proof, 'P(eta >= n+a)' should be 'P(eta >= m+a)' and the term 'n log' should be 'm log'.
  4. [Abstract and Section 2] The abstract says the first and third regimes draw without replacement, but the constant model of Section 2 is an i.i.d. model, not a finite without-replacement urn; it is only described as an asymptotic urn limit with N1/N2 ~ p/q. Please clarify this wording to avoid confusion.
  5. [Section 4, equation (4)] The combinatorial probability (4) is stated without derivation. It is correct (and verified for k=1,2), but a short derivation or reference would improve the exposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all load-bearing results are derived from independent lemmas proved in the paper; self-citations are background only.

full rationale

The paper's derivation chain is self-contained and acyclic. Theorem 2.1 is proved by direct combinatorial enumeration and the polynomial identity in Lemma 5.1; no quantity is fitted from the object it predicts. Theorem 2.4(b) is proved from a Poisson-process representation and Lemma 5.2, with all estimates established in the paper. Theorem 3.1 conditions on the Beta-distributed limit composition of the Pólya urn and then applies Theorem 2.4(b) conditionally; this is an internal dependency on a theorem proved earlier in the same paper, not a circular reduction, and the convergence argument is legitimate because the conditioning variable has no atom at 1/2. Theorem 4.1 is a direct combinatorial calculation followed by Stirling's formula. The self-citations to Engländer-Volkov and Kingman-Volkov are references to general techniques and background models, not load-bearing premises. The only notable defect is that Theorem 3.1 is stated as an exact finite-n identity although the proof establishes an n→∞ limit; this is a correctness issue, not circularity, and therefore does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on classical probabilistic machinery (Poissonization, Rubin's construction, bridge counting) and elementary combinatorics, all either derived in the paper or standard. There are no free parameters and no invented entities. The main structural caveat is Theorem 3.1's unstated n to infinity limit, which is a statement-precision defect rather than a hidden assumption.

assumptions (6)
  • standard math Optional Stopping Theorem
    Used in Section 2.1 to derive heuristic bounds on E tau and E|X_tau|; the rigorous theorems do not rely on it.
  • standard math Rubin's construction: Pólya urn draws are exchangeable and equal to i.i.d. Bernoulli(xi) conditional on xi ~ Beta(N1, N2)
    Invoked in the proof of Theorem 3.1 to reduce the Pólya model to the constant model.
  • standard math Poisson process coupling: the embedded chain of two independent Poisson processes has the same law as the constant-model game
    Used in the proof of Theorem 2.4; the construction is classical and described in the text.
  • standard math Catalan generating function c(z) = (1 - sqrt(1-4z))/(2z)
    Used in Remark 2.2 to identify the limit of the expectation formula.
  • standard math Stirling's formula
    Used in the proof of Theorem 4.1 to evaluate the combinatorial limit 2^{-k-1}.
  • domain assumption Game definition: independent or urn-governed round outcomes, with play stopping when a player reaches n wins
    The model definition in Section 1, treated as the object of study rather than an empirical claim.

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Cite this review

Pith. "Pith review of First to reach $n$ game." pith.science (2026). https://pith.science/paper/ZBE7K2S3

@misc{pith2026250608782,
  author       = {Pith},
  title        = {Pith review of: First to reach $n$ game},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZBE7K2S3}},
  note         = {Machine review of arXiv:2506.08782}
}
abstract

We consider a game with two players, consisting of a number of rounds, where the first player to win $n$ rounds becomes the overall winner. Who wins each individual round is governed by a certain urn having two types of balls (type 1 and type 2). At each round, we randomly pick a ball from the urn, and its type determines which of the two players wins. We study the game under three regimes. In the first and the third regimes, a ball is taken without replacement, whilst in the second regime, it is returned to the urn with one more ball of the same colour. We study the properties of the random variables equal to the properly defined overall net profits of the players, and the results are drastically different in all three regimes.

Figures

Figures reproduced from arXiv: 2506.08782 by the authors.

Figure 1
Figure 1. A path of the process in case τY < τX. Now, X(τY ) is also a negative binomial with parameters 1/2 and n, so again by part (b) of Lemma 5.2 P(|ν| ≥ n 2/3 ) = P(|X(τY ) − n| ≥ n 2/3 ) ≤ 2 exp  − n 4/3 4(n + n2/3 )  ≤ 2 exp  − n 1/3 8  Hence P(|η − ν| ≥ n 2/5 ) ≤ P(|η − ν| ≥ n 2/5 , ν < n2/3 ) + 2 exp  − n 1/3 8  ≤ 2 exp  − n 2/15 8  + 2 exp  − n 1/3 8  → 0. Since η − ν = (Y (τX) − n) − (n − X(τY )) = Y (τX)… view at source ↗

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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    work page arXiv 2025

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    Williams, D., and McIlroy, P. (1998). The OK Corral and the power of the law (a curious Poisson kernel formula for a parabolic equation). Bull.\ London Math.\ Soc., 30 , 166--170

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