Winning Probabilities of Balanced and Nontransitive n-tuples of Dice

Joshua Rooney

REVIEW 2 major objections 5 minor 17 references

Winning Probabilities of Balanced and Nontransitive n-tuples of Dice

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

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Pith's one-line read This paper proves that every rational number in (1/2, π_n] can be realized as the common winning probability of a balanced nontransitive n-tuple of dice, completing the classification.

desk verdict Settles the Kim et al. open problem with a full classification of achievable winning probabilities; proof is sound overall, but Lemma 3.2's transposition argument needs a careful rewrite. read the letter →

arxiv 2505.21950 v1 pith:G6UHKW5S submitted 2025-05-28 math.CO

Joshua Rooney This is my paper · ORCID

classification math.CO MSC 60C05

keywords nontransitivedicebalancedwinningprobabilitycentralwordsadjacenttranspositionsleastupperboundrationaldensitycombinatorial

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

Balanced and nontransitive dice are arranged in a cycle so that each die beats its successor with probability greater than $1/2$, and all these probabilities are equal. This paper proves that for every integer $n \geq 3$ and every rational $p$ strictly between $1/2$ and the bound $\pi_n = 1 - \frac{1}{4\cos^2(\pi/(n+2))}$, there exists a balanced nontransitive $n$-tuple of dice whose common winning probability is exactly $p$. Combined with the known upper bound $w \leq \pi_n$ [2, 9], this fully characterizes the set of achievable winning probabilities and answers the question posed in [7]. The result matters because it turns a long-studied probabilistic paradox into a precisely understood phenomenon: the winning chance can be tuned to any rational value up to a sharp ceiling.

What carries the argument

The carrying object is the central word: a word $\sigma = \sigma(A_1,\dots,A_n)$ in which all faces of $A_n$ appear in a single contiguous block, with type $(a_1,\dots,a_{n-1})$ recording how many of the letters $A_1,\dots,A_{n-1}$ lie before that block. Lemma 3.2 extends a lemma of [7] to $n \geq 4$: for any prescribed counts $s_i$ in the natural intervals, a central word of a given type exists with $N_\sigma(A_i < A_{i+1}) = s_i$ for each $i = 1,\dots,n-2$. The proof proceeds by induction, inserting the new die as a block and then applying adjacent transpositions that change exactly one comparison count at a time. Lemma 3.3 uses the continuity of the bracketing functions $L$ and $U$ and the density of the rationals to produce rational parameters $p_2,\dots,p_k$ that keep the prescribed target $p$ inside the required interval, so that the integer-interval conditions of Lemma 3.2 hold for a suitably large even $m$.

What would settle it

One could enumerate, for a small pair $(n, m)$ and a fixed type $(a_1,\dots,a_{n-1})$, all central words and compare the attainable values of $N_\sigma(A_{j-1} < A_j)$ with the interval claimed in Lemma 3.2; a single missing integer would falsify the lemma. A more targeted test runs the inductive transposition process from Section 4.1 on a concrete example and checks whether the count increases by exactly one at each step or jumps over any value.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: for every $n \geq 3$ and every rational $p \in (1/2, \pi_n]$, there is a positive integer $m$ and a balanced nontransitive $n$-tuple of $m$-sided dice $(A_1,\dots,A_n)$ with $w(A_1,\dots,A_n) = P(A_1 < A_2) = \cdots = P(A_n < A_1) = p$. The proof treats the endpoint separately, showing that $\pi_n$ is rational only when $n = 4$, where $\pi_4 = 2/3$ and the construction follows directly from Lemma 3.2. For $p < \pi_n$, the proof selects rational parameters $p_2,\dots,p_k$ by a continuity and density argument around the trigonometric values defined in [9], then applies the extended word lemma to realize the desired comparison counts.

Load-bearing premise

The load-bearing premise is the induction step of Lemma 3.2, which assumes that adjacent transpositions can move the comparison count $N_\sigma(A_{j-1} < A_j)$ through every integer in its claimed interval while preserving all earlier counts; the informal description of that step does not rigorously rule out skipped values.

Editorial extensions

If this is right

The achievable winning probabilities of balanced nontransitive $n$-tuples are exactly the rationals in $(1/2, \pi_n]$, so the set of attainable values is dense in that interval.
Problem 1.4 of [7] is settled: the least upper bound for $n \geq 4$ is $\pi_n$, and it is attained only for $n = 4$, where $\pi_4 = 2/3$.
Every rational winning probability in the range can be realized using dice that all have the same number of sides $m$, so unequal side counts are never necessary for exact realizations.
Since $\pi_n < 3/4$ for every $n$, no balanced nontransitive collection of dice can have a common winning probability of $3/4$ or more, no matter how many dice are used.
The same word-tuning machinery is pointed out by the author as a likely route toward the open side-count problem posed in [7].

Reading between the lines

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

The informal 'step 2' in the proof of Lemma 3.2 deserves a rigorous justification; if a gap exists, a different rearrangement scheme might still realize every integer count, so the main theorem need not fall.
The trigonometric parameters used in Lemma 3.3 are ratios of consecutive sine values, hinting at a spectral interpretation of the bound $\pi_n$ that could extend to other cyclic comparison problems.
The construction gives no bound on the minimal number of sides $m$ needed for a given rational $p$; numerical experiments near $p = \pi_n$ could reveal how that minimal $m$ grows.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

You should know two things about this paper quickly. First, it solves Kim et al.'s problem on balanced nontransitive dice: for every n≥3 and every rational p in (1/2, π_n], there is a balanced nontransitive n-tuple of dice with winning probability p. Together with the Bogdanov–Komisarski upper bound, this is the full characterization. Second, the proof is mostly checkable, but the key lemma, Lemma 3.2, has an under-specified transposition step that needs a careful rewrite before I would call it rigorous.

The new contribution is genuine. The n=3 case was already classified; here the construction extends to all n, and the paper handles the endpoint p=π_n (which forces n=4) as a separate case. Lemma 3.2 generalizes Kim et al.'s pairwise-count lemma to n dice, and Lemma 3.3 uses Komisarski's p_j^* values plus a density argument to find rational auxiliary parameters. The algebra for the construction lines up cleanly, and the paper does not overclaim: it states the rationals in the interval as the achievable set, consistent with the upper bound.

The soft spot is in Section 4.1. The inductive step says to apply 'as many transpositions A_j A_α -> A_α A_j as possible' and then one A_j A_{j-1} -> A_{j-1} A_j, repeating until the count q_{j-1} reaches the target. That is a bubble-sort idea, but the direction is not specified: which A_j moves, what 'as possible' means, and why an A_{j-1} is always adjacent after the A_α moves. If the reader interprets 'as possible' globally, an A_j could shoot past all the small-index letters and never meet an A_{j-1}. I think the lemma is true and the interval formula is consistent with the invariants, so this is a rigor gap in the write-up, not a fundamental problem. The typesetting is also rough: several fractions in the displayed equations are garbled in the arXiv text, which made checking Lemmas 4.3 and 4.4 slower. Probably a conversion artifact, but it should be fixed.

This paper is for anyone working on nontransitive dice or combinatorial probability, and it is a nice application of a density argument to turn an upper bound into a constructive classification. It deserves a serious referee. My recommendation: send it to review, ask the author to formalize the transposition schedule in Lemma 3.2 and clean up the rendering, then accept.

Referee Report

2 major / 5 minor

Summary. The paper studies balanced and nontransitive n-tuples of dice (BN n-tuples) and characterizes the set of achievable common winning probabilities. For n≥3 and every rational p in (1/2, π_n], where π_n = 1 − 1/(4 cos^2(π/(n+2))), the paper constructs a BN n-tuple of dice, each with the same number of sides, whose winning probability equals p. Together with the known upper bound of Bogdanov and Komisarski, this gives a complete classification of the achievable winning probabilities for BN n-tuples of dice. The proof represents dice by words, reduces the construction to a word-combinatorial lemma (Lemma 3.2), and uses a rational approximation argument based on trigonometric quantities from Komisarski's work (Lemma 3.3).

Significance. If the proof is made fully rigorous, this is a complete resolution of Problem 1.4 posed by Kim et al., extending the known n=3 classification to all n≥4 and also covering n=3. The paper's construction is explicit and combinatorial, and the endpoint case p=π_n is handled with concrete computations. The re-derivation of the needed properties of Komisarski's auxiliary values in Lemmas 4.3 and 4.4 is elementary and checkable. The main value added is the matching lower-bound construction, since the upper bound is external. The paper also ships the key reduction in a clean form, which should make the result usable for further questions such as Problem 5.1.

major comments (2)

[Section 4.1] The proof of Lemma 3.2 is not fully specified. In the inductive step, σ_{i+1} is defined by 'Perform as many transpositions A_jA_α → A_αA_j for α ∈ {1,...,j−2} as possible' followed by 'Perform one transposition A_jA_{j−1} → A_{j−1}A_j'. This does not identify which copy of A_j is moved, whether the swaps are applied in the left or right region relative to the A_{j+1} block, or in what order. The assertion that after step 2 an adjacent A_jA_{j−1} pair is available is not proved; a global reading of 'as many as possible' could move A_j's past A_α's in one region without producing an A_jA_{j−1} adjacency where it is needed. Consequently, the claim that q_{j−1} takes every integer value in the stated interval is an unproven scheduling statement. Since Lemma 3.2 is the engine of Theorem 1.2, this must be repaired, for instance by giving an explicit algorithm that moves a chosen A_j rightward past A_α's until it reaches an A_{j−1}, performs the swap, and then proceeds, in each region, and by proving that this exhausts the interval.
[Sections 3 and 4.2] Lemma 3.3 and its use in the proof of Theorem 1.2 have an indexing gap for n=3 and n=4. For these n we have k=floor((n−1)/2)=1, so the tuple p_2,...,p_k is empty, yet the last line of the inequalities (3) uses p_1, and the proof of Theorem 1.2 goes on to define a_ℓ using p_{n−(k+1)} etc., which is only meaningful for k≥2. Thus the proof as written does not cover the case n=4 with p<π_4, even though n=4 is part of the theorem and of the motivating problem. This is fixable by treating n=3 via Theorem 1.3 and n=4 via the same direct construction used for p=π_4 (that construction works for all p∈(1/2,2/3]), or by extending the definition of p_j to j=1. As written, however, the proof is incomplete for these cases.

minor comments (5)

[Throughout] The typeset text contains many garbled fractions and broken expressions, such as 'greater than1\n2', 'πn := 1 − 1\n4 cos2( π\nn+2 )', and 'p ∈\n 1\n2 , πn\n'. This makes the paper unnecessarily hard to read and should be corrected in the final version.
[Section 4.1] The examples in Observation 4.2 appear garbled. For Case 1, swapping A_3A_1 to A_1A_3 in σ=A3 A1 A3 A4 A3 A2 A3 gives A1 A3^2 A4 A3 A2 A3, not the word printed. Similar corruption appears in the Case 2 and Case 3 examples.
[Lemma 4.3] The sentence 'Since cos(α) < 1 < 1/2−√2' is unintelligible as printed. The inequality being proved for the odd-n case ((1−p*_k)^2 < 1/2) is true, but the printed argument is not understandable and should be rewritten.
[Lemma 3.3] The notation 'there exists rationals p_2,...,p_k' is problematic when k=1, since the list is empty but the displayed inequalities still refer to p_1 in the last line. The statement should either require n≥5 or define p_1 for the small cases.
[Abstract and Introduction] There are small typos: 'nontransitve' in the abstract, and 'P(A1, < A2)' with an erroneous comma in the Introduction.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is self-contained and the upper bound is an independent external input.

full rationale

I walked the derivation chain and found no step in which a claimed prediction reduces by construction to an input or to a self-citation. The main theorem combines two independent halves: the Bogdanov-Komisarski upper bound w <= pi_n is imported as an external theorem from [2,9], while Theorem 1.2 supplies a lower-bound construction. That construction is explicit: the paper sets m_i = m, s_j = pm^2, and a_l to explicit linear functions of p and the auxiliary p_j, then verifies the box inequalities (2) using the k inequalities from Lemma 3.3. Lemma 3.3 is proved inside the paper: the trigonometric p*_j were indeed first defined in Komisarski [9], but the paper re-derives the needed identities in Lemmas 4.3 and 4.4, and the passage to rational p_j uses only continuity and density of the rationals. No fitted parameter is renamed as a prediction. Lemma 3.2 is an extension of Lemma 3.1 of Kim et al. [7], but it is a combinatorial existence lemma, not a fitted input, and the inductive proof in Section 4.1 attempts to construct the desired word from scratch. The informal scheduling step ('Perform as many transpositions A_j A_alpha -> A_alpha A_j ... as possible') is a possible rigor gap in showing that every intermediate count q_{j-1} is attained, but that is a correctness concern, not circularity: the step does not assume the conclusion w = p or derive the conclusion from the same quantity it is meant to predict. There are no self-citations by the author that carry logical weight. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

This is a pure mathematics paper with no empirical fitting. The only freely chosen constants are the die size m and the auxiliary p_i, both of which are existential construction parameters with proofs of existence. The core assumptions are standard background facts plus the cited upper bound of Bogdanov and Komisarski. No new entities are postulated.

free parameters (2)

m (number of sides per die) = any sufficiently large even integer
Chosen in Theorem 1.2 large enough so that mp and mp_i are integers; existence is guaranteed, no specific value is fitted to data.
auxiliary rationals p_2,...,p_k = not explicitly given; existence proven
Produced by Lemma 3.3 through a continuity/density argument; they depend on p and n but are not fitted to data, only required to satisfy inequalities (3).

assumptions (4)

domain assumption Dice are discrete uniform random variables on finite sets of distinct consecutive integers; all win probabilities depend only on relative ordering.
Section 2 conventions; the paper argues this restriction is without loss of generality because scaling or shifting preserves P(A_i<A_j).
standard math Niven's theorem: rational values of cosine at rational multiples of pi are only 0, plus or minus 1/2, plus or minus 1.
Used in the proof of Theorem 1.2 to show pi_n rational implies n=4. Not explicitly cited in the text.
standard math The rationals are dense in the reals and the functions L and U defined in Lemma 3.3 are continuous.
Used in the proof of Lemma 3.3 to replace the real numbers p*_j with rational p_j.
standard math The Bogdanov-Komisarski upper bound w(A1,...,An) ≤ pi_n is taken as a black box.
Cited as Theorems [2,9]; the paper combines it with the new construction but does not re-prove it.

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Cite this review

Pith. "Pith review of Winning Probabilities of Balanced and Nontransitive n-tuples of Dice." pith.science (2026). https://pith.science/paper/G6UHKW5S

@misc{pith2026250521950,
  author       = {Pith},
  title        = {Pith review of: Winning Probabilities of Balanced and Nontransitive n-tuples of Dice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G6UHKW5S}},
  note         = {Machine review of arXiv:2505.21950}
}

abstract

For a positive integer $n$, an $n$-tuple of dice $(A_1,A_2,\dots,A_n)$ is called balanced if $P(A_1<A_2) = P(A_2<A_3) = \cdots = P(A_n<A_1)$ and nontransitive if $P(A_1<A_2), P(A_2<A_3), \dots, P(A_n<A_1)$ are each greater than $\frac{1}{2}$. For a balanced and nontransitive $n$-tuple of dice $(A_1,A_2,\dots,A_n)$, we define the winning probability $w(A_1,A_2,\dots,A_n) := P(A_1 < A_2)$. The works of Trybula and Kim et al. together show that for a balanced and nontransitve triple of dice $(A_1,A_2,A_3)$, the least upper bound on the winning probability is $\frac{-1+\sqrt{5}}{2}$. Kim et al. then asked what the least upper bound on the winning probability was for the $n \geq 4$ cases. Bogdanov and Komisarski independently have shown that for $n\geq 3$ and a balanced and nontransitive $n$-tuple of dice $(A_1,A_2,\dots,A_n)$, the winning probability is less than $\pi_n := 1-\frac{1}{4\cos^2\left( \frac{\pi}{n+2} \right)}$. In this paper, we will show that for $n \geq 3$ and every rational $p \in \left( \frac{1}{2}, \pi_n \right]$, there exists a balanced and nontransitive $n$-tuple of dice with winning probability $p$. Paired with Bogdanov and Komisarski's results, this fully answers the problem posed by Kim et al. and establishes a complete characterization of the winning probabilities for nontransitive and balanced $n$-tuples of dice.

Figures

Figures reproduced from arXiv: 2505.21950 by the authors.

**Figure 2.** Modified Efron Dice Under these conventions, it becomes clear that relative ordering is truly what determines P(A1 < A2), . . . , P(An < A1). Say, for example, that we doubled every integer in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** A triple of BN 6-sided dice with winning probability [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: A triple of BN dice with winning probability [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

A direct construction of non-transitive dice sets, 2016

Levi Angel and Matt Davis. A direct construction of non-transitive dice sets, 2016

work page 2016

[2] [2]

I. I. Bogdanov. Нетранзитивные рулетки. Математическое просвещение, 3(14):240–255, 2010

work page 2010

[3] [3]

Maximally nontransitive dice.The American Mathe- matical Monthly, 125:387–399, 05 2018

Joe Buhler, Ron Graham, and Al Hales. Maximally nontransitive dice.The American Mathe- matical Monthly, 125:387–399, 05 2018

work page 2018

[4] [4]

The paradox of the nontransitive dice and the elusive principle of indifference

Martin Gardner. The paradox of the nontransitive dice and the elusive principle of indifference. Scientific American, 223:110–114, 1970

work page 1970

[5] [5]

Possible probability and irreducibility of balanced nontransitive dice

Injo Hur and Yeansu Kim. Possible probability and irreducibility of balanced nontransitive dice. Journal of Mathematics, 2021(1):6648248, 2021. 13 Winning Probabilities of Balanced and Nontransitiven-tuples of Dice Rooney

work page 2021

[6] [6]

Joyce, A

S. Joyce, A. Schaefer, D. West, and T. Zaslavsky. Strongly connectable digraphs and non- transitive dice. AKCE International Journal of Graphs and Combinatorics, 17(1):480–485, 2020

work page 2020

[7] [7]

Balanced nontransitive dice: Existence and probability.The Electronic Journal of Combinatorics, 31:Not available, 1 2024

Dohyeon Kim, Ringi Kim, Wonjun Lee, Yuhyeon Lim, and Yoojin So. Balanced nontransitive dice: Existence and probability.The Electronic Journal of Combinatorics, 31:Not available, 1 2024

work page 2024

[8] [8]

Balanced nontransitive 4-dice

Ringi Kim. Balanced nontransitive 4-dice. https://papers.ssrn.com/abstract=5214731,

work page