The top two scores of a round-robin tournament: exact distributions, and when the maximum winner's margin is attained
Abstract
In a round-robin tournament of n equally strong players, let s_((n)) and s_((n-1)) be the largest and the second largest score. Both are named as objects of study in the recent literature, which calls the exact distribution computationally intractable even for small n and derives asymptotics instead. It is not intractable: carrying only the sorted multiset of scores gives a dynamic program whose state count is the number of realisable score sequences, and we compute the joint law together with the marginals of the runner-up score, of the winner's margin s_((n))-s_((n-1)), and of the number of co-champions, exactly, for the classical model up to n=13, the chess model up to n=9 and the soccer model (3 points for a win, 1 each for a draw) up to n=7. Two consequences are uniform in n. First, the winner is unique exactly when the margin is positive, so the margin distribution's mass at 0 is the complement of the unique-winner count in every outcome model. Second, our main theorem, a characterisation of the largest margin any tournament realises. Write M for the most one player can take from a single match and μ(k) for the smallest winner's score over all k-player tournaments of the model, the minimax score. On n ≥ 2 players no margin exceeds M(n-1)-μ(n-1), with no hypothesis on the model whatever; and that bound is the largest realised margin if and only if the model contains a total-domination result, one awarding M points to one player and nothing to the other. If moreover some pair of results splits the smallest match total m as evenly as possible in both orders, then μ(k)=⌈ m(k-1)/2⌉ and the bound becomes the closed form M(n-1)-⌈ m(n-2)/2⌉, which is ⌊ kn/2⌋ in the model where a match is worth k points and every split of it is possible — hence ⌊ n/2⌋ classically and n in the chess model — and 2n-1 in the soccer model. Without that extra hypothesis the closed form can be strictly too large: when a win is worth g ≥ 2 and draws are impossible, the largest margin is g⌊ n/2⌋ at every n ≥ 2. Everything asserted here is machine-checked in Lean 4, the margin results from the definitions with no computation anywhere in their proofs.
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Archived files
- Version 3 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
a30298b6988e4c2bc1661c735c0bb08baf30f54c2e3ab6bd1234f804844f0547
Claim ledger
Stated results
score-01routine2026-08-23
The dynamic program computes the exact distribution: the DP over sorted score multisets equals the enumeration over all |O|ᵇinom(n,2) labelled tournaments
score-02routine2026-08-23
Any relabelling-invariant statistic of the score multiset is readable off the table; three instances and the total identity
score-03known data2026-08-23
MacMahon's 1923 exact ranges re-derived: the full classical distribution for n <= 9 and the full chess distribution for n <= 6, plus classical n = 10 and chess n = 7
score-04routine2026-08-23
The winner's-score marginal, the unique-winner count and the number of score sequences: classical n <= 13, chess n <= 9
score-05known data2026-08-23
MacMahon's nine-player erratum reproduced from the definition: the frequency of (2,2,3,3,4,4,6,6,6) is 361307520, not the printed 361297520
score-06routine2026-08-23
Negative controls: non-score-sequences count zero, Landau's condition characterises the support, the DP state cannot be coarsened, and the definition itself is right
score-07routine2026-08-28
The top two order statistics are readable off the dynamic program, and the winner is unique exactly when the margin is positive
score-08routine2026-08-28
The exact joint distribution of (sₙ, sₙ₋₁) and its three marginals — runner-up score, winner's margin, number of co-champions: classical n ≤ 13, chess n ≤ 9
score-09candidate2026-08-28
The maximum winner's margin is ⌊n/2⌋ classically, n in the chess model and 2n − 1 in the soccer model — one formula M(n−1) − ⌈m(n−2)/2⌉ verified in three models
score-10routine2026-08-28
Negative controls for the top-two statistics: the definitions, the non-removable length hypothesis, the marginals against the sample space, and the published first moment A125031
score-11known data2026-08-28
The soccer model (3 for a win, 1 each for a draw): the bridge needs no fixed match total, and the full tables to n = 7 reproduce Zeilberger's published output
score-12routine2026-08-28
The winner's margin is at most M(n−1) − ⌈m(n−2)/2⌉, at every n ≥ 2 and in every outcome model
score-13routine2026-08-28
The bound is attained: an explicit near-regular tournament with a closed-form score list, at every n ≥ 2
score-14candidate2026-08-28
The maximum winner's margin is exactly M(n−1) − ⌈m(n−2)/2⌉ at every n ≥ 2 — the family's published Conjecture 5.2, now a theorem
score-15routine2026-08-28
Every Dₖ at every n ≥ 2 gives ⌊kn/2⌋; classical ⌊n/2⌋, chess n, soccer 2n−1
score-16routine2026-08-28
Negative controls for the margin theorem: neither hypothesis can be DROPPED (a model failing each misses the bound), (ii) is not a necessary CONDITION – asymPair fails it and attains (wording fixed 2026-08-28; MarginControls.lean docstring carries the old contradiction), and the bound is off by one in neither direction
score-17routine2026-08-30
Deleting a player from a tournament, and the runner-up bound s_(n−1) ≥ minMax O (n−1)
score-18candidate2026-08-30
The largest winner's margin is exactly M(n−1) − μ(n−1), where μ(k) is the minimax score — hypothesis (ii) of the published Conjecture 5.2 is not needed
score-19routine2026-08-30
The averaging bound 2μ(k) ≥ m(k−1) in every model, its tightness under a balanced pair, and μ(Dₖ, n) = ⌈k(n−1)/2⌉
score-20routine2026-08-30
The no-draw model [(g,0),(0,g)]: μ(k) = g⌊k/2⌋, largest margin g⌊n/2⌋ at every n ≥ 2, and the published bound strictly larger at every odd n ≥ 3
score-21routine2026-08-30
A player's score decomposed into its n−1 matches: playerRow, and the exact decrement under deletion
score-22candidate2026-08-30
The sharp margin bound is attained iff the model contains a total-domination outcome — hypothesis (i) of score-18 is necessary, not just sufficient
score-23routine2026-08-30
The champion as the oldest player: extendOld, and the reversed model [(0,g)] with μ(k) = g(k−1) and largest margin g at every n
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Exact score-sequence distributions in round-robin tournaments, verified in Lean, with a proved bridge from the dynamic program to the enumeration it replaces.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7