A monotone, Condorcet consistent, 2-nm(1) tournament rule on seven agents
Abstract
A tournament rule turns the outcome of a round robin into a probability distribution over the players. Pennock, Schvartzman and Xue measure a rule's resistance to a two-player collusion by a selfishness parameter λ: a rule is 2-nm(λ) if no two agents, by fixing the match between them, can raise their joint winning probability by more than λ times the loss suffered by the one who throws the match. They prove that no Condorcet consistent rule is 2-nm(λ) for λ<1, conjecture that λ=1 is attainable, and report that a feasibility linear program finds such rules for tournaments on at most six agents before the search becomes too large. The question is restated as open — "a compelling open question" — in a December 2025 follow-up, the one paper on record as citing theirs. We settle the next case. There is a tournament rule on seven agents that is monotone, Condorcet consistent and 2-nm(1). The rule is exhibited: its values are rationals with common denominator 55 440=11 · 7!, given by a table of 456 × 7 integers indexed by the isomorphism classes of tournaments on seven agents, and the four axioms are verified directly at each of the 2 097 152 labelled tournaments and at each of the 22 020 096 single-match flips between them. With the published lower bound this pins the least λ admitting a monotone, Condorcet consistent, 2-nm(λ) rule on seven agents to exactly 1. We also show that strengthening Condorcet consistency to top cycle consistency, which the source does not consider in this model and which the follow-up names as an open direction, costs nothing at four, five and six agents. Explicit rules and the exhaustive checks are machine-verified in Lean 4; the symmetry reduction that produced the numbers is a search device and is deliberately kept off the verified path. Nothing here bears on eight or more agents.
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Claim ledger
Stated results
tn1-01candidate2026-09-02
A monotone, Condorcet consistent, 2-NM₁ tournament rule exists on n = 7 agents: an explicit rule with rational values, verified at all 2²1 labelled tournaments and all 21 single-edge flips
tn1-02known data2026-09-02
Explicit monotone, Condorcet consistent, 2-NM₁ rules on n = 4, 5 and 6 agents (common denominators 24, 120, 720), kernel-verified at all 2⁶, 2¹0 and 2¹5 labelled tournaments
tn1-03known2026-09-02
The feasible set on 4 agents is exactly the source's Figure-2 polytope: six vertices, reproduced by exact rational vertex enumeration
This ledger entry is reported in prose and is not bound to a Lean theorem.tn1-04candidate2026-09-02
Strengthening Condorcet to top-cycle consistency costs nothing at n <= 6: explicit monotone, top-cycle-consistent, 2-NM₁ rules on 4, 5 and 6 agents, so the minimum lambda in the top-cycle column is exactly 1 there
tn1-05routine2026-09-02
Negative controls: the exhibited n = 4 rule fails 2-NM_(1/2); a one-unit perturbation inside a class fails; the uniform rule fails Condorcet consistency; a class list missing one of the four isomorphism classes fails the coverage clause
tn1-06routine2026-09-02
The neutrality reduction: monotonicity linearises 2-NMₗambda, so the feasible set is a polytope on which Sₙ acts, and a feasible rule averages to a neutral one – collapsing the n = 7 search from 14 680 064 variables to 3 040
This ledger entry is reported in prose and is not bound to a Lean theorem.tn1-07measurement2026-09-02
Measured: with cover consistency in place of Condorcet consistency the neutral LP is infeasible at n = 5 and n = 6 for every mu = 1 + lambda tried up to 10⁷, with the violation shrinking as mu grows (-1.8e-2, -9.2e-4, -8.5e-6 at mu = 2, 10, 100)
This ledger entry is reported in prose and is not bound to a Lean theorem.tn1-08routine2026-09-02
No monotone, Condorcet consistent, 2-NM₁ rule on 4 agents can depend only on the score sequence: in the strongly connected 4-tournament the two agents of out-degree 1 must receive probabilities differing by at least 7/39
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- A tournament T on n agents is a complete asymmetric relation: for every pair i!= j exactly one of i >_T j, j >_T i holds (the outcome of their match). A tournament rule r maps each tournament to a probability distribution over the agents; rᵢ(T) is the probability that i is declared the winner. Two tournaments T, T' are i,j-adjacent when they agree on every match except possibly the one between i and j.
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- 2026-09-07 03:53 UTC
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