Five-voter realizability versus tournament predictability

Does the proposed predictability threshold 3/5 characterize tournaments induced by five voters’ rankings? A separating tournament would exceed the threshold but still have no five-voter realization.

Why it matters and what counts as progress

Why it matters: A five-voter obstruction would show that the proposed numerical predictability threshold does not characterize realizability by rankings.

What counts as progress: Certify a tournament’s predictability and rule out every five-voter realization, or prove a corrected sufficient criterion.

How the problem developed

2008 · Definition and result [1]
Dylan Shepardson and Craig Tovey Defined tournament predictability and identified the smallest example with predictability below 2/3; the work set up later comparisons between predictability and voter realizability.
29 Jul 2026 · Reported counterexample [2]
Leonid Chindelevitch and Ararat Harutyunyan Refuted the threshold criterion for five voters: Paley(43) has predictability 181/301>3/5 but is not induced by any five-voter majority profile.

AI contribution

Tournaments determined by three and five voters

Leonid Chindelevitch, Ararat Harutyunyan

The Paley tournament on 43 vertices has predictability 181/301, above the five-voter threshold 3/5, yet cannot be induced by five voters. The proof uses a co-backing constraint for cyclic triangles and an exact finite shell-and-screen enumeration.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (exact combinatorial enumeration and LP/ILP certificate verification + human)

Human understandingFull