`Polymath’ is a set of projects in massive collaborative proof of mathematical results; Terry Tao and Timothy Gowers are two of the famous mathematicians involved. There’s a new potential project on Gowers’s blog, which he describes a being related to intransitive dice. As you know, if you read this blog, (a) I prefer the term non-transitive, and (b) this means it’s about the Wilcoxon test.

The idea of the conjecture is that you define an \(n\)-sided die by sampling uniformly with replacement \(n\) numbers from \(1, 2,3,\dots,n\) as the numbers on the sides, with the constraint that the numbers have to add up to \(n(n+1)/2\). Rolling such a die \(M\) times samples \(M\) observations from a distribution on \(1,2,3,\\dots, n\) with mean \((n+1)/2\). You construct three such dice, \(A\), \(B\), and \(C\). Their distributions have the same mean, so the \(t\)-test would have no ability to distinguish data from the three distributions, no matter how large \(M\) was. But the Wilcoxon test probably would. It’s assumed, but not yet proved, that the probability of a tie according to the Wilcoxon test goes to zero as \(n\to\infty\).

The interesting conjecture is that if you see \(A\) beat \(B\) and \(B\) beat \(C\) according to the Wilcoxon test, the probability that \(A\) beats \(C\) goes to 1/2 as \(n\) goes to infinity. That is, given equal means, the Wilcoxon test is basically as non-transitive as possible.