Wilcoxon and polymath: another update

As I wrote before, there’s a polymath (large-scale collaborative pure maths) project on transitivity of dice. Here’s the latest update from Timothy Gowers’s blog.

Suppose $X$, $Y$, and $Z$ are discrete distributions supported on $1,2,dots,n$.  We can ask about $P(X<Y)$ and $P(Y<Z)$ and $P(X<Z)$, which is what the Wilcoxon/Mann-Whitney rank test does. 

The project has basically proved that under one model for randomly choosing distributions, if $X$,  $Y$, and $Z$ have the same mean and $P(X>Y)>1/2$ and $P(Y>Z)>1/2$, the probability of $P(X>Z)>1/2$ is $1/2+o\left(1\right)$. That is, if three distributions have the same mean, and the Wilcoxon test says $X$ is bigger than $Y$ and $Y$ is bigger than $Z$, you’ve got essentially no information about whether it will say $X$ is bigger than $Z$.

Gowers also says they are close to showing a converse: if the means are different, then $P(X>Y)>1/2$, $P(Y>Z)>1/2$ and $P(X>Z)>1/2$ are true or false they way you’d assume from the ordering of the means. 

That is, we knew the Wilcoxon test does not give a self-consistent ordering on all distributions. Now we know that (for this particular model of discrete distributions) when it does give an ordering, the ordering is typically the same as the ordering by means.