A potential new Polymath project: intransitive dice

A few days ago I received an email from Brian Conrey, who has a suggestion for a possible Polymath project. The problem he wants to see solved is a little different from the problems in most previous projects, in that it is not a well known and long-standing question of acknowledged importance. However, that is not an argument against trying it out, since it is still far from clear what kinds of problems are best suited to the polymathematical approach, and it would be good to get more data. And this problem has other qualities that could make it very suitable indeed. First of all, it is quite new — it arises from a paper published last year, though it appeared on arXiv in 2013 — so we do not yet have a clear idea of how difficult it is, which should give us hope that it may turn out to be doable. Secondly, and more importantly, it is a very attractive question.

Suppose you have a pair of dice with different numbers painted on their sides. Let us say that beats if, thinking of them as random variables, the probability that is greater than the probability that . (Here, the rolls are of course independent, and each face on each die comes up with equal probability.) It is a famous fact in elementary probability that this relation is not transitive. That is, you can have three dice such that beats , beats , and beats .

Brian Conrey, James Gabbard, Katie Grant, Andrew Liu and Kent E. Morrison became curious about this phenomenon and asked the kind of question that comes naturally to an experienced mathematician: to what extent is intransitivity “abnormal”? The way they made the question precise is also one that comes naturally to an experienced mathematician: they looked at -sided dice for large and asked about limiting probabilities. (To give another example where one might do something like this, suppose one asked “How hard is Sudoku?” Well, any Sudoku puzzle can be solved in constant time by brute force, but if one generalizes the question to arbitrarily large Sudoku boards, then one can prove that the puzzle is NP-hard to solve, which gives a genuine insight into the usual situation with a board.)