Intransitive dice VII — aiming for further results

While Polymath13 has (barring a mistake that we have not noticed) led to an interesting and clearly publishable result, there are some obvious follow-up questions that we would be wrong not to try to answer before finishing the project, especially as some of them seem to be either essentially solved or promisingly close to a solution. The ones I myself have focused on are the following.

1. Is it true that if two random elements and of are chosen, then beats with very high probability if it has a sum that is significantly larger? (Here “significantly larger” should mean larger by for some function — note that the standard deviation of the sum has order , so the idea is that this condition should be satisfied one way or the other with probability ).
2. Is it true that the stronger conjecture, which is equivalent (given what we now know) to the statement that for almost all pairs of random dice, the event that beats a random die has almost no correlation with the event that beats , is false?
3. Can the proof of the result obtained so far be modified to show a similar result for the multisets model?

The status of these three questions, as I see it, is that the first is basically solved — I shall try to justify this claim later in the post, for the second there is a promising approach that will I think lead to a solution — again I shall try to back up this assertion, and while the third feels as though it shouldn’t be impossibly difficult, we have so far made very little progress on it, apart from experimental evidence that suggests that all the results should be similar to those for the balanced sequences model. [Added after finishing the post: I may possibly have made significant progress on the third question as a result of writing this post, but I haven’t checked carefully.]

Intransitive dice VI: sketch proof of the main conjecture for the balanced-sequences model

I have now completed a draft of a write-up of a proof of the following statement. Recall that a random -sided die (in the balanced-sequences model) is a sequence of length of integers between 1 and that add up to , chosen uniformly from all such sequences. A die beats a die if the number of pairs such that exceeds the number of pairs such that . If the two numbers are the same, we say that ties with .

Theorem. Let and be random -sided dice. Then the probability that beats given that beats and beats is .

In this post I want to give a fairly detailed sketch of the proof, which will I hope make it clearer what is going on in the write-up.

Intransitive dice V: we want a local central limit theorem

It has become clear that what we need in order to finish off one of the problems about intransitive dice is a suitable version of the local central limit theorem. Roughly speaking, we need a version that is two-dimensional — that is, concerning a random walk on — and completely explicit — that is, giving precise bounds for error terms so that we can be sure that they get small fast enough.

There is a recent paper that does this in the one-dimensional case, though it used an elementary argument, whereas I would prefer to use Fourier analysis. Here I’d like to begin the process of proving a two-dimensional result that is designed with our particular application in mind. If we are successful in doing that, then it would be natural to try to extract from the proof a more general statement, but that is not a priority just yet.

As people often do, I’ll begin with a heuristic argument, and then I’ll discuss how we might try to sharpen it up to the point where it gives us good bounds for the probabilities of individual points of . Much of this post is cut and pasted from comments on the previous post, since it should be more convenient to have it in one place.

Intransitive dice IV: first problem more or less solved?

I hope, but am not yet sure, that this post is a counterexample to Betteridge’s law of headlines. To back up that hope, let me sketch an argument that has arisen from the discussion so far, which appears to get us close to showing that if and are three -sided dice chosen independently at random from the sequences model (I will recap some of these definitions in a moment), then the probability that beats given that beats and beats is . Loosely speaking, the information that beats and beats tells you nothing about how likely is to beat . Having given the argument, I will try to isolate a statement that looks plausible, not impossible to prove, and sufficient to finish off the argument.

Basic definitions

An –sided die in the sequence model is a sequence of elements of such that , or equivalently such that the average value of is , which is of course the average value of a random element of . A random -sided die in this model is simply an -sided die chosen uniformly at random from the set of all such dice.

Given -sided dice and , we say that beats if

If the two sets above have equal size, then we say that ties with .

Intransitive dice III

I now feel more optimistic about the prospects for this project. I don’t know whether we’ll solve the problem, but I think there’s a chance. But it seems that there is after all enough appetite to make it an “official” Polymath project. Perhaps we could also have an understanding that the pace of the project will be a little slower than it has been for most other projects. I myself have various other mathematical projects on the boil, so can’t spend too much time on this one, but quite like the idea of giving it an occasional go when the mood takes me, and trying to make slow but steady progress. So I’ve created a polymath13 category, into which this post now fits. I’ve also retrospectively changed the category for the previous two posts. I don’t think we’ve got to the stage where a wiki will be particularly useful, but I don’t rule that out at some point in the future.

In this post I want to expand on part of the previous one, to try to understand better what would need to be true for the quasirandomness assertion to be true. I’ll repeat a few simple definitions and simple facts needed to make the post more self-contained.

Intransitive dice II

I’m not getting the feeling that this intransitive-dice problem is taking off as a Polymath project. However, I myself like the problem enough to want to think about it some more. So here’s a post with some observations and with a few suggested subproblems that shouldn’t be hard to solve and that should shed light on the main problem. If the rate of comments by people other than me doesn’t pick up, then I think I’ll simply conclude that there wasn’t sufficient interest to run the project. However, if I do that, I have a back-up plan, which is to switch to a more traditional collaboration — that is, done privately with a small number of people. The one non-traditional aspect of it will be that the people who join the collaboration will select themselves by emailing me and asking to be part of it. And if the problem gets solved, it will be a normal multi-author paper. (There’s potentially a small problem if someone asks to join in with the collaboration and then contributes very little to it, but we can try to work out some sort of “deal” in advance.)

But I haven’t got to that point yet: let me see whether a second public post generates any more reaction.

I’ll start by collecting a few thoughts that have already been made in comments. And I’ll start that with some definitions. First of all, I’m going to change the definition of a die. This is because it probably makes sense to try to prove rigorous results for the simplest model for which they are true, and random multisets are a little bit frightening. But I am told that experiments suggest that the conjectured phenomenon occurs for the following model as well. We define an -sided die to be a sequence of integers between 1 and such that . A random -sided die is just one of those chosen uniformly from the set of all of them. We say that beats if

That is, beats if the probability, when you roll the two dice, that shows a higher number than is greater than the probability that shows a higher number than . If the two probabilities are equal then we say that ties with .

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.)