Ran the tests based on the distance metric on the constrained sum case. I used the absolute distance as opposed to relative distance, so the graphs look slightly different. However, it seems to validate your conclusions: file is results_testDice_100s_70_200_650.png at github.com/wbrackenbury/intransDice. I generated 3 dice with distances 70, 200, and 650 from the standard die, and then generated 100000 dice, and measured their distances from each of those dice (red, blue, and green respectively for each of the distances). As seen in the figure, the distance from the standard die correlates strongly with distance from other dice, even with the random walk generation method. Could be useful for evaluating other hypotheses!

]]>Another note that I tried generating dice made of n N(0,1) numbers conditioned on the sum being 0, and those are highly transitive. For 1000-sided dice, something like 1 in 10000 quadruples were non-transitive.

]]>For either the multiset or the n-tuple model, an n-sided die with face values can be considered as a discrete probability measure on [0,1] by putting point masses of weight at . Let be the measure constructed from . Then consider sequences where the are chosen independently. Weak limits of these sequences would be good candidates for your “continuously-valued” dice. However, it must be that almost surely they converge weakly to the uniform (i.e., Lebesgue) measure on [0,1], and so in the limit all the dice are standard. I don’t really have a proof for this claim. It looks like the Glivenko-Cantelli theorem but with conditioning, since the sum restriction implies that the mean of is .

]]>For the graphs above they were proper dice and random selection was done uniformly among the multisets. So this was not done using the code snippet I gave earlier. This was using something akin to the partition model Gowers suggested.

There are actually two distinguishing things here:

– the properties of dice (what values are allowed, and constraints on those values)

– the random distribution we use to select a random die

The snippet of code I shared previously was for:

– proper dice (allowed values are integers 1 to n, sum constrained to n*(n+1)/2)

– random selection was uniform in the “sequence” representation

Your original code was of this (dice type, distribution) as well.

Your random walk code we discussed before was also of this type.

The dice calculated for the graphs above were:

– proper dice

– random selection was uniform in the “multiset” representation.

Also, this would describe the dice and random distribution of the other random walk Gowers discussed.

The original conjecture was about uniformly selecting in the “multiset” representation. I was not being very careful at first, and implicitly assumed that because the conjecture was that in some sense the comparisons are as random as possible, that the sequence distribution should be fine (and it was easier for me to calculate). It is possible as n goes to infinity, the conjectures holds for both distributions. But I don’t know if results in one distribution can be used to prove the other or not.

So my last few results have been from selecting uniformly randomly in the “multiset” representation. I did this using an adapted form of Gowers suggested partition method. This should be slower than the random walk method, but for n up to about 200, I can still get reasonable statistics if i run code for a few days and I wanted some statistics that we knew for sure were selected uniform in multisets. Those calculations should finish today or tomorrow.

Anyway, I’ve been trying to be more clear now when I state results to also mention the distribution I used. If anything is confusing, or if I’m misusing terminology, don’t hesitate to point it out.

]]>That would be nice, right? However, it is not hard to see that the answer in general is no. As a counter example consider e.g. all multisets of order . There are . The first three are fixed points for the above involution whereas the other two get mapped to each other. Since the only group of order , the cyclic group, has exactly idempotent, we cannot equip our set of multi-sets with an operation that has the above properties.

If its not a group, what else can it be? A cocycle? (I.e., there is a (meaningful) map from to our set of multisets and a (meaningful) operation on our multisets such that and ) I do not know that, but while writing this I realize that the question I really have is actually very easy and the answer is probably well-known:

Are there meaningful operations on the set of multi-sets?

]]>1: (0,0,4,4,4,4)

2: (3,3,3,3,3,3)

3: (2,2,2,2,6,6)

4: (1,1,1,5,5,5)

Each die beats the next one with a probability of 2/3, and die 4 beats die 1. Interestingly, the sums of the face numbers are 16, 18, 20 and 18: so the expectation value does not determine whether the next die is beaten or not!

There’s a reference in the article that might be interesting: R.P. Savage “The Paradox of Non-Transitive Dice”, Am. Math. Mo., Vol 101, May 1994, p429-36.

]]>I think an issue with this approach is that the functions will not be continuous, and so doesn’t appear to make the problem nicer. And because it is discontinuous, I’m not sure what it would mean to leap to real values.

]]>First, programming wise, the continual off by one thing is annoying so for the math below consider every value of the die -= 1. This doesn’t change any comparisons, and just makes the index of arrays or values discussed below not have +/- 1 everywhere.

Now, for the sequence representation it would also be nice to avoid even the n log n sorting hit. So convert directly from the sequence representation to the multiset representation, by stepping through the list, counting how many times you’ve seen each value. Now we have a list identifying the die using the multiset representation, where A[i] = (# of dice faces with value i).

We know the total of this list should be n.

Using that, step through the list, creating a new list where:

entry i holds a tuple that is ( multiset count, partially calculated score)

= ( # of dice faces with value i, (# of dice faces with value less than i) – (# of dice faces with value greater than i))

This whole thing took two trips through the list, so O(n) for conversion.

For lack of a better phrase, let’s call this representation the “fastdie” representation.

Now to compare two fastdie representations A and B, just sum up A[i][0] * B[i][1]. So O(n) for comparison.

This takes advantage of there only being n possible integer values in the initial die model, so will also be okay for the “unconstrained sum” dice model, but not improper dice nor the scaled real valued dice. If we pay the n log n to sort up front, then we should be able to compare those other representations in O(n) as well, but I have not written code for that.

As a side note, the partially calculated score in the fastdie representation is essentially Gower’s f_A(j) function.

]]>For each face of A, increment the corresponding element of the array by 1.

For i=2 to n, increase array[i] by array[i-1] for the accumulated totals.

Sum up the corresponding elements for B.

Something like that.

]]>I’m not sure how you define a “distribution over distributions over [0,1]”, which I think you’d have to do.

Finite-sided dice could be modeled as having a spike in density at every possible value of the face (scaled to fit in [0,1]). But since that’s only a subset of possible “real-valued” dice, I don’t think answering this question would necessarily help with the original question.

I think that if you only consider continuously-valued dice, then ties have measure zero. That might not help, though (and takes it even further from the original question).

So I’m not at all sure that this helps answer the original question, but think it’s an interesting related question.

]]>.

In particular, considering the “standard” die to be the center of the space, and using the distance from it as a kind of radial distance.

Selecting a die A, the distribution of scores (number of roll pair wins – losses) from comparing to random dice, will be centered around 0 with the standard deviation depending strongly on the distance of A from the center.

For example here are three difference dice for n=100, at distance 30, 312, and 2450.

http://imgur.com/a/yEQI8

The last is the die with its values closest to average, and so is the furthest distance die from the center. Because it’s entries are all 50 or 51, there are strong undulations in the score due to ties if the comparison die happens to have the number 50 or 51 in it. If I used a larger histogram bin to smooth that out, it would look Gaussian.

The other extreme, dist=0 (the standard die), would be a zero width distribution since everything ties with it.

So the standard deviation of comparison scores for an individual die appear strongly correlated with that die’s distance from the standard die.

And here is a chart showing the distribution of distances when selecting uniformly in the multiset model, for n=100 (tested by selecting 100000 dice).

http://imgur.com/a/YoWL5

I haven’t yet calculated the distribution seen in the tuple model, but I have a feeling this is the measurement that would help distinguish it the most. Since the tuple model sees more ties, likely it means the distance distribution is shifted towards the center.

Will B, can you try your random walk generator and see how well it lines up with that distance distribution?

]]>The last step in my argument (control of maximal hj for a typical dice) seems to be easy. However, to make the argument fully rigorous, we need to refer to something like Berry–Esseen theorem (that is, control of speed of convergence to normal distribution), but for random vectors. I am sure this should be classical. Anybody knows the reference?

]]>