Condorcet Paradoxes and dice

NB: don’t take this as a sign that I’ve brought the blog back to life for real. I’ve made too many unfulfilled promises on that front…

There was a talk at UChicago in the fall (I’m still on the seminar mailing lists) given by Jan Hązła based, I think, on a paper titled “The Probability of Intransitivity in Dice and Close Elections” with Elchanan Mossel, Nathan Ross, and Guangqu Zheng. The abstract was quite interesting and led to a discussion with my colleagues Emina Soljanin and Roy Yates in which I realized I didn’t quite get the result so I promised to come back after reading it more carefully. Fast forward several months and now I am in Berkeley (on a pre-tenure sabbatical) and they are back east so I figured I could at least blog about it.

The problem is about intransitive dice, which was a new term for me. Consider an n-sided die with numbers \mathbf{a} = (a_1, a_2, a_n) and call \sum_{i=1}^{n} a_i the face sum. The die is fair, so the expected face value is \frac{1}{n} \sum_{i=1}^{n} a_i. We can define an ordering on dice by saying $\mathbf{a} \succ \mathbf{b}$ if a uniformly chosen face of \mathbf{a} is larger than a uniformly chosen face of \mathbf{b}. That is, if you roll both dice then on average \mathbf{a} would beat \mathbf{a}.

A collection of dice is intransitive if the relation \succ based on dice beating each other cannot be extended to a linear order. The connection to elections is in ranked voting — an election in which voters rank candidates may exhibit a Condorcet paradox in which people’s pairwise preferences form a cycle: A beats B, B beats C, but C beats A in pairwise contests. (As an aside, in election data we looked at in my paper on instant runoff voting we actually rarely (maybe never?) saw a Condorcet cycle).

Suppose we generate a die randomly with face values drawn from the uniform distribution on [-1,1] and condition on the face sum being equal to 0. Then as the number of faces n \to \infty, three such independently generated dice will become intransitive with high probability (see the Polymath project).

However, it turns out that this is very special to the uniform distribution. What this paper shows (among other things) is that if you generate the faces from any other distribution (but still condition the face sum to be 0), the dice are in fact transitive with high probability. This to me is interesting because it shows the uniform distribution as rather precariously balanced — any shift and a total linear order pops out. But this also makes some sense: the case where the dice become intransitive happens when individuals essentially are choosing a random permutation of the candidates as their preferences. In face, the authors show that if you generate voters/votes this way and then condition on the election being “close” you get a higher chance of turning up a Condorcet paradox.

The details of the proofs are a bit hairy, but I often find ranking problems neat and interesting. Maybe one day I will work on them again…