Abstract
We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for n-sided dice with pairwise ordering induced by the probability, relative to 1/2, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on {1,ldots ,n} and conditioned on the average of faces equal to (n+1)/2 are intransitive with asymptotic probability 1/4. We show that if dice faces are drawn from a non-uniform continuous mean zero distribution conditioned on the average of faces equal to 0, then three dice are transitive with high probability. We also extend our results to stationary Gaussian dice, whose faces, for example, can be the fractional Brownian increments with Hurst index Hin (0,1). Second, we pose an analogous model in the context of Condorcet voting. We consider n voters who rank k alternatives independently and uniformly at random. The winner between each two alternatives is decided by a majority vote based on the preferences. We show that in this model, if all pairwise elections are close to tied, then the asymptotic probability of obtaining any tournament on the k alternatives is equal to 2^{-k(k-1)/2}, which markedly differs from known results in the model without conditioning. We also explore the Condorcet voting model where methods other than simple majority are used for pairwise elections. We investigate some natural definitions of “close to tied” for general functions and exhibit an example where the distribution over tournaments is not uniform under those definitions.
Highlights
A different fascinating aspect of intransitivity arises in the context of games of chance: The striking phenomenon of non-transitive dice
The main motivating question of this paper is: What is the chance of observing intransitivity in natural random setups? We present some quantitative answers to this question
There has been some interest in the quantitative study of intransitive dice
Summary
Some (mostly) experimental results were presented by Conrey, Gabbard, Grant, Liu and Morrison [7] Among others, they conjectured that the model where n-sided dice are sampled uniformly from multisets of integers between 1 and n conditioned on the face-sum equal to n(n + 1)/2 is chaotic. To understand the differing behavior of uniform versus non-uniform dice implied by Theorem 1 and the Polymath result, we first recall that, as shown by Polymath [31], for unconditioned dice with faces uniform in (0, 1), the face-sums determine if a beats b with high probability. The variance calculation uses a CLT calculation with a rather attentive tracking of errors This is interesting in comparison with [32], since it suggests that careful application of central limit theorems is important in establishing both transitivity and intransitivity results.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have