The probability of intransitivity in dice and close elections

• 成果类型: Article
• 署名作者: Hazla, Jan; Mossel, Elchanan; Ross, Nathan; Zheng, Guangqu
• 署名单位: Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Massachusetts Institute of Technology (MIT); University of Melbourne; University of Kansas
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00994-7
• 发表日期: 2020
• 页码: 951-1009
• 关键词: quantitative version; PARADOX

摘要

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, ... , n} and conditioned on the average of faces equal to (n+1)/2 are intransitive with asymptotic probability 1/4. We showthat 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 H is an element of (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), whichmarkedly differs from known results in themodel 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.