The probability of intransitivity in dice and close elections
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The probability of intransitivity in dice and close elections Jan Ha˛zła1 · Elchanan Mossel2 · Nathan Ross3 · Guangqu Zheng4 Received: 30 July 2019 / Revised: 1 April 2020 © The Author(s) 2020
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, . . . , 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 H ∈ (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. Keywords Intransitivity · Dice · Condorcet paradox Mathematics Subject Classification 60C05
E.M. and J.H. are partially supported by awards ONR N00014-16-1-2227, NSF CCF-1665252 and DMS-1737944. N.R. and G.Z were partially supported by ARC DP150101459. E.M. was supported by the Simons Investigator award (622132).
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Jan Ha˛zła [email protected]
Extended author information available on the last page of the article
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J. Ha˛zła et al.
1 Introduction The phenomenon of intransitivity often arises when one ranks three or more alternatives. An early example is the Condorcet paradox, discovered in the eighteenth century in the context of voting. This type of intransitivity is much more general, as proved by Arrow in his social choice theorem [2]. A different fascinating aspect of intransitivity arises in the context of games of chance: The striking phenomenon of non-transitive dice. It was discovered by the statistician Brad Efron [10] and has fans such as Warren Buffet (who reportedly tried to trick Bill Gates [19]). 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. We in
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