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    The probability of intransitivity in dice and close elections

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    Author
    Hazla, J; Mossel, E; Ross, N; Zheng, G
    Date
    2020-08-19
    Source Title
    Probability Theory and Related Fields
    Publisher
    SPRINGER HEIDELBERG
    University of Melbourne Author/s
    Ross, Nathan
    Affiliation
    School of Mathematics and Statistics
    Metadata
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    Document Type
    Journal Article
    Citations
    Hazla, J., Mossel, E., Ross, N. & Zheng, G. (2020). The probability of intransitivity in dice and close elections. PROBABILITY THEORY AND RELATED FIELDS, 178 (3-4), pp.951-1009. https://doi.org/10.1007/s00440-020-00994-7.
    Access Status
    Open Access
    URI
    http://hdl.handle.net/11343/252169
    DOI
    10.1007/s00440-020-00994-7
    ARC Grant code
    ARC/DP150101459
    Abstract
    <jats:title>Abstract</jats:title> <jats:p>We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for <jats:italic>n</jats:italic>-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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$\{1,\ldots ,n\}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and conditioned on the average of faces equal to <jats:inline-formula><jats:alternatives><jats:tex-math>$$(n+1)/2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$H\in (0,1)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. Second, we pose an analogous model in the context of Condorcet voting. We consider <jats:italic>n</jats:italic> voters who rank <jats:italic>k</jats:italic> 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 <jats:italic>k</jats:italic> alternatives is equal to <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^{-k(k-1)/2}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>, 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.</jats:p>

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