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A functional equation of tail-balance for continuous signals in the Condorcet Jury Theorem

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Alpern, Steve, Chen, Bo and Ostaszewski, Adam J. (2021) A functional equation of tail-balance for continuous signals in the Condorcet Jury Theorem. Aequationes Mathematicae, 95 . pp. 67-74. doi:10.1007/s00010-020-00750-1

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Official URL: https://doi.org/10.1007/s00010-020-00750-1

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Abstract

Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability $p$, then the probability of a correct verdict tends to one as the jury size tends to infinity (Condorcet, 1785). Recently, Alpern and Chen (2017a, b) developed a model where jurors sequentially receive independent signals from an interval according to a distribution which depends on the state of Nature and on the juror's "ability", and vote sequentially. This paper shows that, to mimic Condorcet's binary signal, such a distribution must satisfy a functional equation related to tail-balance, that is, to the ratio $\alpha(t)$ of the probability that a mean-zero random variable satisfies $X$ $>t$ given that $|X|>t$. In particular, we show that under natural symmetry assumptions the tail-balances $\alpha(t)$ uniquely determine the signal distribution and so the distributions assumed in Alpern and Chen (2017a, b) are uniquely determined for $\alpha(t)$ linear.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Social Sciences > Warwick Business School > Operational Research & Management Sciences
Faculty of Social Sciences > Warwick Business School
Library of Congress Subject Headings (LCSH): Functional equations, Voting -- Mathematical models, Probabilities
Journal or Publication Title: Aequationes Mathematicae
Publisher: Springer
ISSN: 0001-9054
Official Date: February 2021
Dates:
DateEvent
February 2021Published
7 September 2020Available
4 August 2020Accepted
Date of first compliant deposit: 12 August 2020
Volume: 95
Page Range: pp. 67-74
DOI: 10.1007/s00010-020-00750-1
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access
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