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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
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 | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Social Sciences > Warwick Business School > Operational Research & Management Sciences Faculty of Social Sciences > Warwick Business School |
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| 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: |
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| 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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| Open Access Version: |
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