A functional equation of tail-balance for continuous signals in the Condorcet jury theorem

Alpern, Steve, Chen, Bo and Ostaszewski, Adam ORCID: 0000-0003-2630-8663 (2021) A functional equation of tail-balance for continuous signals in the Condorcet jury theorem. Aequationes Mathematicae, 95 (1). 67 - 74. ISSN 0001-9054

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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 (Marquis de Condorcet in Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix, Imprim. Royale, Paris, 1785). Recently, Alpern and Chen (Eur J Oper Res 258:1072–1081, 2017, Theory Decis 83:259–282, 2017) 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 α(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 α(t) uniquely determine the signal distribution and so the distributions assumed in Alpern and Chen (Eur J Oper Res 258:1072–1081, 2017, Theory Decis 83:259–282, 2017) are uniquely determined for α(t) linear.

Item Type:	Article
Official URL:	https://www.springer.com/journal/10
Additional Information:	© 2020 The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	03 Aug 2020 09:42
Last Modified:	08 Nov 2024 23:42
URI:	http://eprints.lse.ac.uk/id/eprint/105845

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