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- Title
A functional equation of tail-balance for continuous signals in the Condorcet Jury Theorem.
- Authors
Alpern, Steve; Chen, Bo; Ostaszewski, Adam J.
- 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.
- Subjects
PARIS (France); FUNCTIONAL equations; JURY; JURORS; INFINITY (Mathematics); VERDICTS
- Publication
Aequationes Mathematicae, 2021, Vol 95, Issue 1, p67
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-020-00750-1