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

Text (A functional equation of tail-balance for continuous signals in the Condorcet Jury Theorem)
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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 |
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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: | 20 Apr 2021 03:16 |

URI: | http://eprints.lse.ac.uk/id/eprint/105845 |

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