Abstract
AbstractConsider 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 $$\alpha (t)$$
α
(
t
)
of the probability that a mean-zero random variable satisfies X$$>t$$
>
t
given that $$|X|>t$$
|
X
|
>
t
. In particular, we show that under natural symmetry assumptions the tail-balances $$\alpha (t)$$
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 $$\alpha (t)$$
α
(
t
)
linear.
Funder
London School of Economics and Political Science
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,General Mathematics
Reference10 articles.
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3. Alpern, S., Chen, B.: Who should cast the casting vote? Using sequential voting to amalgamate information. Theory Decis. 83, 259–282 (2017)
4. Alpern, S., Chen, B., Ostaszewski, A.J.: A Functional Equation of Tail-Balance for Continuous Signals in the Condorcet Jury Theorem. arXiv:1911.11827 [math.PR], 26 Nov 2019 (2019)
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