A functional equation of tail-balance for continuous signals in the Condorcet Jury Theorem
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Aequationes Mathematicae
A functional equation of tail-balance for continuous signals in the Condorcet Jury Theorem Steve Alpern, Bo Chen
and Adam J. Ostaszewski
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 ` a la probabilit´e des d´ ecisions rendues ` a la pluralit´e 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 tailbalances 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. Mathematics Subject Classification. 91B12, 39B22.
1. Introduction This paper solves a functional equation arising from an extension of the celebrated Condorcet Jury Theorem. In Condorcet’s model, an odd-sized jury must decide whether Nature is in one of two equiprobable states of Nature, A or B. Each juror receives an independent binary signal (for A or for B) which is correct with the same probability p > 1/2. Condorcet[6] [or see [10, Ch. XVII] for a textbook discussion] showed that, when jurors vote simultaneously according to their signal, the probability of a correct majority verdict tends to 1 as the number of jurors tends to infinity. Recently, Alpern and Chen [2,3] considered a related sequential voting model where each juror independently receives a signal S in the interval [−1, +1], rather than a binary signal.
S. Alpern et al.
AEM
Low signals indicate B and high signals indicate A. The strength of this “indication” depends on the “ability” of the juror, a number a between 0 and 1, which is a proxy for Condorcet’s p ranging from 1/2 to 1. When deciding how to vote, each juror notes the previous voting, the abilities of the previous jurors, his own signal S and his own ability. He then votes for the alternative, A or B, that is more likely, conditioned on this information. The mechanism that underlies this determination is the common knowledge of the distributions by which a signal is given as private information to each juror, depending on his ability and the state of Nature. It is not relevant to the discussions of this paper, but we me
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