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 \(\alpha (t)\) 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 \(\alpha (t)\) 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 \(\alpha (t)\) linear.

考虑一个奇数大小的陪审团，该陪审团决定两个自然状态之间的多数票。如果每个陪审员独立地以相同的概率p接收标识正确状态的二进制信号，则随着陪审团人数趋于无穷的可能性，正确判决的可能性也趋向于无穷大（Essai sur l'application de l'analyseàlaprobabilitédesdécisionsrenduesàlavoitédes voix的Marquis de Condorcet，巴黎，皇家， 1785）。最近，Alpern和Chen（Eur J Oper Res 258：1072-1081，2017，Theory Decis 83：259-282，2017）开发了一种模型，其中陪审员根据依赖于状态的分布从间隔中依次接收独立信号性质和对陪审员的“能力”，并依序投票。本文表明，要模拟Condorcet的二进制信号，这种分布必须满足与尾部平衡有关的函数方程，即均值零随机变量的概率之比\（\ alpha（t）\）满足X \（> t \）给定\（| X |> t \）。特别是，我们证明了在自然对称假设下，尾部平衡\（\ alpha（t）\）唯一确定了信号分布，因此在Alpern和Chen（Eur J Oper Res 258：1072–1081，2017， Theory Decis 83：259–282，2017）是唯一确定的线性\（\ alpha（t）\）。