A Functional Equation of Tail-balance for Continuous Signals in the Condorcet Jury Theorem

Steve Alpern; Bo Chen; Adam J. Ostaszewski

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arXiv · 1911.11827

A Functional Equation of Tail-balance for Continuous Signals in the Condorcet Jury Theorem

Steve Alpern·Bo Chen·Adam J. Ostaszewski

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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 (Condorcet, 1785). Recently, the first two authors developed a model where jurors sequentially receive 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 distribution.

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BibTeX RIS

Steve Alpern, Bo Chen, Adam J. Ostaszewski. 2019-11-26. A Functional Equation of Tail-balance for Continuous Signals in the Condorcet Jury Theorem. https://arxiv.org/abs/1911.11827

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