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

TL;DR

This paper explores a functional equation related to tail-balance in the context of continuous signals in a jury decision model, extending Condorcet's binary signal framework to continuous distributions.

Contribution

It establishes that tail-balance functions uniquely determine the distribution of signals under symmetry assumptions, linking continuous models to Condorcet's binary signal theory.

Findings

Tail-balance functions satisfy a specific functional equation.

Under symmetry, tail-balance determines the distribution uniquely.

The model generalizes Condorcet's binary signal to continuous signals.

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 $\alpha(t)$ 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 distribution.