  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.

Probability