The Jury Theorem

The first point concerns what has been called Condorcet’s “jury problem” (Black 1958) or “jury theorem”. Let r (r for “verite”, that is, truth) be the probability for each voter to make the right choice, and e (e for error) the probability of being mistaken: e = (1 - r).

(1) before the vote, what is the probability p to obtain a decision complying with truth?

(2) Once the decision is taken, what is, for an external observer, the probability p* that this decision complies with the truth? In modern parlance (see, for example, Granger 1956: 105-6), probability p is found using Bernoulli’s binomial distribution. It is the sum, for all x, q ≤ x ≤ n, of the probability r^x(1 - r)^{n - x} that a decision is true when it obtains x votes, multiplied by the possible number of occurrences (χ) = _x!,„"^! _x)! of this event:

Probability p* is found using the Bayes-Laplace theorem and is given by:

From the first equation,p → 1 when n → ∞ if r > 0.5, butp → 0 in the opposite case. (Note that in case r = 0.5, p = 0.5 for all n.) This is the “jury theorem”: in an assembly in which the probability for each voter to make the right choice is greater than 0.5, the probability for the outcome to be true increases with the number of voters - and conversely, when r < 0.5, the probability of the outcome to be true is a decreasing function of this number (Condorcet 1785: xxiii-xxiv, 6-9).

These are both positive and negative results. The positive side of the story is the proposition that - under the very restrictive conditions noted above - an assembly could collectively have a degree of wisdom superior to its individual members, and that, if r > 0.5, this degree increases with the number of voters. This is the kind of statement already made by Aristotle when, examining the different possible political regimes, he declared that it is possible that many individuals, of whom no one is “virtuous”, are col­lectively better when they are assembled than the best ones among them (Politics, III, 11, 1281-a). Condorcet’s theorem could thus be taken as a powerful argument in favour of democracy.

The negative aspect arises if r < 0.5. Then the opposite conclusion applies: “it could be dangerous to give a democratic constitution to an unenlightened people: a pure democracy could even only suit a people much more enlightened, much more freed of prejudices than is any of those we know in history” (Condorcet 1785: xxiv). In these circumstances, nevertheless, a pure democracy would be acceptable if decisions are “limited to what regards the maintaining of safety, liberty and property, all objects on

which a direct personal interest can enlighten everybody” (ibid.; see also ibid.: 135) - these being precisely among the “general” or “universal” objects in Rousseau’s approach. Otherwise the assembly, to decide on an issue, could designate a committee composed of its most enlightened members and then judge, not the decision itself, but whether the decision does not hurt justice or some of the fundamental human rights (ibid.: 7).

However, while aware of the novelty and complexity of his developments on the forms of elections or choices made in the various parts of the book, Condorcet in the end rela­tivized the importance of the choice to be made between the different possible devices. For him, the key variable remains the probability for each voter to be right or wrong; hence his tireless action in favour of public instruction.

[T]he happiness of men depends less on the form of assemblies that decide their fate than on the enlightenment of those who compose them, or, in other words,... the progress of reason affects more their happiness than the form of political constitutions. (1785: 136; see also ibid.: lxx)