Condorcet’s Principle (1785)

Named after French mathematician and philosopher MARIE ANTOINE CONDORCET (1743-1794), Condorcet’s principle deals with the complexity of voting and choices, by which the final choice is made by rejection of all other alternatives in a series of paired contests.

Condorcet’s jury theorem is a political science theorem about the relative probability of a given group of individuals arriving at a correct decision. The theorem was first expressed by the Marquis de Condorcet in his 1785 work Essay on the Application of Analysis to the Probability of Majority Decisions.

The assumptions of the simplest version of the theorem are that a group wishes to reach a decision by majority vote. One of the two outcomes of the vote is correct, and each voter has an independent probability p of voting for the correct decision. The theorem asks how many voters we should include in the group. The result depends on whether p is greater than or less than 1/2:

• If p is greater than 1/2 (each voter is more likely to vote correctly), then adding more voters increases the probability that the majority decision is correct. In the limit, the probability that the majority votes correctly approaches 1 as the number of voters increases.

• On the other hand, if p is less than 1/2 (each voter is more likely to vote incorrectly), then adding more voters makes things worse: the optimal jury consists of a single voter.

The probability of a correct majority decision P(n, p), when the individual probability p is close to 1/2 grows linearly in terms of p − 1/2. For n voters each one having probability p of deciding correctly and for odd n (where there are no possible ties):

where

and the asymptotic approximation in terms of n is very accurate. The expansion is only in odd powers and . In simple terms, this says that when the decision is difficult (p close to 1/2), the gain by having n voters grows .

This version of the theorem is correct, given its assumptions, but its assumptions are unrealistic in practice. Some objections that are commonly raised:

• Real votes are not independent, and do not have uniform probabilities. The non-asymptotic part of Condorcet’s jury theorem does not hold for correlated votes in general. This is not necessarily a problem since the theorem may still hold under sufficiently general assumptions. One very strong version of the theorem requires only that the average of the individual competence levels of the voters (i.e. the average of their individual probabilities of deciding correctly) is slightly greater than half. This version of the theorem does not require voter independence, but takes into account the degree to which votes may be correlated.

• The notion of “correctness” may not be meaningful when making policy decisions as opposed to deciding questions of fact. Some defenders of the theorem hold that it is applicable when voting is aimed at determining which policy best promotes the public good, rather than at merely expressing individual preferences. On this reading, what the theorem says is that although each member of the electorate may only have a vague perception of which of two policies is better, majority voting has an amplifying effect. The “group competence level”, as represented by the probability that the majority chooses the better alternative, increases towards 1 as the size of the electorate grows assuming that each voter is more often right than wrong.

• The theorem doesn’t directly apply to decisions between more than two outcomes. This critical limitation was in fact recognized by Condorcet (see Condorcet’s paradox), and in general it is very difficult to reconcile individual decisions between three or more outcomes (see Arrow’s theorem), although List and Goodin present evidence to the contrary. This limitation may also be overcome by means of a sequence of votes on pairs of alternatives, as is commonly realized via the legislative amendment process. (However, as per Arrow’s theorem, this creates a “path dependence” on the exact sequence of pairs of alternatives; e.g., which amendment is proposed first can make a difference in what amendment is ultimately passed, or if the law—with or without amendments—is passed at all.)

• The behaviour that everybody in the jury votes according to his own beliefs might not be a Nash equilibrium under certain circumstances.

Nonetheless, Condorcet’s jury theorem provides a theoretical basis for democracy, even if somewhat idealized, as well as a basis of the decision of questions of fact by jury trial, and as such continues to be studied by political scientists.

CONDORCET observed that ‘majority voting is the best voting rule when only two people can vote’.

Also see: paradox of voting, compensation principle, social welfare function, impossibility theorem