Condorcet's jury theorem

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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^[1].

The assumptions 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 than not 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 than not to vote incorrectly), then adding more voters makes things worse: the optimal jury consists of a single voter.

[edit] Proof

To avoid the need for a tie-breaking rule, we assume n is odd. Essentially the same argument works for even n if ties are broken by fair coin-flips.

Now suppose we start with n voters, and let m of these voters vote correctly.

Consider what happens when we add two more voters (to keep the total number odd). The majority vote changes in only two cases:

m was one vote too small to get a majority of the n votes, but both new voters voted correctly.
m was just equal to a majority of the n votes, but both new voters voted incorrectly.

The rest of the time, either the new votes cancel out, only increase the gap, or don't make enough of a difference. So we only care what happens when a single vote (among the first n) separates a correct from an incorrect majority.

Restricting our attention to this case, we can imagine that the first n-1 votes cancel out and that the deciding vote is cast by the n-th voter. In this case the probability of getting a correct majority is just p. Now suppose we send in the two extra voters. The probability that they change an incorrect majority to a correct majority is (1-p)p², while the probability that they change a correct majority to an incorrect majority is p(1-p)(1-p). The first of these probabilities is greater than the second if and only if p > 1/2, proving the theorem.

[edit] Limitations

The theorem is correct within its assumptions, but its assumptions do not necessarily hold in practice. Some objections that are commonly raised:

Real votes are not independent, and do not have uniform probabilities. This is not necessarily a problem as long as each voter is more likely than not to produce a correct vote, and subsequent work ^[2] has considered the case of correlated votes.
The notion of "correctness" may not be meaningful when making policy decisions as opposed to deciding questions of fact. In this case, what the theorem says is that an electorate is most likely to arrive at the more likely choice of each of its members, and that this amplifying effect increases as the electorate grows.
The theorem doesn't 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.^[3]

Nonetheless, Condorcet's jury theorem provides a theoretical basis for democracy, even if somewhat idealized, and as such continues to be studied by political scientists.