Condorcet Jury Theorem

The French philosopher and mathematician Marquis de Condorcet published the Jury Theorem in a 1785 essay, providing a theoretical foundation for democracy. Although it may be somewhat oversimplified, the tool has been useful in fields apart from political science as well, from medical diagnosis to machine learning.

The Condorcet Jury Theorem assumes a group is wanting to make a decision by majority vote. As is common in probability theory, p represents the independent probability between 0 and 1 that an individual “jury member” or voter will choose the correct answer. The theorem then asks: what is the optimal number of voters in the group?

Here’s where the math gets interesting. It turns out that, if p is greater than 50%, i.e. each voter is more likely to choose the correct option (imagine: an educated, well-informed population), then each additional voter added to the group will continue to increase the probability that the group will choose the correct option — asymptotically approaching 1, for anyone who remembers calculus.

If p is less than 50% though, then adding more voters actually makes things progressively worse. In the case of p < 1/2, the most optimal size of the jury is actually just one person. Which looks a whole lot like autocracy. And offers more pessimism (if anyone needed more) about dictatorship and one-party rule as systems of government — because it essentially implies that the population is not very well educated and/or not very well informed about what’s going on, and about basic issues of civic society.

A mathematical underpinning to democracy

The Condorcet Jury Theorem doesn’t take some very basic factors of voting in real life into account, including vote correlation, non-uniform competence of voters, and indirect majority voting systems. Still, it’s a powerful tool to understand an early inkling of the wisdom of crowds; relating as well to the concepts of statistical significance and the law of large numbers, Condorcet’s work went on to influence both the French Revolution and the authors of the American Constitution, most namely Thomas Jefferson and James Madison.