Condorcet's jury theorem

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.

Essentially the same argument works for even n if ties are broken by adding a single voter.

The majority vote changes in only two cases: 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.

In this case the probability of getting a correct majority is just p. Now suppose we send in the two extra voters.

Each person decides independently, so the probabilities of their decisions multiply.

is a shorthand for x multiplications of p. Committee or jury accuracies can be easily estimated by using this approach in computer spreadsheets or programs.

It is used to solve hard problems that cannot be solved exactly and to provide simpler forms of complicated results, from early results like Taylor's and Stirling's formulas to the prime number theorem.

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.

In simple terms, this says that when the decision is difficult (p close to 1/2), the gain by having n voters grows proportionally to

[2] The Condorcet jury theorem has recently been used to conceptualize score integration when several physician readers (radiologists, endoscopists, etc.)

This task arises in central reading performed during clinical trials and has similarities to voting.

According to the authors, the application of the theorem can translate individual reader scores into a final score in a fashion that is both mathematically sound (by avoiding averaging of ordinal data), mathematically tractable for further analysis, and in a manner that is consistent with the scoring task at hand (based on decisions about the presence or absence of features, a subjective classification task)[3] The Condorcet jury theorem is also used in ensemble learning in the field of machine learning.

[4] An ensemble method combines the predictions of many individual classifiers by majority voting.

Many political theorists and philosophers use the Condorcet’s Jury Theorem (CJT) to defend democracy, see Brennan[5] and references therein.

Nevertheless, it is an empirical question whether the theorem holds in real life or not.

Note that the CJT is a double-edged sword: it can either prove that majority rule is an (almost) perfect mechanism to aggregate information, when

For instance, Bryan Caplan has extensively argued that voters' knowledge is systematically biased toward (probably) wrong options.

Recently, another approach to study the applicability of the CJT was taken.

This case was previously studied by Daniel Berend and Jacob Paroush[7] and includes the classical theorem of Condorcet (when

Then, following a Bayesian approach, the prior probability (in this case, a priori) of the thesis predicted by the theorem is estimated.

With this new approach, proponents of the CJT should present strong evidence of competence, to overcome the low prior probability.

That is, it is not only the case that there is evidence against competence (posterior probability), but also that we cannot expect the CJT to hold in the absence of any evidence (prior probability).