Graph of cumulative probabilities of success (y axis) for a few binomial distributions with given individual chance of success (x axis) and number of “jurors” (color).

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 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.

Since Condorcet, many other researchers have proved various other jury theorems, relaxing some or all of Condorcet's assumptions.

Proofs

Source:^[2]

Asymptotics

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):

${\displaystyle P(n,p)=1/2+c_{1}(p-1/2)+c_{3}(p-1/2)^{3}+O\left((p-1/2)^{5}\right),}$

where

${\displaystyle c_{1}={n \choose {\lfloor n/2\rfloor }}{\frac {\lfloor n/2\rfloor +1}{4^{\lfloor n/2\rfloor }}}={\sqrt {\frac {2n+1}{\pi }}}\left(1+{\frac {1}{16n^{2}}}+O(n^{-3})\right),}$

and the asymptotic approximation in terms of n is very accurate. The expansion is only in odd powers and ${\displaystyle c_{3}<0}$. 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 ${\displaystyle {\sqrt {n}}}$.

The theorem in other disciplines

The Condorcet jury theorem has recently been used to conceptualize score integration when several physician readers (radiologists, endoscopists, etc.) independently evaluate images for disease activity. 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. Assuming that each of the individual classifiers predict with slightly greater than 50% accuracy and their predictions are independent, then the ensemble of their predictions will be far greater than their individual predictive scores.

Applicability to democratic processes

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 ${\displaystyle p>1/2}$, or an (almost) perfect disaster, when ${\displaystyle p<1/2}$. A disaster would mean that the wrong option is chosen systematically. Some authors have argued that we are in the latter scenario. For instance, Bryan Caplan has extensively argued that voters' knowledge is systematically biased toward (probably) wrong options. In the CJT setup, this could be interpreted as evidence for ${\displaystyle p<1/2}$.

Recently, another approach to study the applicability of the CJT was taken.^[6] Instead of considering the homogeneous case, each voter is allowed to have a probability ${\displaystyle p_{i}\in [0,1]}$, possibly different from other voters. This case was previously studied by Daniel Berend and Jacob Paroush^[7] and includes the classical theorem of Condorcet (when ${\displaystyle p_{i}=p~~\forall ~i\in \mathbb {N} }$) and other results, like the Miracle of Aggregation (when ${\displaystyle p_{i}=1/2}$ for most voters and ${\displaystyle p_{i}=1}$ for a small proportion of them). Then, following a Bayesian approach, the prior probability (in this case, a priori) of the thesis predicted by the theorem is estimated. That is, if we choose an arbitrary sequence of voters (i.e., a sequence ${\displaystyle (p_{i})_{i\in \mathbb {N} }}$ ), will the thesis of the CJT hold? The answer is no. More precisely, if a random sequence of ${\displaystyle p_{i}}$ is taken following an unbiased distribution that does not favor competence, ${\displaystyle p_{i}>1/2}$, or incompetence, ${\displaystyle p_{i}<1/2}$, then the thesis predicted by the theorem will not hold almost surely. 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).

Further reading

• Condorcet method
• Condorcet paradox
• Jury theorem
• Wisdom of the crowd

Notes