Condorcet's jury theorem is apolitical science theorem about therelative probability of a given group of individuals arriving at a correct decision. The theorem was first expressed by theMarquis de Condorcet in his 1785 workEssay 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 bymajority vote. One of the two outcomes of the vote iscorrect, and each voter has an independent probabilityp of voting for the correct decision. The theorem asks how many voters we should include in the group. The result depends on whetherp is greater than or less than 1/2:

• Ifp 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, ifp 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 otherjury theorems, relaxing some or all of Condorcet's assumptions.

To avoid the need for a tie-breaking rule, we assumen is odd. Essentially the same argument works for evenn if ties are broken by adding a single voter.

Now suppose we start withn voters, and letm 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 then votes, but both new voters voted correctly.
• m was just equal to a majority of then 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 firstn) separates a correct from an incorrect majority.

Restricting our attention to this case, we can imagine that the firstn-1 votes cancel out and that the deciding vote is cast by then-th voter. In this case the probability of getting a correct majority is justp. 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 isp(1-p)². The first of these probabilities is greater than the second if and only ifp > 1/2, proving the theorem.

This proof is direct; it just sums up the probabilities of the majorities. Each term of the sum multiplies the number ofcombinations of a majority by theprobability of that majority. Each majority is counted using acombination,n items takenk at a time, wheren is the jury size, andk is the size of the majority. Probabilities range from 0 (= the vote is always wrong) to 1 (= always right). Each person decides independently, so the probabilities of their decisions multiply. The probability of each correct decision isp. The probability of an incorrect decision,q, is the opposite ofp, i.e. 1 −p. The power notation, i.e.${\displaystyle p^x}$ is a shorthand forx multiplications ofp.

Committee or jury accuracies can be easily estimated by using this approach in computer spreadsheets or programs.

As an example, let us take the simplest case ofn = 3,p = 0.8. We need to show that 3 people have higher than 0.8 chance of being right. Indeed:

0.8 × 0.8 × 0.8 + 0.8 × 0.8 × 0.2 + 0.8 × 0.2 × 0.8 + 0.2 × 0.8 × 0.8 = 0.896.

Asymptotics is “The Calculus of Approximations”. 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 theprime number theorem. An important topic in the study of asymptotic isasymptotic distribution which is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. The probability of a correct majority decisionP(n, p), when the individual probabilityp is close to 1/2 grows linearly in terms ofp − 1/2. Forn voters each one having probabilityp of deciding correctly and for oddn (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

${\displaystyle c_1=(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor })\frac{\lfloor n/2\rfloor +1}{4^{\lfloor n/2\rfloor }}=\sqrt{\frac{2n+1}{\pi }}\left(1+\frac{1}{16n^2}+O(n^{-3})\right),}$

and the asymptotic approximation in terms ofn 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 havingn voters grows proportionally to${\displaystyle \sqrt{n}}$.^[2]

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 inensemble learning in the field ofmachine 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.

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 adouble-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. 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 extensivelyargued that voters' knowledge is systematically biased toward (probably) wrong options. In the CJT setup, this could be interpreted as evidence for.

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${\displaystyle p_i=p\mathrm{\forall }i\in \mathbb{N}}$) 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, theprior 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,, then the thesis predicted by the theorem will not holdalmost 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).

• Condorcet method
• Condorcet paradox
• Jury theorem
• Law of large numbers
• Wisdom of the crowd

• Theorems in probability theory
• Voting theory

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