Condorcet's jury theorem

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 ${\displaystyle {\sqrt {n}}}$.

Condorcet's theorem assumes that all voters have the same competence, i.e., the probability of deciding correctly is uniform among all voters. In practice, different voters have different competence levels.

A stronger version of the theorem requires only that the average of the individual competence levels of the voters (i.e. the average of their individual probabilities of deciding correctly) is slightly greater than half.[2]

Condorcet's theorem assumes that the votes are statistically independent. But real votes are not independent: voters are often influenced by other voters, causing a peer pressure effect.

The non-asymptotic part of Condorcet's jury theorem does not hold for correlated votes in general.[3] This is not necessarily a problem since the theorem may still hold under sufficiently general assumptions.[4] A strong version of the theorem does not require voter independence, but takes into account the degree to which votes may be correlated.[5]

In a jury comprising an odd number of jurors ${\displaystyle n}$, let ${\displaystyle p}$ be the probability of a juror voting for the correct alternative and ${\displaystyle c}$ be the (second-order) correlation coefficient between any two correct votes. If all higher-order correlation coefficients in the Bahadur representation[6] of the joint probability distribution of votes equal to zero, and ${\displaystyle (p,c)\in {\mathcal {B}}_{n}}$ is an admissible pair, then:

The probability of the jury collectively reaching the correct decision (Condorcet probability) under simple majority is given by:

${\displaystyle P(n,p,c)=I_{p}\left({\frac {n+1}{2}},{\frac {n+1}{2}}\right)+0.5c(n-1)(0.5-p){\frac {\partial I_{p}({\frac {n+1}{2}},{\frac {n+1}{2}})}{\partial p}},}$

where ${\displaystyle I_{p}}$ is the regularized incomplete beta function.

Example: Take a jury of three jurors ${\displaystyle (n=3)}$, with individual competence ${\displaystyle p=0.55}$ and second-order correlation ${\displaystyle c=0.4}$. Then ${\displaystyle P(3,0.55,0.4)=0.54505}$. The competence of the jury is lower than the competence of a single juror, which equals to ${\displaystyle 0.55}$. Moreover, enlarging the jury by two jurors ${\displaystyle (n=5)}$ decreases the jury competence ${\displaystyle P(5,0.55,0.4)=0.5196194}$.

Note that ${\displaystyle p=0.55}$ and ${\displaystyle c=0.4}$ is an admissible pair of parameters. For ${\displaystyle n=5}$ and ${\displaystyle p=0.55}$, the maximum admissible second-order correlation coefficient equals ${\displaystyle \approx 0.43}$.

The above example shows that when the individual competence is low but the correlation is high

1. The collective competence under simple majority may fall below that of a single juror,
2. Enlarging the jury may decrease its collective competence.

The above result is due to Kaniovski and Zaigraev, who discuss optimal jury design for homogenous juries with correlated votes.[3]

Condorcet's theorem considers a direct majority system, in which all votes are counted directly towards the final outcome. Many countries use an indirect majority system, in which the voters are divided into groups. The voters in each group decide on an outcome by an internal majority vote; then, the groups decide on the final outcome by a majority vote among them. For example,[7] suppose there are 15 voters. In a direct majority system, a decision is accepted whenever at least 8 votes support it. Suppose now that the voters are grouped into 3 groups of size 5 each. A decision is accepted whenever at least 2 groups support it, and in each group, a decision is accepted whenever at least 3 voters support it. Therefore, a decision may be accepted even if only 6 voters support it.

Boland, Proschan and Tong[8] prove that, when the voters are independent and p>1/2, a direct majority system - as in Condorcet's theorem - always has a higher chance of accepting the correct decision than any indirect majority system.

Berg and Paroush[9] consider multi-tier voting hierarchies, which may have several levels with different decision-making rules in each level. They study the optimal voting structure, and compares the competence against the benefit of time-saving and other expenses.

Condorcet's theorem is correct given its assumptions, but its assumptions are unrealistic in practice. Besides the issue of correlated votes, some objections that are commonly raised are:

1. The notion of "correctness" may not be meaningful when making policy decisions, as opposed to deciding questions of fact. Some defenders of the theorem hold that it is applicable when voting is aimed at determining which policy best promotes the public good, rather than at merely expressing individual preferences. On this reading, what the theorem says is that although each member of the electorate may only have a vague perception of which of two policies is better, majority voting has an amplifying effect. The "group competence level", as represented by the probability that the majority chooses the better alternative, increases towards 1 as the size of the electorate grows assuming that each voter is more often right than wrong.

2. The theorem doesn't directly 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.[10] This limitation may also be overcome by means of a sequence of votes on pairs of alternatives, as is commonly realized via the legislative amendment process. (However, as per Arrow's theorem, this creates a "path dependence" on the exact sequence of pairs of alternatives; e.g., which amendment is proposed first can make a difference in what amendment is ultimately passed, or if the law—with or without amendments—is passed at all.)

3. The behaviour that everybody in the jury votes according to his own beliefs might not be a Nash equilibrium under certain circumstances.[11]

Despite these objections, Condorcet's jury theorem provides a theoretical basis for democracy, even if somewhat idealized, as well as a basis of the decision of questions of fact by jury trial, and as such continues to be studied by political scientists.

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)[12]

The Condorcet jury theorem is also used in ensemble learning in the field of machine learning. 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.

• Non-asymptotic Condorcet jury theorem.[13]
• Majority systems and the Condorcet jury theorem:[7] discusses non-homogeneous and correlated voters, and indirect majority systems.
• Evolution in collective decision making.[14]