It serves as a formal argument for the idea of wisdom of the crowd, for decision of questions of fact by jury trial, and for democracy in general.
If all higher-order correlation coefficients in the Bahadur representation[2] of the joint probability distribution of votes equal to zero, and
is an admissible pair, then the probability of the jury collectively reaching the correct decision under simple majority is given by: where
The above example shows that when the individual competence is low but the correlation is high: The above result is due to Kaniovski and Zaigraev.
[1] One criticism of Conditional Competence is that it depends on the way the decision question is formulated.
Moreover, if the state is very specific, then the probability of voting correctly might be below 1/2, so Conditional Competence might not hold.
In addition to the dependence on the true option, there are many other reasons for which voters' opinions may be correlated.
For example: It is possible to weaken the Conditional Independence assumption, and conditionalize on all common causes of the votes (rather than just the state).
A jury theorem by Pivato[9] shows that, if the average covariance between voters becomes small as the population becomes large, then Crowd Infallibility holds (for some voting rule).
[10][11] Other ways to cope with voter correlation include causal networks, dependence structures, and interchangeability.
A jury theorem by Ben Yashar and Paroush[15] shows that, under certain conditions, the correctness probability of a jury, or of a subset of it chosen at random, is larger than the correctness probability of a single juror selected at random.
[17] A jury theorem by Owen, Grofman and Feld[18] analyzes a setting where the competence level is random.
They show what distribution of individual competence maximizes or minimizes the probability of correctness.
Grofman and Shapley[20] analyze the effect of interdependencies between voters on the optimal decision rule.
Dietrich[22] generalizes this result to a setting that does not require prior probabilities of the 'correctness' of the two alternative.
Dietrich shows that Epistemic Monotonicity implies that the optimal decision rule is weighted majority with a threshold.
In the same paper, he generalizes the optimal decision rule to a setting that does not require the input to be a vote for one of the alternatives.
A general problem with the weighted majority rules is that they require to know the competence levels of the different voters, which is usually hard to compute in an objective way.
Baharad, Goldberger, Koppel and Nitzan[23] present an algorithm that solves this problem using statistical machine learning.
If the list is sufficiently large, then its probability of correctness converges to 1 even if the individual voters' competence levels are close to 1/2.
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.
They study the optimal voting structure, and compares the competence against the benefit of time-saving and other expenses.
Goodin and Spiekermann[29] compute the amount by which a small group of experts should be better than the average voters, in order for them to accept better decisions.
Surprisingly, strategic voting might occur even with two alternatives and when all voters have the same preference, which is to reveal the truth.
In practice, this problem may not be very severe, since most voters care not only about the final outcome, but also about voting correctly by their conscience.
[1]: 4.7 The notion of "correctness" may not be meaningful when making policy decisions, which are based on values or preferences, rather than just on facts.
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.
Several papers show that, under reasonable conditions, large groups are better trackers of the majority preference.