Example

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^[2] 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 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 even further, ${\displaystyle P(5,0.55,0.4)=0.5196194}$. 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:

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

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

There are several jury theorems that weaken the Independence assumption in various ways.

Truth-sensitive independence and competence

In binary decision problems, there is often one option that is easier to detect that the other one. For example, it may be easier to detect that a defendant is guilty (as there is clear evidence for guilt) than to detect that he is innocent. In this case, the probability that the opinion of a single voter is correct is represented by two different numbers: probability given that option #1 is correct, and probability given that option #2 is correct. This also implies that opinions of different voters are correlated. This motivates the following relaxations of the above assumptions:

1. Conditional Independence: for each of the two options, the voters' opinions given that this option is the true one are independent random variables.
2. Conditional Competence: for each of the two options, the probability that a single voter's opinion is correct given that this option is true is larger than 1/2.
3. Conditional Uniformity: for each of the two options, all voters have the same probability of being correct given that this option is true.

Growing Reliability and Crowd Infallibility continue to hold under these weaker assumptions.^[1]

One criticism of Conditional Competence is that it depends on the way the decision question is formulated. For example, instead of asking whether the defendant is guilty or innocent, one can ask whether the defendant is guilty of exactly 10 charges (option A), or guilty of another number of charges (0..9 or more than 11). This changes the conditions, and hence, the conditional probability. Moreover, if the state is very specific, then the probability of voting correctly might be below 1/2, so Conditional Competence might not hold.^[4]

Effect of an opinion leader

Another cause of correlation between voters is the existence of an opinion leader. Suppose each voter makes an independent decision, but then each voter, with some fixed probability, changes his opinion to match that of the opinion leader. Jury theorems by Boland^[5] and Boland, Proschan and Tong^[6] shows that, if (and only if) the probability of following the opinion leader is less than 1-1/2p (where p is the competence level of all voters), then Crowd Infallibility holds.

Problem-sensitive independence and competence

In addition to the dependence on the true option, there are many other reasons for which voters' opinions may be correlated. For example:

• Deliberation among voters;
• Peer pressure;
• False evidence (e.g. a guilty defendant that excels at pretending to be innocent);
• External conditions (e.g. poor weather affecting their judgement).
• Any other common cause of votes

It is possible to weaken the Conditional Independence assumption, and conditionalize on all common causes of the votes (rather than just the state). In other words, the votes are now independent conditioned on the specific decision problem. However, in a specific problem, the Conditional Competence assumption may not be valid. For example, in a specific problem with false evidence, it is likely that most voters will have a wrong opinion. Thus, the two assumptions - conditional independence and conditional competence - are not justifiable simultaneously (under the same conditionalization).^[7]

A possible solution is to weaken Conditional Competence as follows. For each voter and each problem x, there is a probability p(x) that the voter's opinion is correct in this specific problem. Since x is a random variable, p(x) is a random variable too. Conditional Competence requires that p(x) > 1/2 with probability 1. The weakened assumption is:

• Tendency to Competence: for each voter, and for each r>0, the probability that p(x) = 1/2+r is at least as large as the probability that p(x) = 1/2-r.

A jury theorem by Dietrich and Spiekerman^[8] says that Conditional Independence, Tendency to Competence, and Conditional Uniformity, together imply Growing Reliability. Note that Crowd Infallibility is not implied. In fact, the probability of correctness tends to a value which is below 1, if and only of Conditional Competence does not hold.

Bounded correlation

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). There are other jury theorems that take into account the degree to which votes may be correlated.^[10][11]

Other solutions

Other ways to cope with voter correlation include causal networks, dependence structures, and interchangeability.^[1]: 2.2

Stronger competence requirements

Uniformity can be dismissed if the Competence assumption is strengthened. There are several ways to strengthen it:

• Strong Competence: for each voter i, the probability of correctness p_i is at least 1/2+e, where e>0 is fixed for all voters. In other words: the competence is bounded away from a fair coin toss. A jury theorem by Paroush^[12] shows that Strong Competence and Conditional Independence together imply Crowd Infallibility (but not Growing Reliability).
• Average Competence: 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, or converges to a value above 1/2. Jury theorems by Grofman, Owen and Feld,^[13] and Berend and Paroush,^[14] show that Average Competence and Conditional Independence together imply Crowd Infallibility (but not Growing Reliability).

Random voter selection

instead of assuming that the voter identity is fixed, one can assume that there is a large pool of potential voters with different competence levels, and the actual voters are selected at random from this pool (as in sortition).

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. A more general jury theorem by Berend and Sapir^[16] proves that Growing Reliability holds in this setting: the correctness probability of a random committee increases with the committee size. The theorem holds, under certain conditions, even with correlated votes.^[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.

Weighted majority rule

When the competence levels of the voters are known, the simple majority rule may not be the best decision rule. There are various works on identifying the optimal decision rule - the rule maximizing the group correctness probability. Nitzan and Paroush^[19] show that, under Unconditional Independence, the optimal decision rule is a weighted majority rule, where the weight of each voter with correctness probability p_i is log(p_i/(1-p_i)), and an alternative is selected iff the sum of weights of its supporters is above some threshold. Grofman and Shapley^[20] analyze the effect of interdependencies between voters on the optimal decision rule. Ben-Yashar and Nitzan^[21] prove a more general result.

Dietrich^[22] generalizes this result to a setting that does not require prior probabilities of the 'correctness' of the two alternative. The only required assumption is Epistemic Monotonicity, which says that, if under certain profile alternative x is selected, and the profile changes such that x becomes more probable, then x is still selected. 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. It can be, for example, a subjective degree of belief. Moreover, competence parameters do not need to be known. For example, if the inputs are subjective beliefs x₁,...,x_n, then the optimal decision rule sums log(x_i/(1-x_i)) and checks whether the sum is above some threshold. Epistemic Monotonicity is not sufficient for computing the threshold itself; the threshold can be computed by assuming expected-utility maximization and prior probabilities.

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. It requires as input only a list of past votes; it does not need to know whether these votes were correct or not. 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.