A jury theorem is a mathematical theorem proving that, under certain assumptions, a decision attained using majority voting in a large group is more likely to be correct than a decision attained by a single expert. 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.
The first and most famous jury theorem is Condorcet's jury theorem. It assumes that all voters have independent probabilities to vote for the correct alternative, these probabilities are larger than 1/2, and are the same for all voters. Under these assumptions, the probability that the majority decision is correct is strictly larger when the group is larger; and when the group size tends to infinity, the probability that the majority decision is correct tends to 1.
There are many other jury theorems, relaxing some or all of these assumptions.
Setting The premise of all jury theorems is that there is an objective truth, which is unknown to the voters. Most theorems focus on binary issues (issues with two possible states), for example, whether a certain defendant is guilty or innocent, whether a certain stock is going to rise or fall, etc. There are voters (or jurors), and their goal is to reveal the truth. Each voter has an opinion about which of the two options is correct. The opinion of each voter is either correct (i.e., equals the true state), or wrong (i.e., differs than the true state). This is in contrast to other settings of voting, in which the opinion of each voter represents his/her subjective preferences and is thus always "correct" for this specific voter. The opinion of a voter can be considered a random variable: for each voter, there is a positive probability that his opinion equals the true state.
The group decision is determined by the majority rule. For example, if a majority of voters says "guilty" then the decision is "guilty", while if a majority says "innocent" then the decision is "innocent". To avoid ties, it is often assumed that the number of voters is odd. Alternatively, if is even, then ties are broken by tossing a fair coin.
Jury theorems are interested in the probability of correctness - the probability that the majority decision coincides with the objective truth. Typical jury theorems make two kinds of claims on this probability:
Growing Reliability: the probability of correctness is larger when the group is larger. Crowd Infallibility: the probability of correctness goes to 1 when the group size goes to infinity. Claim 1 is often called the non-asymptotic part and claim 2 is often called the asymptotic part of the jury theorem.
Obviously, these claims are not always true, but they are true under certain assumptions on the voters. Different jury theorems make different assumptions.
Independence, competence, and uniformity
Condorcet's jury theorem makes the following three assumptions:
Unconditional Independence: the voters make up their minds independently. In other words, their opinions are independent random variables. Unconditional Competence: the probability that the opinion of a single voter coincides with the objective truth is larger than 1/2 (i.e., the voter is smarter than a random coin-toss). Uniformity: all voters have the same probability of being correct. The jury theorem of Condorcet says that these three assumptions imply Growing Reliability and Crowd Infallibility.
Correlated votes: weakening the independence assumption The opinions of different voters are often correlated, so Unconditional Independence may not hold. In this case, the Growing Reliability claim might fail.
Example Let be the probability of a juror voting for the correct alternative and be the (second-order) correlation coefficient between any two correct votes. If all higher-order correlation coefficients in the Bahadur representation 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 is the regularized incomplete beta function.
Example: Take a jury of three jurors , with individual competence and second-order correlation . Then . The competence of the jury is lower than the competence of a single juror, which equals to . Moreover, enlarging the jury by two jurors decreases the jury competence even further, . Note that and is an admissible pair of parameters. For and , the maximum admissible second-order correlation coefficient equals .
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.
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 than the other. 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:
Conditional Independence: for each of the two options, the voters' opinions given that this option is the true one are independent random variables. 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. 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.
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.
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 and Boland, Proschan and Tong 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.
More recently, Karge et al. proved Crowd Infallibility for an opinion-leader model with finitely many alternatives, approval voting, and heterogeneous voter competences, provided that the leader’s influence is bounded relative to those competences; they also bound the number of voters needed to attain a prescribed success probability under these conditions.
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).
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 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 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.see for example:
Other solutions Other ways to cope with voter correlation include causal networks, dependence structures, and interchangeability.
Diverse capabilities: weakening the uniformity assumption Different voters often have different competence levels, so the Uniformity assumption does not hold. In this case, both Growing Reliability and Crowd Infallibility may not hold. This may happen if new voters have much lower competence than existing voters, so that adding new voters decreases the group's probability of correctness. In some cases, the probability of correctness might converge to 1/2 (- a random decision) rather than to 1.
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 pi 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 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, and Berend and Paroush, 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).