Jury theorem

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, 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 $n$ 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 $n$ is odd. Alternatively, if $n$ 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:

1. Growing Reliability: the probability of correctness is larger when the group is larger.
2. 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

Main article: Condorcet's jury theorem

Condorcet's jury theorem makes the following three assumptions:

1. Unconditional Independence: the voters make up their minds independently. In other words, their opinions are independent random variables.
2. 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).
3. 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

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.

Failure for large correlation

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 excells at pretending to be innocent);
• External conditions (e.g. poor weather affecting their judgement).
• Any other common cause of votes.

In these cases, even Conditional Independence may not hold, and the Growing Reliability claim might fail. As an example, let $p$ be the probability of a juror voting for the correct alternative and $c$ 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 $(p,c)\in {\mathcal {B}}_{n}$ is an admissible pair, then the probability of the jury collectively reaching the correct decision under simple majority is given by:

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

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

Example: Take a jury of three jurors $(n=3)$, with individual competence $p=0.55$ and second-order correlation $c=0.4$. Then $P(3,0.55,0.4)=0.54505$. The competence of the jury is lower than the competence of a single juror, which equals to $0.55$. Moreover, enlarging the jury by two jurors $(n=5)$ decreases the jury competence even further, $P(5,0.55,0.4)=0.5196194$. Note that $p=0.55$ and $c=0.4$ is an admissible pair of parameters. For $n=5$ and $p=0.55$, the maximum admissible second-order correlation coefficient equals $\approx 0.43$.

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.

Bounded correlation

One way to weaken the Independence Axiom is to allow correlation, but require that the correlation be bounded. 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.

Problem-sensitive independence and competence

One strategy to handle voter correlation is to weaken the Conditional Independence assumption, and conditionalize on all common causes of the votes. 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.

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 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 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).

A jury theorem by Ben Yashar and Paroush 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 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.

A jury theorem by Owen, Grofman and Feld 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 show that, under Unconditional Independence, the optimal decision rule is a weighted majority rule, where the weight of each voter with correctness probability pi 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 analyze the effect of interdependencies between voters on the optimal decision rule. Ben-Yashar and Nitzan prove a more general result.

Dietrich 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.

Indirect majority systems

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, 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 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 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.

More than two options

Most theorems do not 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 and Everaere, Konieczny and Marquis present evidence to the contrary. 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.)

Strategic voting

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

Limitations

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.

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.

Further reading

• Majority systems and the Condorcet jury theorem: discusses non-homogeneous and correlated voters, and indirect majority systems.
• Evolution in collective decision making.
• Law of large numbers: a mathematical generalization of jury theorems.
• Realizing Epistemic Democracy: a criticism on the assumptions of jury theorems.
• The Epistemology of Democracy: a comparison of jury theorems to two other epistemic models of democracy: experimentalism and Diversity trumps ability.