If n independent voters are each right with probability p about a question that has a correct answer, the probability that a majority is right can be computed exactly. It rises with n when p exceeds one half, falls when p is below one half, and stays at one half when p equals it. The probability that a single vote decides the outcome shrinks as the electorate grows. The wrong model is that a large electorate is wise whatever its voters' competence.
Core ideas
The earlier lessons treated votes as expressions of preference. This one treats them as estimates of a fact, and asks what a majority of estimates is worth.
Each voter answers a yes-or-no question that has a correct answer, and is right with probability p, independently of the others. The theorem is silent on questions of taste, where there is no answer to be right about.
With an odd number n of voters the majority is right exactly when at least (n + 1)/2 of them are, and the probability is the sum of the exact probabilities of each such count.
One voter decides the result only if the other n − 1 voters divide exactly half and half. The probability of that is small, and it falls as the electorate grows.
The material
The majority's chance of being right, worked out for three voters, compared with eleven, with voters below one half, and with the probability that one vote is the one that counts.
Condorcet's question was whether a group votes better than a person. The case that makes it computable is a jury deciding a question of fact, such as whether the accused did it. Suppose each of n jurors is right with probability p, whatever the others think, and the verdict is whatever more than half of them say.
Take three jurors, each right with probability 3/5. The majority is right when at least two of them are. All three are right with probability (3/5)·(3/5)·(3/5), which is 27/125. Exactly two are right in three ways, according to which juror is wrong, and each way has probability (3/5)·(3/5)·(2/5), so together they have probability 54/125.
So the majority of three is right with probability 81/125, which is 0.648, while a single juror is right with 3/5, which is 75/125. The jury is better than its members, by a little. The lab prints the exact fraction and works for any number of jurors up to 101.
With eleven jurors, still each right with probability 3/5, the majority is right with probability 36791901/48828125, about 0.75. The gain from the size of the jury has been substantial. The pattern is general: if p is above 1/2, the probability that the majority is right increases with every added pair of jurors, and tends to certainty.
The pattern reverses on the other side. Three jurors each right with probability 2/5 produce a majority that is right with probability 44/125, which is less than 2/5. Eleven such jurors produce 12036224/48828125, about 0.25. With probability below 1/2 each, the crowd does worse than its members, and the larger the crowd, the worse it does. At exactly 1/2 the crowd gains nothing: eleven jurors, each a coin flip, are right with probability 1/2.
If voters answer a yes-or-no question independently, each right with the same probability p, then the probability that a majority is right increases with the number of voters when p is above 1/2, decreases when it is below, and stays at 1/2 when it equals 1/2.
Two assumptions carry the result, and both are strong. The first is independence: each voter's chance of being right does not depend on how the others voted. Voters who read the same newspaper are not independent; if each follows a newspaper that is right with probability 3/5, a million voters are right together with probability 3/5 and not higher. “Testimony and Independent Witnesses” made the same point about witnesses.
The second is that there is a fact to be right about, and that voters are better than chance at finding it. A vote on a tax rate or a way of life is not an estimate of anything. The theorem's weight rests on the question being of the first kind.
The probability that one juror decides the verdict is a different quantity. With three jurors, a given juror decides the result only if the other two split, one right and one wrong, which can happen in two ways: 2·(3/5)·(2/5), which is 12/25. In general, for odd n, the others must divide exactly evenly, and the lab prints that probability.
With eleven jurors at 3/5 it is 1959552/9765625, about 0.20; with eleven coin-flipping jurors, 63/256, about 0.25. The probability shrinks as the electorate grows, and in a national election it is very small. That is the paradox of the pivotal voter: the benefit from any one vote is the probability that it decides, multiplied by whatever is at stake, and the first factor is tiny. The lab computes the first factor and not the second, and what the stakes are worth is a question about welfare, not about probability.
Interactive
six voting rules, the majority relation, judgment aggregation and the jury theorem
Method
Five moves for an odd number of jurors. The first four give the majority and the last gives the pivotal vote.
The question has a correct answer and each voter is right with probability p, independently. Write down n and p.
For three voters the majority is right when two or three are. In general it is every count of right voters from (n + 1)/2 to n.
A given set with k right and the rest wrong has probability p to the power k times 1 − p to the power n − k, and there are as many such sets as ways to choose which k are right.
The sum is the probability that the majority is right. If it exceeds p, the jury has improved on its members.
The others must split exactly evenly, (n − 1)/2 right and (n − 1)/2 wrong, in as many ways as there are ways to choose which are right. Multiply by that count.
The majority is right when at least two of the three are.
The majority is right more often than a juror, and the margin is small: 81 against 75 out of 125. The pivotal probability is 12/25 for three jurors and is already 1959552/9765625 for eleven.
Given n and p, write the probability that a majority is right as a sum, evaluate it for three voters, say which way it moves as n grows when p is above, at and below one half, and compute the pivotal probability.
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