30  The Condorcet Jury Theorem

In the previous chapter, we saw how May’s Theorem justifies Majority Rule by showing it uniquely satisfies certain principles of fair decision-making. In this chapter, we will explore a different justification for majority rule: the Condorcet Jury Theorem shows that under certain conditions, groups using Majority Rule are remarkably good at discovering the truth.

30.1 The Setup

Consider a group trying to determine the truth about some factual question - for instance, whether a defendant is guilty or innocent, or whether a medical treatment will be effective. We model this situation as follows:

  • There is a group of experts numbered \(1\) through \(n\).
  • There is a question with exactly two possible answers, labeled as \(0\) and \(1\).
  • Each expert must vote for either \(0\) or \(1\)
  • There is an objectively correct answer (though the experts don’t know it when voting).
  • Each expert has some probability of voting for the correct answer. That is, there is a number \(r\) between \(0\) and \(1\) such that: if the true answer is \(1\), then the probability the expert votes \(1\) is \(r\); and if the true answer is \(0\), then the probability the expert votes \(0\) is \(r\).

To formalize this mathematically, we introduce the following notation:

  • Let \(\mathbf{x}\) be the true state (the correct answer), which is either \(0\) or \(1\).
  • For each expert \(i\), let \(\mathbf{v}_i\) be their vote (either \(0\) or \(1\)).
  • We say expert \(i\) voted correctly if their vote matches the truth (i.e., if \(\mathbf{v}_i = \mathbf{x}\)).
  • Let \(R_i\) be the event that expert \(i\) votes correctly

30.2 The Theorem

Condorcet’s theorem relies on two crucial assumptions:

  • Independence: The experts vote independently - each expert’s probability of being correct does not depend on how others vote. This means that we are assuming experts make up their own minds rather than influencing each other.
  • Competence: Each expert is:
    • Better than random guessing (their probability of being correct exceeds 1/2).
    • Equally competent (all experts have the same probability of being correct).

Under these assumptions, Condorcet proved two striking conclusions about majority decision-making:

  1. Growing Reliability: As the group gets larger, the probability that the majority vote is correct increases.
  2. Asymptotic Infallibility: As the group size approaches infinity, the probability of the majority being correct approaches 100%.

This means that, under the right conditions, large groups using majority rule are extremely good at discovering the truth - better than even the smartest individual in the group.

The Condorcet Jury Theorem is an epistemic justification of majority rule showing that under the assumption that the voters are competent in the sense that each voters has a greater than 50% chance of voting correctly and that the events that the voters are correct are independent, then the probability that the majority is correct increases to 1 as the size of the group increases. The following simulation illustrates this remarkable result.