Jury Theorems

POLS 3220: How to Predict the Future

Warmup

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Warmup

Ponder: What if I asked the class to vote on the answers? How many correct answers do you think the majority would get?

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Warmup

Ponder: What if I only took the majority vote of students who were really confident in their answers (>95%)? Would that result be better or worse?

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Warmup

Wisdom of Crowds

  • This phenomenon is known as the Wisdom of Crowds.

    • Groups of people often outperform the individuals that make them up.

  • Over the next few weeks, we’ll discuss how to harness the Wisdom of Crowds to make better predictions.

  • We’ll also explore the conditions under which groups of people perform worse than individuals (the Madness of Crowds).

Marquis de Condorcet (1743-1794)

Condorcet Jury Theorem

Assumptions:

  • A group is voting on a binary (Yes/No) decision.

  • Each voter has probability \(p > \frac{1}{2}\) of making the correct choice.

  • Individual votes are independent of one another.

Theorem:

  • As the size of the group \(n\) gets large:

    • The probability that the majority makes the correct choice approaches 100%.

Condorcet Jury Theorem

Ponder: Does the Condorcet Jury Theorem remind you of anything?

Condorcet Jury Theorem

Example: A jury with 5 members, where each individual has a 60% probability of choosing the right answer.

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Start Voting

1R,0W

0R,1W

2R,0W

1R,1W

0R,2W

3R,0W

2R,1W

1R,2W

0R,3W

4R,0W

3R,1W

2R,2W

1R,3W

0R,4W

5R,0W (7.8%)

4R,1W (25.9%)

3R,2W (34.6%)

2R,3W (23%)

1R,4W (7.7%)

0R,5W (1%)

Condorcet Jury Theorem

Condorcet Jury Theorem

Condorcet Jury Theorem

Condorcet Jury Theorem

Condorcet Jury Theorem

Condorcet Jury Theorem

Condorcet Jury Theorem

Takeaways

  • The Condorcet Jury Theorem shows us when we can expect majority rule to yield good judgments.

  • Majority rule is a powerful method for finding the right answer, if:

    • Individuals are competent \((p > \frac{1}{2})\).
    • Choices are made independently.
    • The group is large.

  • The theorem does not imply that non-independent groups will always make the wrong choice.

    • Or small groups

    • Or incompetent groups

  • Rather, it shows that these three assumptions are sufficient to guarantee the group will make the correct choice.

  • Next time: what if a group is making a non-binary choice?

References