Jury Theorems
POLS 3220: How to Predict the Future
Warmup
Warmup
Warmup
Ponder: What if I asked the class to vote on the answers? How many correct answers do you think the majority would get?
Warmup
Warmup
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?
Warmup
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.
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?