This is a summary of the article by David Austin-smith and Jeffrey S.Banks, Information Aggregation, Rationality, and the Condorcet Jury Theorem (see reference at the bottom). We were encouraged to read it before Monday’s course. Of course I had a painful time reading it and the summary may not accurately convey the article’s intention.

What is Condorcet Jury theorem?

The Condorcet Jury theorem states that under certain conditions, the probability that a collective chooses the correct alternative by majority votes exceeds the probability that any constituent member of the collective would unilaterally choose that alternative. That is to say, majorities are more likely to select the “better” alternatives under uncertainty.

What is the problem?

This theorem has an important but largely implicit assumption that an individual behaves in exactly the same manner as when that individual alone selects the outcome. Actually, sincere behavior is not rational even has a common preference. And sincere voting does not constitute a Nash equilibrium.

You may ask, what is sincere voting? OK, there are three sorts of voting behaviors:

• Sincere voting: each individual selects the alternative yielding his or her highest expected payoff conditional on their signal.
• Informative voting: each individual votes for A if and only if receiving a signal s = 0, and B when s = 1.
• Rational voting: individual’s decision constitute a Nash equilibrium.
  Attention: these three voting behaviors are not mutually exclusive.

The authors describe two assumptions:

Sincere voting is informative when one receives a signal that makes him/her think A/B is the true state.
The common prior belief that the true state is A is sufficiently strong that there is only one condition that someone will select B that he/she observes all three individuals’ signals as B. (Why they talk about prior belief here? Otherwise how can you use Bayes’ method!)

How to prove the claim?

The authors looked at the role the “sincerity” assumption in Jury Theroems with three variations of an extremely simple model.

(I will not introduce the model here because I have to admit I totally can’t understand the definition and formula. After all, they use the model to explain it and it seems quite interesting. I think I need to learn more about game theory to understand it).

Reference

Austen-Smith, D., & Banks, J. (1996). Information Aggregation, Rationality, and the Condorcet Jury Theorem. The American Political Science Review, 90(1), 34-45. doi:10.2307/2082796