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Condorcet’s Jury Theorem

October 25, 2011

In 1785 the French philosopher and mathematician Marquise de Condorcet published his influential Essay on the Application of Analysis to the Probability of Majority Decisions.  Perhaps the most famous result of the work is to show how voting can be similar to a rock-paper-scissors game: With three candidates for president (A, B, and C), it is possible to obtain a situation where A can be called a better choice than B, B a better choice than C, and C better than A.

In the same work Condorcet also introduced his Jury Theorem (CJT) – I’ll paraphrase from D. Austen-Smith and J. S. Banks, The American Political Science Review, Vol. 90, No. 1, p. 34: “Suppose there are two mutually exclusive alternatives, A, B.  One of these alternatives is unequivocally better for all of n individuals in a group but the identity of the better alternative (i.e., A or B) is unknown. Suppose further that for all individuals the probability that i votes for the better alternative is p > 1/2 and is independent of the other votes. The Theorem states that the probability that a majority votes for the better alternative approaches 1 as n goes to infinity.”

The CJT has been taken as evidence for decision making by majority rule.  The argument is that the larger the jury (electorate), the more likely they are to vote for the better decision according to the CJT.  One can quibble with the assumption of independence, but the theorem can be proved even when the votes are cast dependently.

There is another thing I’d like to point out (although I am certainly not the first to do so): Note that the statement of the CJT may be interpreted as assuming that there are two states of the world, and that we are trying to infer which is the true state.  For instance, if we look at an actual jury the defendant is either guilty or innocent.  The assumptions of Condorcet’s Theorem are that individuals in the Jury make independent choices and that their probability of making the correct choice is p.  Here is where we should be careful – there are actually two distinct probabilities lumped into one here.  There is the probability of correctly declaring the defendant guilty (call it p) and correctly declaring the defendant not guilty (call it q).  These two probabilities may be the same, but may be quite different.

Suppose you have a person that has such a negative view of the world that they believe everybody to be guilty.  In that case, they would always correctly identify the guilty parties (p=1), but fail to identify the not guilty parties (q=0).  I believe it is more pertinent to ask what is the best choice to make given a particular vote, and if we know p and q for this electorate (or the individual probabilities if they differ between the voters).  Under some circumstances this will the vote of the majority will give the correct decision.  However, frequently it will not.  The optimal decision can be found using the theorem named after Condorcet’s contemporary Thomas Bayes  (And a side note about using Bayes’ Theorem.  It is necessary to assign prior probabilities on the defendant being guilty or not.  Assuming literally that a “defendant is innocent until proven guilty” would actually imply that everybody is innocent – it is necessary to assume a nonzero probability of the defendant being guilty before any evidence is presented.)

There is another situation where the classical statement of the CJT makes sense – if there is one state of the world, which is fixed, and people infer it correctly with probability p.  I think this is the less interesting case: If there is only one possible state of the world, then why would there be disagreement about it, and why would people have to vote on it?  There must be at least two probable states.  Otherwise nobody would believe the alternative.

There are other problems with calling this a Jury Theorem: In a jury people communicate and exchange information.  The theorem implicitly assumes that every individual makes a choice that is dependent only on their individual information.  In any case, I am unconvinced that the theorem says anything useful about the democratic process.  However, it does bring up some thorny issues about decision making in groups.

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