A Model of Jury Decisions Where All Jurors Have the Same Evidence

In the classical Condorcet jury model, different jurors' votes are independent random variables, where each juror has the same probability p>1/2 of voting for the correct alternative. The probability that the correct alternative will win under majority voting converges to 1 as the number of jurors increases. Hence the probability of an incorrect majority vote can be made arbitrarily small, a result that may seem unrealistic. A more realistic model is obtained by relaxing the assumption of independence and relating the vote of every juror to the same "body of evidence". In terms of Bayesian trees, the votes are direct descendants not of the true state of the world ('guilty' or 'not guilty'), but of the "body of evidence", which in turn is a direct descendant of the true state of the world. This permits the possibility of a misleading body of evidence. Our main theorem shows that the probability that the correct alternative will win under majority voting converges to the probability that the body of evidence is not misleading, which may be strictly less than 1.