What is special about the proportion? A research report on special majority voting and the classical Condorcet jury theorem

It is known that, in Condorcet’s classical model of jury decisions, the proportion of jurors supporting a decision is not a significant indicator of that decision’s reliability: the probability that a particular majority decision is correct given the size of the majority depends only on the absolute margin between the majority and the minority, and is invariant under changes of the proportion in the majority if the absolute margin is held fixed. Here I show that, if we relax the assumption that juror competence is independent of the jury’s size, the proportion can be made significant: there are then conditions in which the probability that a given majority decision is correct depends only on the proportion of jurors supporting that decision, and is invariant under changes of the jury size. The proportion is significant in this way if and only if juror competence is a particular decreasing function of the jury size. However, the required condition on juror competence is not only highly special – thereby casting doubt on the significance of the proportion in realistic conditions – but it also has an adverse implication for the Condorcet jury theorem. If the proportion is significant, then the Condorcet jury theorem fails to hold; and if the Condorcet jury theorem holds, the proportion is not significant. I discuss the implications of these results for defining and justifying special majority voting from the perspective of an epistemic account of voting.