In a homogeneous jury, the votes are exchangeable correlated Bernoulli random variables. We derive the bounds on a homogeneous jury’s competence as the minimum and maximum probability of the jury being correct, which arise due to unknown correlations among the votes. The lower bound delineates the downside risk associated with entrusting decisions to the jury. In large and not-too-competent juries the lower bound may fall below the success probability of a fair coin flip—one half, while the upper bound may not (...) reach a certainty. We also derive the bounds on the voting power of an individual juror as the minimum and maximum probability of her/his casting a decisive vote. The maximum is less than one, while the minimum of zero can be attained for infinitely many combinations of distribution moments. (shrink)

According to the skeptical argument from disagreement, we are mandated to suspend judgement about a question if we discover that others disagree with us. Critics, however, have proposed that this skeptical argument fails if there are not equally many people on either side of the debate: numbers matter. The present paper explicates this as the argument that a group can be more likely to arrive at the correct view by majority rule than the members are on their own. Defenders of (...) the skeptical argument have resisted that numbers matter by observing that if group members depend on each other when forming their beliefs, then the group can be less competent than its members. However, neither side of the debate has accompanied their views with quantitative estimates of how detrimental dependence is for group competence. This paper tries to improve this situation by drawing on jury theorems from social choice theory. The paper cannot settle the debate, but it shows that even the lower limit on group competence will exceed the average individual competence when dependence among voters remains moderate. This should give confidence to those who propose that asymmetry between the disputing parties can counter the skeptical argument from disagreement since being in the majority can thus be higher-order evidence for the disputed proposition. (shrink)

The purpose of this paper is to illustrate, formally, an ambiguity in the exercise of political influence. To wit: A voter might exert influence with an eye toward maximizing the probability that the political system (1) obtains the correct (e.g. just) outcome, or (2) obtains the outcome that he judges to be correct (just). And these are two very different things. A variant of Condorcet's Jury Theorem which incorporates the effect of influence on group competence and interdependence is developed. Analytic (...) and numerical results are obtained, the most important of which is that it is never optimal--from the point-of-view of collective accuracy--for a voter to exert influence without limit. He ought to either refrain from influencing other voters or else exert a finite amount of influence, depending on circumstance. Philosophical lessons are drawn from the model, to include a solution to Wollheim's "Paradox in the Theory of Democracy". (shrink)