On the probability of the Condorcet Jury Theorem or the Miracle of Aggregation

The Condorcet Jury Theorem or the Miracle of Aggregation are frequently invoked to ensure the competence of some aggregate decision-making processes. In this article we explore the probability of the thesis predicted by the theorem (if there are enough voters, majority rule is a competent decision procedure) in different settings. We use tools from measure theory to conclude that it will happen almost surely or almost never, depending on the probability measure. In particular, it will fail almost surely for measures estimating the prior probability. To update this prior either more evidence in favor of competence would be needed (so that a large likelihood term compensates a small prior term in Bayes' Theorem) or a modification of the decision rule. The former includes the case of (rational) agents reversing their vote if its probability of voting the right option is less than 1/2. Following the latter, we investigate how to obtain an almost sure competent information aggregation mechanism for almost any evidence on voter competence (including the less favorable ones). To do so, we substitute simple majority rule by weighted majority rule based on some stochastic weights correlated with epistemic rationality such that every voter is guaranteed a minimal weight equal to one. We also explore how to obtain these weights in a real setting.