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

TL;DR

This paper critically examines the probability of the Condorcet Jury Theorem's validity, showing it almost surely fails under measure-theoretic analysis, and proposes weighted majority rules to ensure competence.

Contribution

It introduces a measure-theoretic approach to assess the prior probability of the theorem and proposes a weighted aggregation method to guarantee competence.

Findings

The prior probability of the theorem's thesis fails almost surely.

Weighted majority rules can ensure almost sure competence.

Measure theory reveals limitations of simple majority rule.

Abstract

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 an estimation of the prior probability of the thesis predicted by the theorem (if there are enough voters, majority rule is a competent decision procedure). We use tools from measure theory to conclude that, prima facie, it will fail almost surely. To update this prior either more evidence in favor of competence would be needed or a modification of the decision rule. 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 weights correlated with epistemic rationality such that every voter is guaranteed a minimal weight equal to one.