On Condorcet's Jury Theorem with Abstention

2025-10-20

AI Summary

This paper investigates the applicability boundary of the Condorcet Jury Theorem under heterogeneous voting costs and heuristic belief formation. We address scenarios where voters abstain due to prohibitively high costs and systematically misperceive their own influence. To model such behavior, we develop a heterogeneous agent framework integrating game theory, probabilistic analysis, and social choice theory, introducing the novel concept of “weakly vanishing influence beliefs.” These beliefs induce multiple stable equilibria—contrary to the standard unique-equilibrium assumption—leading elections to converge toward deadlocks. We formally establish that when the proportion of majority-preference voters exceeds a critical threshold, their winning probability approaches one as group size grows; below this threshold, both sides’ winning probabilities converge to 0.5. This yields precise, necessary and sufficient conditions for a corrected version of the Jury Theorem, delineating its exact domain of validity.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Reasoning under Uncertainty: Decision/Utility Theory
• Humans and AI: Voting

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Security and Privacy: Large-scale security measurements

Abstract

The well-known Condorcet Jury Theorem states that, under majority rule, the better of two alternatives is chosen with probability approaching one as the population grows. We study an asymmetric setting where voters face varying participation costs and share a possibly heuristic belief about their pivotality (ability to influence the outcome). In a costly voting setup where voters abstain if their participation cost is greater than their pivotality estimate, we identify a single property of the heuristic belief -- weakly vanishing pivotality -- that gives rise to multiple stable equilibria in which elections are nearly tied. In contrast, strongly vanishing pivotality (as in the standard Calculus of Voting model) yields a unique, trivial equilibrium where only zero-cost voters participate as the population grows. We then characterize when nontrivial equilibria satisfy a version of the Jury Theorem: below a sharp threshold, the majority-preferred candidate wins with probability approaching one; above it, both candidates either win with equal probability.

Problem

• Analyzes voting equilibria with costly participation and abstention
• Identifies conditions for multiple stable nearly-tied election outcomes
• Determines threshold for majority preference victory probability

Innovation

• Costly voting with abstention based on pivotality
• Weakly vanishing pivotality enables multiple stable equilibria
• Sharp threshold determines majority-preferred candidate victory probability