On Condorcet's Jury Theorem with Abstention

TL;DR

This paper extends Condorcet's Jury Theorem to settings with voter abstention and costs, revealing conditions for stable equilibria and when the majority's choice aligns with the probability approaching certainty.

Contribution

It introduces a model with asymmetric voter costs and heuristic beliefs, identifying conditions for multiple equilibria and the applicability of the Jury Theorem in abstention scenarios.

Findings

Weakly vanishing pivotality leads to multiple near-tie equilibria.

Strongly vanishing pivotality results in a trivial equilibrium with limited participation.

A threshold determines when the majority wins with high probability versus equal chances.

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.