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On Condorcet's Jury Theorem with Abstention

Reshef Meir, Ganesh Ghalme

TL;DR

This work extends the Condorcet Jury Theorem to settings with abstention and heterogeneous participation costs by introducing a general Perceived Pivotality Model $p(n,m)$ that captures how voters estimate their influence via population size $n$ and margin of victory $m$. It shows that when pivotality is tie-sensitive but only weakly vanishing, multiple nontrivial equilibria emerge with pivot points $c^*$ as limits; depending on the rate at which $m(c_N)$ shrinks with $N$, the local jury outcomes range from CJT-like convergence (probability to elect the better candidate approaching 1) to strong non-jury behavior (probability approaching 1/2). Conversely, strong vanishing pivotality yields only a trivial equilibrium where almost no one votes as $N$ grows. The paper provides a taxonomy of PPMs (networked, altruist, polynomial heuristics) and derives sharp thresholds (e.g., $\alpha=2\beta$ in polynomial PPMs) that delineate when CJT-like outcomes hold, offering insights into turnout dynamics and the robustness of collective wisdom under realistic voting heuristics. These results have practical implications for understanding turnout patterns, political participation, and the limits of the CJT under imperfect rationality.

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.

On Condorcet's Jury Theorem with Abstention

TL;DR

This work extends the Condorcet Jury Theorem to settings with abstention and heterogeneous participation costs by introducing a general Perceived Pivotality Model that captures how voters estimate their influence via population size and margin of victory . It shows that when pivotality is tie-sensitive but only weakly vanishing, multiple nontrivial equilibria emerge with pivot points as limits; depending on the rate at which shrinks with , the local jury outcomes range from CJT-like convergence (probability to elect the better candidate approaching 1) to strong non-jury behavior (probability approaching 1/2). Conversely, strong vanishing pivotality yields only a trivial equilibrium where almost no one votes as grows. The paper provides a taxonomy of PPMs (networked, altruist, polynomial heuristics) and derives sharp thresholds (e.g., in polynomial PPMs) that delineate when CJT-like outcomes hold, offering insights into turnout dynamics and the robustness of collective wisdom under realistic voting heuristics. These results have practical implications for understanding turnout patterns, political participation, and the limits of the CJT under imperfect rationality.

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.
Paper Structure (43 sections, 24 theorems, 52 equations, 4 figures, 1 table)

This paper contains 43 sections, 24 theorems, 52 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Calculus of Voting model:
  3. Enter heuristics
  4. Exit rationality
  5. Our Contribution
  6. Our Results:
  7. Model
  8. Calculus of Voting
  9. Rational equilibrium
  10. Support functions
  11. Perceived Pivotality
  12. Pivot functions
  13. Election equilibrium
  14. Issues and Elections
  15. Trivial equilibria
  16. ...and 28 more sections

Key Result

Proposition 1

Let voter costs and types are sampled i.i.d. from a distribution $\mathcal{D}$. Then any distribution $\mathcal{D}$ induces a unique pair of support functions $s_A,s_B$ with $s_A(1)+s_B(1)=1$, and vice-versa.

Figures (4)

  • Figure 1: Position of our work within the literature: EQ=Equilibrium; R=Rational; and H=Heuristic.
  • Figure 2: (L) demonstrates election instance from Section \ref{['sec:simulation']}. For large value $N$, the pivot point $c^*$, two non-trivial equilibria $c_N^+$ and $c_N^-$, and the trivial equilibrium $c^0$ are shown. The bold arrows indicate that $c^+_N, c^0_N$ are stable equilibria whereas $c^-_N$ is not stable. (C) Win probability for different values of $N$ under respective induced equilibria. (R) Win probability of $A$ for $\beta = 0.5$ and different values of $\alpha$ in polynomial PPM model for different values of $N$. The trend reversal can be observed at $\alpha =1$.
  • Figure 3: Pivot points and equilibria when the support functions are partially overlapping.
  • Figure 4: (L) demonstrates election instance from Section \ref{['sec:simulation']}. For large value $N$, the pivot point $c^*$, two non-trivial equilibria $c_N^+$ and $c_N^-$, and the trivial equilibrium $c^0$ are shown. For $c^+_N$, the probability of a random voter to vote $A$ is proportional to $s_A(c^+_N)$. The $m(c^+_N)$ is proportional to the margin of victory. The bold arrows indicate that $c^+_N, c^0_N$ are stable equilibria whereas $c^-_N$ is not stable. (C) Win probability for different values of $N$ under respective induced equilibria. (R) Win probability of $A$ for $\beta = 0.5$ and different values of $\alpha$ in polynomial PPM model for different values of $N$. The trend reversal can be observed at $\alpha =1$.

Theorems & Definitions (60)

  • Definition 1: Threshold profile
  • proof
  • Proposition 1
  • Definition 2
  • Proposition 2
  • Definition 3: Issue equilibrium
  • Proposition 3
  • Example 1: Binomial PPM
  • Definition 4: Vanishing Pivotality
  • Definition 5: Tie-sensitive pivotality
  • ...and 50 more