Condorcet's Jury Theorem with Abstention

TL;DR

The paper investigates Condorcet's Jury Theorem under abstention in an asymmetric two-candidate election by introducing a boundedly rational model where voters rely on heuristic perceived pivotality $p(n,m)$. It characterizes equilibrium limits via pivot points where $s_A(c^*)=s_B(c^*)$, and identifies a phase transition in the asymptotic win probability of the better candidate governed by the parameters $(\alpha,\beta)$ of a polynomial pivot model: CJT holds when $\alpha>2\beta$, fails (yielding a strong non-jury) when $\alpha<2\beta$, and exhibits a margin-dependent limit when $\alpha=2\beta$. The results show that nontrivial equilibria can persist with the winning probability bounded away from 1, independent of population size, and depend mainly on the distribution of voting costs and the pivot function. The paper also demonstrates stability properties of equilibria, proposes unbiased participation via small random samples or multi-round voting, and provides numerical illustrations, highlighting practical implications for turnout dynamics and democratic decision-making.

Abstract

The well-known Condorcet's Jury theorem posits that the majority rule selects the best alternative among two available options with probability one, as the population size increases to infinity. We study this result under an asymmetric two-candidate setup, where supporters of both candidates may have different participation costs. When the decision to abstain is fully rational i.e., when the vote pivotality is the probability of a tie, the only equilibrium outcome is a trivial equilibrium where all voters except those with zero voting cost, abstain. We propose and analyze a more practical, boundedly rational model where voters overestimate their pivotality, and show that under this model, non-trivial equilibria emerge where the winning probability of both candidates is bounded away from one. We show that when the pivotality estimate strongly depends on the margin of victory, victory is not assured to any candidate in any non-trivial equilibrium, regardless of population size and in contrast to Condorcet's assertion. Whereas, under a weak dependence on margin, Condorcet's Jury theorem is restored.

Figures (10)

• Figure 1: A schematic illustation of stability of the equilibrium point $c^+$. When $c' > c^+$ is an equilibrium estimate, the $A$ supporters from right shaded region are incentivized to abstain whereas under $c"<c^+$ the $A$ supporters from left shaded region are incetivized to participate.
• Figure 2: The figure illustrates win probability of the popular candidate in any given intermediate round. For majority, $\Pr(A \ wins) = 0.84134$ and similarly for supermajority with margin $m \times 0.3$ (shown by orange color) it is $0.7580$ (the red area).
• Figure 3: Win probability of candidate $A$ under different values of $\alpha$ in $p(n,m) = \min( 1, \frac{1}{m^{\alpha}\sqrt{N}})$. The blowup shows that win probability is still below 1 for $\alpha=1.25$.
• Figure 4: Win probability for different values of $N$ under respective induced equilibria. The equilibrium win probability for a popular candidate $A$ increases with $N$ whereas it decreases for the unpopular candidate $B$.
• Figure 5: The pivot point $\hat{c}$ is marked by a circle at the intersection of the support functions, with the two non-trivial equilibria on its sides (dashed lines). For the upper equilibrium $c^+$, the probability of a random voter to vote $A$ or $B$ is proportional to $s_A(c^+)$ and $s_B(c^+)$, respectively. The $m'$ is proportional to the margin of victory. The bold arrows above indicate that $c^+, c^0$ are stable equilibria whereas $c^-$ is often not stable.

Theorems & Definitions (35)

• Proposition 1
• Proposition 2
• Proposition 3
• Definition 1: Issue equilibrium
• Example 1: Binomial PPM
• Example 2: Poisson PPM
• Definition 2: Vanishing Pivotality
• Definition 3: Tie-sensitive pivotality
• Example 3: Polynomial PPM
• Proposition 4