Robustness of voting mechanisms to external information in expectation

TL;DR

The paper analyzes how external anchoring information, encoded as a point $w\in\Delta_m$ with weight $\alpha$, biases voting outcomes under ordinal reporting. It introduces an intermediary scoring mechanism $\mathcal{M}$ that replicates anchored behavior without eliciting full cardinal utilities, enabling tractable comparison via histograms and balls-and-bins bounds. Key results show that anchoring can increase the probability that $w$'s preferred alternative is selected and that social welfare tends to rise in expectation when $w$ aligns with population preferences, with precise bounds for Plurality, Borda, and related rules. The framework provides practical tools for estimating anchored effects and offers avenues for extending to multiple information sources and fairness considerations.

Abstract

Analyses of voting algorithms often overlook informational externalities shaping individual votes. For example, pre-polling information often skews voters towards candidates who may not be their top choice, but who they believe would be a worthwhile recipient of their vote. In this work, we aim to understand the role of external information in voting outcomes. We study this by analyzing (1) the probability that voting outcomes align with external information, and (2) the effect of external information on the total utility across voters, or social welfare. In practice, voting mechanisms elicit coarse information about voter utilities, such as ordinal preferences, which initially prevents us from directly analyzing the effect of informational externalities with standard voting mechanisms. To overcome this, we present an intermediary mechanism for learning how preferences change with external information which does not require eliciting full cardinal preferences. With this tool in hand, we find that voting mechanisms are generally more likely to select the alternative most favored by the external information, and when external information reflects the population's true preferences, social welfare increases in expectation.

Figures (4)

• Figure 1: Plurality: Voters whose utilites lie in the red region (lower left) vote for alternative $a$ by ascribing score $(1,0,0)$, and similar for yellow voters voting for alternative $b$, and blue voting for alternative $c$.
• Figure 2: Fully ordinal prefs: Voters whose utility lies in the orange region vote $a \succ b \succ c$ by ascribing score $(2,1,0)$ and so on by permutations of $[m]$.
• Figure 3: Anchoring Plurality votes in the point $w$ yields a menu $\mathcal{M}$ with the level sets of $\Gamma^{\mathcal{M}}$ laid over the level sets of $\Gamma^{\mathcal{R}}$. Utilities in red and yellow intersection (hatched) will vote for candidate $b$ under standard preferences and candidate $a$ with the anchoring information. They are close to indifferent between $a$ and $b$ under standard preferences.
• Figure 4: Hatched regions subsets of $\Delta_3$ where any $w$ in the hatched region increases ensures that reports where $a^{\star}$ is the favorite are precisely the top-$(m-1)!$ most preferred. In turn, for any $w$ in the hatched regions, the bounds on the probability that $w$'s most favored alternative is selected increase.

Theorems & Definitions (15)

• Example 1: Plurality voting
• Example 2: Changing vote via $w$
• Example 3: Induced histograms
• Example 4: Determining the winner of an election
• Proposition 5
• Corollary 6
• Corollary 7
• Corollary 9
• Corollary 10
• proof