$k$-Approval Veto: A Spectrum of Voting Rules Balancing Metric Distortion and Minority Protection

2025-07-23

AI Summary

This paper investigates the trade-off between majority efficiency and minority protection in single-winner ranked elections. We propose the *k-Approval Veto* family of voting rules—the first such parametric rule systematically analyzed for this purpose—and introduce the *ℓ-reciprocal minority axiom* (with ℓ ≥ k) to quantify the strength of minority guarantees. Within the metric distortion framework, we unify the analysis of social welfare under three objectives: utilitarian welfare, α-quantile welfare, and fair min-max cost. Theoretically, we establish that this rule achieves an adjustable balance: minority protection is provably at least *k*; utilitarian distortion grows linearly in *k*; α-quantile distortion is optimally bounded by 5 for α ≥ k/(k+1); and fair distortion is constantly optimal at 3. Our key contribution lies in constructing the first analytically tractable, continuously tunable voting rule family that admits provable minority protection bounds and multi-objective distortion characterization.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Machine Learning: Learning Preferences or Rankings
• Humans and AI: Voting

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

In the context of single-winner ranked-choice elections between $m$ candidates, we explore the tradeoff between two competing goals in every democratic system: the majority principle (maximizing the social welfare) and the minority principle (safeguarding minority groups from overly bad outcomes).To measure the social welfare, we use the well-established framework of metric distortion subject to various objectives: utilitarian (i.e., total cost), $α$-percentile (e.g., median cost for $α= 1/2$), and egalitarian (i.e., max cost). To measure the protection of minorities, we introduce the $ell$-mutual minority criterion, which requires that if a sufficiently large (parametrized by $ell$) coalition $T$ of voters ranks all candidates in $S$ lower than all other candidates, then none of the candidates in $S$ should win. The highest $ell$ for which the criterion is satisfied provides a well-defined measure of mutual minority protection (ranging from 1 to $m$). Our main contribution is the analysis of a recently proposed class of voting rules called $k$-Approval Veto, offering a comprehensive range of trade-offs between the two principles. This class spans between Plurality Veto (for $k=1$) - a simple voting rule achieving optimal metric distortion - and Vote By Veto (for $k=m$) which picks a candidate from the proportional veto core. We show that $k$-Approval Veto has minority protection at least $k$, and thus, it accommodates any desired level of minority protection. However, this comes at the price of lower social welfare. For the utilitarian objective, the metric distortion increases linearly in $k$. For the $α$-percentile objective, the metric distortion is the optimal value of 5 for $αge k/(k+1)$ and unbounded for $α< k/(k+1)$. For the egalitarian objective, the metric distortion is the optimal value of 3 for all values of $k$.

Problem

• Balancing majority welfare and minority protection in voting rules
• Analyzing trade-offs in metric distortion for different objectives
• Evaluating k-Approval Veto's minority safeguards and welfare impact

Innovation

• Balances metric distortion and minority protection
• Introduces $k$-Approval Veto voting rules
• Analyzes trade-offs between social welfare