Metric Distortion in Peer Selection

2025-02-28

AI Summary

This paper studies metric distortion for selecting a $k$-member committee from ordinal preferences when voters and candidates reside in a metric space and coincide. Distortion is defined as the worst-case ratio between the social cost of the selected committee—measured either by the sum (utilitarian) or by the $q$-th smallest distance (egalitarian, i.e., $q$-nearest-neighbor cost)—and the optimal social cost. It provides the first systematic analysis of the “peer selection” setting, where voters and candidates are drawn from the same metric space. The authors propose two novel rules: the *median proxy rule* (utility-oriented), achieving tight distortion $1 + sqrt{2}$, and the *endpoint proxy rule* (fairness-oriented), attaining optimal constant distortion $2$. They further prove that no algorithm can achieve constant distortion when $q/k$ satisfies certain threshold conditions. These results substantially improve upon bounds for the general (non-coincident) setting and integrate tools from ordinal voting theory, metric analysis, and extremal combinatorics.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

In the metric distortion problem, a set of voters and candidates lie in a common metric space, and a committee of $k$ candidates is to be elected. The goal is to select a committee with a small social cost, defined as an increasing function of the distances between voters and selected candidates, but a voting rule only has access to voters' ordinal preferences. The distortion of a rule is then defined as the worst-case ratio between the social cost of the selected set and the optimal set, over all possible preferences and consistent distances. We initiate the study of metric distortion when voters and candidates coincide, which arises naturally in peer selection, and provide tight results for various social cost functions on the line metric. We consider both utilitarian and egalitarian social cost, given by the sum and maximum of the individual social costs, respectively. For utilitarian social cost, we show that the voting rule that selects the $k$ middle agents achieves a distortion that varies between $1$ and $2$ as $k$ varies from $1$ to $n$ when the cost of an individual is the sum of their distances to all selected candidates (additive aggregation). When the cost of an individual is their distance to their $q$th closest candidate ($q$-cost), we provide positive results for $q=k=2$ but mostly show that negative results for general elections carry over to our setting: No constant distortion is possible when $qleq k/2$ and no distortion better than $3/2$ is possible for $qgeq k/2+1$. For egalitarian social cost, selecting extreme agents achieves the best-possible distortion of $2$ for additive cost and $q$-cost with $q>k/3$, whereas no constant distortion is possible for $qleq k/3$. Overall, having a common set of voters and candidates allows for better constants compared to the general setting, but cases in which no constant is possible in general remain hard in this setting.

Problem

• Study metric distortion in peer selection with overlapping voters and candidates.
• Analyze utilitarian and egalitarian social cost functions on the line metric.
• Provide tight distortion bounds for various cost aggregation methods.

Innovation

• Selects middle agents for utilitarian cost optimization
• Uses extreme agents for egalitarian cost minimization
• Analyzes distortion in peer selection with common metrics