Metric Distortion in Peer Selection

TL;DR

This paper studies metric distortion in peer selection, analyzing how well simple voting rules approximate optimal social costs in a common metric space, revealing persistent hardness and some improved bounds in specific cases.

Contribution

It introduces the metric distortion problem in peer selection, providing bounds and hardness results for various objectives and metrics, including the line metric case.

Findings

Lower bounds on distortion remain above one even on the line metric.

Selecting the middle agents yields low distortion for utilitarian cost under additive aggregation.

Choosing extremes achieves optimal distortion for egalitarian cost under certain conditions.

Abstract

In the metric distortion problem, a set of voters and candidates lie in a common metric space, and a committee of $k$ candidates must be elected. The objective is to minimize a social cost, defined as a function of the distances between voters and their chosen representatives, while the voting rule only has access to ordinal preferences. The distortion of a rule is the worst-case ratio between the social cost of its outcome and that of the optimal committee, taken over all consistent preferences and metrics. We initiate the study of metric distortion in peer selection, where voters and candidates coincide. We consider four objectives, obtained by combining two aggregation rules with two types of social cost. Under additive aggregation, an individual's cost is the sum of their distances to all committee members; under $q$-cost, it is their distance to the $q$th closest member. The overall social cost is either utilitarian, given by the sum of all individual costs, or egalitarian, given by the maximum individual cost. Surprisingly, we find that even on the line metric, peer selection retains much of the hardness of the general case: Lower bounds remain strictly larger than one for all objectives, and cases where bounded distortion is impossible in general remain so here as well. On a positive note, cases with bounded distortion in the general setting achieve better constants in peer selection. For utilitarian cost, selecting the $k$ middle agents achieves a distortion between $1$ and $2$ under additive aggregation. Under $q$-cost, we show positive results for $q=k=2$, but impossibility results largely carry over. For egalitarian cost, selecting the extremes yields an optimal distortion of $2$ under additive aggregation and for $q$-cost with $q>k/3$. Thus, while peer selection on the line metric allows better constants, fundamental hardness barriers persist.