Metric Distortion of Social Welfare Functions

2026-08-22

AI Summary

本文研究了社会福利函数在不同信息条件下的度量扭曲问题，通过递归应用单胜者规则来构造排名，并分析了其在已知权重、未知相同权重及未知异构权重情况下的扭曲界限。

Abstract

Metric distortion has primarily been studied for social choice functions, which select a single winner from ordinal preferences. We extend this framework to social welfare functions, which output a ranking of $m$ candidates. We associate each voter $v$ with a monotone weight vector $\mathbf{w}_v = (w_{v1},\ldots,w_{vm})$, specifying the importance of the $i$-th position for voter $v$, and define the cost of a ranking as the position-weighted sum of their distances to the ranked candidates. This model generalizes both single-winner voting and committee selection. We consider three information regimes. First, we study the setting where the positional weight vectors are known. A natural approach recursively applies a single-winner rule with distortion $β$ to construct the ranking one position at a time. We show that this yields distortion at most $3β$ in general. This gap is not merely an artifact of the analysis: we show that no analysis based solely on per-round guarantees can certify a bound better than $2β$. By exploiting structural properties specific to Fractional Veto of Kizilkaya and Kempe, we show that its recursive extension achieves the optimal distortion of 3. Second, when all voters share the same unknown weight vector, recursively applying any social choice function with distortion $β$ achieves distortion at most $1+(β-1)\text{range}(\mathbf{w})$, where $\text{range}(\mathbf{w})=(w_1-w_m)/w_1$ denotes the normalized range of the common weight vector $\mathbf{w}$. Finally, we study unknown heterogeneous weights. Without further assumptions, every rule has unbounded distortion. We therefore consider two natural normalizations: unit-sum, where each voter distributes one unit of value across the ranking, and unit-top, where every voter assigns unit value to the first position. Under both models, we show that the optimal distortion is $Θ(m)$.

Problem

• metric distortion
• social welfare functions
• ordinal preferences
• position-weighted sum
• information regimes

Innovation

• metric distortion
• social welfare functions
• monotone weight vector
• recursive application
• fractional veto