AI 中文总结
该研究将度量失真框架从社会选择函数扩展至社会福利函数，分析了三种信息场景下的失真度，得出递归应用单一胜者规则等方法的失真度界及最优失真度结论。
AI 中文摘要
度量失真主要针对从序数偏好中选择单一胜者的社会选择函数展开研究，我们将该框架扩展至输出m个候选者排名的社会福利函数。我们为每个投票者v关联一个单调权重向量w_v=(w_{v1},…,w_{vm})，指定投票者v对第i个位置的重视程度，并将排名的成本定义为其与所排名候选者的距离的位置加权和。该模型推广了单一胜者投票和委员会选择两种场景。我们考虑三种信息场景：首先，研究位置权重向量已知的情况，一种自然的方法是递归应用失真度为β的单一胜者规则，逐位构建排名，我们证明该方法总体上产生的失真度至多为3β，该差距并非分析的人为产物，我们还证明仅基于每轮保证的分析无法证明优于2β的界，通过利用Kizilkaya和Kempe提出的分数否决（Fractional Veto）的特定结构性质，我们证明其递归扩展达到最优失真度3；其次，当所有投票者共享相同的未知权重向量时，递归应用任何失真度为β的社会选择函数，产生的失真度至多为1+(β-1)range(w)，其中range(w)=(w_1-w_m)/w_1表示共同权重向量w的归一化范围；最后，研究未知异质权重的情况，若无进一步假设，任何规则的失真度都无界，因此我们考虑两种自然归一化：单位和（unit-sum，即每个投票者在排名中分配1个单位的价值）与单位顶部（unit-top，即每个投票者将1个单位价值分配给第一个位置），在这两种模型下，我们证明最优失真度为Θ(m)。
英文摘要
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 $Θ(m)$.
Comments11 pages, 1 table
