社会选择规则的度量失真与公平性
Relating Metric Distortion and Fairness of Social Choice Rules
摘要：
评估社会选择（投票）规则的一种方法是通过功利主义扭曲框架。在这个模型中，我们假设代理提交完整的排名的替代品，这些排名是从潜在的，但未知的，定量成本。因此，社会选择规则的\emph{distortion}是所选方案的总社会成本与任何方案的最优社会成本之比;由于真实成本是未知的，我们考虑所有可能的潜在成本的最坏情况下的扭曲。类似地，我们可以考虑社会选择规则的最坏情况\emph{fairness ratio}，通过比较所选方案与最优方案的公平性（基于近似优化）。通过一个额外的度量假设--成本等于某个度量空间中的代理-替代距离--已知科普兰规则实现了最多5的失真和公平比。对于其他规则，仅失真的界限是已知的，例如，流行的单一可转移投票（STV）规则具有失真$O(\log m)$，其中$m$是备选项的数目。我们证明，失真和公平比的不同概念实际上是密切相关的-在任何投票规则的加性因子为2-因此STV也实现了$O(\log m)$公平比。我们进一步扩展的失真和公平比的概念，选择一个\emph{set}的替代品的社会选择规则。通过将单赢家规则的失真与多赢家规则相关联，我们确定递归Copeland在选择一组备选方案时实现了5的失真和至多7的公平比。
英文摘要：
One way of evaluating social choice (voting) rules is through a utilitarian distortion framework. In this model, we assume that agents submit full rankings over the alternatives, and these rankings are generated from underlying, but unknown, quantitative costs. The \emph{distortion} of a social choice rule is then the ratio of the total social cost of the chosen alternative to the optimal social cost of any alternative; since the true costs are unknown, we consider the worst-case distortion over all possible underlying costs. Analogously, we can consider the worst-case \emph{fairness ratio} of a social choice rule by comparing a useful notion of fairness (based on approximate majorization) for the chosen alternative to that of the optimal alternative. With an additional metric assumption -- that the costs equal the agent-alternative distances in some metric space -- it is known that the Copeland rule achieves both a distortion and fairness ratio of at most 5. For other rules, only bounds on the distortion are known, e.g., the popular Single Transferable Vote (STV) rule has distortion $O(\log m)$, where $m$ is the number of alternatives. We prove that the distinct notions of distortion and fairness ratio are in fact closely linked -- within an additive factor of 2 for any voting rule -- and thus STV also achieves an $O(\log m)$ fairness ratio. We further extend the notions of distortion and fairness ratio to social choice rules choosing a \emph{set} of alternatives. By relating the distortion of single-winner rules to multiple-winner rules, we establish that Recursive Copeland achieves a distortion of 5 and a fairness ratio of at most 7 for choosing a set of alternatives.
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