标题： 合理代表：从哈雷到多普 显示英文标题

标题： Justified Representation: From Hare to Droop

摘要： 在近年来，关于批准选票的多胜者投票中的比例性研究受到了广泛关注。 通常，比例性通过“正当代表”公理的不同变体来体现，这些公理指出，至少由$\ell\cdot\frac{n}{k}$名选民组成的团结群体（其中$n$是总选民数，$k$是期望的获胜者人数）应获得$\ell$名代表。 量$\frac{n}{k}$在社会选择文献中被称为哈雷配额。 另一种——更为严格——的配额选择是 多鲁普配额，定义为$\lfloor\frac{n}{k+1}\rfloor+1$。 这种配额常用于具有排序选票的多胜者投票：例如在单转移投票算法中，以及在多鲁普比例准则等比例性公理中。 一些作者曾在批准选票的背景下考虑过它，但现有的分析远不够全面。 我们工作的贡献是对使用多鲁普配额而非哈雷配额定义的 JR 类公理（以及满足它们的投票规则）进行系统研究。 对于每个标准的 JR 公理（即 JR、PJR、EJR、FPJR、FJR、PJR+ 和 EJR+），我们确定了一个满足该公理的多鲁普版本的投票规则。 在某些情况下，只需考虑已知规则（修改相应的哈雷证明，有时需要大幅修改），而在其他情况下则需要修改先前工作的规则。 当使用多鲁普配额定义时，每个公理都更难满足，因此我们的结果扩展了可满足比例性公理的边界。 我们还通过实验研究补充了理论结果，表明在许多选民批准的概率模型中，多鲁普 JR/EJR+ 比标准（哈雷）JR/EJR+ 要严格得多。 显示英文摘要

摘要： The study of proportionality in multiwinner voting with approval ballots has received much attention in recent years. Typically, proportionality is captured by variants of the Justified Representation axiom, which say that cohesive groups of at least $\ell\cdot\frac{n}{k}$ voters (where $n$ is the total number of voters and $k$ is the desired number of winners) deserve $\ell$ representatives. The quantity $\frac{n}{k}$ is known as the Hare quota in the social choice literature. Another -- more demanding -- choice of quota is the Droop quota, defined as $\lfloor\frac{n}{k+1}\rfloor+1$. This quota is often used in multiwinner voting with ranked ballots: in algorithms such as Single Transferable Voting, and in proportionality axioms, such as Droop's Proportionality Criterion. A few authors have considered it in the context of approval ballots, but the existing analysis is far from comprehensive. The contribution of our work is a systematic study of JR-style axioms (and voting rules that satisfy them) defined using the Droop quota instead of the Hare quota. For each of the standard JR axioms (namely, JR, PJR, EJR, FPJR, FJR, PJR+ and EJR+), we identify a voting rule that satisfies the Droop version of this axiom. In some cases, it suffices to consider known rules (modifying the corresponding Hare proof, sometimes quite substantially), and in other cases it is necessary to modify the rules from prior work. Each axiom is more difficult to satisfy when defined using the Droop quota, so our results expand the frontier of satisfiable proportionality axioms. We complement our theoretical results with an experimental study, showing that for many probabilistic models of voter approvals, Droop JR/EJR+ are considerably more demanding than standard (Hare) JR/EJR+.