The goal of multi-winner elections is to choose a fixed-size committee based on voters' preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. In an influential paper, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). We observe that EJR is incompatible with the highly desirable Perfect Representation (PR) criterion, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with PR, as well as with a stronger variant of this axiom, which we term Fractional Perfect Representation (FPR). Moreover, just like EJR, PJR can be used to characterise the classic Proportional Approval Voting (PAV) rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.

翻译：多赢家选举的目标是根据选民的偏好选择一个固定规模的委员会。在此背景下，代表性问题至关重要：具有凝聚性偏好的大规模选民群体应通过选举获胜者得到充分代表。在一篇具有影响力的论文中，Aziz等人提出了两个旨在捕捉这一理念的公理：合理性代表（JR）及其强化版本扩展合理性代表（EJR）。我们观察到EJR与高度理想的完美代表（PR）准则不相容，因此提出EJR的一种松弛形式，称为比例合理性代表（PJR）。PJR比JR要求更严格，但与EJR不同，它与PR以及我们称之为分数完美代表（FPR）的该公理强化变体相容。此外，与EJR类似，PJR可用于在加权比例批准投票规则类别中刻画经典的比例批准投票（PAV）规则。另一方面，我们证明EJR在平均选民满意度方面比PJR提供更强的保证。