In multiwinner approval elections with many candidates, voters may struggle to determine their preferences over the entire slate of candidates. It is therefore of interest to explore which (if any) fairness guarantees can be provided under reduced communication. In this paper, we consider voters with one-dimensional preferences: voters and candidates are associated with points in $\mathbb R$, and each voter's approval set forms an interval of $\mathbb R$. We put forward a probabilistic preference model, where the voter set consists of $\sigma$ different groups; each group is associated with a distribution over an interval of $\mathbb R$, so that each voter draws the endpoints of her approval interval from the distribution associated with her group. We present an algorithm for computing committees that provide Proportional Justified Representation + (PJR+), which proceeds by querying voters' preferences, and show that, in expectation, it makes $\mathcal{O}(\log( \sigma\cdot k))$ queries per voter, where $k$ is the desired committee size.

翻译：在候选人众多的多赢家认可选举中，选民可能难以确定其对全体候选人的偏好。因此，探索在减少沟通的情况下能够提供何种（若有）公平性保证具有重要意义。本文考虑具有一维偏好的选民：选民与候选人均关联于$\mathbb R$中的点，每位选民的认可集构成$\mathbb R$上的一个区间。我们提出一种概率偏好模型，其中选民集合包含$\sigma$个不同群体；每个群体关联于$\mathbb R$上某个区间的分布，使得每位选民从其所属群体关联的分布中抽取其认可区间的端点。我们提出一种通过查询选民偏好来计算满足比例合理性表征+（PJR+）的委员会的算法，并证明该算法在期望意义上对每位选民进行$\mathcal{O}(\log( \sigma\cdot k))$次查询，其中$k$为目标委员会规模。