In many institutional settings, $k$ items are selected with the goal of representing the underlying distribution of claims, opinions, or characteristics in a large population. We study environments with two adversarial parties whose preferences over the selected items are commonly known and opposed. We propose the Quantile Mechanism: one party partitions the population into $k$ disjoint subsets, and the other selects one item from each subset. We show that this procedure is optimally representative among all feasible mechanisms, and illustrate its use in jury selection, multi-district litigation, and committee formation.

翻译：在许多制度性场景中，需要选择$k$个条目以代表大规模群体中主张、观点或特征的潜在分布。我们研究存在两方对抗者的环境，其对于被选条目的偏好是共同知识且相互对立的。我们提出分位数机制：一方将群体划分为$k$个互不相交的子集，另一方从每个子集中选择一个条目。我们证明该程序在所有可行机制中具有最优代表性，并阐述其在陪审团遴选、跨地区诉讼及委员会组建中的应用。