We study fairness in social choice settings under single-peaked preferences. Construction and characterization of social choice rules in the single-peaked domain has been extensively studied in prior works. In fact, in the single-peaked domain, it is known that unanimous and strategy-proof deterministic rules have to be min-max rules and those that also satisfy anonymity have to be median rules. Further, random social choice rules satisfying these properties have been shown to be convex combinations of respective deterministic rules. We non-trivially add to this body of results by including fairness considerations in social choice. Our study directly addresses fairness for groups of agents. To study group-fairness, we consider an existing partition of the agents into logical groups, based on natural attributes such as gender, race, and location. To capture fairness within each group, we introduce the notion of group-wise anonymity. To capture fairness across the groups, we propose a weak notion as well as a strong notion of fairness. The proposed fairness notions turn out to be natural generalizations of existing individual-fairness notions and moreover provide non-trivial outcomes for strict ordinal preferences, unlike the existing group-fairness notions. We provide two separate characterizations of random social choice rules that satisfy group-fairness: (i) direct characterization (ii) extreme point characterization (as convex combinations of fair deterministic social choice rules). We also explore the special case where there are no groups and provide sharper characterizations of rules that achieve individual-fairness.

本研究探讨在单峰偏好下社会选择设置中的公平性。我们提出了群体公平的概念，包括组内隐私和跨组公平性，证明了满足这些特性的社会选择规则可以分解为确定性规则的凸组合。此外，我们还提出了一些群体公正性概念的特殊情况下的特征。

单峰偏好条件下的群体公平社会选择规则刻画