  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.
