  We consider social resource allocations that deliver an array of scarce
supports to a diverse population. Such allocations pervade social service
delivery, such as provision of homeless services, assignment of refugees to
cities, among others. At issue is whether allocations are fair across
sociodemographic groups and intersectional identities. Our paper shows that
necessary trade-offs exist for fairness in the context of scarcity; many
reasonable definitions of equitable outcomes cannot hold simultaneously except
under stringent conditions. For example, defining fairness in terms of
improvement over a baseline inherently conflicts with defining fairness in
terms of loss compared with the best possible outcome. Moreover, we demonstrate
that the fairness trade-offs stem from heterogeneity across groups in
intervention responses. Administrative records on homeless service delivery
offer a real-world example. Building on prior work, we measure utilities for
each household as the probability of reentry into homeless services if given
three homeless services. Heterogeneity in utility distributions (conditional on
received services) for several sociodemographic groups (e.g. single women with
children versus without children) generates divergence across fairness metrics.
We argue that such heterogeneity, and thus, fairness trade-offs pervade many
social policy contexts.
