  There are growing concerns that algorithms, which increasingly make or
influence important decisions pertaining to individuals, might produce outcomes
that discriminate against protected groups. We study such fairness concerns in
the context of a two-sided market, where there are two sets of agents, and each
agent has preferences over the other set. The goal is producing a matching
between the sets. This setting has been the focus of a rich body of work. The
seminal work of Gale and Shapley formulated a stability desideratum, and showed
that a stable matching always exists and can be found efficiently. We study
this question through the lens of metric-based fairness notions (Dwork et al.,
Kim et al.). We formulate appropriate definitions of fairness and stability in
the presence of a similarity metric, and ask: does a fair and stable matching
always exist? Can such a matching be found in polynomial time? Our
contributions are as follows: (1) Composition failures for classical
algorithms: We show that composing the Gale-Shapley algorithm with fair
hospital preferences can produce blatantly unfair outcomes. (2) New algorithms
for finding fair and stable matchings: Our main technical contributions are
efficient new algorithms for finding fair and stable matchings when: (i) the
hospitals' preferences are fair, and (ii) the fairness metric satisfies a
strong "proto-metric" condition: the distance between every two doctors is
either zero or one. In particular, these algorithms also show that, in this
setting, fairness and stability are compatible. (3) Barriers for finding fair
and stable matchings in the general case: We show that if the hospital
preferences can be unfair, or if the metric fails to satisfy the proto-metric
condition, then no algorithm in a natural class can find a fair and stable
matching. The natural class includes the classical Gale-Shapley algorithms and
our new algorithms.
