  Balancing fairness and efficiency in resource allocation is a classical
economic and computational problem. The price of fairness measures the
worst-case loss of economic efficiency when using an inefficient but fair
allocation rule; for indivisible goods in many settings, this price is
unacceptably high. One such setting is kidney exchange, where needy patients
swap willing but incompatible kidney donors. In this work, we close an open
problem regarding the theoretical price of fairness in modern kidney exchanges.
We then propose a general hybrid fairness rule that balances a strict
lexicographic preference ordering over classes of agents, and a utilitarian
objective that maximizes economic efficiency. We develop a utility function for
this rule that favors disadvantaged groups lexicographically; but if cost to
overall efficiency becomes too high, it switches to a utilitarian objective.
This rule has only one parameter which is proportional to a bound on the price
of fairness, and can be adjusted by policymakers. We apply this rule to real
data from a large kidney exchange and show that our hybrid rule produces more
reliable outcomes than other fairness rules.
