# Balancing Lexicographic Fairness and a Utilitarian Objective with Application to Kidney Exchange

## TL;DR

This paper introduces a hybrid fairness rule for kidney exchange that balances lexicographic fairness and utilitarian efficiency, with adjustable parameters, improving outcome reliability over existing methods.

## Contribution

It proposes a novel hybrid fairness rule that balances fairness and efficiency in kidney exchanges, with a tunable parameter based on the price of fairness, and demonstrates its effectiveness on real data.

## Key findings

- The hybrid rule outperforms other fairness rules in real kidney exchange data.
- The rule's parameter effectively controls the trade-off between fairness and efficiency.
- Application shows improved reliability of outcomes.

## Abstract

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.