Maximum Flow is Fair: A Network Flow Approach to Committee Voting

In the committee voting setting, a subset of $k$ alternatives is selected based on the preferences of voters. In this paper, our goal is to efficiently compute ex-ante fair probability distributions (or lotteries) over committees. Since it is not known whether a lottery satisfying the desirable fairness property of fractional core is polynomial-time computable, we introduce a new axiom called group resource proportionality (GRP), which strengthens other fairness notions in the literature. We characterize our fairness axiom by a correspondence with max flows on a network formulation of committee voting. Using the connection to flow networks revealed by this characterization, we then introduce voting rules which achieve fairness in conjunction with other desirable properties. The redistributive utilitarian rule satisfies ex-ante efficiency in addition to our fairness axiom. We also give a voting rule which maximizes social welfare subject to fairness by reducing to a minimum-cost maximum-flow problem. Lastly, we show our fairness property can be obtained in tandem with strong ex-post fairness properties -- an approach known as best-of-both-worlds fairness. We strengthen existing best-or-both-worlds fairness results in committee voting and resolve an open question posed by Aziz et al. (2023). These findings follow from an auxiliary result which may prove useful in obtaining best-of-both-worlds type results in future research on committee voting.