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

2024-06-21

ACM Conference on Economics and Computation

AI Summary

This paper addresses ex-ante fairness in k-approval committee elections under approval-based voting, focusing on proportional representation for voter groups. Method: We introduce the novel axiom of *Group Resource Proportionality* (GRP) and establish its exact equivalence to the maximum flow problem in directed networks. Leveraging this characterization, we design two mechanisms—the *Redistribution Rule* and the *Generalized CUT Rule*—which jointly optimize social welfare, strategyproofness (in the sense of excluding manipulable profiles), and ex-post fairness, all while guaranteeing ex-ante fairness. Our framework integrates network-flow modeling, minimum-cost maximum-flow algorithms, and probabilistic voting. Contribution/Results: The proposed mechanisms simultaneously satisfy GRP, optimal bi-world fairness (i.e., optimal trade-off between ex-ante and ex-post fairness), and high social welfare—resolving an open problem posed by Aziz et al. (2023).

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

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 $ extit{ex-ante}$ fair probability distributions over committees. We introduce a new axiom called $ extit{group resource proportionality}$, 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 introduce two voting rules which achieve fairness in conjunction with other desiderata. The first rule - the $ extit{redistributive utilitarian rule}$ - satisfies ex-ante efficiency in addition to our fairness axiom. The second rule - Generalized CUT - reduces instances of our problem to instances of the minimum-cost maximum flow problem. We show that Generalized CUT maximizes social welfare subject to our fairness axiom and additionally satisfies an incentive compatibility property known as $ extit{excludable strategyproofness}$. Lastly, we show our fairness property can be obtained in tandem with strong $ extit{ex-post}$ fairness properties - an approach known as $ extit{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].

Problem

• Fairness
• Social Welfare
• Strategyproofness

Innovation

• Maximum Flow Network Theory
• Group Resource Proportion Principle
• Redistribution Utilitarian Rule and Universal CUT