Utilitarian Guarantees for the Method of Equal Shares

TL;DR

This paper analyzes the trade-off between fairness and utilitarian welfare in Participatory Budgeting, providing formal guarantees for the Method of Equal Shares under specific utility models and demonstrating its optimality among proportional rules.

Contribution

It derives the first formal utilitarian welfare guarantees for MES under DNS satisfaction functions, improving previous negative results and establishing MES's optimality among proportional rules.

Findings

Provides minimum utilitarian welfare guarantees for MES under DNS functions.

Shows guarantees are asymptotically tight for rules satisfying Extended Justified Representation.

Demonstrates MES's optimality among proportional rules in welfare guarantees.

Abstract

In recent years, research in Participatory Budgeting (PB) has put a greater emphasis on rules satisfying notions of fairness and proportionality, with the Method of Equal Shares (MES) being a prominent example. However, proportionality can come at a cost to the total utilitarian welfare. Our work formalizes this relationship, by deriving minimum utilitarian welfare guarantees for MES for a subclass of satisfaction functions called DNS functions, which includes two of the most popular ways of measuring a voter's utility in the PB setting: considering (1) the total cost of approved projects or (2) the total number of those projects. Our results are parameterized in terms of minimum and maximum project costs, which allows us to improve on the mostly negative results found in prior studies, and reduce to the existing multiwinner guarantee when project costs are equal. We show that our guarantees are asymptotically tight for rules satisfying Extended Justified Representation up to one project, showing that no proportional rule can achieve a better utilitarian guarantee than MES.