Utilitarian Guarantees for the Method of Equal Shares
Anton Baychkov, Markus Brill, Jannik Peters
TL;DR
This work analyzes the trade-off between fairness-proportionality and utilitarian welfare in participatory budgeting by deriving utilitarian guarantees for the Method of Equal Shares (MES) under DNS satisfaction functions. The authors parameterize guarantees using the budget $b$ and the range of project costs $c_{\min}$ and $c_{\max}$, obtaining a tight lower bound for MES welfare relative to the optimal: $2\sqrt{\frac{c_{\min}}{b}} - \frac{c_{\min}+c_{\max}}{b}$, which also translates to $2\sqrt{\frac{1}{k_2}} - \frac{1}{k_2} - \frac{1}{k_1}$ when recast in the committee-size framework. They prove that this bound is asymptotically tight for any proportional rule satisfying Extended Justified Representation up to one project (EJR1), implying MES cannot be outperformed in worst-case welfare by proportional rules under these conditions. The paper also examines the impact of using incorrect satisfaction functions for Greedy, yielding a separate bound, and demonstrates strong negative results outside the DNS class, underscoring the importance of cost-based parameterization. Overall, the results establish meaningful, instance-parameterized welfare guarantees for MES in PB and point to future work on bounded-over-spending variants and empirical performance studies.
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.