Social Welfare in Budget Aggregation

TL;DR

This paper investigates proportional mechanisms for budget aggregation under $1$-utilities, introducing new mechanisms that optimize welfare and proportionality, and analyzing their theoretical properties and computational complexity.

Contribution

It introduces UtilProp, a new truthful proportional mechanism with optimal worst-case welfare ratio, and GreedyDecomp, a decomposable mechanism with strong welfare guarantees.

Findings

UtilProp achieves the optimal worst-case welfare ratio among proportional mechanisms.

UtilProp welfare-dominates all previously known single-minded proportional mechanisms.

GreedyDecomp provides a 2-approximation to the welfare-optimal decomposable mechanism.

Abstract

We study budget aggregation under $\ell_1$-utilities, a model for collective decision making in which agents with heterogeneous preferences must allocate a public budget across a set of alternatives. Each agent reports their preferred allocation, and a mechanism selects an allocation. Early work focused on social welfare maximization, which in this setting admits truthful mechanisms, but may underrepresent minority groups, motivating the study of proportional mechanisms. However, the dominant proportionality notion, single-minded proportionality, is weak, as it only constrains outcomes when agents hold extreme preferences. To better understand proportionality and its interaction with welfare and truthfulness, we address three questions. First, how much welfare must be sacrificed to achieve proportionality? We formalize this via the price of proportionality, the best worst-case welfare ratio between a proportional mechanism and Util, the welfare-maximizing mechanism. We introduce a new single-minded proportional and truthful mechanism, UtilProp, and show that it achieves the optimal worst-case ratio. Second, how do proportional mechanisms compare in terms of welfare? We define an instance-wise welfare dominance relation and use it to compare mechanisms from the literature. In particular, we show that UtilProp welfare-dominates all previously known single-minded proportional and truthful mechanisms. Third, can stronger notions of proportionality be achieved without compromising welfare guarantees? We answer this question in the affirmative by studying decomposability and proposing GreedyDecomp, a decomposable mechanism with optimal worst-case welfare ratio. We further show that computing the welfare-dominant decomposable mechanism, UtilDecomp, is NP-hard, and that GreedyDecomp provides a 2-approximation to UtilDecomp in terms of welfare.