Social Welfare in Budget Aggregation

We study budget aggregation under $\ell_1$-utilities, a model in which agents with heterogeneous preferences allocate a public budget across alternatives. Each agent reports their preferred allocation, and a mechanism selects an allocation. Early work has focused on social welfare maximization, which 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 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 the welfare-maximizing allocation. We introduce a single-minded proportional and truthful mechanism, UtilProp, and show it achieves the optimal worst-case ratio. Second, how do proportional mechanisms compare in welfare? We define an instance-wise welfare dominance relation and compare existing mechanisms, finding that UtilProp welfare-dominates all previously known single-minded proportional and truthful mechanisms. Third, can stronger proportionality notions be achieved without compromising welfare guarantees? We answer affirmatively by studying decomposability and proposing GreedyDecomp, a decomposable mechanism with optimal worst-case welfare ratio.