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.

翻译：我们研究基于ℓ1效用的预算聚合模型，这是一种用于异质偏好主体在多个备选方案间分配公共预算的集体决策模型。每个主体报告其偏好分配方案，机制则选择一个最终分配方案。早期研究聚焦于社会福利最大化，该框架下存在真实机制，但可能使少数群体代表性不足，从而推动了对比例机制的研究。然而，主流比例性概念——单一偏好比例性——存在局限性，因其仅约束主体持有极端偏好时的结果。为深入理解比例性及其与福利和真实性的交互关系，我们探讨了三个核心问题。首先，实现比例性需要牺牲多少社会福利？我们通过比例性代价——比例性机制与福利最大化机制Util之间的最优最坏情况福利比率——对此进行形式化。我们提出新型单一偏好比例性真实机制UtilProp，证明其达到最优最坏情况比率。其次，比例性机制在福利表现上如何比较？我们定义实例层面的福利支配关系，并以此比较文献中的现有机制。特别地，我们证明UtilProp在福利上支配所有已知的单一偏好比例性真实机制。第三，能否在不损害福利保证的前提下实现更强的比例性概念？我们通过研究可分解性并提出具有最优最坏情况福利比率的可分解机制GreedyDecomp，对此问题给出肯定回答。进一步证明计算福利最优可分解机制UtilDecomp是NP难问题，且GreedyDecomp在福利上为UtilDecomp提供2倍近似解。