We study the problem of aggregating distributions, such as budget proposals, into a collective distribution. An ideal aggregation mechanism would be Pareto efficient, strategyproof, and fair. Most previous work assumes that agents evaluate budgets according to the $\ell_1$ distance to their ideal budget. We investigate and compare different models from the larger class of star-shaped utility functions - a multi-dimensional generalization of single-peaked preferences. For the case of two alternatives, we extend existing results by proving that under very general assumptions, the uniform phantom mechanism is the only strategyproof mechanism that satisfies proportionality - a minimal notion of fairness introduced by Freeman et al. (2021). Moving to the case of more than two alternatives, we establish sweeping impossibilities for $\ell_1$ and $\ell_\infty$ disutilities: no mechanism satisfies efficiency, strategyproofness, and proportionality. We then propose a new kind of star-shaped utilities based on evaluating budgets by the ratios of shares between a given budget and an ideal budget. For these utilities, efficiency, strategyproofness, and fairness become compatible. In particular, we prove that the mechanism that maximizes the Nash product of individual utilities is characterized by group-strategyproofness and a core-based fairness condition.

翻译：我们研究将分布（如预算提案）聚合成集体分布的问题。理想的聚合机制应满足帕累托效率、策略证明性与公平性。以往研究大多假设参与者根据其理想预算的$\ell_1$距离评估预算方案。本文从更广泛的星形效用函数类别——单峰偏好多维推广形式——中探究并比较不同模型。针对两种备选方案的情形，我们通过证明在非常一般的假设下，均匀幻象机制是唯一满足比例性（Freeman等人于2021年提出的最小公平概念）的策略证明机制，从而扩展了现有结论。对于超过两种备选方案的情形，我们针对$\ell_1$与$\ell_\infty$负效用建立了全面的不可能性定理：不存在同时满足效率、策略证明性与比例性的机制。随后我们提出基于给定预算与理想预算份额比率评估预算的新型星形效用函数。在此类效用下，效率、策略证明性与公平性可达成兼容。特别地，我们证明最大化个体效用纳什积的机制具有群组策略证明性及基于核心的公平性特征。