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）聚合为集体分布的问题。理想的聚合机制应满足帕累托效率、防策略性和公平性。先前研究大多假设代理人根据其理想预算的$\ell_1$距离评估预算。本文从更广的星形效用函数类（单峰偏好的多维推广）中研究并比较不同模型。针对两个备选方案的情形，我们通过证明在非常一般的假设下，均匀幻影机制是唯一满足比例性（Freeman等人(2021)提出的最小公平性概念）的防策略机制，从而扩展了现有结果。转向两个以上备选方案的情形，我们为$\ell_1$和$\ell_\infty$负效用建立了广泛的不可能性：没有任何机制能同时满足效率、防策略性和比例性。随后，我们提出一类新的星形效用函数，该函数基于给定预算与理想预算之间的份额比率来评估预算。对于这些效用函数，效率、防策略性和公平性变得兼容。特别地，我们证明最大化个体效用纳什乘积的机制可由群体防策略性和基于核的公平性条件刻画。