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 l1 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 l1 and l∞ 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.

Charity is typically done either by individual donors, who donate money to the charities that they support, or by centralized organizations such as governments or municipalities, which collect the individual contributions and distribute them among a set of charities. On the one hand, individual charity respects the will of the donors but may be inefficient due to a lack of coordination. On the other hand, centralized charity is potentially more efficient but may ignore the will of individual donors.