Optimal Budget Aggregation with Star-Shaped Preference Domains

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 [Formula: see text] distance to their ideal budget. We investigate and compare different models from the larger class of star-shaped utility functions—a multidimensional 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 in prior work. Moving to the case of more than two alternatives, we establish sweeping impossibilities for [Formula: see text] and [Formula: see text] disutilities: no mechanism satisfies efficiency, strategyproofness, and proportionality. We then propose a new kind of star-shaped utility 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. History: This manuscript was accepted for the Special Issue on Mathematics of Market Design. Funding: This work was supported by the Israel Science Foundation [Grants 712/20 and 1092/24], Singapore Ministry of Education [Grant MOE-T2EP20221-0001], Deutsche Forschungsgemeinschaft [Grants BR 2312/11-2 and BR 2312/12-1], and an NUS start-up grant.