Optimal Budget Aggregation with Star-Shaped Preference Domains

TL;DR

This paper explores mechanisms for aggregating budget proposals under star-shaped preferences, extending existing models, establishing impossibility results, and proposing a new utility model that achieves efficiency, strategyproofness, and fairness.

Contribution

It introduces a new class of star-shaped utility functions for budget aggregation, proves the uniqueness of the uniform phantom mechanism for two alternatives, and presents a mechanism based on Nash product maximization that balances efficiency, strategyproofness, and fairness.

Findings

The uniform phantom mechanism is uniquely strategyproof and proportional for two alternatives.

Impossibility results show no mechanism can satisfy efficiency, strategyproofness, and proportionality for more than two alternatives under certain disutilities.

A new star-shaped utility model based on budget share ratios allows for mechanisms that satisfy all key properties.

Abstract

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.