🤖 AI Summary
This paper studies collective aggregation of individual budget distributions in multi-dimensional budget allocation, focusing on mechanism design within the star-shaped preference domain.
Method: We introduce a novel star-shaped utility function based on share ratios and analyze mechanisms—including the Nash product maximization mechanism and the uniform phantom mechanism—under various distance metrics (e.g., ℓ₁, ℓ₂).
Contribution/Results: We establish the first mechanism achieving simultaneous Pareto efficiency, group strategyproofness, and core fairness in multi-option settings. We characterize the Nash product maximization mechanism as both group strategyproof and core fair. For two alternatives, we prove the uniform phantom mechanism is the unique rule satisfying all three properties; however, under ℓ₁ or ℓ₂ distances, no mechanism can satisfy all three in settings with three or more alternatives. Finally, we construct a computationally tractable mechanism that ensures both fairness and efficiency, offering a new paradigm for budget aggregation that balances theoretical rigor with practical implementability.
📝 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 [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.