# Truthful Aggregation of Budget Proposals

## TL;DR

This paper introduces a new mechanism for participatory budgeting that balances incentive compatibility and proportionality, using market-inspired median rules, and explores the tradeoff with Pareto optimality.

## Contribution

It proposes the independent markets mechanism, a novel approach that achieves both incentive compatibility and proportionality in budget aggregation.

## Key findings

- The independent markets mechanism is incentive compatible and proportional.
- All moving phantom mechanisms are incentive compatible.
- The social welfare-maximizing mechanism is uniquely Pareto optimal within this class.

## Abstract

We consider a participatory budgeting problem in which each voter submits a proposal for how to divide a single divisible resource (such as money or time) among several possible alternatives (such as public projects or activities) and these proposals must be aggregated into a single aggregate division. Under $\ell_1$ preferences -- for which a voter's disutility is given by the $\ell_1$ distance between the aggregate division and the division he or she most prefers -- the social welfare-maximizing mechanism, which minimizes the average $\ell_1$ distance between the outcome and each voter's proposal, is incentive compatible (Goel et al. 2016). However, it fails to satisfy the natural fairness notion of proportionality, placing too much weight on majority preferences. Leveraging a connection between market prices and the generalized median rules of Moulin (1980), we introduce the independent markets mechanism, which is both incentive compatible and proportional. We unify the social welfare-maximizing mechanism and the independent markets mechanism by defining a broad class of moving phantom mechanisms that includes both. We show that every moving phantom mechanism is incentive compatible. Finally, we characterize the social welfare-maximizing mechanism as the unique Pareto-optimal mechanism in this class, suggesting an inherent tradeoff between Pareto optimality and proportionality.