Project-Fair and Truthful Mechanisms for Budget Aggregation

TL;DR

This work proposes a novel and simple moving phantom mechanism that provides optimal project fairness guarantees and shows that it minimizes the $\ell_1$ distance to the mean for three projects and gives the first non-trivial bounds on this quantity for more than three projects.

Abstract

We study the budget aggregation problem in which a set of strategic voters must split a finite divisible resource (such as money or time) among a set of competing projects. Our goal is twofold: We seek truthful mechanisms that provide fairness guarantees to the projects. For the first objective, we focus on the class of moving phantom mechanisms [Freeman et al., 2021], which are -- to this day -- essentially the only known truthful mechanisms in this setting. For project fairness, we consider the mean division as a fair baseline, and bound the maximum difference between the funding received by any project and this baseline. We propose a novel and simple moving phantom mechanism that provides optimal project fairness guarantees. As a corollary of our results, we show that our new mechanism minimizes the $\ell_1$ distance to the mean (a measure suggested by Caragiannis et al. [2022]) for three projects and gives the first non-trivial bounds on this quantity for more than three projects.

Figures (3)

• Figure 1: Example from \ref{['lem:overfund-upperbound']} for $n=4$ and $m=3$. See \ref{['sec:prelim']} for an explanation of how to read our figures. Note that the black line segments each represent two voters who both make the same report.
• Figure 2: Example execution of the Ladder mechanism with $n=4$ voters and $m=3$ projects. The left panel shows the positions of the phantoms at $t=\frac{1}{2}$ (before normalization is reached) while the right panel shows them at $t=\frac{11}{12}$ (exactly when normalization is reached).
• Figure 3: Proof sketch of \ref{['thm:main']}: We assume for contradiction that the Ladder mechanism underfunds project $1$ by more than $\frac{1}{2}(1-\frac{1}{m})$. We divide the proof into four steps: Using that the mean for project $1$ is high, in step (i), we derive a lower bound for the number of voters reporting a value in the interval $(\mathcal{A}^{\mathcal{F}}\xspace(P)_1,1]$ (indicated by green). (ii) Since we know that the total number of phantoms and voters strictly above $\mathcal{A}^{\mathcal{F}}\xspace(P)_1$ is at most $n$, we derive an upper bound for the highest phantom at the point of normalization, i.e., $f_0(t^*)$. (iii) Building upon (ii), we can upper bound the number of phantoms within each interval $[\mathcal{A}^{\mathcal{F}}\xspace(P)_j,1]$ (indicated by orange) and thereby lower bound the number of voters reporting a value in the same interval. This in turn allows us to derive a lower bound on the mean of each project $j \neq 1$. (iv) Summing over all lower bounds on the mean implies a contradiction to the fact that the means sum up to $1$.

Theorems & Definitions (19)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Corollary 4
• proof
• Definition 5: Ladder Mechanism
• Lemma 6