Truthful Budget Aggregation: Beyond Moving-Phantom Mechanisms

TL;DR

This work addresses truthful budget-aggregation with $\ell_1$-based disutility, where prior work identified moving-phantom mechanisms as truthful CTAN rules in the two-alternative case. The authors introduce cutoff-phantom mechanisms, pair them with new moving-phantom rules like GreedyMax, and prove that there exist truthful, anonymous, neutral, and continuous mechanisms that are not moving-phantoms for $m\ge3$. They further show that mean-approximation lower bounds extend to all CTAN mechanisms, and they provide partial results on unanimity, including a unanimous non-phantom construction for $n=2$, $m=3$. The results significantly broaden the understood landscape of CTAN mechanisms, highlighting the tradeoffs between truthfulness and fairness, and leaving open the full characterization of the CTAN space and its interaction with unanimity and fairness goals.

Abstract

We study a budget-aggregation setting in which a number of voters report their ideal distribution of a budget over a set of alternatives, and a mechanism aggregates these reports into an allocation. Ideally, such mechanisms are truthful, i.e., voters should not be incentivized to misreport their preferences. For the case of two alternatives, the set of mechanisms that are truthful and additionally meet a range of basic desiderata (anonymity, neutrality, and continuity) exactly coincides with the so-called moving-phantom mechanisms, but whether this space is richer for more alternatives was repeatedly stated as an open question. We answer this question in the affirmative by presenting a class of truthful mechanisms that are not moving-phantoms but satisfy the three properties. Since moving-phantom mechanisms can only provide limited fairness guarantees (measured as the worst-case distance to a fair share solution), one motivation for broadening the class of truthful mechanisms is the hope for improved fairness guarantees. We dispel this hope by showing that lower bounds holding for the class of moving-phantom mechanisms extend to all truthful, anonymous, neutral, and continuous mechanisms.

Figures (4)

• Figure 1: Illustration of $c_\tau$ for $m = 3$ and $\tau = \frac{1}{2}$. Circles are example inputs and crosses are the corresponding outputs. Solid black lines correspond to points where one coordinate is exactly $\tau$.
• Figure 2: Illustrative example of the GreedyMax mechanism. The votes on each alternative are marked by (gray) lines. Since the mechanism only depends on the maximum vote on each alternative, we emphasize the maximum votes and order the alternatives in increasing order of their maxima. The phantom positions are shown as (orange) lines crossing all alternatives and the upper bounds of the hatched areas mark the allocations to each of the alternatives.
• Figure 3: Proof sketch of \ref{['thm:truthful_uniform']} for $m = 3$ and $n = 2$. In the figure, circles denote voters and triangles denote aggregates. For a value $\alpha \in [0,1]$ we write the vote $(\alpha, \frac{1-\alpha}{2}, \frac{1-\alpha}{2})$ as ${ p(\alpha)}$. All voters and aggregates in the white circle in the center are positioned at $p(\frac{1}{3}) = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$. The goal of the proof is to show that for the profile $P$ (orange profile in \ref{['fig:lower_bound_n2_two_profiles']}), the aggregate of any CTAN mechanism has to be in the center, i.e., $\beta=\frac{1}{3}$. To do so, we first show that for all profiles in \ref{['fig:proof-sketch-main-theorem-two']}, the aggregate must lie on the line segment connecting their two voter reports. Then, we suppose for contradiction that in profile $P$ it holds that $\beta > \frac{1}{3}$ and consider profile $P'$ (blue profile, \ref{['fig:lower_bound_n2_two_profiles']}) with aggregate $p(\gamma)$, with $\gamma\leq\frac{1}{3}$. If $\gamma < \frac{1}{3}$, we can then modify $P$ and $P'$ in such a way that their aggregates do not change but that the modified profiles are identical -- a contradiction. If $\gamma=\frac{1}{3}$ then we consider the profile $\hat{P}$ (blue profile, \ref{['fig:lower_bound_n2_parametric_profile']}) along with $P^\star$ (orange profile, \ref{['fig:lower_bound_n2_parametric_profile']}), a modified version of $P$ which is chosen so that its aggregate $p(\beta')$ is sufficiently close to the center. Accordingly, the voter at $\hat{p}$ prefers the orange aggregate $p(\beta')$ over the blue aggregate $p(\frac{1}{m})$ and therefore can manipulate by switching to $p(\alpha)$, effectively turning profile $\hat{P}$ into profile $P^\star$ and obtaining a preferred outcome, a contradiction to truthfulness.
• Figure 4: Visualization of the profiles used in the proof of \ref{['thm:truthful_uniform']} for $m = 3$ and even $n$. Circles denote voter groups and triangles denote aggregates. All voters and aggregates in the white circle are positioned at $p(\frac{1}{m})$. \ref{['fig:lower_bound_two_profiles']} shows the profiles from Case 1 of the proof and \ref{['fig:lower_bound_parametric_profile']} the profiles of Case 2.

Theorems & Definitions (47)

• Theorem 3: freeman2021truthful
• Theorem 1: formal
• Proposition 1
• Proposition 2
• proof
• Lemma 1
• Proposition 3
• Proposition 4
• Theorem 4
• Theorem 5