Discrete Budget Aggregation: Truthfulness and Proportionality

TL;DR

This paper studies budget aggregation under integral-output constraints, contrasting with the fractional-output literature to understand truthfulness and proportionality when budgets are indivisible. It develops integral moving-phantom mechanisms that preserve truthfulness, and shows that naive rounding of fractional mechanisms fails to do so, while FloorIM achieves single-minded quota-proportionality. Conversely, when ballots can be fractional, the authors prove a dictatorial impossibility: any onto, truthful fractional-input mechanism must be dictatorial, extending Gibbard–Satterthwaite-type results to this domain. The work further connects to approval-based committee voting, deriving JR and EJR+ insights and showing incompatibilities between these strong proportionality notions and truthfulness under integration constraints. Overall, the paper clarifies how integrality changes the landscape of truthful budget aggregation and highlights both constructive mechanisms and fundamental impossibilities for representation.

Abstract

We study a budget aggregation setting where voters express their preferred allocation of a fixed budget over a set of alternatives, and a mechanism aggregates these preferences into a single output allocation. Motivated by scenarios in which the budget is not perfectly divisible, we depart from the prevailing literature by restricting the mechanism to output allocations that assign integral amounts. This seemingly minor deviation has significant implications for the existence of truthful mechanisms. Specifically, when voters can propose fractional allocations, we demonstrate that the Gibbard-Satterthwaite theorem can be extended to our setting. In contrast, when voters are restricted to integral ballots, we identify a class of truthful mechanisms by adapting moving-phantom mechanisms to our context. Moreover, we show that while a weak form of proportionality can be achieved alongside truthfulness, (stronger) proportionality notions derived from approval-based committee voting are incompatible with truthfulness.

Figures (3)

• Figure 1: Example of an integral moving-phantom mechanism with $n=2$ voters, $m=3$ alternatives, and a budget of $b = 4$. The votes on each alternative are marked by (black) solid lines. The phantom positions are shown as (orange) dashed lines. The median vote on each alternative is marked by a rectangle. There are two voters with reports $(4, 0, 0)$ and $(3,1,0)$. The figure shows the positions of the phantoms at a time where normalization is reached, i.e., the sum of the median votes is $4$. The returned budget distribution is $(2,1,1)$.
• Figure 2: Illustration showing how to construct the integral phantom system $\Phi$ from a fractional phantom system $\mathcal{F}$. In this example, we have $n = 2$, $m = 3$, $b = 4$, the fractional phantom system is IndependentMarkets, and rounding is done using the floor function. Each fractional phantom $f_k$ is drawn as a (blue) line spanning all alternatives and each integral phantom $\phi_{k,j}$ is drawn as an (orange) dashed line. In the left figure (discrete time step $\tau$), no fractional phantom is crossing an integer value and all integral phantoms correspond to a rounded fractional phantom. As time progresses, the upper fractional phantom $f_0$ reaches $3$, at which point the corresponding integral phantoms should move from $2$ to $3$. To guarantee a time of normalization, they move one after another, as illustrated in the middle and right figures.
• Figure 3: Example showing for $m=3$ and $b=4$ how a vote $p_i \in I^{m}_b$ can be interpreted as an approval ballot, i.e., $p_i = (3,1,0)$ is translated to $A_i = \{c_{1,1}, c_{1,2}, c_{1,3}, c_{2,1}\}$. We apply a similar translation when mapping an allocation $a$ to a committee $W$.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Proposition 1
• Theorem 3
• Theorem 4
• Theorem 5
• proof
• Theorem 6: ACS03b
• Theorem 7
• proof : Proof Sketch of \ref{['thm:continuous_impossibility']}