Social Welfare in Budget Aggregation

Abstract

We study budget aggregation under $\ell_1$-utilities, a model for collective decision making in which agents with heterogeneous preferences must allocate a public budget across a set of alternatives. Each agent reports their preferred allocation, and a mechanism selects an allocation. Early work focused on social welfare maximization, which in this setting admits truthful mechanisms, but may underrepresent minority groups, motivating the study of proportional mechanisms. However, the dominant proportionality notion, single-minded proportionality, is weak, as it only constrains outcomes when agents hold extreme preferences. To better understand proportionality and its interaction with welfare and truthfulness, we address three questions. First, how much welfare must be sacrificed to achieve proportionality? We formalize this via the price of proportionality, the best worst-case welfare ratio between a proportional mechanism and Util, the welfare-maximizing mechanism. We introduce a new single-minded proportional and truthful mechanism, UtilProp, and show that it achieves the optimal worst-case ratio. Second, how do proportional mechanisms compare in terms of welfare? We define an instance-wise welfare dominance relation and use it to compare mechanisms from the literature. In particular, we show that UtilProp welfare-dominates all previously known single-minded proportional and truthful mechanisms. Third, can stronger notions of proportionality be achieved without compromising welfare guarantees? We answer this question in the affirmative by studying decomposability and proposing GreedyDecomp, a decomposable mechanism with optimal worst-case welfare ratio. We further show that computing the welfare-dominant decomposable mechanism, UtilDecomp, is NP-hard, and that GreedyDecomp provides a 2-approximation to UtilDecomp in terms of welfare.

Figures (11)

• Figure 1: Mechanisms that are instance-wise welfare-optimal within their respective class are marked with stars. The solid lines correspond to a worst-case welfare loss of $\frac{n}{2\sqrt{n}-1}$ compared to Util. The dotted line indicates a welfare loss of at most $2 - \frac{1}{n-1}$.
• Figure 2: Visualization of the profiles from the proof of \ref{['prop:decomp-truthfulness']}. Voters are drawn as horizontal (blue and orange) lines and aggregates are drawn as (gray) hatched areas.
• Figure 3: Example instance used for the proof of \ref{['prop:proportional_social_welfare_lower_bound']}. Only non-zero votes are drawn. The numbers to the left of the votes ("$\sqrt{n} \, \times$") indicate the number of voters voting that value. The aggregate of any single-minded proportional mechanism is drawn in dark gray (downwards hatching) and the welfare-optimal aggregate in light gray (upwards hatching).
• Figure 4: Illustration of the proportional spending property on an instance with $n = 4$ voters and $m = 3$ alternatives. For $k = 2$ (left figure), we have $\frac{4}{10} = \sum_{j \in [m]} \mu^k_j < \frac{n-k+1}{n} = \frac{3}{4}$, thus proportional spending requires that $a_j \ge \mu^k_j$ for each alternative $j$. For $k = 3$ (right figure), we have $\frac{13}{10} = \sum_{j \in [m]} \mu^k_j > \frac{n-k+1}{n} = \frac{1}{2}$, thus proportional spending requires that $a_j$ spends a total of at least $\frac{1}{2}$ below $\mu^3$.
• Figure 5: Welfare domination relationships among known moving-phantom mechanisms. The mechanisms UtilProp, Fan, and the trivial Constant mechanism are defined in this paper.

Theorems & Definitions (66)

• Proposition 1
• proof
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• proof
• Theorem 1
• Lemma 1
• proof