Discrete Budget Aggregation: Truthfulness and Proportionality

TL;DR

This paper investigates how restricting budget allocations to integral amounts impacts the design of truthful mechanisms in budget aggregation, revealing fundamental differences from fractional settings and exploring proportionality constraints.

Contribution

It extends the Gibbard-Satterthwaite theorem to integral budget settings and identifies truthful mechanisms compatible with integral constraints.

Findings

Fractional proposals allow Gibbard-Satterthwaite extension.

Integral proposals admit specific truthful mechanisms.

Proportionality and truthfulness are incompatible under certain conditions.

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.