Multiwinner Voting with Interval Preferences under Incomplete Information
Drew Springham, Edith Elkind, Bart de Keijzer, Maria Polukarov
TL;DR
The paper addresses fair representation in multiwinner voting when voter preferences are only partially known, modeling one-dimensional CI approvals via the Random Interval Voter (RIV) framework. It develops a two-type-queries elicitation approach (point and interval) and proves that full preferences can be learned with $\mathcal{O}(\log m)$ queries per voter, while a dedicated PJR+ algorithm achieves proportional representation with $\mathcal{O}(\log(\sigma k))$ queries per voter in expectation, independent of the candidate pool size $m$. The PJR+ algorithm can guarantee the property with probability $1$, and the framework provides a MES-based fallback to core-complete outcomes when needed, under CI preferences. The results have practical impact for large candidate sets (e.g., participatory budgeting or deliberation platforms) by reducing communication while ensuring strong fairness guarantees, and point to natural extensions to higher dimensions via $d$-REV models.
Abstract
In multiwinner approval elections with many candidates, voters may struggle to determine their preferences over the entire slate of candidates. It is therefore of interest to explore which (if any) fairness guarantees can be provided under reduced communication. In this paper, we consider voters with one-dimensional preferences: voters and candidates are associated with points in $\mathbb R$, and each voter's approval set forms an interval of $\mathbb R$. We put forward a probabilistic preference model, where the voter set consists of $σ$ different groups; each group is associated with a distribution over an interval of $\mathbb R$, so that each voter draws the endpoints of her approval interval from the distribution associated with her group. We present an algorithm for computing committees that provide Proportional Justified Representation + (PJR+), which proceeds by querying voters' preferences, and show that, in expectation, it makes $\mathcal{O}(\log( σ\cdot k))$ queries per voter, where $k$ is the desired committee size.