Constant-Factor Distortion Mechanisms for $k$-Committee Election
Haripriya Pulyassary, Chaitanya Swamy
TL;DR
This work addresses selecting a $k$-candidate committee under the Top$_\ell$ objective when only ordinal preferences are available. It develops constant-factor distortion mechanisms that solicit limited cardinal information via value queries, achieving $O(1)$ distortion with either per-agent $O(\log k\log n)$ queries or per-agent complexity that is independent of $n$ (via adaptive sampling), plus a total-query bound of $\widetilde{O}\bigl(k^2\log(\min\{\ell,n/\ell\})\log^2 n\bigr)$. The core techniques include a black-box reduction that simulates a near-cardinal metric from ordinal data and an adaptive-sampling approach adapted to the $\ell$-centrum objective, with sparsification to improve efficiency. The results extend to the setting $A \neq \mathcal{C}$, showing how to compute OPT estimates and adapt the mechanisms accordingly. Overall, the paper significantly advances low-distortion mechanisms for multiwinner voting under limited cardinal information, with implications for fairness-aware clustering and metric-based social choice.
Abstract
In the $k$-committee election problem, we wish to aggregate the preferences of $n$ agents over a set of alternatives and select a committee of $k$ alternatives that minimizes the cost incurred by the agents. While we typically assume that agent preferences are captured by a cardinal utility function, in many contexts we only have access to ordinal information, namely the agents' rankings over the outcomes. As preference rankings are not as expressive as cardinal utilities, a loss of efficiency is inevitable, and is quantified by the notion of \emph{distortion}. We study the problem of electing a $k$-committee that minimizes the sum of the $\ell$-largest costs incurred by the agents, when agents and candidates are embedded in a metric space. This problem is called the $\ell$-centrum problem and captures both the utilitarian and egalitarian objectives. When $k \geq 2$, it is not possible to compute a bounded-distortion committee using purely ordinal information. We develop the first algorithms (that we call mechanisms) for the $\ell$-centrum problem (when $k \geq 2$), which achieve $O(1)$-distortion while eliciting only a very limited amount of cardinal information via value queries. We obtain two types of query-complexity guarantees: $O(\log k \log n)$ queries \emph{per agent}, and $O(k^2 \log^2 n)$ queries \emph{in total} (while achieving $O(1)$-distortion in both cases). En route, we give a simple adaptive-sampling algorithm for the $\ell$-centrum $k$-clustering problem.