The Distortion of Threshold Approval Multiwinner Voting
We study multiwinner voting with $n$ voters and $m$ candidates, where the goal is to select a committee of size $k$. Voters have unit-sum cardinal utilities but report only coarse information induced by $t$ utility thresholds. In particular, each voter partitions the candidates into approval sets, with the utilities of all candidates in the same set lying between the same pair of consecutive thresholds. We model the utility of each voter for a committee as a normalized monotone symmetric norm, capturing, among others, additive welfare, best-representative utility, top-$r$ welfare, $\ell_p$ welfare, and normalized non-increasing ordered weighted averages. We design a deterministic rule with distortion at most $2(2m)^{1/t}$ and a randomized rule with distortion at most $2\min\{m^{1/(t+1)},m/k\}$. These guarantees hold for every committee size $k$ and every norm in our framework. Consequently, a single threshold yields distortion $O(m)$ for deterministic rules and $O(\sqrt{m})$ for randomized rules, while $O(\log m)$ thresholds suffice for constant distortion. We complement these results with norm-dependent lower bounds that asymptotically match our upper bounds for constant committee sizes, including the single-winner case.