Close distortion bounds for growing committees under general norms

Close the gap between the upper and lower distortion bounds for growing committee sizes under general normalized monotone symmetric norms, thereby characterizing how committee size and the chosen norm affect distortion.

Background

The paper studies threshold-approval multiwinner voting in which each voter reports candidates through common utility thresholds, while committee utility is represented by a normalized monotone symmetric norm. It establishes matching asymptotic deterministic and randomized distortion bounds for single-winner elections and committees of constant size, but the upper and lower bounds do not generally coincide when the committee size grows with the number of candidates.

The unresolved problem is to determine whether the remaining gap can be closed for arbitrary normalized monotone symmetric norms. A solution would clarify the joint effects of committee size and the selected norm on the optimal distortion achievable from threshold approvals.