Tightness of the rank score veto core lower bound

Determine whether the worst-case distortion of every rank score veto core is independent of the final-position score s_m, thereby proving that the lower bound 1 + 2(\|s\|_1 - s_m) is tight for every positional scoring vector.

Background

The paper introduces the rank score veto core by assigning candidate weights according to normalized aggregate scores from a non-increasing positional scoring vector s = (s_1, ..., s_m), normalized so that s_1 = 1. It proves that the worst-case distortion of this veto core is bounded below by 1 + 2(|s|_1 - s_m) and above by 1 + 2|s|_1.

The proportional veto core and the (m−1)-approval veto core provide evidence that the upper bound is not always tight: their scoring vectors differ only in the last-position score s_m, yet both have distortion 2m−1. The unresolved question is whether this insensitivity to s_m holds for every rank score veto core, which would establish tightness of the paper’s lower bound throughout the family.