  In this research note, we lay some groundwork for analyzing the
manipulability of core-selecting payment rules in combinatorial auctions. In
particular, we focus on payment rules based on the bidders' Shapley values. We
define a sensitivity metric, and provide analytical results for this metric in
LLG, for six different payment vectors used as reference points for
minimum-revenue core-selecting payment rules. We furthermore show how this
sensitivity affects the derivative of the resulting payment rules.
