  Combinatorial auctions (CAs) allow bidders to express complex preferences for
bundles of goods being auctioned. However, the behavior of bidders under
different payment rules is often unclear. In this paper, we aim to understand
how core constraints interact with different core-selecting payment rules. In
particular, we examine the natural and desirable non-decreasing property of
payment rules, which states that bidders cannot decrease their payments by
increasing their bids. Previous work showed that, in general, the widely used
VCG-nearest payment rule violates the non-decreasing property in single-minded
CAs. We prove that under a single effective core constraint, the VCG-nearest
payment rule is non-decreasing. In order to determine in which auctions single
effective core constraints occur, we introduce a conflict graph representation
of single-minded CAs and find sufficient conditions for the single effective
core constraint in CAs. Finally, we study the consequences on the behavior of
the bidders and show that no over-bidding exists in any Nash equilibrium for
non-decreasing core-selecting payment rules.
