Understanding the Relationship Between Core Constraints and Core-Selecting Payment Rules in Combinatorial Auctions

TL;DR

This paper investigates how core constraints influence core-selecting payment rules in combinatorial auctions, proving non-decreasing properties under certain conditions and analyzing bidder behavior and equilibrium properties.

Contribution

It introduces a conflict graph framework to identify single effective core constraints and proves the non-decreasing property of VCG-nearest payments under these constraints.

Findings

VCG-nearest payment rule is non-decreasing under a single effective core constraint.

Conflict graph representation helps identify conditions for core constraints.

No over-bidding occurs in Nash equilibrium with non-decreasing core-selecting rules.

Abstract

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.