Bundling Equilibrium in Combinatorial auctions

Abstract

This paper analyzes individually-rational ex post equilibrium in the VC (Vickrey-Clarke) combinatorial auctions. If $Σ$ is a family of bundles of goods, the organizer may restrict the participants by requiring them to submit their bids only for bundles in $Σ$. The $Σ$-VC combinatorial auctions (multi-good auctions) obtained in this way are known to be individually-rational truth-telling mechanisms. In contrast, this paper deals with non-restricted VC auctions, in which the buyers restrict themselves to bids on bundles in $Σ$, because it is rational for them to do so. That is, it may be that when the buyers report their valuation of the bundles in $Σ$, they are in an equilibrium. We fully characterize those $Σ$ that induce individually rational equilibrium in every VC auction, and we refer to the associated equilibrium as a bundling equilibrium. The number of bundles in $Σ$ represents the communication complexity of the equilibrium. A special case of bundling equilibrium is partition-based equilibrium, in which $Σ$ is a field, that is, it is generated by a partition. We analyze the tradeoff between communication complexity and economic efficiency of bundling equilibrium, focusing in particular on partition-based equilibrium.