Algorithms for Recognizing Economic Properties in Matrix Bid Combinatorial Auctions

A combinatorial auction is an auction where multiple items are for sale simultaneously to a set of buyers. Furthermore, buyers are allowed to place bids on subsets of the available items. This paper focuses on a combinatorial auction where a bidder can express his preferences by means of a so-called ordered matrix bid. Ordered matrix bids are a bidding language that allows a compact representation of a bidder's preferences and was developed by Day [Day, R. W. 2004. Expressing preferences with price-vector agents in combinatorial auctions. Ph.D. thesis, University of Maryland, College Park]. We give an overview of how a combinatorial auction with matrix bids works. We discuss the relevance of recognizing whether a given matrix bid has properties related to elements of economic theory such as free disposal, subadditivity, submodularity, and the gross substitutes property. We show that verifying whether a matrix bid has these properties can be done in polynomial time by solving one or more shortest-path problems. Finally, we investigate to what extent randomly generated matrix bids satisfy these properties.