Existence of competitive equilibria in combinatorial auctions

Abstract

Competitive equilibrium provides a natural steady state for iterative combinatorial auctions that maximize social welfare, and therefore the first step in auction design is to establish its existence. Recent work by Baldwin and Klemperer (2012) has proved that the "demand type" of valuations being "unimodular" is a necessary and sufficient condition for the existence of a competitive equilibrium, but under the general setting where both buyers and sellers as well as multiple copies of items may exist, and the supply could be any combination of items available. In this work, we investigate the same condition under the more restrictive but standard setting for combinatorial auctions, where only buyers and a single copy of each distinct item are allowed and the supply is fixed to be the set of all available items. First, we provide an alternative proof of the sufficiency result for unimodular "complements" demand type, which defines a subclass of valuations for which a competitive equilibrium exists according to Baldwin and Klemperer (2012). While their original proof and analysis use tools from tropical geometry, our approach is based on linear programming. Relying on a result from Bikhchandani and Mamer (1999) that a competitive equilibrium exists if and only if a related linear program has an integral optimal solution, we provide a direct proof that the linear program has an integral optimal solution. Our analysis provides a fundamental understanding of the structure of the linear program and leads to various properties which may be helpful in auction design. Second, we provide an algorithm to determine the demand types of sign-consistent tree graphical valuations, for which competitive equilibria are known to exist due to Candogan et al. (2013). We then analyze the relationship between the set of the demand types of sign-consistent tree graphical valuations and the set of unimodular demand types. Our analysis implies that the unimodularity of demand type is not necessary for the existence of a competitive equilibrium in combinatorial auctions.