Abstract
This paper presents a combinatorial auction, which is of particular interest when short completion times are of importance. It is based on a method for approximating the bidders’ preferences over two types of item when complementarity between the two may exist. The resulting approximated preference relation is shown to be complete and transitive at any given price vector. It is shown that an approximated Walrasian equilibrium always exists if all bidders either view the items as substitutes or complements. If the approximated preferences of the bidders comply with the gross substitutes condition, then the set of approximated Walrasian equilibrium prices forms a complete lattice. A process is proposed that is shown to always reach the smallest approximated Walrasian price vector. Simulation results suggest that the approximation procedure works well as the difference between the approximated and true minimal Walrasian prices is small.
Highlights
Auctions are extensively used as a way to determine who gets to buy which good and at what price
This paper presents a combinatorial auction, which is of particular interest when short completion times are of importance
If the approximated preferences of the bidders comply with the gross substitutes condition, the set of approximated Walrasian equilibrium prices forms a complete lattice
Summary
Auctions are extensively used as a way to determine who gets to buy which good and at what price. Extending to heterogeneous items, Gul and Stacchetti (2000) designed a generalized version of Demange et al (1986)’s auction, which terminates at the unique minimal Walrasian equilibrium price vector.1 In their setting, the existence of a Walrasian equilibrium is guaranteed when bidders have gross substitute preferences. It is further shown that imposing the gross substitutes condition on the bidders’ approximated preference relations is sufficient for the set of approximated Walrasian equilibrium prices to form a complete lattice and, to contain unique minimal element. Using the bidders’ approximated preferences as input, the process is a structured method for finding the unique minimal approximated Walrasian equilibrium price vector This price vector may be of particular importance when the auctioneer is concerned with consumer welfare.
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