Substitute valuations: Generation and structure

Authors:
Bruce Hajek
Affiliations:
Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois, 1308 W. Main Street, Urbana, IL 61801-2307, United States
Venue:
Performance Evaluation
Year:
2008

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Abstract

Substitute valuations (in some contexts called gross substitute valuations) are prominent in combinatorial auction theory. An algorithm is given in this paper for generating a substitute valuation using a random number generator. In addition, the geometry of the set of all substitute valuations for a fixed number of goods K is investigated. The set consists of a union of polyhedrons, and the maximal polyhedrons are identified for K=4. It is shown that the maximum dimension of the polyhedrons increases with K nearly as fast as two to the power K. Consequently, under broad conditions, if a combinatorial algorithm can present an arbitrary substitute valuation given a list of input numbers, the list must grow nearly as fast as two to the power K.