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Abstract: Substitute valuations in some contexts called gross substitute valuationsare prominent in combinatorial auction theory. An algorithm is given in thispaper for generating a substitute valuation through Monte Carlo simulation. Inaddition, the geometry of the set of all substitute valuations for a fixednumber 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 themaximum dimension of the maximal polyhedrons increases with K nearly as fast astwo to the power K. Consequently, under broad conditions, if a combinatorialalgorithm can present an arbitrary substitute valuation given a list of inputnumbers, the list must grow nearly as fast as two to the power K.



Author: Bruce Hajek

Source: https://arxiv.org/







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