Ultra valuations

Abstract: This paper proposes an original exchange property of valuations.This property is shown to be equivalent to a property described by Dress and Terhalle in the context of discrete optimization and matroids and shown there to characterize the valuations for which the demand oracle can be implemented by a greedy algorithm. The same exchange property is also equivalent to a property described independently by Reijnierse, van Gellekom and Potters and by Lehmann, Lehmann and Nisan and shown there to be satisfied by substitutes valuations. It studies the family of valuations that satisfy this exchange property, the ultra valuations. Any substitutes valuation is an ultra valuation, but ultra valuations may exhibit complementarities. Any symmetric valuation is an ultra valuation. Substitutes valuations are exactly the submodular ultra valuations. Ultra valuations define ultrametrics on the set of items. The maximum of an ultra valuation on $n$ items can be found in $O(n^2)$ steps. Finding an efficient allocation among ultra valuations is NP-hard.