Ultra valuations

12/12/2017

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by   Daniel Lehmann, et al.

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This paper defines and studies a family of valuations, the ultra valuations, that satisfy an exchange property. It shows that this exchange property is implied by a property discovered in [LLN:GEB] and satisfied by substitutes valuations. Any substitutes valuation is therefore 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^4) steps. Finding an efficient allocation among ultra valuations is NP-hard.