Geometry of gross substitutes valuations

Abstract We consider the set of gross substitutes valuations which is one of the most important classes of valuation functions in combinatorial auctions. This is due the fact that the existence of a Walrasian equilibrium is guaranteed if all bidders have gross substitutes valuations. We interpret normalized valuation functions v : 2 N → R with v ( 0 ) = 0 on N ≔ { 1 , … , n } as points of R 2 n − 1 . The set of gross substitutes valuations turns out to be the union of finitely many polyhedral cones. It has been proved that the set of gross substitutes valuations has Lebesgue-measure zero in R 2 n − 1 , but the question whether the dimension of each of these cones is polynomial in the number of items n is open. We answer this question negatively by showing that the dimension of each cone is actually exponential in n and provide a lower bound of ⌈ 1 n + 1 ⋅ ( 2 n − n − 2 ) ⌉ + 2 n − 1 for their dimension. By explicit calculation we verify that our lower bound matches the dimension in the case of n ≤ 3 , but becomes loose for larger n .