The Finite Matroid-Based Valuation Conjecture is False

Ngoc Mai Tran

Abstract: The matroid-based valuation (MBV) conjecture of Ostrovsky and Paes Leme [Theoret. Econom., 10 (2015), pp. 853--865] states that all gross substitutes valuations on $n$ items can be produced from merging and endowments of weighted ranks of matroids defined on at most $m(n)$ items. We show that if $m(n) = n$, then this statement holds for $n \leq 3$ and fails for all $n \geq 4$. In particular, the set of gross substitutes valuations on $n \geq 4$ items is strictly larger than the set of MBVs defined on the ground set $[n]$. Our proof uses matroid theory and discrete convex analysis to explicitly construct a large family of counter-examples. It indicates that merging and endowment by themselves are poor operations to generate gross substitutes valuations. We also connect the general MBV conjecture and related questions to longstanding open problems in matroid theory and conclude with open questions at the intersection of this field and economics.