# The Finite Matroid-Based Valuation Conjecture is False

## TL;DR

This paper disproves a conjecture that all gross substitutes valuations can be generated from matroid-based valuations for more than three items, revealing limitations of matroid operations in economic valuation models.

## Contribution

It demonstrates that the matroid-based valuation conjecture fails for n ≥ 4, providing explicit counterexamples and connecting the problem to open questions in matroid theory.

## Key findings

- The conjecture holds for n ≤ 3 but fails for n ≥ 4.
- Explicit counterexamples are constructed using matroid theory.
- Merging and endowment operations are insufficient to generate all gross substitutes valuations.

## Abstract

The matroid-based valuation conjecture of Ostrovsky and Paes Leme 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 matroid based valuations 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 long-standing open problems in matroid theory, and conclude with open questions at the intersection of this field and economics.