  Given an assignment of real weights to the ground elements of a matroid, the
min-max weight of a ground element $e$ is the minimum, over all circuits
containing $e$, of the maximum weight of an element in that circuit with the
element $e$ removed. We use this concept to answer the following structural
questions for the minimum weight basis problem. Which elements are persistent
under a given weighting (belong to all or to none of the optimal bases)? What
changes of the weights are allowed while preserving optimality of optimal
bases? How does the minimum weight of a basis change when the weight of a
single ground element is changed, or when a ground element is contracted or
deleted? Our answer to this latter question gives the tropical (min,+,-)
analogue of Kirchhoff's arithmetic (+,x,/) effective conductance formula for
electrical networks.
