  The problem of computing the local hidden variable (LHV) value of a Bell
inequality plays a central role in the study of quantum nonlocality. In
particular, this problem is the first step towards characterizing the LHV
polytope of a given scenario. In this work, we establish a relation between the
LHV value of bipartite Bell inequalities and the mathematical notion of excess
of a matrix. Inspired by the well developed theory of excess, we derive several
results that directly impact the field of quantum nonlocality. We show infinite
families of bipartite Bell inequalities for which the LHV value can be computed
exactly, without needing to solve any optimization problem, for any number of
measurement settings. We also find tight Bell inequalities for a large number
of measurement settings.
