  We consider three different approaches to define natural Riemannian metrics
on polytopes of stochastic matrices. First, we define a natural class of
stochastic maps between these polytopes and give a metric characterization of
Chentsov type in terms of invariance with respect to these maps. Second, we
consider the Fisher metric defined on arbitrary polytopes through their
embeddings as exponential families in the probability simplex. We show that
these metrics can also be characterized by an invariance principle with respect
to morphisms of exponential families. Third, we consider the Fisher metric
resulting from embedding the polytope of stochastic matrices in a simplex of
joint distributions by specifying a marginal distribution. All three approaches
result in slight variations of products of Fisher metrics. This is consistent
with the nature of polytopes of stochastic matrices, which are Cartesian
products of probability simplices. The first approach yields a scaled product
of Fisher metrics; the second, a product of Fisher metrics; and the third, a
product of Fisher metrics scaled by the marginal distribution.
