  The article considers the multivariate stochastic orders of upper orthants,
lower orthants and positive quadrant dependence (PQD) among simple max-stable
distributions and their exponent measures. It is shown for each order that it
holds for the max-stable distribution if and only if it holds for the
corresponding exponent measure. The finding is non-trivial for upper orthants
(and hence PQD order). From dimension $d\geq 3$ these three orders are not
equivalent and a variety of phenomena can occur. However, every simple
max-stable distribution PQD-dominates the corresponding independent model and
is PQD-dominated by the fully dependent model. Among parametric models the
asymmetric Dirichlet family and the H\"usler-Reiss family turn out to be
PQD-ordered according to the natural order within their parameter spaces. For
the H\"usler-Reiss family this holds true even for the supermodular order.
