  Given two multivariate copulas with corresponding tail dependence functions,
we investigate the relation between a natural tail dependence ordering
$\leq_{tdo}$ and the order $\leq_{loc}$ of local stochastic dominance. We show
that, although the two orderings are not equivalent in general, they coincide
for various important classes of copulas, among them all multivariate
Archimedean and bivariate lower extreme value copulas. We illustrate the
relevance of our results by an implication to risk management.
