  We prove the optimal estimate between the Szlenk and $w^*$-dentability
indices of an arbitrary $w^*$-compact subset of the dual of a Banach space. For
a given $w^*$-compact, convex subset $K$ of the dual of a Banach space, we
introduce a two player game the winning strategies of which determine the
Szlenk index of $K$. We give applications to the $w^*$-dentability index of a
Banach space and of an operator.
