  We construct (\alpha ,\beta) and \alpha -winning sets in the sense of
Schmidt's game, played on the support of certain measures (very friendly and
awfully friendly measures) and show how to derive the Hausdorff dimension for
some. In particular we prove that if K is the attractor of an irreducible
finite family of contracting similarity maps of R^N satisfying the open set
condition then for any countable collection of non-singular affine
transformations \Lambda_i:R^N \to R^N, dimK=dimK\cap (\cap
^{\infty}_{i=1}(\Lambda_i(BA))) where BA is the set of badly approximable
vectors in R^N.
