  Winning sets of Schmidt's game enjoy a remarkable rigidity. Therefore, this
game (and modifications of it) have been applied to many examples of complete
metric spaces (X, d) to show that the set of "badly approximable points", with
respect to a given collection of resonant sets in X, is a winning set. For
these examples, strategies were deduced that are, in most cases, strongly
adapted to the specific dynamics and properties of the underlying setting. We
introduce a new modification of Schmidt's game which is a combination and
generalization of the ones of [18] and [20]. This modification allows us to
axiomatize conditions on the collection of resonant sets under which there
always exists a winning strategy. Moreover, we discuss properties of winning
sets of this modification and verify our conditions for several examples -
among them, the set of badly approximable vectors in the Euclidian space and
the p-adic integers with weights and, as a main example, the set of geodesic
rays in proper geodesic CAT(-1) spaces which avoid a suitable collection of
convex subsets.
