  Schmidt's game is a powerful tool for studying properties of certain sets
which arise in Diophantine approximation theory, number theory, and dynamics.
Recently, many new results have been proven using this game. In this paper we
address determinacy and indeterminacy questions regarding Schmidt's game and
its variations, as well as more general games played on complete metric spaces
(e.g. fractals). We show that except for certain exceptional cases, these games
are undetermined on Bernstein sets.
