  A new approach to play games quantum mechanically is proposed. We consider
two players who perform measurements in an EPR-type setting. The payoff
relations are defined as functions of *correlations*, i.e. without reference to
classical or quantum mechanics. Classical bi-matrix games are reproduced if the
input states are classical and perfectly anti-correlated, that is, for a
classical correlation game. However, for a quantum correlation game, with an
entangled singlet state as input, qualitatively different solutions are
obtained. For example, the Prisoners' Dilemma acquires a Nash equilibrium if
the players both apply a mixed strategy. It appears to be conceptually
impossible to reproduce the properties of quantum correlation games within the
framework of classical games.
