  Playing a symmetric bi-matrix game is usually physically implemented by
sharing pairs of 'objects' between two players. A new setting is proposed that
explicitly shows effects of quantum correlations between the pairs on the
structure of payoff relations and the 'solutions' of the game. The setting
allows a re-expression of the game such that the players play the classical
game when their moves are performed on pairs of objects having correlations
that satisfy the Bell's inequalities. If players receive pairs having quantum
correlations the resulting game cannot be considered another classical
symmetric bi-matrix game. Also the Nash equilibria of the game are found to be
decided by the nature of the correlations.
