  Effect of replacing the classical game object with a quantum object is
analyzed. We find this replacement requires a throughout reformation of the
framework of Game Theory. If we use density matrix to represent strategy state
of players, they are full-structured density matrices with off-diagonal
elements for the new games, while reduced diagonal density matrix will be
enough for the traditional games on classical objects. In such formalism, the
payoff function of every player becomes Hermitian Operator acting on the
density matrix. Therefore, the new game looks really like Quantum Mechanics
while the traditional game becomes Classical Mechanics.
