Abstract: A simple and full Hilbert space formulation of the game theory is presented,
which for the two strategy case is completely solvable.
The quantum game admits an interpretation that it consists of a family of classical games
and the quantum interference invoked by the correlation between the players' strategies.
Examination of the Nash equilibria for the Prisoner's Dilemma shows that the quantum solution found previously is optimal in the classical family,
while there exists another quantum Nash equilibrium supported by the quantum interference.
Novel aspects of the quantum game such as
stone-scissor-paper phase game and quantum moderation are also revealed.