  We present a novel formulation of quantum game theory based on the Schmidt
decomposition, which has the merit that the entanglement of quantum strategies
is manifestly quantified. We apply this formulation to 2-player, 2-strategy
symmetric games and obtain a complete set of quantum Nash equilibria. Apart
from those available with the maximal entanglement, these quantum Nash
equilibria are extensions of the Nash equilibria in classical game theory. The
phase structure of the equilibria is determined for all values of entanglement,
and thereby the possibility of resolving the dilemmas by entanglement in the
game of Chicken, the Battle of the Sexes, the Prisoners' Dilemma, and the Stag
Hunt, is examined. We find that entanglement transforms these dilemmas with
each other but cannot resolve them, except in the Stag Hunt game where the
dilemma can be alleviated to a certain degree.
