  Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in
detail by using a scheme in which all pure states in the 2-qubit Hilbert space
are utilized for strategies. We consider two different types of symmetric games
exemplified by the familiar games, the Battle of the Sexes (BoS) and the
Prisoners' Dilemma (PD). These two types of symmetric games are shown to be
related by a duality map, which ensures that they share common phase structures
with respect to the equilibria of the strategies. We find eight distinct phase
structures possible for the symmetric games, which are determined by the
classical payoff matrices from which the quantum games are defined. We also
discuss the possibility of resolving the dilemmas in the classical BoS, PD and
the Stag Hunt (SH) game based on the phase structures obtained in the quantum
games. It is observed that quantization cannot resolve the dilemma fully for
the BoS, while it generically can for the PD and SH if appropriate correlations
for the strategies of the players are provided.
