  Quantum games with incomplete information can be studied within a Bayesian
framework. We consider a version of prisoner's dilemma (PD) in this framework
with three players and characterize the Nash equilibria. A variation of the
standard PD game is set up with two types of the second prisoner and the first
prisoner plays with them with probability p and 1-p respectively. The Bayesian
nature of the game manifests in the uncertainty that the first prisoner faces
about his opponent's type which is encoded either in a classical probability or
in the amplitudes of a wave function. Here, we consider scenarios with
asymmetric payoffs between the first and second prisoner for different values
of the probability, p, and the entanglement. Our results indicate a class of
Nash equilibria (NE) with rich structures, characterized by a phase
relationship on the strategies of the players. The rich structure that can be
exploited by the referee to set up rules of the game to push the players
towards a specific class of NE. These results provide a deeper insight into the
quantum advantages of Bayesian games over their classical counterpart.
