  Quantum games with incomplete information can be studied within a Bayesian
framework. We analyze games quantized within the EWL framework [Eisert,
Wilkens, and Lewenstein, Phys Rev. Lett. 83, 3077 (1999)]. We solve for the
Nash equilibria of a variety of two-player quantum games and compare the
results to the solutions of the corresponding classical games. We then analyze
Bayesian games where there is uncertainty about the player types in two-player
conflicting interest games. The solutions to the Bayesian games are found to
have a phase diagram-like structure where different equilibria exist in
different parameter regions, depending both on the amount of uncertainty and
the degree of entanglement. We find that in games where a Pareto-optimal
solution is not a Nash equilibrium, it is possible for the quantized game to
have an advantage over the classical version. In addition, we analyze the
behavior of the solutions as the strategy choices approach an unrestricted
operation. We find that some games have a continuum of solutions, bounded by
the solutions of a simpler restricted game. A deeper understanding of Bayesian
quantum game theory could lead to novel quantum applications in a multi-agent
setting.
