  Quantum entanglement has been recently demonstrated as a useful resource in
conflicting interest games of incomplete information between two players, Alice
and Bob [Pappa et al., Phys. Rev. Lett. 114, 020401 (2015)]. General setting
for such games is that of correlated strategies where the correlation between
competing players is established through a trusted common adviser; however,
players need not reveal their input to the adviser. So far, quantum advantage
in such games has been revealed in a restricted sense. Given a quantum
correlated equilibrium strategy, one of the players can still receive a higher
than quantum average payoff with some classically-correlated equilibrium
strategy. In this work, by considering a class of asymmetric Bayesian games, we
show the existence of games with quantum correlated equilibrium where average
payoff of both the players exceed respective individual maximum for each player
over all classically-correlated equilibriums.
