Classical games get fundamentally modified in the quantum realm because of non-locality and entanglement, that bypass some of the crucial features of the classical problem that define a dilemma. We will analyze how the dilemma can be shunted and even completely eliminated by the players using quantum strategies from the viewpoint of joint probabilities. In this approach, the game information (entropy) needs to be incorporated into the game strategies. We also connect the potential of the formalism of quantum games with the transmission of quantum information in quantum noisy channels and recent considerations of the connection between thermalization mechanisms in statistical mechanics, the many body problem and cooperative games considered here in the quantum regime.
A joint probability formalism for quantum games with noise is proposed, inspired by the formalism of non-factorizable probabilities that connects the joint probabilities to quantum games with noise. Using this connection, we show that the joint probabilities are non-factorizable; thus, noise does not generically destroy entanglement. This formalism was applied to the Prisoner's Dilemma, the Chicken Game, and the Battle of the Sexes, where noise is coupled through a single parameter μ. We find that for all the games except for the Battle of the Sexes, the Nash inequalities are maintained up to a threshold value of the noise. Beyond the threshold value, the inequalities no longer hold for quantum and classical strategies. For the Battle of the sexes, the Nash inequalities always hold, no matter the noise strength. This is due to the symmetry and anti-symmetry of the parameters that determine the joint probabilities for that game. Finally, we propose a new correlation measure for the games with classical and quantum strategies, where we obtain that the incorporation of noise, when we have quantum strategies, does not affect entanglement, but classical strategies result in behavior that approximates quantum games with quantum strategies without the need to saturate the system with the maximum value of noise. In this manner, these correlations can be understood as entanglement for our game approach.