The effect of entanglement and correlated noise in a four-player quantum Minority game (QMG) is investigated. Different time correlated quantum memory channels are considered to analyze the Nash equilibrium (NE) payoff of the first player. It is seen that the NE payoff is substantially enhanced due to the presence of correlated noise. The behavior of damping channels (amplitude damping (AD) and phase damping) is approximately similar. However, bit-phase flip channel heavily influences the Minority game as compared to other channels in the presence of correlated noise. On the other hand, phase flip channel has a symmetrical behavior around 50 noise threshold. The significant reduction in payoffs due to decoherence is well compensated due to the presence of correlated noise. However, the NE of the game does not change in the presence of noise. It is seen that in case of generalized AD channel, entanglement plays a significant role at lower level of decoherence. The channel has less dominant effects on the payoff at higher values of decoherence. Furthermore, AD and generalized amplitude damping channels have almost comparable effects at lower level of decoherence (p < 0.5). Therefore, the game deserves careful study during its implementation due to prominent role of noise for different channels.
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