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Iranian Journal of Physics Research

Study of Nash equilibrium by increasing game parameters in 3-player quantum game

H. Goudarzi, Sohrab Minaei Beirami

Published Mar 15, 2013• 0 citations
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Abstract

Using the quantum game formalism in 3-player system, we calculate the Nash equilibrium in quantum Prisoners’ Dilemma by increasing parameters of unitary operator. Since, the entanglement plays an important role in quantum states of particles quantum game, actually its effect on the obtained results of Nash equilibrium is investigated. It is shown that increasing the parameters enhances the game payoff function.

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Using the quantum game formalism in 3-player system, we calculate the Nash equilibrium in quantum Prisoners’ Dilemma by increasing parameters of unitary operator. Since, the entanglement plays an important role in quantum states of particles quantum game, actually its effect on the obtained results of Nash equilibrium is investigated. It is shown that increasing the parameters enhances the game payoff function.
The effect of increasing quantum bits and Unruh effect on quantum Prisoners’ dilemma has been investigated for both entangled and unentangled initial states. The Nash equilibrium, as an important result of quantum game theory, was obtained through the different payoffs resulted from choosing various strategies. It has been shown that the non-inertial frame disturbs the symmetry of the game. Actually, selection of the basic quantum strategy by players and calculating the payoff of the game via the density matrix of 4-qubit quantum states can represent a scale of influence of entanglement in a quantum mechanical system
We study the three-player prisoner's dilemma game under the effect of decoherence and correlated noise. It is seen that the quantum player is always better off than the classical players. It is also seen that the game's Nash equilibrium does not change in the presence of correlated noise in contradiction to the effect of decoherence in the multiplayer case. Furthermore, it is shown that for maximum correlation the game does not behave as a noiseless game and the quantum player is still better off for all values of the decoherence parameter p which is not possible in the two-player case. In addition, the payoffs reduction due to decoherence is controlled by the correlated noise throughout the course of the game.
In this paper, we extend the quantum game theory of Prisoner’s Dilemma to the N-player case. The final state of quantum game theory of N-player Prisoner’s Dilemma is derived, which can be used to investigate the payoff of each player. As demonstration, two cases (2-player and 3-player) are studied to illustrate the superiority of quantum strategy in the game theory. Specifically, the non-unique entanglement parameter is found to maximize the total payoff, which oscillates periodically. Finally, the optimal strategic set is proved to depend on the selection of initial states.
The Prisoner's Dilemma plays a very important part in the game theory.It is related to economics,political science,sociology,philosophy,ethnics,etc.when study in such field quantized the Prisoner's Dilemma and found that the payoffs for the players are better than that in the classical game.Then,people developed the two-player Quantum Prisoner's Dilemma into multiplayer quantum game.Also,they found that the quantum game does better than classical game.But now,the quantum advantage arising in a simplified multiplayer quantum game is found to be disadvantage when the game's qubit source is corrupted.