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arXiv:1102.1353 · Chinese Physics Letters

Quantum Stackelberg Duopoly in a Noninertial Frame

Salman Khan, M. Khalid Khan

Published Jul 1, 2011• 16 citations
arXivHTMLPDF

Abstract

We study the influence of the Unruh effect on quantum Stackelberg duopoly. It is shown that the acceleration of a noninertial frame strongly affects the payoffs of the Arms. The validation of the subgame perfect Nash equilibrium is limited to a particular range of acceleration of the noninertial frame. The benefit of the initial state entanglement in the quantum form of the duopoly in the inertial frame is adversely affected by the acceleration. The duopoly can become a follower advantage only in a small region of the acceleration.

DOI
10.1088/0256-307x/28/7/070202

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We study the influence of Unruh effect on quantum Stackelberg duopoly. We show that the acceleration of noninertial frame strongly effects the payoffs of the firms. The validation of subgame perfect Nash equilibrium is limited to a particular range of acceleration of the noninertial frame. The benefit of initial state entanglement in the quantum form of the duopoly is adversely affected by the acceleration. The duopoly can become as a follower advantage only in a small region of the acceleration.
We investigate the influence of the Unruh effect on the quantum Battle of the Sexes (BOF). We show that the acceleration of a noninertial frame greatly affects the payoffs of the players. In the case of Battle of the Sexes, the Nash Equilibra in the inertial frame may be equilibrium but in other games this conclusion may not be true anymore. Furthermore, we also show that the new Nash Equilibria of the game played in the noninertial frame and the probability distributions of the players in the new Nash Equilibria vary with the acceleration of moving player and the degree of entanglement in the initial state shared by the players.
In this paper, we investigate the influence of entanglement and decoherence on the quantum Stackelberg duopoly (QSD) game. It is shown that the first-mover advantage can be weakened or enhanced due to the existence of entanglement for the QSD game without decoherence. The influence of decoherence induced by the amplitude damping and the phase damping are explicitly studied in the formalism of Kraus operator representations. We show that the amplitude damping drastically changes the Nash equilibrium of the QSD game and the profits of the two players while the phase damping does not affect the Nash equilibrium and the profits of the two players. It is found that under certain conditions there exists a 'critical point' of the damping parameter for the amplitude damping environment. At the 'critical point' the two players have the same moves and payoffs. The QSD game can change from the first-mover advantage game into the follower-mover advantage game when the damping parameter varies from the left-hand-side regime of the 'critical point' to the right-hand-side regime.
We study the influence of entanglement and correlated noise using correlated amplitude damping, depolarizing and phase damping channels on the quantum Stackelberg duopoly. Our investigations show that under the influence of an amplitude damping channel a critical point exists for an unentangled initial state at which firms get equal payoffs. The game becomes a follower advantage game when the channel is highly decohered. Two critical points corresponding to two values of the entanglement angle are found in the presence of correlated noise. Within the range of these limits of the entanglement angle, the game is a follower advantage game. In the case of a depolarizing channel, the payoffs of the two firms are strongly influenced by the memory parameter. The presence of quantum memory ensures the existence of the Nash equilibrium for the entire range of decoherence and entanglement parameters for both the channels. A local maximum in the payoffs is observed which vanishes as the channel correlation increases. Moreover, under the influence of the depolarizing channel, the game is always a leader advantage game. Furthermore, it is seen that the phase damping channel does not affect the outcome of the game.
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 investigate the influence of the Unruh effect on three-qubit quantum games. In particular, we interpret the quantum Prisoners' Dilemma, which is a famous, non-zero sum game both for entangled and unentangled initial states and show that the acceleration of non-inertial frames disturbs the symmetry of the game. Using the various strategies, the novel Nash equilibrium is obtained at infinite acceleration ( r = π/4). As a remarkable point, it is shown that in our three-player system, in contrast to the two-player quantum game in non-inertial frames (see Khan et al 2011 J. Phys. A: Math. Theor. 44 355302), there is not a dominant strategy (even classical strategy) in the game and choosing the quantum strategy by each player can be the dominant strategy depending on the kind of strategy chosen by others. Since the entangled states of particles play an important role in the quantum game, finally we argue that the results of the players depend on the degree of entanglement in the initial state of the game.