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Communications in Theoretical Physics

Quantum Stackelberg Duopoly Game in Depolarizing Channel

Zhu Xia, Kuang Le-Man

Published Jan 1, 2008• 31 citations
Publisher

Abstract

In this paper, we investigate the quantum Stackelberg duopoly (QSD) game in the noise environment with the depolarizing channel expressed by the Kraus-operator representation. It is found that the presence of the damping in the depolarizing channel always leads to the decrease of the quantities of the moves and payoffs of the two players in the QSD game. It is indicated that under certain conditions the first-mover advantage in the QSD game can be weakened due to the presence of the damping in the depolarizing channel.

DOI
10.1088/0253-6102/49/1/24

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In this paper, we investigate the quantum Stackelberg duopoly (QSD) game in the noise environment with the depolarizing channel expressed by the Kraus-operator representation. It is found that the presence of the damping in the depolarizing channel always leads to the decrease of the quantities of the moves and payoffs of the two players in the QSD game. It is indicated that under certain conditions the first-mover advantage in the QSD game can be weakened due to the presence of the damping in the depolarizing channel.
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.
Abstract In this work, we make an attempt to understand how noise, memory and entangling operators collectively decide the profit of the firms Here we have studied the quantum version of Stackelberg-Bertrand duopoly game using modified EWL scheme in both correlated and uncorrelated noise channels. When this game is analyzed for amplitude damping channels following interesting results are obtained: Firstly, decoherence in channel 2 effects the profit function of the firms more than that of channel 1. Secondly, in the case of correlated noise memory prevents the death of entanglement at maximum noise. Finally, the profit function of the firms depends upon noise, memory, strategies, and entangling operator in any given game setting.
Abstract We study the influence of memory effects on the quantum Stackelberg duopoly game through an amplitude-damping channel where successive uses of the channels are correlated. It is shown that the memory effects can drastically change the Nash equilibrium and the payoffs of the two firms. As the degree of channel memory increases, there exists a Nash equilibrium for the entire range of the entanglement parameter. Similarly, the presence of memory ensures the existence of the Nash equilibrium for the whole range of the decoherence parameter both for entangled and unentangled initial states. Moreover, quantum memory leads to a ‘critical point’ of the damping parameter, at which both firms are equally benefited. With the maximum-memory effect, the position of the ‘critical point’ moves to the place of fully decohered.
We study the effect of quantum noise in 3 × 3 entangled quantum games. By taking into account different noisy quantum channels, we analyze how a two-player, three-strategy Rock–Scissor–Paper game is influenced by the quantum noise. We consider the winning non-transitive strategies R , S and P such that R beats S , S beats P and P beats R . The game behaves as a noiseless game for the maximum value of the quantum noise. It is seen that Alice's payoff is heavily influenced by the depolarizing noise as compared to the amplitude damping noise. A depolarizing channel causes a monotonic decrease in players' payoffs as we increase the amount of quantum noise. In the case of the amplitude damping channel, Alice's payoff function reaches its minimum for α = 0.5 and is symmetrical. This means that larger values of quantum noise influence the game weakly. On the other hand, the phase damping channel does not influence the game. Furthermore, the Nash equilibrium and non-transitive character of the game are not affected under the influence of quantum noise.