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Journal of the Physical Society of Japan

Influence of the Asymmetry of the Strategy Spaces on the Properties of the Quantum Prisoner’s Dilemma

Abd El Hakeem Abd El Naby, Heba A. Elrayes, Ahmed M. K. Tarabia, Ahmed S. Elgazzar

Published Aug 4, 2022• 5 citations
Publisher

Abstract

When it comes to global issues, interactions among participants are always asymmetric due to resource availability and environmental impacts. Therefore, it is important to analyze decision-making among participants with different strategies. We introduce quantum prisoner’s dilemma in asymmetric strategy spaces. The formulation with two dilemma strength parameters is considered, and the Eisert–Wilkens–Lewenstein quantization scheme is applied. The properties of Nash equilibrium are studied for all possible cases of the values of the dilemma strength parameters and for each level of asymmetry. A solution for the dilemma can be obtained only in some extreme limits where the environmental influence is the largest. Consequently, it is unlikely to find a solution to the dilemma in the quantum domain with asymmetric strategy spaces. Therefore, strategies need to be standardized when dealing with global issues.

DOI
10.7566/jpsj.91.094801

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When it comes to global issues, interactions among participants are always asymmetric due to resource availability and environmental impacts. Therefore, it is important to analyze decision-making among participants with different strategies. We introduce quantum prisoner’s dilemma in asymmetric strategy spaces. The formulation with two dilemma strength parameters is considered, and the Eisert–Wilkens–Lewenstein quantization scheme is applied. The properties of Nash equilibrium are studied for all possible cases of the values of the dilemma strength parameters and for each level of asymmetry. A solution for the dilemma can be obtained only in some extreme limits where the environmental influence is the largest. Consequently, it is unlikely to find a solution to the dilemma in the quantum domain with asymmetric strategy spaces. Therefore, strategies need to be standardized when dealing with global issues.
The choice of a unique Nash equilibrium (NE) is crucial in theoretical classical and quantum games. The Eiswer-Wilkens-Lewenstein quantization scheme solves the prisoner's dilemma only for high entanglement. At medium entanglement, there are multiple NEs. We investigate the selection of a unique NE in the quantum prisoner's dilemma with variable dilemma strength parameters. The risk-dominance criterion is used. The influence of the dilemma strength parameters and entanglement is emphasized. We found that entanglement completely controls the risk-dominant equilibrium. Entanglement promotes quantum-cooperation in the risk-dominant equilibrium and thus improves its outcome.
Decision-making is important especially during a crisis such as the novel COVID-19 pandemic. The quantum prisoner's dilemma with two dilemma strength parameters is introduced as a model for the interaction between pharmaceutical and other related enterprises during the pandemic. Novel Nash equilibria are identified. The coopetition equilibrium (simultaneous cooperation and competition) is emphasized. Motivated by the novel equilibria of the quantum version, a classical mixed-strategy formulation that can be applied to real-world situations is proposed. Suitable values of the dilemma strength parameters and quantum entanglement can encourage coopetition, which can be considered as a route to full cooperation.
This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic behaviour in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In the case of the standard Prisoner's Dilemma, the resulting Nash equilibria of quantum extensions are found to be closer to Pareto optimal solutions than those of the classical equilibrium.
The concept of Nash equilibria in pure strategies for quantum extensions of the general form of the Prisoner's Dilemma game is investigated. The process of quantization involves incorporating two additional unitary strategies, which effectively expand the classical game. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. The resulting Nash equilibria are found to be more closely aligned with Pareto-optimal solutions than those of the conventional Nash equilibrium outcome of the classical game. Our results demonstrate the complexity and diversity of strategic behavior in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In particular, we provide a detailed characterization of strategy profiles and their corresponding Nash equilibria, thereby extending the understanding of quantum strategies' impact on traditional game-theoretical problems.
A quantized version of the prisoner’s dilemma game based on the Marinatto–Weber quantization scheme is introduced. A four-qubit-basis initial entangled quantum state is assumed. Two domains for Nash equilibria are observed in terms of the degree of entanglement of the game. Each Nash equilibrium point is found to be Pareto optimal in its corresponding domain. Therefore, there is a unique solution to the game dilemma in each domain of the entanglement parameter. The maximally entangled limit is studied, and the players’ payoffs are found to equal the average of the payoff matrix independently of their choices. Consequently, any combination of players’ choices is a solution to the dilemma of the game. The nonentangled limits are also studied and found to coincide with the classical mixed-strategy game. Our results are equivalent to those of the noisy schemes of the quantum prisoner’s dilemma game. In our scheme, entanglement can be interpreted as a noise.
Decision-making by the two negotiating parties is simulated by a prisoner's dilemma game.The game is formulated in a quantum manner, where players strategies are unitary transformations of qubits built over the basis of opposite decision options.Quantum strategies are correlated through the mechanism of quantum entanglement and the result of the game is obtained by the collapse of the resulting transformed state.The range of strategies allowed for quantum players is richer than in case of a classical game and therefore the result of the game can be better optimized.On the other hand, the quantum game is save against eavesdropping and the players can be assured that this type of quantum arbitration is fair.We show that quantum prisoner's dilemma has more favorable Nash equilibria than its classical analog and they are close to the Pareto optimal solutions.Some economical examples of utilizing quantum game Nash equilibria are proposed.