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Modern Physics Letters A

Implementing quantum prisoner dilemma via different classes of entangled and separable states

A. T. M. Makram-Allah, Mahmoud Youssef Abd‐Rabbou, NASSER METWALLY

Published Jul 18, 2025• 2 citations
Publisher

Abstract

Abstract The possibility of implementing the Prisoner's Dilemma game using different classes of initial states between the players is discussed. Different behaviors of the payoffs have been exhibited, may it increases, decreases or fixed. These behaviors depend on the used initial state either maximum/partial entangled, the chosen entangling gate, and the strategies employed by each player. The increased payoff is displayed when the players adopt different defect strategies, while the fixed payoff is shown when both players apply the same defect strategy. The maximum bounds of the payoff can be attained if the players initially share a maximum entangled state. The maximum /minimum fixed payoff depends on the used entangling gate, where the maximum one is shown if the players send their final states to entangling gate. It is shown that, the nature of the payoff, whether stationary, increasing or decreasing is contingent upon the polarization of the initial state and the used defect strategy. However, if both states are polarized on the same axis direction, the predicted maximum payoff is smaller than that obtained if the initial state are polarized in different polarization of the same axis. Therefore, the initial state settings, entangling parameter and the choice of the defect strategy can be used as control tools to maximize/minimize the obtained payoff.

DOI
10.1142/s0217732325501093

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Abstract The possibility of implementing the Prisoner's Dilemma game using different classes of initial states between the players is discussed. Different behaviors of the payoffs have been exhibited, may it increases, decreases or fixed. These behaviors depend on the used initial state either maximum/partial entangled, the chosen entangling gate, and the strategies employed by each player. The increased payoff is displayed when the players adopt different defect strategies, while the fixed payoff is shown when both players apply the same defect strategy. The maximum bounds of the payoff can be attained if the players initially share a maximum entangled state. The maximum /minimum fixed payoff depends on the used entangling gate, where the maximum one is shown if the players send their final states to entangling gate. It is shown that, the nature of the payoff, whether stationary, increasing or decreasing is contingent upon the polarization of the initial state and the used defect strategy. However, if both states are polarized on the same axis direction, the predicted maximum payoff is smaller than that obtained if the initial state are polarized in different polarization of the same axis. Therefore, the initial state settings, entangling parameter and the choice of the defect strategy can be used as control tools to maximize/minimize the obtained payoff.
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.
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.
The possibility of implementing a time dependent version of Eisert, Wilkens, and Lewenstein (EWL), quantum game is discussed. It is assumed that the players initial states are time dependent, while the entangling quantum gate either time dependent or independent during the game. The obtained results shows that the players payoff depend on their initial state, while The time-independent entangling gate has a minor impact on the game compared to its time-dependent counterpart. However, one can improve the lower bounds of the payoffs by using time dependent gate. Moreover, the exponential form of the entangling power is more powerful than the linear one, where the payoffs are much better. However, the maximum potential rewards in this game may be lower than those attainable in the classical game, the minimum guaranteed outcomes are much better.
Cooperation is a fundamental aspect of biological systems and human society. In this work, we propose a generalized prisoner's dilemma quantum game and study the game on Watts-Strogatz small-world network (WS model) with different values of entanglement. Here entanglement is used as a metaphor to study relationship between players. In a single step, each player has three strategies to choose: cooperation, defection and adopting a special quantum strategy first introduced by Eisert (denoted by Q). The person who chooses the quantum strategy Q can be regarded as a super cooperator. In prisoner's dilemma game, cooperation is very hard to emerge on relatively homogeneous networks, such as random graphs and WS model. Our experimental results show that in generalized prisoners dilemma quantum game, entanglement can guarantee emergence of super cooperators and greatly increase the payoff of the whole network. Even on homogeneous networks with very large inducement value(b is close to 2), the frequency of cooperators can still exceeds 70%. This work provides a new insight into understanding cooperation when considering the role of relationship between cooperators.
A well-known model in game theory, the Volunteer's Dilemma describes a group of nnn players who decide whether to volunteer for a collective benefit at a personal cost, or to abstain and risk forfeiting the benefit altogether. A quantum version of this dilemma, developed within the Eisert-Wilkens-Lewenstein framework, allows each player to manipulate one qubit of a shared entangled state, leading to symmetric Nash equilibria with higher expected payoffs than in the classical game. Existing analyses, however, assume maximal entanglement, which is experimentally demanding for hardware-constrained quantum systems. Within the same framework, we introduce a generalized Quantum Volunteer's Dilemma with a tunable entanglement parameter γγγ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating γγγ, the number of players, and the players' strategies under which symmetric Nash equilibria exist, focusing on two canonical strategy profiles: one for 2≤n≤92\leq n\leq 92≤n≤9, and one for even nnn. We find that maximal entanglement is not required to sustain symmetric equilibria. Instead, equilibrium behavior persists above a threshold value, which we compute analytically in both cases. We also demonstrate that the threshold value directly depends on system size.