It is known that quantum strategies change the range of possible payoffs for the players in the prisoner's dilemma. In this paper, we examine the effect of the degree of entanglement in determining the payoffs. When both players play quantum strategies, we show that the payoff for both players is unaffected by the entanglement value and it leads to a new Nash equilibrium.
It is known that quantum strategies change the range of possible payoffs for the players in the prisoner's dilemma. In this paper, we examine the effect of the degree of entanglement in determining the payoffs. When both players play quantum strategies, we show that the payoff for both players is unaffected by the entanglement value and it leads to a new Nash equilibrium.
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
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.
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.
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.
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.
What happens when an infinite number of players play a quantum game? In this paper, we will answer this question by looking at the emergence of cooperation, in the presence of noise, in a one-shot quantum prisoner’s dilemma (QuPD). We will use the numerical agent-based model (ABM) and compare it with the analytical Nash equilibrium mapping (NEM) technique. To measure cooperation, we consider five indicators, i.e., game magnetization, entanglement susceptibility, correlation, player’s payoff average, and payoff capacity, respectively. In quantum social dilemmas, entanglement plays a non-trivial role in determining the behavior of the quantum players (or qubits) in the thermodynamic limit, and for QuPD, we consider the existence of bipartite entanglement between neighboring quantum players. For the five indicators in question, we observe first-order phase transitions at two entanglement values, and these phase transition points depend on the payoffs associated with the QuPD game. We numerically analyze and study the properties of both the quantum and the defect phases of the QuPD via the five indicators. The results of this paper demonstrate that both ABM and NEM, in conjunction with the chosen five indicators, provide insightful information on cooperative behavior in an infinite-player one-shot quantum prisoner’s dilemma.