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.
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.
Abstract The maximally entangled (ME) Eisert–Wilkens–Lewenstein (EWL) quantization scheme efficiently solves the classical prisoner’s dilemma (PD). Despite the intensive research on the EWL scheme, little is known about the role of the arbiter in controlling the scheme properties. We redefine the EWL scheme by introducing an arbitrary phase shift into the ME initial state. Four basic initial states were analyzed in this study. It is found that the arbiter completely controls the characteristics of the arbitrary ME quantum PD. The arbiter can adjust the ME initial state to obtain an efficient symmetric Nash equilibrium. In contrast, the arbiter can nullify the quantization advantages. Moreover, it can preserve/violate the symmetry of the game or even turn it into a potential game.
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.
A well-known model in game theory, the Volunteer's Dilemma describes a group of
n 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
γ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating
γ, the number of players, and the players' strategies under which symmetric Nash equilibria exist, focusing on two canonical strategy profiles: one for
2≤n≤9, and one for even
n. 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.