The Dilemma

Wanda Strauven

Angsheng Li, Xi Yong

Recently, the authors proposed a quantum prisoner's dilemma game based on the spatial game of Nowak and May, and showed that the game can be played classically. By using this idea, we proposed three generalized prisoner's dilemma (GPD, for short) games based on the weak Prisoner's dilemma game, the full prisoner's dilemma game and the normalized Prisoner's dilemma game, written by GPDW, GPDF and GPDN respectively. Our games consist of two players, each of which has three strategies: cooperator (C), defector (D) and super cooperator (denoted by Q), and have a parameter γ to measure the entangled relationship between the two players. We found that our generalised prisoner's dilemma games have new Nash equilibrium principles, that entanglement is the principle of emergence and convergence (i.e., guaranteed emergence) of super cooperation in evolutions of our generalised prisoner's dilemma games on scale-free networks, that entanglement provides a threshold for a phase transition of super cooperation in evolutions of our generalised prisoner's dilemma games on scale-free networks, that the role of heterogeneity of the scale-free networks in cooperations and super cooperations is very limited, and that well-defined structures of scale-free networks allow coexistence of cooperators and super cooperators in the evolutions of the weak version of our generalised prisoner's dilemma games.

Daniel Kleppner

Abd El Hakeem Abd El Naby, Heba A. Elrayes, Ahmed M. K. Tarabia +1 more

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.

Ahmed S. Elgazzar

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.