Cooperation is essential to biological organizations and human society.In this paper, the conditions of dilemmas resolving by quantum strategies in the three games, Prisoners' Dilemma, Chicken Game and Stag Hunt, are presented.The rule for the evolution of cooperation in general quantum games, b/c > coth 2 γ, is put forward.The formation of the rule is similar to those of the five rules in the classical game theory summarized by Nowak.b/c is still the benefit-to-cost ratio of the altruistic act, while γ is a measure for the game's entanglement.The rule is the prerequisite for the realization of cooperation in quantum games.
Cooperation is essential to biological organizations and human society.In this paper, the conditions of dilemmas resolving by quantum strategies in the three games, Prisoners' Dilemma, Chicken Game and Stag Hunt, are presented.The rule for the evolution of cooperation in general quantum games, b/c > coth 2 γ, is put forward.The formation of the rule is similar to those of the five rules in the classical game theory summarized by Nowak.b/c is still the benefit-to-cost ratio of the altruistic act, while γ is a measure for the game's entanglement.The rule is the prerequisite for the realization of cooperation in quantum games.
It was known that cooperation of evolutionary prisoner's dilemma games fails to emerge in homogenous networks such as random graphs. Here we proposed a quantum prisoner's dilemma game. The game consists of two players, in which each player has three choices of strategy: cooperator (C), defector (D) and super cooperator (denoted by Q). We found that quantum entanglement guarantees emergence of a new cooperation, the super cooperation of the quantum prisoner's dilemma games, and that entanglement is the mechanism of guaranteed emergence of cooperation of evolutionary prisoner's dilemma games on networks. We showed that for a game with temptation b, there exists a threshold arccos √b/b for a measurement of entanglement, beyond which, (super) cooperation of evolutionary quantum prisoner's dilemma games is guaranteed to quickly emerge, giving rise to stochastic convergence of the cooperations, that if the entanglement degree γ is less than the threshold arccos √b/b, then the equilibrium frequency of cooperations of the games is positively correlated to the entanglement degree γ, and that if γ is less than arccos √b/b and b is beyond some boundary, then the equilibrium frequency of cooperations of the games on random graphs decreases as the average degree of the graphs increases.
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.
We present a novel formulation of quantum game theory based on the Schmidt decomposition, which has the merit that the entanglement of quantum strategies is manifestly quantified. We apply this formulation to 2-player, 2-strategy symmetric games and obtain a complete set of quantum Nash equilibria. Apart from those available with the maximal entanglement, these quantum Nash equilibria are extensions of the Nash equilibria in classical game theory. The phase structure of the equilibria is determined for all values of entanglement, and thereby the possibility of resolving the dilemmas by entanglement in the game of Chicken, the Battle of the Sexes, the Prisoners' Dilemma, and the Stag Hunt, is examined. We find that entanglement transforms these dilemmas with each other but cannot resolve them, except in the Stag Hunt game where the dilemma can be alleviated to a certain degree.
Since the establishment of von Neumann and Nash's concept of equilibrium, game theory has become a universal language in economics and social sciences. However, classical game theory is deeply rooted in the mechanistic worldview, and its deterministic assumptions, static equilibrium, and individualistic methodology are becoming increasingly inadequate in explaining the emergence of cooperation, uncertainty, and complex social interactions. This book directly faces this dilemma and introduces the core concepts of quantum mechanics into game analysis, constructing a complete system of quantum game theory. Starting from the century long evolution of game theory, the entire book deeply criticizes the mechanistic nature of classical game theory, and re establishes the philosophical foundation of game theory through practical ontology. On this basis, this book proposes the hypothesis of a "quantum gamer" and constructs a complete analytical framework from superposition strategy space, quantum entanglement to measurement collapse. The system reconstructs core areas such as quantum zero sum games, quantum non-zero sum games, quantum prisoner's dilemma, quantum coordination games, quantum evolutionary games, quantum cooperative games, and quantum incomplete information games, and extends them to cutting-edge directions such as quantum repetitive games, quantum game learning, and quantum mechanism design. At the application level, this book comprehensively demonstrates the broad prospects of quantum game theory in fields such as economics, political science, sociology, biology, computer science, quantum computing, artificial intelligence, and ethics. This book emphasizes that quantum entanglement enables cooperation to emerge, and quantum measurement leads equilibrium from static to dynamic collapse, thus surpassing the explanatory boundaries of classical game theory. The book has a rigorous structure and in-depth arguments, combining theoretical originality with interdisciplinary perspectives, providing a groundbreaking work for researchers, graduate students, and cutting-edge explorers to understand the quantum game revolution.
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.