By incorporating payoff matrix and population evolution of open systems and proposing density games, pioneering effort has been made to break the following two basic assumptions of evolutionary game theory over the past decades: the payoff from the game determines the reproductive rate of individuals, and the total population size remains constant by a variable called the death rate. On the other hand, most studies on strategic interactions in game theories consider independent individual strategies as the basic unit of analyses. This is in fact not always true in human interaction strategies. It has been unclear that how subjective non-independent species with strategic interactions in open dynamic systems evolve. In this paper, we create the new concept of hybrid quantum games, and propose an open quantum dynamic symbiosis model for exploring the evolution of this type of species under unclosed social settings. We show that the stability conditions of the model with non-independent strategic interactions are influenced by the specific symbiosis states, game payoff matrix and quantum entanglement. The stability conditions of the competition scenario are more stringent than those of the cooperation one. Our system dynamics simulation study demonstrates and verifies the dynamic changes for the population sizes of two different species in the theoretical model, on the basis of the stability conditions of the competition and cooperation scenarios. Our model combines quantum games, game payoff matrix and symbiosis equations organically, and can provide insights into competition, cooperation and symbiosis mechanisms in open systems with subjective non-independent strategies considered.
By incorporating payoff matrix and population evolution of open systems and proposing density games, pioneering effort has been made to break the following two basic assumptions of evolutionary game theory over the past decades: the payoff from the game determines the reproductive rate of individuals, and the total population size remains constant by a variable called the death rate. On the other hand, most studies on strategic interactions in game theories consider independent individual strategies as the basic unit of analyses. This is in fact not always true in human interaction strategies. It has been unclear that how subjective non-independent species with strategic interactions in open dynamic systems evolve. In this paper, we create the new concept of hybrid quantum games, and propose an open quantum dynamic symbiosis model for exploring the evolution of this type of species under unclosed social settings. We show that the stability conditions of the model with non-independent strategic interactions are influenced by the specific symbiosis states, game payoff matrix and quantum entanglement. The stability conditions of the competition scenario are more stringent than those of the cooperation one. Our system dynamics simulation study demonstrates and verifies the dynamic changes for the population sizes of two different species in the theoretical model, on the basis of the stability conditions of the competition and cooperation scenarios. Our model combines quantum games, game payoff matrix and symbiosis equations organically, and can provide insights into competition, cooperation and symbiosis mechanisms in open systems with subjective non-independent strategies considered.
A new representation of Game Theory is developed in this paper. State of players is represented by a density matrix, and payoff function is a set of hermitian operators, which when applied onto the density matrix give the payoff of players. By this formulism, a new way to find the equilibria of games is given by generalizing the thermodynamical evolutionary process leading to equilibrium in Statistical Mechanics. And in this formulism, when quantum objects instead of classical objects are used as the objects in the game, it's naturally leads to the so-called Quantum Game Theory, but with a slight difference in the definition of strategy state of players: the probability distribution is replaced with a density matrix. Further more, both games of correlated and independent players can be reached in this single framework, while traditionally, they are treated separately by Non-cooperative Game Theory and Coalitional Game Theory. Because of the density matrix is used as state of players, besides classical correlated strategy, quantum entangled states can also be used as strategies, which is an entanglement of strategies between players, and it is different with the entanglement of objects' states as in the so-called Quantum Game Theory. At last, in the form of density matrix, a class of quantum games, where the payoff matrixes are commutative, can be reduced into classical games. In this sense, it will put the classical game as a special case of our quantum game.
Coopetition is defined as the existence of simultaneous competition and cooperation between the same set of players, leading to entanglement of payoffs and actions of the players. This paper provides insights into the game theoretical application of quantum games to model simultaneity and entanglement that occur in coopetition. Modeling as a quantum game also allows for a larger action space and hence new equilibrium that may not have existed earlier. The impact of the level of entanglement on the equilibrium of the game can also be studied. We demonstrate the same through an example of two players who currently compete in the domestic market and are considering cooperating simultaneously in the international market. They need to determine the equilibrium strategy to adopt under coopetition that maximizes their payoffs. We also arrive at how to ensure that the quantum strategy is the equilibrium strategy for both players, namely, how to design the quantum strategy and how to define the unitary operator.
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.
A new representation of Game Theory is developed in this paper. State of players is represented by a density matrix, and payoff function is a set of hermitian operators, which when applied onto the density matrix give the payoff of players. By this formulism, a new way to find the equilibria of games is given by generalizing the thermodynamical evolutionary process leading to equilibrium in Statistical Mechanics. And in this formulism, when quantum objects instead of classical objects are used as the objects in the game, it's naturally leads to the so-called Quantum Game Theory, but with a slight difference in the definition of strategy state of players: the probability distribution is replaced with a density matrix. Further more, both games of correlated and independent players can be reached in this single framework, while traditionally, they are treated separately by Non-cooperative Game Theory and Coalitional Game Theory. Because of the density matrix is used as state of players, besides classical correlated strategy, quantum entangled states can also be used as strategies, which is an entanglement of strategies between players, and it is different with the entanglement of objects' states as in the so-called Quantum Game Theory. At last, in the form of density matrix, a class of quantum games, where the payoff matrixes are commutative, can be reduced into classical games. In this sense, it will put the classical game as a special case of our quantum game.
Game theory is the study of strategic competition used in many branches of science. In general, a game is a match between two or more players and each wants to maximize his or her score. In a game, the player’s score depends on their strategies. The goal is to find the optimized decision, with regard to the payoff, in conformity with the rules of the game. This thesis considers the games with two players who know both their own and their rival’s payoff. However, they do not know the strategy chosen by the opponent.\nIn a classical game, the options of players are referred to as cooperation and defect. Quantum game theory is a generalization of classical game theory. In quantum games a linear combination of two options specify the strategy based on qubit. The quantum domain is characterized by entanglement. The quantum entanglement coefficient (y) varies from 0 to . The game is in the classical regime when y is smaller than a threshold. Quantum entanglement is intrinsically non-local.\nIn an iterated game tournament, 63 different strategies contest with four known strategies. Depending on the result, they are divided to a number of groups. The competition specifies which group of strategies can obtain a higher score.\nOther types of games are discussed in the thesis. They are known as incomplete information games. In these games one player has two payoff matrixes, and the rival does not know which of them is used. We discuss the advantage of using quantum strategy over the classical game, particular in avoiding the dilemma in classical games.
Game theory is the mathematical framework for analyzing strategic interactions in conflict and competition situations. In recent years quantum game theory has earned the attention of physicists, and has emerged as a branch of quantum information theory [1]. With the aid of entanglement and linear superposition of strategies, quantum games are shown to yield signifcant advantage over their classical counterparts. In this paper we explore optimal and equilibrium solutions to quantum minority games. Initial states with different level of entanglement are investigated. Focus will be on 4 and 6 player games with some N-player generalizations.