With nearly one decade of life, the theory of quantum games has nowadays become a very reach and prolific field of research. Multiplayer and multistrategy setups, a quantum approach to Evolutionary Game Theory, quantum game-like simulations of market phenomena, quantum duels, the effect of decoherence and noise during the implementation of a quantum game, are just some of the many scenarios where the quantum aspects of games were analyzed [2–4, 6, 7, 11–13, 16–18, 23]. Here we do not attempt to review the vast universe of quantum game theory, but just to give a comprehensive and didactic introduction to the pioneering work of Eisert [5]. This introduction will allow us to expose a recent approach developed to geometrically understand quantum games [20, 21], which represents the main focus this chapter. For a complete review of quantum games, we refer the reader to [9].
With nearly one decade of life, the theory of quantum games has nowadays become a very reach and prolific field of research. Multiplayer and multistrategy setups, a quantum approach to Evolutionary Game Theory, quantum game-like simulations of market phenomena, quantum duels, the effect of decoherence and noise during the implementation of a quantum game, are just some of the many scenarios where the quantum aspects of games were analyzed [2–4, 6, 7, 11–13, 16–18, 23]. Here we do not attempt to review the vast universe of quantum game theory, but just to give a comprehensive and didactic introduction to the pioneering work of Eisert [5]. This introduction will allow us to expose a recent approach developed to geometrically understand quantum games [20, 21], which represents the main focus this chapter. For a complete review of quantum games, we refer the reader to [9].
Game theory is a well established field with applications in many areas including the social sciences, biology, and computer science [6]. Recently, however, researchers in quantum computation have begun to study the implications of games set in the quantum domain (i.e. game theoretical situations where either one or more players have quantum computational power or where the implementation of strategies uses some form of quantum information). Such a setting allows for entanglement and superpositions of strategies and evaluation of payoffs using quantum measurement. This approach to game theory, according to some scholars in the field, could help us understand certain physical systems, improve economic markets, and develop quantum algorithms. I have conducted an investigation into quantum game theory. I have constructed simulation tools using Maple that, along with research into related work, have helped gain some insight into this new field. The important solutions in classical game theory are usually Nash equilibria. What do they look like in the quantum realm? In what
We give a concise and self-contained introduction to the theory of Quantum Games by reviewing the seminal works of Meyer, Eisert-Wilkens-Lewenstein, Marinatto-Weber and Landsburg, which initiated the study of this field. By generalizing this body of work, we formulate a protocol to Quantumize any finite classical n-player game, and use a novel approach of describing such a Quantum Game in terms of commuting Payoff Operators. We describe what advantages can be gained by players by quantumizing such a game, particularly, what additional Nash Equilibria the players can achieve and the Pareto-Optimality of these additional equilibria.
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.
The authors briefly review quantum game theory and its application in economics. This review is addressed at newcomers and some basic ideas of quantum theory are necessary to follow the text—the short introduction in chapter “A Brief Introduction to Quantum Formalism” will be sufficient. Due to the lack of space only the flagship issues will be discussed. Quantum game theory, whatever opinions may be held due to its abstract physical formalism, have already found various applications even outside the orthodox physics domain. We are aware that the implementation of genuine quantum models is not an easy task. Nevertheless, such models are already an interesting although sophisticated theoretical tool. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research in the field of quantum computation, in particular, quantum game theory is in its foundational stage. Primarily, quantum game theory adds a new dimension to the well - developed classical game theory by making possible the superposition of strategies. By projecting the classical Prisoner's Dilemma game into the quantum domain, experts in this regime proved that, for an ideally closed quantum system, superposed quantum strategies offer more promising alternatives and solutions than discrete classical ones for this peculiar game. To present a more realistic picture of a quantum game system, this project seeks to examine the efficiency of quantum strategies in a less than perfect environment where decoherence of pure quantum states can occur. The findings advocate that quantum strategies remain superior to classical ones despite the deterioration of solutions due to decoherence.
In these lecture notes we investigate the implications of the identification of strategies with quantum operations in game theory beyond the results presented in [J. Eisert, M. Wilkens, and M. Lewenstein, Phys. Rev. Lett. 83, 3077 (1999)]. After introducing a general framework, we study quantum games with a classical analogue in order to flesh out the peculiarities of game theoretical settings in the quantum domain. Special emphasis is given to a detailed investigation of different sets of quantum strategies.