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arXiv:quant-ph/0004076 · Journal of Modern Optics

Quantum games

Jens Eisert, Martin Wilkens

Published Nov 1, 2000• 222 citations
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Abstract

In these lecture notes we investigate the implications of the identification of strategies with quantum operations in game theory beyond the results presented Eisert et al. (1999, Phys. Rev. Lett., 83, 3077). 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.

DOI
10.1080/09500340008232180

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In these lecture notes we investigate the implications of the identification of strategies with quantum operations in game theory beyond the results presented Eisert et al. (1999, Phys. Rev. Lett., 83, 3077). 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.
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.
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 develop a rigorous mathematical framework for quantum game theory applied to static 2x2 games, extending classical concepts to the quantum setting where players may employ arbitrary unitary operations (pure strategies) or probability measures over the continuous group SU(2) (mixed strategies). The Eisert-Wilkens-Lewenstein protocol is introduced as the standard implementation of quantum 2x2 games. We prove the existence of Nash equilibria for continuous quantum mixed strategies via a fixed-point argument, generalising the classical Nash theorem to the quantum case.
Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player two-strategy (2 × 2) dilemma containing classical games into the quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2 × 2 games, it became generally accepted that quantum versions can be easily extended to N -player situations by simply allowing N -partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in the quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.
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.