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arXiv:1008.3366

Quantum realization of extensive games

Piotr Frąckiewicz

Published Aug 19, 2010• 0 citations
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Abstract

We generalize a concept of classical finite extensive game to make it useful for application of quantum objects. The generalization extends a quantum realization scheme of static games to any finite extensive game. It represents an extension of any classical finite extensive games to the quantum domain. In addition our model is compatible with well-known quantum schemes of static games. The paper is summed up by two examples.

DOI
10.48550/arxiv.1008.3366

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We generalize a concept of classical finite extensive game to make it useful for application of quantum objects. The generalization extends a quantum realization scheme of static games to any finite extensive game. It represents an extension of any classical finite extensive games to the quantum domain. In addition our model is compatible with well-known quantum schemes of static games. The paper is summed up by two examples.
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.
We give a strict mathematical description for a refinement of the Marinatto–Weber quantum game scheme. The model allows the players to choose projector operators that determine the state on which they perform their local operators. The game induced by the scheme generalizes finite strategic-form game. In particular, it covers normal representations of extensive games, i.e., strategic games generated by extensive ones. We illustrate our idea with an example of extensive game and prove that rational choices in the classical game and its quantum counterpart may lead to significantly different outcomes.
We modify the concept of quantum strategic game to make it useful for extensive form games. We prove that our modification allows us to consider the normal representation of any finite extensive game using the fundamental concepts of quantum information. The Selten's Horse game and the general form of two-stage extensive game with perfect information are studied to illustrate a potential application of our idea. In both examples we use the Eisert–Wilkens–Lewenstein approach as well as the Marinatto–Weber approach to quantization of games.
We modify the concept of quantum strategic game to make it useful for extensive form games. We prove that our modification allows to consider the normal representation of any finite extensive game using the fundamental concepts of quantum information. The Selten's Horse game and the general form of two-stage extensive game with perfect information are studied to illustrate a potential application of our idea. In both examples we use Eisert-Wilkens-Lewenstein approach as well as Marinatto-Weber approach to quantization of games.
A number of recent studies have focused on novel features in game theory when the games are played using quantum mechanical toolbox (entanglement, unitary operators, measurement). Researchers have concentrated in two-player-two strategy, 2x2, dilemma containing classical games, and transferred them into quantum realm showing that in quantum pure strategies dilemmas in such games can be resolved if entanglement is distributed between the players armed with quantum operations. Moreover, it became clear that the 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 2x2 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 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.
Main papers on quantum games are written by physicists for physicists, and the inevitable exploitation of physics jargon may create difficulties for mathematicians or economists. Our goal here is to make clear the physical content and to stress the new features of the games that may be revealed in their quantum versions. We first introduce from scratch the most fundamental facts of finite-dimensional quantum mechanics. In the main sections the foundations of quantum games are built via the basic examples. We omit sometimes the lengthy calculations (referring to the original papers) once the physical part is sorted out and the problem is reformulated as pure game-theoretic problem of calculating the Nash or dominated equilibria. Finally we touch upon general theory of finite quantum static games and provide further links and references.