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arXiv:1109.3753 · Journal of Physics A Mathematical and Theoretical

Quantum repeated games revisited

Piotr Frąckiewicz

Published Feb 14, 2012• 20 citations
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Abstract

We present a scheme for playing quantum repeated 2 × 2 games based on Marinatto and Weber's approach to quantum games. As a potential application, we study the twice repeated Prisoner's Dilemma game. We show that results not available in the classical game can be obtained when the game is played in the quantum way. Before we present our idea, we comment on the previous scheme of playing quantum repeated games proposed by Iqbal and Toor. We point out the drawbacks that make their results unacceptable.

DOI
10.1088/1751-8113/45/8/085307

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arXiv:1303.4019•Acta Physica Polonica B•Jan 1, 2013•17 citations
We present a scheme for playing quantum repeated 2 × 2 games based on Marinatto and Weber's approach to quantum games. As a potential application, we study the twice repeated Prisoner's Dilemma game. We show that results not available in the classical game can be obtained when the game is played in the quantum way. Before we present our idea, we comment on the previous scheme of playing quantum repeated games proposed by Iqbal and Toor. We point out the drawbacks that make their results unacceptable.
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A quantum approach to twice-repeated 2×22\times 22×2 game
Abstract This work proposes an extension of the well-known Eisert–Wilkens–Lewenstein scheme for playing a twice repeated 2×22\times 22×2 2 × 2 game using a single quantum system with ten maximally entangled qubits. The proposed scheme is then applied to the Prisoner’s Dilemma game. Rational strategy profiles are examined in the presence of limited awareness of the players. In particular, the paper considers two cases of a classical player against a quantum player game: the first case when the classical player does not know that his opponent is a quantum one and the second case, when the classical player is aware of it. To this end, the notion of unawareness was used, and the extended Nash equilibria were determined.
The Marinatto–Weber approach to quantum games is a straightforward way to apply the power of quantum mechanics to classical game theory. In the simplest case, the quantum scheme is that players manipulate their own qubits of a two-qubit state either with the identity or the Pauli operator σ x . However, such a simplification of the scheme raises doubt as to whether it could really reflect a quantum game. In this paper we put forward examples which may constitute arguments against the present form of the Marinatto–Weber scheme. Next, we modify the scheme to eliminate the undesirable properties of the protocol by extending the players’ strategy sets.
Quantization has become a new way to investigate classical game theory since quantum strategies and quantum games were proposed. In the existing studies, many typical game models, such as the prisoner's dilemma, battle of the sexes, Hawk-Dove game, have been extensively explored by using quantization approach. Along a similar method, here several game models of opinion formations will be quantized on the basis of the Marinatto-Weber quantum game scheme, a frequently used scheme of converting classical games to quantum versions. Our results show that the quantization can fascinatingly change the properties of some classical opinion formation game models so as to generate win-win outcomes.
Iqbal and Toor [Phys. Rev. A {\bf 65}, 022306 (2002)] and [Commun. Theor. Phys. {\bf 42}, 335 (2004)] generalized the Marinatto-Weber quantum scheme for 2×22 \times 22×2 games in order to study bimatrix games of 3×33 \times 33×3 dimension, in particular the Rock-Paper-Scissors game. In our paper we show that Iqbal and Toor's generalization exhibits certain undesirable property that can considerably influence the game result. To support our argumentation, in the further part of the paper we construct the protocol corresponding to the MW concept for any finite bimatrix game that is free from the fault.