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arXiv:1303.4019 · Acta Physica Polonica B

A Comment on the Generalization of the Marinatto–Weber Quantum Game Scheme

Piotr Frąckiewicz

Published Jan 1, 2013• 17 citations
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Abstract

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 $2 \times 2$ games in order to study bimatrix games of $3 \times 3$ 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.

DOI
10.5506/aphyspolb.44.29

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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.
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.
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.
Our purpose is to focus attention on a new criterion for quantum schemes by bringing together the notions of quantum game and game isomorphism. A quantum game scheme is required to generate the classical game as a special case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We show how this isomorphism condition influences the players’ strategy sets. We are concerned with the Marinatto-Weber type quantum game scheme and the strong isomorphism between games in strategic form.
Players’ choices in quantum game schemes are often correlated by a quantum state. This enables players to obtain payoffs that may not be achievable when classical pure or mixed strategies are used. On the other hand, players’ choices can be correlated due to a classical probability distribution, and if no player benefits by a unilateral deviation from the vector of recommended strategies, the probability distribution is a correlated equilibrium. The aim of this paper is to investigate relation between correlated equilibria and Nash equilibria in the MW-type schemes for quantum games.
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.