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arXiv:1507.07966 · Europhysics Letters (EPL)

Quantum games of opinion formation based on the Marinatto-Weber quantum game scheme

Xinyang Deng, Yong Deng, Qi Liu, Lei Shi, Zhen Wang

Published Jun 1, 2016• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1209/0295-5075/114/50012

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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.
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.
A quantized version of the prisoner’s dilemma game based on the Marinatto–Weber quantization scheme is introduced. A four-qubit-basis initial entangled quantum state is assumed. Two domains for Nash equilibria are observed in terms of the degree of entanglement of the game. Each Nash equilibrium point is found to be Pareto optimal in its corresponding domain. Therefore, there is a unique solution to the game dilemma in each domain of the entanglement parameter. The maximally entangled limit is studied, and the players’ payoffs are found to equal the average of the payoff matrix independently of their choices. Consequently, any combination of players’ choices is a solution to the dilemma of the game. The nonentangled limits are also studied and found to coincide with the classical mixed-strategy game. Our results are equivalent to those of the noisy schemes of the quantum prisoner’s dilemma game. In our scheme, entanglement can be interpreted as a noise.
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.
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.
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.