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arXiv:1109.2169

Quantum information approach to the ultimatum game

Piotr Frąckiewicz

Published Sep 9, 2011• 0 citations
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Abstract

The paper is devoted to quantization of extensive games with the use of both the Marinatto-Weber and the Eisert-Wilkens-Lewenstein concept of quantum game. We revise the current conception of quantum ultimatum game and we show why the proposal is unacceptable. To support our comment, we present the new idea of the quantum ultimatum game. Our scheme also makes a point of departure for a protocol to quantize extensive games.

DOI
10.48550/arxiv.1109.2169

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The paper is devoted to quantization of extensive games with the use of both the Marinatto-Weber and the Eisert-Wilkens-Lewenstein concept of quantum game. We revise the current conception of quantum ultimatum game and we show why the proposal is unacceptable. To support our comment, we present the new idea of the quantum ultimatum game. Our scheme also makes a point of departure for a protocol to quantize extensive games.
We give a concise and self-contained introduction to the theory of Quantum Games by reviewing the seminal works of Meyer, Eisert-Wilkens-Lewenstein, Marinatto-Weber and Landsburg, which initiated the study of this field. By generalizing this body of work, we formulate a protocol to Quantumize\textit{Quantumize}Quantumize any finite classical nnn-player game, and use a novel approach of describing such a Quantum Game in terms of commuting Payoff Operators. We describe what advantages can be gained by players by quantumizing such a game, particularly, what additional Nash Equilibria the players can achieve and the Pareto-Optimality of these additional equilibria.
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.
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 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.
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.