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A Quantum Approach to Static Games of Complete Information

L. MARINATTO, WEBER, TULLIO

Published Jan 1, 2000• 221 citations
OpenAlex

Abstract

We extend the concept of a classical two-person static game to the quantum domain, by giving an Hilbert structure to the space of classical strategies and studying the {\em Battle of the Sexes} game. We show that the introduction of entangled strategies leads to a unique solution of this game.

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arXiv:quant-ph/0110096•Journal of Physics A Mathematical and General•Mar 29, 2004•61 citations

The battle of sex game on the quantum games

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Quantum games include classical games.Some quantum games can get precise results.If we take the same strategies,we can get the same results.This article analyzes the battle of sexes and the results by using the way of quantum theory and the classical games.

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A multi-sector probability matrix formalism for games with incomplete information are formulated. Classical and quantum strategies can be treated in the same footing. A multi-sector extension of the game of Battle of Sexes is analyzed to clarify the two distinct roles of nonlocal strategies that break Bell inequalities.

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Battle of the Sexes: A Quantum Games Theory Approach

J. R.

Real life situations where there are strategies to be chosen in order to obtain a profit can be reproduced by games, so game theory is a way to describe the evolution of possible scenarios where players can select a scheme of play. Game theory takes importance in many areas, such as people decisions making, where their choices do affect others benefit (Davis, 1970; Myerson, 1991). It is to remark that the principles of game theory were initiated by trying to understand the behavior of economic strategies, however Von Neumann presented the concept of modern game theory in 1944 (Von Neumann & Morgenstern, 1947). Quantum mechanics is a tool that creates another point of view for the traditional game theory due to multiple strategies offered for the players, whom possibilities are numerously expanded in contrast of classical ones (Eisert & Wilkens, 2000). There are also games where the player who uses quantum strategies, enhances his payoffs or even always wins against a player who only uses classical moves (Meyer, 1999). It is to remark that there are plenty of applications for quantum game theory such as quantum cryptography and computation, economics and biology (Piotrowski & Sladkowski, 2003; Hanauske et al.,2009). The Battle of the Sexes game is a largely analyzed problem, based on two players: Alice and Bob and their choice about an activity for a Saturday night with each other. It is pretty important to remark that both want the best possible payoff in the decision, so the game can be developed normally, otherwise it would not be our case. Alice, really loves Opera, but wants to be with Bob; Bob likes Football but he wants to have Alice’s company along the activity. This game has a lot of applications in real life scenarios such as the spread of some type of genes from a reproduction between two organisms (Dawkins, 2006); another interesting application is neuroeconomics (Montague & Berns, 2002), where brain studies have been done in order to incite neurons to choose either to “work” for a reward or to “shirk” (Glimcher, 2003).

InTech eBooks•Feb 22, 2012•0 citations

Remark On Quantum Battle of The Sexes Game

Jiangfeng Du, Hui Li, Xiaodong Xu +3 more

Recently Quantum Battle of The Sexes Game has been studied by Luca Marinatto and Tullio Weber. Yet some important problems exist in their scheme. Here we propose a new scheme to quantize Battle of The Sexes Game, and this scheme will truly remove the dilemma that exists in the classical form of the game.

arXiv:quant-ph/0103004•Mar 2, 2001•11 citations

Application of Game-theoretic Techniques to Quantum Information Theory

Muhammad Asif Ramzan

This work is mainly based on quantum game-theoretic techniques and their application to quantum information theory.Quantum game theory is an important development in quantum computation and may have implications for quantum information and quantum communication.The select of quantum memory on quantum games have been studied.Three di􀀞erent games such as, Prisoner Dilemma, Battle of the Sexes and Chicken have been analyzed in the quantum domain. By considering the restricted game \nsituation, it is shown that the ects of quantum memory and decoherence become e􀀞ective in a maximally entangled case.For which the quantum player can out perform the classical player in all the three games. It is also shown that the quantum player enjoys an advantage over the classical player in Battle of the Sexes game for the amplitude-damping and the depolarizing channels.The quantum memory compensates the reduction in player pay is due to decoherence. It has no ect on the Nash equilibria of the three games.A generalization of two-player quantization scheme to a three-player Prisoners Dilemma game under the ect of correlated noise is also presented.In a restricted game scenario, it is shown the quantumplayer is always better; for all values of the decoherence parameter for the entire range of memory parameter.It is seen that for maximum degree of correlation, the game does not become \nnoiseless and quantum player can still outscore the classical players for the entire range of the decoherence parameter producing an interesting result in comparison with a two-player quantum game.It is also shown that correlated noise has not on the Nash equilibrium of the game.By exploiting the three-player quantization scheme, we have studiedthe communication aspects of a three-player Prisoner Dilemma quantum game. It is shown that entanglement plays a dominant role in a three-player quantum game.On the baois of initial state and the measurement basis entanglement parameters, a relation among different pay is established.It is investigated that the strategies of the players act as information carriers in quantum games. A relationship for the information shared among the parties is also established.A scheme for quantum key distribution is proposed in which a secret key can be generated from the data coming through a partially entanglement breaking channel.This scheme is rather deterministic andcient in the sense that two classical bits can be transferred \nper entangled pair of qubits.Furthermore, it is important to point out that, in this scheme, same symbol can be used for Eve detection and key generation. It is also worth noting that the eavesdropper, Eve, can be detected very easily from the disturbance of the elements of the decoding bimatrix. We have checked for the \nsecurity of the scheme and found it secure against individual attacks.It is further investigated that quantum games are useful in developing quantum cryptographic protocols.A multiparty quantum cryptographic protocol using tripartite entangled GHZ states is devised using game-theoretic techniques.In this protocol, two classical bits can be transferred per entangled pair of qubits to the communicating parties.Unitary operators applied by the sender on a tripartite entangled state encodes a classical symbol that can be decoded at receiver end with the help of a decoding matrix. In this protocol, if Eve interferes during the transmission, she can be detected by the disturbance of the decoding matrix. Our security analysis shows that this protocol is also secure against intercept/re-send attacks.

•Jan 1, 2010•0 citations
Real life situations where there are strategies to be chosen in order to obtain a profit can be reproduced by games, so game theory is a way to describe the evolution of possible scenarios where players can select a scheme of play. Game theory takes importance in many areas, such as people decisions making, where their choices do affect others benefit (Davis, 1970; Myerson, 1991). It is to remark that the principles of game theory were initiated by trying to understand the behavior of economic strategies, however Von Neumann presented the concept of modern game theory in 1944 (Von Neumann & Morgenstern, 1947). Quantum mechanics is a tool that creates another point of view for the traditional game theory due to multiple strategies offered for the players, whom possibilities are numerously expanded in contrast of classical ones (Eisert & Wilkens, 2000). There are also games where the player who uses quantum strategies, enhances his payoffs or even always wins against a player who only uses classical moves (Meyer, 1999). It is to remark that there are plenty of applications for quantum game theory such as quantum cryptography and computation, economics and biology (Piotrowski & Sladkowski, 2003; Hanauske et al.,2009). The Battle of the Sexes game is a largely analyzed problem, based on two players: Alice and Bob and their choice about an activity for a Saturday night with each other. It is pretty important to remark that both want the best possible payoff in the decision, so the game can be developed normally, otherwise it would not be our case. Alice, really loves Opera, but wants to be with Bob; Bob likes Football but he wants to have Alice’s company along the activity. This game has a lot of applications in real life scenarios such as the spread of some type of genes from a reproduction between two organisms (Dawkins, 2006); another interesting application is neuroeconomics (Montague & Berns, 2002), where brain studies have been done in order to incite neurons to choose either to “work” for a reward or to “shirk” (Glimcher, 2003).
This work is mainly based on quantum game-theoretic techniques and their application to quantum information theory.Quantum game theory is an important development in quantum computation and may have implications for quantum information and quantum communication.The select of quantum memory on quantum games have been studied.Three di􀀞erent games such as, Prisoner Dilemma, Battle of the Sexes and Chicken have been analyzed in the quantum domain. By considering the restricted game \nsituation, it is shown that the ects of quantum memory and decoherence become e􀀞ective in a maximally entangled case.For which the quantum player can out perform the classical player in all the three games. It is also shown that the quantum player enjoys an advantage over the classical player in Battle of the Sexes game for the amplitude-damping and the depolarizing channels.The quantum memory compensates the reduction in player pay is due to decoherence. It has no ect on the Nash equilibria of the three games.A generalization of two-player quantization scheme to a three-player Prisoners Dilemma game under the ect of correlated noise is also presented.In a restricted game scenario, it is shown the quantumplayer is always better; for all values of the decoherence parameter for the entire range of memory parameter.It is seen that for maximum degree of correlation, the game does not become \nnoiseless and quantum player can still outscore the classical players for the entire range of the decoherence parameter producing an interesting result in comparison with a two-player quantum game.It is also shown that correlated noise has not on the Nash equilibrium of the game.By exploiting the three-player quantization scheme, we have studiedthe communication aspects of a three-player Prisoner Dilemma quantum game. It is shown that entanglement plays a dominant role in a three-player quantum game.On the baois of initial state and the measurement basis entanglement parameters, a relation among different pay is established.It is investigated that the strategies of the players act as information carriers in quantum games. A relationship for the information shared among the parties is also established.A scheme for quantum key distribution is proposed in which a secret key can be generated from the data coming through a partially entanglement breaking channel.This scheme is rather deterministic andcient in the sense that two classical bits can be transferred \nper entangled pair of qubits.Furthermore, it is important to point out that, in this scheme, same symbol can be used for Eve detection and key generation. It is also worth noting that the eavesdropper, Eve, can be detected very easily from the disturbance of the elements of the decoding bimatrix. We have checked for the \nsecurity of the scheme and found it secure against individual attacks.It is further investigated that quantum games are useful in developing quantum cryptographic protocols.A multiparty quantum cryptographic protocol using tripartite entangled GHZ states is devised using game-theoretic techniques.In this protocol, two classical bits can be transferred per entangled pair of qubits to the communicating parties.Unitary operators applied by the sender on a tripartite entangled state encodes a classical symbol that can be decoded at receiver end with the help of a decoding matrix. In this protocol, if Eve interferes during the transmission, she can be detected by the disturbance of the decoding matrix. Our security analysis shows that this protocol is also secure against intercept/re-send attacks.