In this paper I quantize the stag hunt game in the framework proposed by Marinatto and Weber which, is introduced to quantize the Battle of the Sexes game and gives a general quntization scheme of various game theories. Then I discuss the Nash equibilium solution in the cases of which starting strategies are taken in both non entangled state and entangled state and uncover the structure of Nash Equilibrium solutions and compare the case of the Battle of the Sexes game. Since the game has 4 parameters in the payoff matrix has rather rich structure than the Battle of the Sexes game with 3-parameters in the payoff matrix, the relations of the magnitude of these payoff values in Nash Equilibriums are much involuved. This structure is uncovered completly and it is found that the best strategy which give the maximal sum of the payoffs of both players strongly depends on the initial quntum state. As the bonus of the formulation the stag hunt game with four parameters we can discuss various types of symmetric games played by two players by using the latter formulation, i.e. Chicken game. As result some common properties are found between them and the stag hunt game. Lastly a little remark is made on Prisoner's Dillemma.
In this paper I quantize the stag hunt game in the framework proposed by Marinatto and Weber which, is introduced to quantize the Battle of the Sexes game and gives a general quntization scheme of various game theories. Then I discuss the Nash equibilium solution in the cases of which starting strategies are taken in both non entangled state and entangled state and uncover the structure of Nash Equilibrium solutions and compare the case of the Battle of the Sexes game. Since the game has 4 parameters in the payoff matrix has rather rich structure than the Battle of the Sexes game with 3-parameters in the payoff matrix, the relations of the magnitude of these payoff values in Nash Equilibriums are much involuved. This structure is uncovered completly and it is found that the best strategy which give the maximal sum of the payoffs of both players strongly depends on the initial quntum state. As the bonus of the formulation the stag hunt game with four parameters we can discuss various types of symmetric games played by two players by using the latter formulation, i.e. Chicken game. As result some common properties are found between them and the stag hunt game. Lastly a little remark is made on Prisoner's Dillemma.
We present a novel formulation of quantum game theory based on the Schmidt decomposition, which has the merit that the entanglement of quantum strategies is manifestly quantified. We apply this formulation to 2-player, 2-strategy symmetric games and obtain a complete set of quantum Nash equilibria. Apart from those available with the maximal entanglement, these quantum Nash equilibria are extensions of the Nash equilibria in classical game theory. The phase structure of the equilibria is determined for all values of entanglement, and thereby the possibility of resolving the dilemmas by entanglement in the game of Chicken, the Battle of the Sexes, the Prisoners' Dilemma, and the Stag Hunt, is examined. We find that entanglement transforms these dilemmas with each other but cannot resolve them, except in the Stag Hunt game where the dilemma can be alleviated to a certain degree.
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.
In a normal form, prisoners' dilemma (PD) is represented by a payoff matrix showing players' strategies and payoffs. To obtain the distinguishing trait and strategic form of PD, certain constraints are imposed on the elements of its payoff matrix. We quantize PD by a generalized quantization scheme to analyze its strategic behavior in the quantum domain. The game starts with a general entangled state of the form and the measurement for payoffs is performed in entangled and product bases. We show that for both measurements, there exist respective cutoff values of entanglement of the initial quantum state up to which the strategic form of the game remains intact. Beyond these cutoffs the quantized PD behaves like the chicken game (CG) up to another cutoff value. For the measurement in the entangled basis the dilemma is resolved for sinξ > 1/7 with Q ⊗ Q as a Nash Equilibrium (NE). However, the quantized game behaves like PD when sinξ > 1/3; whereas in the range 1/7 < sinξ < 1/3 it behaves like CG with Q ⊗ Q as an NE. For the measurement in the product basis the quantized PD behaves like classical PD for with D ⊗ D as an NE. In region , the quantized PD behaves like classical CG with C ⊗ D and D ⊗ C as NEs.
This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game GC has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game GQ with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γc such that for γγc there is a pure strategy Nash equilibrium and for γ≥γc there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.
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 dierent 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 eective 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.
We study quantum games with correlated noise through a generalized quantization scheme. We investigate the effects of memory on quantum games, such as Prisoner's Dilemma, Battle of the Sexes and Chicken, through three prototype quantum-correlated channels. It is shown that the quantum player enjoys an advantage over the classical player for all nine cases considered in this paper for the maximally entangled case. However, the quantum player can also outperform the classical player for subsequent cases that can be noted in the case of the Battle of the Sexes game. It can be seen that the Nash equilibria do not change for all the three games under the effect of memory.