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arXiv:1307.5508

Werner-like States and Strategic Form of Quantum Games

Ahmad Ahsan Nawaz

Published Jul 21, 2013• 8 citations
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Abstract

We quantize prisoners dilemma, chicken game and battle of sexes to explore the effect of quantization on their strategic form. The games start with Werner-like state as an initial state. We show that for the measurement in entangled basis the strategic forms of these games remain unaffected by quantization. On the other hand when measurement is performed in product basis then these games could not retain their strategic forms.

DOI
10.48550/arxiv.1307.5508

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We quantize prisoners dilemma, chicken game and battle of sexes to explore the effect of quantization on their strategic form. The games start with Werner-like state as an initial state. We show that for the measurement in entangled basis the strategic forms of these games remain unaffected by quantization. On the other hand when measurement is performed in product basis then these games could not retain their strategic forms.
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.
We quantize prisoners dilemma and chicken game by our generalized quantization scheme to explore the role of quantum discord in quantum games. In order to establish this connection we use Werner-like state as an initial state of the game. In this quantization scheme measurement can be performed in entangled as well as in product basis. For the measurement in entangled basis the dilemma in both the games can be resolved by separable states with non-zero quantum discord. Similarly for product basis measurement the payoffs are quantum mechanical only for nonzero values of quantum discord.
In its normal form prisoners' dilemma (PD) is represented by a payoff matrix showing players strategies and payoffs. To obtain distinguishing trait and strategic form of PD certain constraints are imposed on the elements of its payoff matrix. We quantize PD by generalized quantization scheme to analyze its strategic behavior in quantum domain. The game starts with general entangled state of the form $\left}ψ\right\rangle =\cos\fracξ% {2}\left|00\right\rangle +i\sin\fracξ{2}\left|11\right\rangle $ 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 initial quantum state up to which strategic form of game remains intact. Beyond these cutoffs the quantized PD behaves like chicken game up to another cutoff value. For the measurement in entangled basis the dilemma is resolved for\ sin⁡ξ>17\sinξ>\frac{1}{7}sinξ>71​ with Q⊗QQ\otimes QQ⊗Q as a NE but the quantized game behaves like PD when sin⁡ξ>13\sinξ>\frac{1}{3}sinξ>31​; whereas in the range $\frac{1}{7}
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.
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 quantization scheme for a three-player Prisoner's Dilemma game. It is shown that entanglement plays a dominant role in the three-player quantum game. Four different types of payoffs are identified on the basis of different combinations of initial state and measurement basis entanglement parameters. A relation among these different payoffs is also established. We also study the communication aspects of the three-player game. By exploiting different combinations of initial state and measurement basis entanglement parameters, we establish a relationship for the information shared among the parties. It is seen that the strategies of the players act as carriers of information in quantum games.