Research On Influencing Factors And Derivation Algorithm For Nash Equilibrium In Quantum Games

With the development of quantum mechanics and quantum technology,quantum information has gradually become one of the front directions in scientific researches.Quantum game theory is the combination of the classical game theory and quantum information,which receives more and more scholars' attentions.Based on quantum entanglement characteristic of quantum mechanics and quantization schemes,this thesis discusses the impact of several elements for Nash equilibrium in quantum games.Furthermore,we not only build an algorithm to obtain pure strategy Nash equilibrium for complete information static games,but also design programs in MATLAB.The main content of this thesis is presented as follows:(1)We first explore the impact of distributional fairness degree and quantum entanglement on finding Nash equilibrium between different players by investigating quantization of a modified prisoner's dilemma game.Firstly,we introduce a new concept of distributional fairness degree and generalize the classical prisoner's dilemma.Afterward,based on Eisert quantization scheme,we obtain the Nash equilibrium conditions for overcoming dilemma in terms of fairness and entanglement inequalities in the quantum prisoner's dilemma.It has been demonstrated that distributional fairness can be of fundamental importance to promote cooperation with the help of quantum entanglement.Finally,it is discussed that the relationship between entanglement and distributional fairness depends on the payoff distribution by way of plotting method.(2)Furthermore,based on semi-tensor product,we introduce an algorithm for verifying the existence of pure strategy Nash equilibrium for complete information static games.By establishing the structural matrix,the payoff function of each player can be expressed into algebraic form.We discussed the necessary and sufficient conditions for the pure strategy Nash equilibrium.Moreover,we make some programs to automatically get Nash equilibrium by means of MATLAB software.Finally,we apply the algorithm to two-player finite strategy quantum games,two-player infinite strategy quantum games and multi-player finite strategy quantum games respectively.