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 ph...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 G<sub>C</sub> has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game G<sub>Q</sub> 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 0γ<sub>c</sub> such that for γγ<sub>c</sub> there is a pure strategy Nash equilibrium and for γ≥γ<sub>c </sub>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.展开更多
在实际分类决策中,序贯三支决策模型为决策者提供了一个渐进式的决策方法.然而,现有序贯三支决策模型的研究从提高分类精度或减少不确定性的动机来求取每一粒层的决策阈值,缺乏对二者的综合考虑.为了解决这个问题,本文结合博弈论的思想...在实际分类决策中,序贯三支决策模型为决策者提供了一个渐进式的决策方法.然而,现有序贯三支决策模型的研究从提高分类精度或减少不确定性的动机来求取每一粒层的决策阈值,缺乏对二者的综合考虑.为了解决这个问题,本文结合博弈论的思想构建了基于错误分类率与边界域不确定性博弈的序贯三支决策模型.首先,分析了序贯三支决策模型中边界域不确定性与决策区域错误分类率的变化关系并构建了二者之间的博弈;其次,从博弈终止的条件出发,基于纯策略纳什均衡原理,提出了求取每一粒层自适应决策阈值的优化模型;再次,为了比较不同模型的效果,从多目标决策的角度,设计了基于TOPSIS(Technique for Order Preference by Similarity to an Ideal Solution)的阈值选取方法;最后,通过UCI数据集进行了两种模型的对比实验.实验结果表明:基于博弈论的序贯三支决策模型求取的决策阈值具有更小的错误分类率以及更合理的阈值结构.展开更多
文摘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 G<sub>C</sub> has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game G<sub>Q</sub> 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 0γ<sub>c</sub> such that for γγ<sub>c</sub> there is a pure strategy Nash equilibrium and for γ≥γ<sub>c </sub>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.
文摘在实际分类决策中,序贯三支决策模型为决策者提供了一个渐进式的决策方法.然而,现有序贯三支决策模型的研究从提高分类精度或减少不确定性的动机来求取每一粒层的决策阈值,缺乏对二者的综合考虑.为了解决这个问题,本文结合博弈论的思想构建了基于错误分类率与边界域不确定性博弈的序贯三支决策模型.首先,分析了序贯三支决策模型中边界域不确定性与决策区域错误分类率的变化关系并构建了二者之间的博弈;其次,从博弈终止的条件出发,基于纯策略纳什均衡原理,提出了求取每一粒层自适应决策阈值的优化模型;再次,为了比较不同模型的效果,从多目标决策的角度,设计了基于TOPSIS(Technique for Order Preference by Similarity to an Ideal Solution)的阈值选取方法;最后,通过UCI数据集进行了两种模型的对比实验.实验结果表明:基于博弈论的序贯三支决策模型求取的决策阈值具有更小的错误分类率以及更合理的阈值结构.
