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
This paper studies the problem of coordinated motion generation for a group of rigid bodies. Two classes of coordinated motion primitives, relative equilibria and ma- neuvers, are given as building blocks for generati...This paper studies the problem of coordinated motion generation for a group of rigid bodies. Two classes of coordinated motion primitives, relative equilibria and ma- neuvers, are given as building blocks for generating coordi- nated motions. In a motion-primitive based planning frame- work, a control method is proposed for the robust execution of a coordinated motion plan in the presence of perturba- tions. The control method combines the relative equilibria stabilization with maneuver design, and results in a close- loop motion planning framework. The performance of the control method has been illustrated through a numerical sim- ulation.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.
基金supported by the National Natural Science Foundation of China (11072002,10832006)
文摘This paper studies the problem of coordinated motion generation for a group of rigid bodies. Two classes of coordinated motion primitives, relative equilibria and ma- neuvers, are given as building blocks for generating coordi- nated motions. In a motion-primitive based planning frame- work, a control method is proposed for the robust execution of a coordinated motion plan in the presence of perturba- tions. The control method combines the relative equilibria stabilization with maneuver design, and results in a close- loop motion planning framework. The performance of the control method has been illustrated through a numerical sim- ulation.
