On Topics in Quantum Games

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.

The effect of quantum noise on the restricted quantum game

It has recently been established that quantum strategies have great advantage over classical ones in quantum games. However, quantum states are easily affected by the quantum noise resulting in decoherence. In this paper, we investigate the effect of quantum noise on the restricted quantum game in which one player is restricted in classical strategic space, another in quantum strategic space and only the quantum player is affected by the quantum noise. Our results show that in the maximally entangled state, no Nash equilibria exist in the range of 0 〈 p ≤ 0.422 （p is the quantum noise parameter）, while two special Nash equilibria appear in the range of 0.422 〈 p 〈 1. The advantage that the quantum player diminished only in the limit of maximum quantum noise. Increasing the amount of quantum noise leads to the increase of the classical player＇s payoff and the reduction of the quantum player＇s payoff, but is helpful in forming two Nash equilibria.

Linear optics implementation for quantum game under quantum noise

It has recently been shown that linear optics alone would suffice to implement efficient quantum computation. Quantum computation circuits using coherent states as the logical qubits can be constructed from very simple linear networks, conditional measurements and coherent superposition resource states. We present the quantum game under quantum noise and a proposal for implementing the noisy quantum game using only linear optics.

Theoretical analysis of quantum game in cavity QED

Recent years, several ways of implementing quantum games in different physical systems have been presented. In this paper, we perform a theoretical analysis of an experimentally feasible way to implement a two player quantum game in cavity quantum electrodynamic（QED）. In the scheme, the atoms interact simultaneously with a highly detuned cavity mode with the assistance of a classical field. So the scheme is insensitive to the influence from the cavity decay and the thermal field, and it does not require the cavity to remain in the vacuum state throughout the procedure.

Ensemble feature selection integrating elitist roles and quantum game model

To accelerate the selection process of feature subsets in the rough set theory （RST）, an ensemble elitist roles based quantum game （EERQG） algorithm is proposed for feature selec- tion. Firstly, the multilevel elitist roles based dynamics equilibrium strategy is established, and both immigration and emigration of elitists are able to be self-adaptive to balance between exploration and exploitation for feature selection. Secondly, the utility matrix of trust margins is introduced to the model of multilevel elitist roles to enhance various elitist roles＇ performance of searching the optimal feature subsets, and the win-win utility solutions for feature selec- tion can be attained. Meanwhile, a novel ensemble quantum game strategy is designed as an intriguing exhibiting structure to perfect the dynamics equilibrium of multilevel elitist roles. Finally, the en- semble manner of multilevel elitist roles is employed to achieve the global minimal feature subset, which will greatly improve the fea- sibility and effectiveness. Experiment results show the proposed EERQG algorithm has superiority compared to the existing feature selection algorithms.

Scheme for Implementation of Quantum Game in Cavity QED

We propose an experimentally feasible scheme to implement two-player quantum game in cavity quantum electrodynamics （QED）. During the process, the cavity is only virtually excited, thus our scheme is insensitive to the cavity field states and cavity decay. The scheme can be realized in the range of current cavity QED techniques.

A scheme for implementing quantum game in cavity QED

In this paper, we propose a scheme for implementing quantum game （QG） in cavity quantum electrodynamics（QED）. In the scheme, the cavity is only virtually excited and thus the proposal is insensitive to the cavity fields states and cavity decay. So our proposal can be experimentally realized in the range of current cavity QED techniques.

The effect of quantum noise on multiplayer quantum game

It has recently been realized that quantum strategies have a great advantage over classical ones in quantum games. However, quantum states are easily affected by the quantum noise, resulting in decoherence. In this paper, we investigate the effect of quantum noise on a multiplayer quantum game with a certain strategic space, with all players affected by the same quantum noise at the same time. Our results show that in a maximally entangled state, a special Nash equilibrium appears in the range of 0≤p≤0.622 （p is the quantum noise parameter）, and then disappears in the range of 0.622 〈 p≤ 1. Increasing the amount of quantum noise leads to the reduction of the quantum player＇s payoff.

Proposal of Realization Restricted Quantum Game with Linear Optic Method

We present a quantum game with the restricted strategic space and its realization with linear optical system, which can be played by two players who are separated remotely. This game can also be realized on any other quantum computers. We find that the constraint brings some interesting properties that are useful for making game models.

Influence of Constraint in Parameter Space on Quantum Games

We study the influence of the constraint in the parameter space on quantum games.Decomposing SU(2)operator into product of three rotation operators and controlling one kind of them,we impose a constraint on the parameter space of the players' operator.We find that the constraint can provide a tuner to make the bilateral payoffs equal,so that the mismatch of the players' action at multi-equilibrium could be avoided.We also find that the game exhibits an intriguing structure as a function of the parameter of the controlled operators,which is useful for making game models.

Decoherence Effects on Multiplayer Cooperative Quantum Games

We study the behavior of cooperative multiplayer quantum games [Q. Chen, Y. Wang, J.T. Liu, and K.L. Wang, Phys. Lett. A 327 （2004） 98; A.P. Flitney and L.C.L. Hollenberg, Quantum Inf. Comput. 7 （2007） 111] in the presence of decoherence using different quantum channels such as amplitude damping, depolarizing and phase damping. It is seen that the outcomes of the games for the two damping channels with maximum values of decoherence reduce to same value. However, in comparison to phase damping channel, the payoffs of cooperators are strongly damped under the influence amplitude damping channel for the lower values of decoherence parameter. In the case of depolarizing channel, the game is a no-payoff game irrespective of the degree of entanglement in the initial state for the larger values of decoherence parameter. The deeoherenee gets the cooperators worse off.

On fairness,full cooperation,and quantum game with incomplete information

Quantum entanglement has emerged as a new resource to enhance cooperation and remove dilemmas.This paper aims to explore conditions under which full cooperation is achievable even when the information of payoff is incomplete.Based on the quantum version of the extended classical cash in a hat game,we demonstrate that quantum entanglement may be used for achieving full cooperation or avoiding moral hazards with the reasonable profit distribution policies even when the profit is uncertain to a certain degree.This research further suggests that the fairness of profit distribution should play an important role in promoting full cooperation.It is hopeful that quantum entanglement and fairness will promote full cooperation among distant people from various interest groups when quantum networks and quantum entanglement are accessible to the public.

Generalized Quantum Games with Nash Equilibrium

We define generalized quantum games by introducing the coherent payoff operators and propose a simple scheme to illustrate it.The scheme is implemented with a single spin qubit system and a two-entangled-qubit system.The Nash Equilibrium Theorem is proved for the models.

Best Response Analysis in Two Person Quantum Games

In this paper, we find particular use for a maximally entangled initial state that produces a quantized version of two player two strategy games. When applied to a variant of the well-known game of Chicken, our construction shows the existence of new Nash equilibria with the players receiving better payoffs than those found in literature.

Constructing Entanglers in 2-Players–N-Strategies Quantum Game

In quantum games based on 2-player-N-strategies classical games, each player has a quNit (a normalized vector in an N-dimensional Hilbert space HN) upon which he applies his strategy (a matrix U∈SU(N)). The players draw their payoffs from a state . Here ?and J (both determined by the game’s referee) are respectively an unentangled 2-quNit (pure) state and a unitary operator such that ?is partially entangled. The existence of pure strategy Nash equilibrium in the quantum game is intimately related to the degree of entanglement of . Hence, it is practical to design the entangler J= J(β) to be dependent on a single real parameter β that controls the degree of entanglement of , such that its von-Neumann entropy SN(β) is continuous and obtains any value in . Designing J(β) for N=2 is quite standard. Extension to N>2 is not obvious, and here we suggest an algorithm to achieve it. Such construction provides a special quantum gate that should be a useful tool not only in quantum games but, more generally, as a special gate in manipulating quantum information protocols.

The impact of honesty and trickery on a Bayesian quantum prisoners’ dilemma game

To explore the influence of quantum information on the common social problem of honesty and trickery,we propose a Bayesian model for the quantum prisoners’dilemma game.In this model,the players’strategy formation is regarded as a negotiation of their move contract based on their types of decision policies,honesty or trickery.Although the implementation of quantum information cannot eliminate tricky players,players in our model can always end up with higher payoffs than in the classical game.For a good proportion of a credibility parameter value,a rational player will take an honest action,which is in remarkable contrast to the observation that players tend to defect in the classical prisoners’dilemma game.This research suggests that honesty will be promoted to enhance cooperation with the assistance of quantum information resources.

Quantum Stackelberg Duopoly Game in Depolarizing Channel

In this paper, we investigate the quantum Stackelberg duopoly （QSD） game in the noise environment with the depolarizing channel expressed by the Kraus-operator representation. It is found that the presence of the damping in the depolarizing channel always leads to the decrease of the quantities of the moves and payoffs of the two players in the QSD game. It is indicated that under certain conditions the first-mover advantage in the QSD game can be weakened due to the presence of the damping in the depolarizing channel.

量子博弈——“PQ”问题

研究了量子博弈中的“PQ”问题,重点探讨了参与者策略制定、收益分配和量子效应对博弈结果的影响.首先回顾了量子博弈理论的发展,并对经典博弈与量子博弈进行了对比,详细定义了参与者和策略空间,并利用收益矩阵和量子操作符进行了建模.为解决“PQ”问题,提出了量子策略选择和优化算法,并设计了针对该问题的量子算法.在实际应用和案例研究方面,通过评述经典双人博弈——“PQ翻硬币”问题,表明当其中一个参与人采用量子策略时,他可以打败他的经典对手,获得更高的收益.通过量化手段,把整个“PQ”问题放在一个更为普遍和公平的条件下,对其推广并从多个方面和角度再次对其进行研究.同时,给出了“不完全公平博弈”和“完全公平博弈”的定义,对量子硬币翻转博弈(Meyer 1999 Phys.Rev.lett.821052)进行了修改,使博弈公平,并研究了量子硬币翻转的多轮版本.最后对“PQ”问题的局限性与扩展性进行了讨论,并展望了量子博弈研究的未来发展方向.

Reverse Engineering, Some Board Games as a Key to the Development of Artificial Intelligence and Construction of Economic Models

Reverse engineering, also called back engineering, is the process of extracting knowledge or designing information from anything： mechanical device electronic components, computer programs, or biological, chemical matters, and--subsequently--analyzing its components. Reverse engineering can be applied for the sake of creating artificial intelligence, e.g. when the used methods are based on very old games, such as Go and chess. Many generations of computers are able to play at the same level as human grandmasters. A computers＇ arrival at that level is due to imitation of human Go or chess play. A particular emphasis shall be placed upon the Go game, known for 5,500 years. Invented in China, it may be classified as the oldest board game, having its ardent enthusiasts until present times. Old physical issues can be easily projected upon the modeling of new economic phenomena and quantum games. The aim of this paper is to discuss the applications of reverse engineering, traditional social games and other domains, such as quantum physics, to the analysis of utterly new social as well as economic phenomena.

“以邻为壑”如何迈向“同舟共济”——基于量子博弈的区域大气污染府际间协同治理研究

区域大气污染协同治理往往存在搭便车现象,针对如何有效解决这一问题,本文借助量子博弈理论构建了双异质政府的量子博弈模型,对是否考虑态的纠缠分别讨论,并通过策略拓展将博弈结果锁定于量子版的双方合作策略,最后基于量子策略和量子纠缠的解读,对府际协同的“纠缠协议”做了明确规定。研究结果表明:纠缠态和量子策略可以有效解决政府“搭便车”行为,因为量子纠缠和量子策略通过利益关联和指标量化,避免了“道德风险”问题,强化了政府双方履行自身职责的约束力。根据研究结论,本文提出推动地方政府数字化转型,建立空气质量信息共享平台;完善大气污染协同治理绩效评估及问责机制和培养区域认同感、建立府际信任三点管理启示,为我国推进区域大气污染协同治理提供了决策参考。