Experimental implementation of a three qubit quantum game with corrupt source using nuclear magnetic resonance quantum information processor

In a three player quantum 'Dilemma' game each player takes independent decisions to maximize his/her individual gain. The optimal strategy in the quantum version of this game has a higher payoff compared to its classical counterpart. However, this advantage is lost if the initial qubits provided to the players are from a noisy source. We have experimentally implemented the three player quantum version of the 'Dilemma' game as described by Johnson, [N.F. Johnson, Phys. Rev. A 63 (2001) 020302(R)] using nuclear magnetic resonance quantum information processor and have experimentally verified that the payoff of the quantum game for various levels of corruption matches the theoretical payoff.     

Arbiter as the Third Man in Classical and Quantum Games

We study the possible influence of a not necessarily sincere arbiter on the course of classical and quantum 2×2 games and we show that this influence in the quantum case is much bigger than in the classical case. Extreme sensitivity of quantum games on initial states of quantum objects used as carriers of information in a game shows that a quantum game, contrary to a classical game, is not defined by a payoff matrix alone but also by an initial state of objects used to play a game. Therefore, two quantum games that have the same payoff matrices but begin with different initial states should be considered as different games.     

A competitive game whose maximal Nash-equilibrium payoff requires quantum resources for its achievement

While it is known that shared quantum entanglement can offer improved solutions to a number of purely cooperative tasks for groups of remote agents, controversy remains regarding the legitimacy of quantum games in a competitive setting. We construct a competitive game between four players based on the minority game where the maximal Nash-equilibrium payoff when played with the appropriate quantum resource is greater than that obtainable by classical means, assuming a local hidden variable model.     

Quantum Version of a Generalized Monty Hall Game and Its Possible Applications to Quantum Secure Communications

In this work, the authors propose a quantum version of a generalized Monty Hall game, that is, one in which the parameters of the game are left free and not fixed on its regular values. The developed quantum scheme is then used to study the expected payoff of the player, using both a separable and an entangled initial‐state. In the two cases, the classical mixed‐strategy payoff is recovered under certain conditions. Lastly, the authors extend the quantum scheme to include multiple independent players, and use this extension to sketch two possible application of the game mechanics to quantum networks, specifically, two validated, multi‐party, key‐distribution quantum protocols.     

Quantum Game with Restricted Matrix Strategies

We study a quantum game played by two players with restricted multiple strategies. It is found that in this restricted quantum game Nash equilibrium does not always exist when the initial state is entangled. At the same time,we find that when Nash equilibrium exists the payoff function is usually different from that in the classical counterpart except in some special cases. This presents an explicit example showing quantum game and classical game may differ.When designing a quantum game with limited strategies, the allowed strategy should be carefully chosen according to the type of initial state.     

Quantum Game with Restricted Matrix Strategies

We study a quantum game played by two players with restricted multiple strategies. It is found that in this restricted quantum game Nash equilibrium does not always exist when the initial state is entangled. At the same time,we find that when Nasli equilibrium exists the payoff function is usually different from that in the classical counterpart except in some special cases. This presents an explicit example showing quantum game and classical game may differ.When designing a quantum game with limited strategies, the allowed strategy should be carefully chosen according to the type of initial state.     

Constructing quantum games from symmetric non-factorizable joint probabilities

We construct quantum games from a table of non-factorizable joint probabilities, coupled with a symmetry constraint, requiring symmetrical payoffs between the players. We give the general result for a Nash equilibrium and payoff relations for a game based on non-factorizable joint probabilities, which embeds the classical game. We study a quantum version of Prisoners' Dilemma, Stag Hunt, and the Chicken game constructed from a given table of non-factorizable joint probabilities to find new outcomes in these games. We show that this approach provides a general framework for both classical and quantum games without recourse to the formalism of quantum mechanics.     

Quantum Prisoners’ Dilemma in Fluctuating Massless Scalar Field

Quantum systems are easily affected by external environment. In this paper, we investigate the influences of external massless scalar field to quantum Prisoners’ Dilemma (QPD) game. We firstly derive the master equation that describes the system evolution with initial maximally entangled state. Then, we discuss the effects of a fluctuating massless scalar field on the game’s properties such as payoff, Nash equilibrium, and symmetry. We find that for different game strategies, vacuum fluctuation has different effects on payoff. Nash equilibrium is broken but the symmetry of the game is not violated.     

Equivalence between Bell inequalities and quantum minority games

We show that, for a continuous set of entangled four-partite states, the task of maximizing the payoff in the symmetric-strategy four-player quantum Minority game is equivalent to maximizing the violation of a four-particle Bell inequality. We conclude the existence of direct correspondences between (i) the payoff rule and Bell inequalities, and (ii) the strategy and the choice of measured observables in evaluating these Bell inequalities. We also show that such a correspondence is unique to minority-like games.     

Two-Player 2 × 2 Quantum Game in Spin System

In this work, we study the payoffs of quantum Samaritan’s dilemma played with the thermal entangled state of XXZ spin model in the presence of Dzyaloshinskii-Moriya (DM) interaction. We discuss the effect of anisotropy parameter, strength of DM interaction and temperature on quantum Samaritan’s dilemma. It is shown that although increasing DM interaction and anisotropy parameter generate entanglement, players payoffs are not simply decided by entanglement and depend on other game components such as strategy and payoff measurement. In general, Entanglement and Alice’s payoff evolve to a relatively stable value with anisotropy parameter, and develop to a fixed value with DM interaction strength, while Bob’s payoff changes in the reverse direction. It is noted that the augment of Alice’s payoff compensates for the loss of Bob’s payoff. For different strategies, payoffs have different changes with temperature. Our results and discussions can be analogously generalized to other 2 × 2 quantum static games in various spin models.     

On subgame perfect equilibria in quantum Stackelberg duopoly with incomplete information

The Li–Du–Massar quantum duopoly model is one of the generally accepted quantum game schemes. It has applications in a wide range of duopoly problems. Our purpose is to study Stackelberg's duopoly with incomplete information in the quantum domain. The result of Lo and Kiang has shown that the correlation of players' quantities caused by the quantum entanglement enhances the first-mover advantage in the game. Our work demonstrates that there is no first-mover advantage if the players' actions are maximally correlated. Furthermore, we proved that the second mover gains a higher equilibrium payoff than the first one.     

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 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.     

Coexistence of cooperators and defectors in well mixed populations mediated by limiting resources

Traditionally, resource limitation in evolutionary game theory is assumed just to impose a constant population size. Here we show that resource limitations may generate dynamical payoffs able to alter an original prisoner's dilemma, and to allow for the stable coexistence between unconditional cooperators and defectors in well-mixed populations. This is a consequence of a self-organizing process that turns the interaction payoff matrix into evolutionary neutral, and represents a resource-based control mechanism preventing the spread of defectors. To our knowledge, this is the first example of coexistence in well-mixed populations with a game structure different from a snowdrift game.     

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 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.     

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.     

Adjusting learning motivation to promote cooperation

An evolutionary prisoner’s dilemma game with players adjusting their learning motivation is studied. At each time step, each player can adjust his/her learning motivation according to the difference between the current payoff and payoff aspiration. Greater payoff aspiration means stronger learning motivation, and vice versa. We find that the density of cooperation in a spatial prisoner’s dilemma game is enhanced when the learning motivation mechanism is considered. Meanwhile, we show that proper noise can not only induce the highest cooperation level but also can maintain the cooperation phenomenon even though there is more temptation to defect.     

基于鹰鸽量子博弈理论评价科研院所的科技自主创新能力

科研院所的科技自主创新能力是推动国家科技进步和经济发展,应对国际经济危机的主要动力,创建科学、完善的科技创新能力评价方法有助于提升科研院所科技创新能力,并为国家制定科技创新决策提供参考依据。本文基于鹰鸽量子博弈理论,提出了一种评价科研院所自主创新能力的方法。介绍了量子博弈论的各基本要素在科技自主创新体系中所对应的物理内涵,根据鹰鸽量子博弈理论建立了科技自主创新能力评价模型,分析了纠缠度与收益矩阵之间的关系,确立了依靠各参与者在鹰鸽量子博弈中的纠缠度来表征科技自主创新能力的方法。给出了科研院所科技自主创新能力的量子博弈论解释,构建了科技自主创新能力评价指标体系,并确定了评价的合成计算方法,即量子纠缠度的计算方法。最后,以中科院部分研究所为实例进行了科技自主创新能力的评价,并利用主成份分析法和中物院的简单统计方法对得到的数据进行了对比分析,结果证明了提出的方法合理且有可操作性。     

第四讲 量子对策论

量子对策论是量子信息学的新兴分支,是经典对策论与量子信息学两门学科的交叉学科.由 于引入了量子力学中的量子叠加性和纠缠态,量子对策得出了与经典对策迥然不同的结果.     

Evolutionarily stable strategies in quantum games

Evolutionarily stable strategy (ESS) in classical game theory is a refinement of Nash equilibrium concept. We investigate the consequences when a small group of mutants using quantum strategies try to invade a classical ESS in a population engaged in symmetric bimatrix game of prisoner's dilemma. Secondly we show that in an asymmetric quantum game between two players an ESS pair can be made to appear or disappear by resorting to entangled or unentangled initial states used to play the game even when the strategy pair remains a Nash equilibrium in both forms of the game.     

Mesoscopic approach to minority games in herd regime

We study minority games in efficient regime. By incorporating the utility function and aggregating agents with similar strategies we develop an effective mesoscale notion of the state of the game. Using this approach, the game can be represented as a Markov process with substantially reduced number of states with explicitly computable probabilities. For any payoff, the finiteness of the number of states is proved. Interesting features of an extensive random variable, called aggregated demand, viz. its strong inhomogeneity and presence of patterns in time, can be easily interpreted. Using Markov theory and quenched disorder approach, we can explain important macroscopic characteristics of the game: behavior of variance per capita and predictability of the aggregated demand. We prove that in the case of linear payoff many attractors in the state space are possible.