Parrondo Games as Lattice Gas Automata

Parrondo games are coin flipping games with the surprising property that alternating plays of two losing games can produce a winning game. We show that this phenomenon can be modelled by probabilistic lattice gas automata. Furthermore, motivated by the recent introduction of quantum coin flipping games, we show that quantum lattice gas automata provide an interesting definition for quantum Parrondo games.     

Maximal Parrondo’s Paradox for Classical and Quantum Markov Chains

Parrondo’s paradox refers to the situation where two, multi-round games with a fixed winning criteria, both with probability greater than one-half for one player to win, are combined. Using a possibly biased coin to determine the rule to employ for each round, paradoxically, the previously losing player now wins the combined game with probability greater than one-half. In this paper, we will analyze classical observed, classical hidden, and quantum versions of a game that displays this paradox. The game we have utilized is simpler than games for which this behavior has been previously noted in the classical and quantum cases. We will show that in certain situations the paradox can occur to a greater degree in the quantum version than is possible in the classical versions.     

Parrondo?s game using a discrete-time quantum walk

We present a new form of a Parrondo game using discrete-time quantum walk on a line. The two players A and B with different quantum coins operators, individually losing the game can develop a strategy to emerge as joint winners by using their coins alternatively, or in combination for each step of the quantum walk evolution. We also present a strategy for a player A (B) to have a winning probability more than player B (A). Significance of the game strategy in information theory and physical applications are also discussed.     

Properly quantized history-dependent Parrondo games, Markov processes, and multiplexing circuits

In the context of quantum information theory, “quantization” of various mathematical and computational constructions is said to occur upon the replacement, at various points in the construction, of the classical randomization notion of probability distribution with higher order randomization notions from quantum mechanics such as quantum superposition with measurement. For this to be done “properly”, a faithful copy of the original construction is required to exist within the new quantum one, just as is required when a function is extended to a larger domain. Here procedures for extending history-dependent Parrondo games, Markov processes and multiplexing circuits to their quantum versions are analyzed from a game theoretic viewpoint, and from this viewpoint, proper quantizations developed.     

Parity effect of the initial capital based on Parrondo’s games and the quantum interpretation

In our previous study [Zhu et al., Quantum game interpretation for a special case of Parrondo’s paradox, Physica A 390 (2011) 579], the capital-dependent Parrondo’s game where one game depends on the capital modulus $M=4$ was shown not to have a definite stationary probability distribution and that payoffs of the game depended on the parity of the initial capital. This paper presents a generalization of these results to even $M$ greater than 4. An intuitive explanation for producing this phenomenon is that the discrete-time Markov chain of the game is divided into two completely unrelated inner and outer rings. The process taking the inner ring or outer ring of the game is determined by the initial capital of parity and then a win or loss of the game is determined. Quantum game theory is used to further analyze the phenomenon. The results show that the explanation of the game corresponding to a stationary probability distribution is that the probability of the initial capital has reached parity.     

Collective decision making and paradoxical games

We study an ensemble of individuals playing the two games of the so-called Parrondo paradox. In our study, players are allowed to choose the game to be played by the whole ensemble in each turn. The choice cannot conform to the preferences of all the players and, consequently, they face a simple frustration phenomenon that requires some strategy to make a collective decision. We consider several such strategies and analyze how fluctuations can be used to improve the performance of the system.     

Reversals of chance in paradoxical games

We present two collective games with new paradoxical features when they are combined. Besides reproducing the so-called Parrondo effect, where a winning game is obtained from the alternation of two fair games, there also exists a current inversion when varying the mixing probability between the games. We show that this is a new effect insofar one of the games is an unbiased random walk without internal structure. We present a detailed study by means of a discrete-time Markov chain analysis, obtaining analytical expressions for the stationary probabilities for a finite number of players. We also provide qualitative insight into this current inversion effect.     

Illusion of control in a Brownian game

Both single-player Parrondo games (SPPG) and multi-player Parrondo games (MPPG) display the Parrondo effect (PE) wherein two or more individually fair (or losing) games yield a net winning outcome if alternated periodically or randomly. (There is a more formal, less restrictive definition of the PE.) We illustrate that, when subject to an elementary optimization rule, the PG displays degraded rather than enhanced returns. Optimization provides only the illusion of control, when low-entropy strategies (i.e., which use more information) under-perform random strategies (with maximal entropy). This illusion is unfortunately widespread in many human attempts to manage or predict complex systems. For the PG, the illusion is especially striking in that the optimization rule reverses an already paradoxical-seeming positive gain—the Parrondo effect proper—and turns it negative. While this phenomenon has been previously demonstrated using somewhat artificial conditions in the MPPG [L. Dinis, J.M.R. Parrondo, Europhys. Lett. 63 (2003) 319; J.M.R. Parrondo, L. Dinis, J. Buceta, K. Lindenberg, Advances in Condensed Matter and Statistical Mechanics, E. Korutcheva, R. Cuerno (Eds.), Nova Science Publishers, New York, 2003], we demonstrate it in the natural setting of a history-dependent SPPG.     

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.     

A quantum extension to inspection game

Quantum game theory is a new interdisciplinary field between game theory and system engineering research. In this paper, we extend the classical inspection game into a quantum game version by quantizing the strategy space and importing entanglement between players. Our results show that the quantum inspection game has various Nash equilibria depending on the initial quantum state of the game. It is also shown that quantization can respectively help each player to increase his own payoff, yet fails to bring Pareto improvement for the collective payoff in the quantum inspection game.     

Preferences in quantum games

A quantum game can be viewed as a state preparation in which the final output state results from the competing preferences of the players over the set of possible output states that can be produced. It is therefore possible to view state preparation in general as being the output of some appropriately chosen (notional) quantum game. This reverse engineering approach in which we seek to construct a suitable notional game that produces some desired output state as its equilibrium state may lead to different methodologies and insights. With this goal in mind we examine the notion of preference in quantum games since if we are interested in the production of a particular equilibrium output state, it is the competing preferences of the players that determine this equilibrium state. We show that preferences on output states can be viewed in certain cases as being induced by measurement with an appropriate set of numerical weightings, or payoffs, attached to the results of that measurement. In particular we show that a distance-based preference measure on the output states is equivalent to a having a strictly-competitive set of payoffs on the results of some measurement.     

From Quantum State Targeting to Bell Inequalities

Quantum state targeting is a quantum game which results from combining traditional quantum state estimation with additional classical information. We consider a particular version of the game and show how it can be played with maximally entangled states. The optimal solution of the game is used to derive a Bell inequality for two entangled qutrits. We argue that the nice properties of the inequality are direct consequences of the method of construction.     

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.     

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.     

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.     

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.     

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。     

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.     

Two Party Non-Local Games

In this work we have introduced two party games with respective winning conditions. One cannot win these games deterministically in the classical world if they are not allowed to communicate at any stage of the game. Interestingly we find out that in quantum world, these winning conditions can be achieved if the players share an entangled state. We also introduced a game which is impossible to win if the players are not allowed to communicate in classical world (both probabilistically and deterministically), yet there exists a perfect quantum strategy by following which, one can attain the winning condition of the game.     

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.