Adrian P. Flitney, Derek Abbott
The application of the methods of quantum mechanics to game theory provides us with the ability to achieve results not otherwise possible. Both linear superpositions of actions and entanglement between the players' moves can be exploited. We provide an introduction to quantum game theory and review the current status of the subject.
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A framework for discussing relationships between different types of games is proposed. Within the framework, quantum simultaneous games, finite quantum simultaneous games, quantum sequential games, and finite quantum sequential games are defined. In addition, a notion of equivalence between two games is defined. Finally, the following three theorems are shown: (1) For any quantum simultaneous game G, there exists a quantum sequential game equivalent to G. (2) For any finite quantum simultaneous game G, there exists a finite quantum sequential game equivalent to G. (3) For any finite quantum se…
Playing Bayesian games better with separable quantum states than with any classical correlation
研究贝叶斯游戏中关联对玩家的影响，发现可分离量子态能创造优于经典关联均衡的新量子均衡，证明非经典关联是资源，让游戏中的量子优势更易实现。
We consider a coalitional game with the same payoff for all players. To maximize the payoff, the players need to use one collective strategy, if all players are in certain states, and the other strategy otherwise. The current state of each player changes according to external conditions and is not known to the other players. In one example of such a game, quantum entanglement between players results in the optimal payoff thrice the maximal payoff for unentangled players.
The noncooperative Nash equilibrium solution of classical games corresponds to a rational expectations attitude on the part of the players. However, in many cases, games played by human players have outcomes very different from Nash equilibria. A restricted version of quantum games is proposed to implement, in mathematical games, the interplay of self-interest and internalized social norms that rules human behavior.
An example of the macroscopic game of two partners consisting of two classical games played simultaneously with special dependence of strategies is considered. The average profit of each partner is equal to the average profit obtained in the quantum game with two noncommuting operators for the spin one half system with strategies defined by the wave function. Nash equilibria in the macroscopic game coincide with Nash equilibria of the quantum game.
We study possible influence of not necessarily sincere arbiter on the course of classical and quantum 2x2 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 consi…
Several variants of nonlocal games have been considered in the study of quantum entanglement and nonlocality. This paper concerns two of these variants, called quantum-classical games and extended nonlocal games. We give a construction of an extended nonlocal game from any quantum-classical game that allows one to translate certain facts concerning quantum- classical games to extended nonlocal games. In particular, based on work of Regev and Vidick, we conclude that there exist extended nonlocal games for which no finite-dimensional entangled strategy can be optimal. While this conclusion is a…
In this paper we show the connection between the supersymmetry and quantum games.
Quantumizing Classical Games: An Introduction to Quantum Game Theory
本文回顾量子博弈论的开创性工作，提出将任意有限经典n人博弈量子化的协议，采用可对易收益算子描述量子博弈，分析量子化带来的额外纳什均衡及帕累托最优性等优势。
A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question, is to understand the conditions under which a classical game-theoretic setting can be transformed to a quantum game, and under which conditions there is an equivalence. In this paper, we consider quantum games as those that allow non-factorizable probabilities. We discuss two approaches for obtaining a non-factorizable game and study the o…
A number of recent studies have focused on novel features in game theory when the games are played using quantum mechanical toolbox (entanglement, unitary operators, measurement). Researchers have concentrated in two-player-two strategy, 2x2, dilemma containing classical games, and transferred them into quantum realm showing that in quantum pure strategies dilemmas in such games can be resolved if entanglement is distributed between the players armed with quantum operations. Moreover, it became clear that the players receive the highest sum of payoffs available in the game, which are otherwise…
Quantum Game Jam -- Making Games with Quantum Physicists
本文介绍了2014-2019年举办五届的Quantum Game Jam跨学科游戏创作活动，评估其组织模式与原型效用，证实其能促进协作学习并产出可后续开发的游戏原型。
Quantum pseudo-telepathy games are good examples of explaining the strangeness of quantum mechanics and demonstrating the advantage of quantum resources over classical resources. Most of the quantum pseudo-telepathy games are common interest games, nevertheless conflicting interest games are more widely used to model real world situations. Recently Pappa et al. (Phys. Rev. Lett. 114, 020401, 2015) proposed the first two-party conflicting interest game where quantum advice enhances social optimality. In the present paper we give two new three-party conflicting interest games and show that quant…
We pursue a general theory of quantum games. We show that quantum games are more efficient than classical games, and provide a saturated upper bound for this efficiency. We demonstrate that the set of finite classical games is a strict subset of the set of finite quantum games. We also deduce the quantum version of the Minimax Theorem and the Nash Equilibrium Theorem.
Conway's classic game of life is a two-dimensional cellular automaton in which each cell, either alive or dead, evolves according to rules based on its local environment. The semi-quantum game of life (SQGOL) is an adaptation in which each cell is in a superposed state of both dead and alive and evolves according to modified rules. Computer simulation of the SQGOL reveals remarkable complexity and previously unseen game behaviors. Systems evolve to a "quantum cloud" with a liveness distribution of mean $<\!a\!>=0.3480\pm0.0001$ and standard deviation $σ=0.0071$ which is dependent solely on the…
We present a perspective on quantum games that focuses on the physical aspects of the quantities that are used to implement a game. If a game is to be played, it has to be played with objects and actions that have some physical existence. We call such games playable. By focusing on the notion of playability for games we can more clearly see the distinction between classical and quantum games and tackle the thorny issue of what it means to quantize a game. The approach we take can more properly be thought of as gaming the quantum rather than quantizing a game and we find that in this perspectiv…
Quantum games have proposed a new point of view for the solution of the classical problems and dilemmas in game theory. Certain quantization relationships can be proposed with the objective that a game can be generalized into a quantum domain where the linear superposition of actions is allowed. This quantization let us describe and solution problems originated by conflicting or cooperative behaviors among the members of a system from the point of view of quantum mechanical interactions. This leads us to interesting aspects which only can be observed through the quantization of a game like the…
Coordination and cooperation are among the most important issues of game theory. Recently, the attention turned to game theory on graphs and social networks. Encouraged by interesting results obtained in quantum evolutionary game analysis, we study cooperative Parrondo's games in a quantum setup. The game is modeled using multidimensional quantum random walks with biased coins. We use the GHZ and W entangled states as the initial state of the coins. Our analysis shows than an apparent paradox in cooperative quantum games and some interesting phenomena can be observed.
We quantize prisoners dilemma and chicken game by our generalized quantization scheme to explore the role of quantum discord in quantum games. In order to establish this connection we use Werner-like state as an initial state of the game. In this quantization scheme measurement can be performed in entangled as well as in product basis. For the measurement in entangled basis the dilemma in both the games can be resolved by separable states with non-zero quantum discord. Similarly for product basis measurement the payoffs are quantum mechanical only for nonzero values of quantum discord.
We pursue the possible connections between classical games and quantum computation. The Parrondo game is one in which a random combination of two losing games produces a winning game. We introduce novel realizations of this Parrondo effect in which the player can `win' via random reflections and rotations of the state-vector, and connect these to known quantum algorithms.
Quantum Extensive Form Games
本文提出量子展开型博弈概念，利用量子跃迁实现博弈树路径的成对湮灭，并将其用于量子化Conway的Angel问题，使已解决的经典博弈产生非平凡的量子版本。
A quantum logic gate of particular interest to both electrical engineers and game theorists is the quantum multiplexer. This shared interest is due to the facts that an arbitrary quantum logic gate may be expressed, up to arbitrary accuracy, via a circuit consisting entirely of variations of the quantum multiplexer, and that certain one player games, the history dependent Parrondo games, can be quantized as games via a particular variation of the quantum multiplexer. However, to date all such quantizations have lacked a certain fundamental game theoretic property. The main result in this disse…
We use the example of playing a 2-player game with entangled quantum objects to investigate the effect of quantum correlation. We find that for simple game scenarios it is classical correlation that is the central feature and that these simple quantum games are not sensitive to the quantum part of the correlation. In these games played with quantum objects it is possible to transform a game such as Prisoner's Dilemma into the game of Chicken. We show that this behaviour, and the associated enhanced equilibrium payoff over playing the game with quantum objects in non-entangled states, is entire…
A quantum version of the Matching Pennies (MP) game is proposed that is played using an Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) setting. We construct the quantum game without using the state vectors, while considering only the quantum mechanical joint probabilities relevant to the EPR-Bohm setting. We embed the classical game within the quantum game such that the classical MP game results when the quantum mechanical joint probabilities become factorizable. We report new Nash equilibria in the quantum MP game that emerge when the quantum mechanical joint probabilities maximally violate the Clau…
Categorical quantum mechanics, which examines quantum theory via dagger-compact closed categories, gives satisfying high-level explanations to the quantum information procedures such as Bell-type entanglement or complementary observables (\cite{AC}, \cite{Co}, \cite{Co2}). Inspired by the fact that Quantum Game Theory can be seen as branch of quantum information, we express Quantum Game Theory procedures using the topological semantics provided by Categorical Quantum Mechanics. We also investigate Bayesian Games with correlation from this novel point of view while considering the connection be…
The physical world obeys the rules of quantum, as opposed to classical, physics. Since the playing of any particular game requires physical resources, the question arises as to how Game Theory itself would change if it were extended into the quantum domain. Here we provide a general formalism for {\em quantum} games, and illustrate the explicit application of this new formalism to a quantized version of the well-known prisoner's dilemma game.
This paper has been withdrawn by the author(s), due to some technical problem.
In this work we propose and develop modified quantum games (zero and non-zero sum) in which payoffs and strategies are entangled. For the games studied, Nash and Pareto equilibriums are always obtained indicating that there are some interesting cases where quantum games can be applied.
Quantum guessing games form a versatile framework for studying different tasks of information processing. A quantum guessing game with posterior information uses quantum systems to encode messages and classical communication to give partial information after a quantum measurement has been performed. We present a general framework for quantum guessing games with posterior information and derive structure and reduction theorems that enable to analyze any such game. We formalize symmetry of guessing games and characterize the optimal measurements in cases where the symmetry is related to an irred…
The game of Prisoner Dilemma is analyzed to study the role of measurement basis in quantum games. Four different types of payoffs for quantum games are identified on the basis of different combinations of initial state and measurement basis. A relation among these different payoffs is established.
Methods of exploring Nash equilibrium in quantum games are studied. Analytical conditions of the existence, the uniqueness or the multiplicity of the equilibria are found.
Recent development in quantum computation and quantum information theory allows to extend the scope of game theory for the quantum world. The paper presents the history, basic ideas and recent development in quantum game theory. On grounds of the discussed material, we reason about possible future development of quantum game theory and its impact on information processing and the emerging information society.
We address the question of when quantum entanglement is a useful resource for information processing tasks by presenting a new class of nonlocal games that are simple, direct, generalizations of the Clauser Horne Shimony Holt game. For some ranges of the parameters that specify the games, quantum mechanics offers an advantage, while, surprisingly, for others quantum mechanics is no more powerful than classical mechanics in performing the nonlocal task. This sheds new light on the difference between classical, quantum and super-quantum correlations.
The so called \emph{quantum game theory} has recently been proclaimed as one of the new branches in the development of both quantum information theory and game theory. However, the notion of a quantum game itself has never been strictly defined, which has led to a lot of conceptual confusion among different authors. In this paper we introduce a new conceptual framework of a \emph{scenario} and an \emph{implementation} of a game. It is shown that the procedures of "quantization" of games proposed in the literature lead in fact to several different games which can be defined within the same scen…
This essay gives a self-contained introduction to quantum game theory, and is primarily oriented to economists with little or no acquaintance with quantum mechanics. It assumes little more than a basic knowledge of vector algebra. Quantum mechanical notation and results are introduced as needed. It is also shown that some fundamental problems of quantum mechanics can be formulated as games.
The influence of spontaneous emission channel and generalized Pauli channel on quantum Monty Hall Game is analysed. The scheme of Flittney and Abbott is reformulated using the formalism of density matrices. Optimal classical strategies for given quantum strategies are found. The whole presented scheme illustrates how quantum noise may change the odds of a zero-sum game.
We investigate the quantization of non-zero sum games. For the particular case of the Prisoners' Dilemma we show that this game ceases to pose a dilemma if quantum strategies are allowed for. We also construct a particular quantum strategy which always gives reward if played against any classical strategy.
In recent years methods have been proposed to extend classical game theory into the quantum domain. This paper explores further extensions of these ideas that may have a substantial potential for further research. Upon reformulating quantum game theory as a theory of classical games played by "quantum players" I take a constructive approach. The roles of the players and the arbiter are investigated for clues on the nature of the quantum game space. Upon examination of the role of the arbiter, a possible non-commutative nature of pay-off operators can be deduced. I investigate a sub-class of ga…
Determining Quantum Correlation through Nash Equilibria in Constant-Sum Games
本文利用常和博弈中的纯策略纳什均衡作为检测量子关联的机制，并通过贝叶斯博弈论框架给出了识别量子效应的充分条件。
Quantum Checkers: The Development and Analysis of a Quantum Combinatorial Game
本研究开发了融合量子力学概念的新型量子组合游戏量子跳棋（Cheqqers），通过多等级量子规则实现经典到量子游戏的插值，可作为AI量子游戏基准与人类量子直觉培养工具，并初步分析了游戏复杂度。
The aim of the paper is to examine the notion of simple Kantian equilibrium in $2 \times 2$ symmetric games and their quantum counterparts. We focus on finding the Kantian equilibrium strategies in the general form of the games. As a result, we derive a formula that determines the reasonable strategies for any payoffs in the bimatrix game. This allowed us to compare the payoff results for classical and quantum way of playing the game. We showed that a very large part of $2\times 2$ symmetric games have more beneficial Kantian equilibria when they are played with the use of quantum strategies.
Non-transitivity can arise in games with three or more strategies $A,B,C$, when $A$ beats $B$, $B$ beats $C$, and $C$ beats $A$, ($A>B>C>A$). An example is the children's game \textquotedblleft rock, scissors, paper" ($R,S,P$) where $R>S>P>R$. We discuss the conditions under which quantum versions of $R,S,P$ retain the non-transitive characteristics of the corresponding classical game. Some physical implications of non-transitivity in quantum game theory are also considered.
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 pay off function is usually different from that in the classical counterpart except in some special cases. This presents an explicit example where 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 s…
Attention to the very physical aspects of information characterizes the current research in quantum computation, quantum cryptography and quantum communication. In most of the cases quantum description of the system provides advantages over the classical approach. Game theory, the study of decision making in conflict situation has already been extended to the quantum domain. We would like to review the latest development in quantum game theory that is relevant to information processing. We will begin by illustrating the general idea of a quantum game and methods of gaining an advantage over "c…
Effect of replacing the classical game object with a quantum object is analyzed. We find this replacement requires a throughout reformation of the framework of Game Theory. If we use density matrix to represent strategy state of players, they are full-structured density matrices with off-diagonal elements for the new games, while reduced diagonal density matrix will be enough for the traditional games on classical objects. In such formalism, the payoff function of every player becomes Hermitian Operator acting on the density matrix. Therefore, the new game looks really like Quantum Mechanics w…
Quantum bounds for compiled XOR games and $d$-outcome CHSH games
本研究拓展现有证明技术，证实Kalai等人的非局域博弈编译流程可保留XOR博弈与d结果CHSH博弈的量子界，还证明任意量子比特测量对都存在对应XOR博弈，其最优胜率可作为自检测依据。
In these lecture notes we investigate the implications of the identification of strategies with quantum operations in game theory beyond the results presented in [J. Eisert, M. Wilkens, and M. Lewenstein, Phys. Rev. Lett. 83, 3077 (1999)]. After introducing a general framework, we study quantum games with a classical analogue in order to flesh out the peculiarities of game theoretical settings in the quantum domain. Special emphasis is given to a detailed investigation of different sets of quantum strategies.
In the article we analyse how noisiness of quantum channels can influence the magic squares quantum pseudo-telepathy game. We show that the probability of success can be used to determine characteristics of quantum channels. Therefore the game deserves more careful study aiming at its implementation.
We propose a scheme for a quantum game based on performing an EPR type experiment and in which each player's spatial directional choices are considered as their strategies. A classical mixed-strategy game is recovered by restricting the players' choices to specific spatial trajectories. We show that for players' directional choices for which the Bell-CHSH inequality is violated, the players' payoffs in the quantum game have no mapping within the classical mixed-strategy game. The scheme provides a more direct link between classical and quantum games.
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 q…
Quantum games are usually considered as games with strategies defined not by the standard Kolmogorovian probabilistic measure but by the probability amplitude used in quantum physics. The reason for the use of the probability amplitude or "quantum probabilistic measure", is the nondistributive lattice occurring in physical situations with quantum microparticles. In our paper we give examples of getting nondistributive orthomodular lattices in some special macroscopic situations without use of quantum microparticles.
We study optimal and nearly-optimal quantum strategies for non-local XOR games. First, we prove the following general result: for every non-local XOR game, there exists a set of relations with the properties: (1) a quantum strategy is optimal for the game if and only if it satisfies the relations, and (2) a quantum strategy is nearly optimal for the game if and only if it approximately satisfies the relations. Next, we focus attention on a specific infinite family of XOR games: the CHSH(n) games. This family generalizes the well-known CHSH game. We describe the general form of CHSH(n) optimal …
This paper studies a simple class of zero-sum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players' payoffs. We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that one-turn quantum refereed games, wherein the referee is specified by a quantum circuit, can be simulated in polynomial space.
Quantum Parrondo Games in Low-Dimensional Hilbert Spaces
本文研究低维Hilbert空间上的量子Parrondo博弈，将组成博弈实现为环上量子行走，并利用量子硬币引入随机序列，发现长时间下因纠缠与自干涉产生复杂的系统性输赢结构。
Our purpose is to focus attention on a new criterion for quantum schemes by bringing together the notions of quantum game and game isomorphism. A quantum game scheme is required to generate the classical game as a special case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We show how this isomorphism condition influences the players' strategy sets. We are concerned with the Marinatto-Weber type quantum game scheme and the strong isomorphism between games in strategic form.
The aim of this paper is to bring together the notions of quantum game and game isomorphism. The work is intended as an attempt to introduce a new criterion for quantum game schemes. The generally accepted requirement forces a quantum scheme to generate the classical game in a particular case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We are concerned with the Eisert-Wilkens-Lewenstein quantum game scheme and the strong isomorphism between games in strategic form.
We present a scheme for playing quantum repeated 2x2 games based on the Marinatto and Weber's approach to quantum games. As a potential application, we study twice repeated Prisoner's Dilemma game. We show that results not available in classical game can be obtained when the game is played in the quantum way. Before we present our idea, we comment on the previous scheme of playing quantum repeated games.
Citizen Science Games on the Timeline of Quantum Games
本文概述用于量子物理公民科学研究的量子游戏，探索其与量子计算机游戏的联系，指出领域受严肃游戏应用、量子计算机发展及开放游戏开发活动驱动，并展望未来协作等前景。
We generalize a concept of classical finite extensive game to make it useful for application of quantum objects. The generalization extends a quantum realization scheme of static games to any finite extensive game. It represents an extension of any classical finite extensive games to the quantum domain. In addition our model is compatible with well-known quantum schemes of static games. The paper is summed up by two examples.
The History of Quantum Games
该研究收集260余款量子游戏，从三个维度梳理其发展历程，反思量子游戏的定义与内涵，指出量子主题正拓展至商业游戏，且量子硬件发展催生了新型量子计算机游戏。
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