Quantum Game Theory Research Articles

Quantum Entanglement and the Emergence of Collaboration in Social Media

In this paper, we devise a slightly modified version of the vote with the wallet game used by Becchetti et al. [L. Becchetti and F. Salustri, The vote with the wallet as a multiplayer prisoner’s dilemma, CEIS Tor Vergata Research Paper No. 359, Vol. 13, Issue 10, Centre for Economic and International Studies, Rome, Italy (2015); L. Becchetti, V. Pelligra and F. Salustri, Testing for heterogeneity of preferences in randomized experiments: A satisfaction-based approach applied to multiple prisoner dilemmas, Appl. Econ. Lett. 24(10) (2017) 722–726] for the use of social media, where the player decides whether to responsibly share social knowledge or not. We follow the point of view of Bennet and Bennet [D. Bennet and A. Bennet, Social learning from the inside out: The creation and sharing of knowledge from the mind/brain perspective, in Social Knowledge: Using Social Media to Know What You Know, eds. J. P. Girard and J. L. Girard (IGI Global, 2011), pp. 1–23] according to which another social settings may emerge through the so-called “process of collaborative entanglement.” In this environment, members of a community interact continuously with strong emotional feelings to combine the sources of knowledge and the beneficiaries of that knowledge and move toward a common direction. The application of the model to the quantum game theory substantially confirms that the cooperative strategy becomes the optimal one depending on the frequency of interactions and people’s cultural, geographical and social reachability and traceability.

Correspondence between quantization schemes for two-player nonzero-sum games and CNOT complexity

The well-known quantization schemes for two-player nonzero-sum games are Eisert–Wilkens–Lewenstein scheme and Marinatto–Weber scheme. In this work, we establish the connection between the two schemes from the perspective of quantum circuits. Further, we provide the correspondence between any game quantization schemes and the CNOT complexity, where CNOT complexity is up to the local unitary operations. While CNOT complexity is known to be useful in the analysis of universal quantum circuit, in this work, we find its applicability in quantum game theory.

Some Remarks on Quantum Logics and Quantum Mechanics

Some remarks on the use of quantum logic and corresponding nondistributive lattices in macroscopic situations—in the theory of automata and macroscopic quantum games—are discussed. Comparison with microscopic quantum mechanics is made.

Quantum gambling based on Nash-equilibrium

The problem of establishing a fair bet between spatially separated gambler and casino can only be solved in the classical regime by relying on a trusted third party. By combining Nash-equilibrium theory with quantum game theory, we show that a secure, remote, two-party game can be played using a quantum gambling machine which has no classical counterpart. Specifically, by modifying the Nash-equilibrium point we can construct games with arbitrary amount of bias, including a game that is demonstrably fair to both parties. We also report a proof-of-principle experimental demonstration using linear optics.

Quantum voting and violation of Arrow's impossibility theorem

We propose a quantum voting system, in the spirit of quantum games such as the quantum Prisoner's Dilemma. Our scheme enables a constitution to violate a quantum analog of Arrow's Impossibility Theorem. Arrow's Theorem is a claim proved deductively in economics: Every (classical) constitution endowed with three innocuous-seeming properties is a dictatorship. We construct quantum analogs of constitutions, of the properties, and of Arrow's Theorem. A quantum version of majority rule, we show, violates this Quantum Arrow Conjecture. Our voting system allows for tactical-voting strategies reliant on entanglement, interference, and superpositions. This contribution to quantum game theory helps elucidate how quantum phenomena can be harnessed for strategic advantage.

Quantum load balancing in ad hoc networks

This paper presents a novel approach in targeting load balancing in ad hoc networks utilizing the properties of quantum game theory. This approach benefits from the instantaneous and information-less capability of entangled particles to synchronize the load balancing strategies in ad hoc networks. The quantum load balancing (QLB) algorithm proposed by this work is implemented on top of OLSR as the baseline routing protocol; its performance is analyzed against the baseline OLSR, and considerable gain is reported regarding some of the main QoS metrics such as delay and jitter. Furthermore, it is shown that QLB algorithm supports a solid stability gain in terms of throughput which stands a proof of concept for the load balancing properties of the proposed theory.

Quantum game application to spectrum scarcity problems

Recent spectrum-sharing research has produced a strategy to address spectrum scarcity problems. This novel idea, named cognitive radio, considers that secondary users can opportunistically exploit spectrum holes left temporarily unused by primary users. This presents a competitive scenario among cognitive users, making it suitable for game theory treatment. In this work, we show that the spectrum-sharing benefits of cognitive radio can be increased by designing a medium access control based on quantum game theory. In this context, we propose a model to manage spectrum fairly and effectively, based on a multiple-users multiple-choice quantum minority game. By taking advantage of quantum entanglement and quantum interference, it is possible to reduce the probability of collision problems commonly associated with classic algorithms. Collision avoidance is an essential property for classic and quantum communications systems. In our model, two different scenarios are considered, to meet the requirements of different user strategies. The first considers sensor networks where the rational use of energy is a cornerstone; the second focuses on installations where the quality of service of the entire network is a priority.

A quantum extension to inspection game

Quantum game theory is a new interdisciplinary field between game theory and physical 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 result shows that the quantum inspection game has various Nash equilibrium 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.

The effect of entanglement and non-inertial frame on four-qubit quantum game

(Received 24 June 2014 ; in final form 6 January 2016) Abstract The effect of increasing quantum bits and Unruh effect on quantum Prisoners’ dilemma has been investigated for both entangled and unentangled initial states. The Nash equilibrium, as an important result of quantum game theory, was obtained through the different payoffs resulted from choosing various strategies. It has been shown that the non-inertial frame disturbs the symmetry of the game. Actually, selection of the basic quantum strategy by players and calculating the payoff of the game via the density matrix of 4-qubit quantum states can represent a scale of influence of entanglement in a quantum mechanical system.

Quantum-mechanical machinery for rational decision-making in classical guessing game.

In quantum game theory, one of the most intriguing and important questions is, “Is it possible to get quantum advantages without any modification of the classical game?” The answer to this question so far has largely been negative. So far, it has usually been thought that a change of the classical game setting appears to be unavoidable for getting the quantum advantages. However, we give an affirmative answer here, focusing on the decision-making process (we call ‘reasoning’) to generate the best strategy, which may occur internally, e.g., in the player’s brain. To show this, we consider a classical guessing game. We then define a one-player reasoning problem in the context of the decision-making theory, where the machinery processes are designed to simulate classical and quantum reasoning. In such settings, we present a scenario where a rational player is able to make better use of his/her weak preferences due to quantum reasoning, without any altering or resetting of the classically defined game. We also argue in further analysis that the quantum reasoning may make the player fail, and even make the situation worse, due to any inappropriate preferences.

A quantum derivation of a reputational risk premium

Using quantum modeling, we propose a novel approach to reputational risk management arguing that taking care of corporate reputation can be considered a coalitional strategy framed into a quantum game theory schema. Following a stochastic mechanics approach, and assuming that the revenues of a firm can be modeled as an Ornstein–Uhlenbeck process represented by the well-known Langevin equation for the diffusion of a particle with unit mass under non-linear friction, we offer a mathematical derivation for the discount rate of a firm that does not manage its reputational risk and for a company that optimally manages its reputational risk. By comparison, we derive an analytical expression for the reputational risk premium. Our approach provides useful managerial and financial implications, suggesting that the use of the quantum ideology may result useful in enlarging the body of knowledge of corporate management.

Maximum Entropy and Bayesian Inference for the Monty Hall Problem

We devise an approach to Bayesian statistics and their applications in the analysis of the Monty Hall problem. We combine knowledge gained through applications of the Maximum Entropy Principle and Nash equilibrium strategies to provide results concerning the use of Bayesian approaches unique to the Monty Hall problem. We use a model to describe Monty’s decision process and clarify that Bayesian inference results in an “irrelevant, therefore invariant” hypothesis. We discuss the advantages of Bayesian inference over the frequentist inference in tackling the uneven prior probability Monty Hall variant. We demonstrate that the use of Bayesian statistics conforms to the Maximum Entropy Principle in information theory and Bayesian approach successfully resolves dilemmas in the uneven probability Monty Hall variant. Our findings have applications in the decision making, information theory, bioinformatics, quantum game theory and beyond.

Do quantum strategies always win?

In a seminal paper, Meyer [David Meyer, Phys. Rev. Lett. 82, 1052 (1999)] described the advantages of quantum game theory by looking at the classical penny flip game. A player using a quantum strategy can win against a classical player almost 100\% of the time. Here we make a slight modification to the quantum game, with the two players sharing an entangled state to begin with. We then analyze two different scenarios, first in which quantum player makes unitary transformations to his qubit while the classical player uses a pure strategy of either flipping or not flipping the state of his qubit. In this case the quantum player always wins against the classical player. In the second scenario we have the quantum player making similar unitary transformations while the classical player makes use of a mixed strategy wherein he either flips or not with some probability "p". We show that in the second scenario, 100\% win record of a quantum player is drastically reduced and for a particular probability "p" the classical player can even win against the quantum player. This is of possible relevance to the field of quantum computation as we show that in this quantum game of preserving versus destroying entanglement a particular classical algorithm can beat the quantum algorithm.

The Pagerank-Index: Going beyond Citation Counts in Quantifying Scientific Impact of Researchers.

Quantifying and comparing the scientific output of researchers has become critical for governments, funding agencies and universities. Comparison by reputation and direct assessment of contributions to the field is no longer possible, as the number of scientists increases and traditional definitions about scientific fields become blurred. The h-index is often used for comparing scientists, but has several well-documented shortcomings. In this paper, we introduce a new index for measuring and comparing the publication records of scientists: the pagerank-index (symbolised as π). The index uses a version of pagerank algorithm and the citation networks of papers in its computation, and is fundamentally different from the existing variants of h-index because it considers not only the number of citations but also the actual impact of each citation. We adapt two approaches to demonstrate the utility of the new index. Firstly, we use a simulation model of a community of authors, whereby we create various ‘groups’ of authors which are different from each other in inherent publication habits, to show that the pagerank-index is fairer than the existing indices in three distinct scenarios: (i) when authors try to ‘massage’ their index by publishing papers in low-quality outlets primarily to self-cite other papers (ii) when authors collaborate in large groups in order to obtain more authorships (iii) when authors spend most of their time in producing genuine but low quality publications that would massage their index. Secondly, we undertake two real world case studies: (i) the evolving author community of quantum game theory, as defined by Google Scholar (ii) a snapshot of the high energy physics (HEP) theory research community in arXiv. In both case studies, we find that the list of top authors vary very significantly when h-index and pagerank-index are used for comparison. We show that in both cases, authors who have collaborated in large groups and/or published less impactful papers tend to be comparatively favoured by the h-index, whereas the pagerank-index highlights authors who have made a relatively small number of definitive contributions, or written papers which served to highlight the link between diverse disciplines, or typically worked in smaller groups. Thus, we argue that the pagerank-index is an inherently fairer and more nuanced metric to quantify the publication records of scientists compared to existing measures.

A connection between quantum decision theory and quantum games: The Hamiltonian of Strategic Interaction

Experimental economics and studies in psychology show incompatibilities between human behavior and the perfect rationality assumption which do not fit in classical decision theory, but a more general representation in terms of Hilbert spaces can account for them. This paper integrates previous theoretical works in quantum game theory, Yukalov and Sornette’s quantum decision theory and Pothos and Busemeyer’s quantum cognition model by postulating the Hamiltonian of Strategic Interaction which introduces entanglement in the strategic state of the decision-maker. The Hamiltonian is inherited from the algebraic structure of angular momentum in quantum mechanics and the only required parameter, θ̃∈[0,π], represents the strength of the interaction. We consider it as a non-revealed type of the decision-maker. Considering θ̃ to be a continuous random variable, phenomena like learning when participating in repeated games and the influence of the amount of disposable information could be considered as an evolution in the mode and shape of the distribution function fθ̃(t,I). This modeling is motivated by the Eisert–Wilkens–Lewenstein quantization scheme for Prisoner’s Dilemma game and then it is applied in the Ultimatum game, which is not a simultaneous but a sequential game. Even when this non-revealed type θ̃ cannot be directly observed, we can compute observable outcomes: the probabilities of offering different amounts of coins and the probability of the different offers being accepted or not by the other player.

Full characterization of quantum correlated equilibria

Quantum game theory aims to study interactions of people (or other agents) using quantum devices with possibly conflicting interests. Recently Zhang studied some quantitative questions in general quantum strategic games of growing sizes~\cite{Zha12}. However, a fundamental question not addressed there is the characterization of quantum correlated equilibria (QCE). In this paper, we answer this question by giving a sufficient and necessary condition for an arbitrary state $\rho$ being a QCE. In addition, when the condition fails to hold for some player $i$, we give an explicit positive-operator valued measurement (POVM) for that player to achieve a strictly positive gain of payoff. Finally, we give some upper bounds for the maximum gain by playing quantum strategies over classical ones, and the bounds are tight for some games.

A Novel Multi Pursuers-one Evader Game based on Quantum Game Theory

A Novel Multi Pursuers-one Evader Game based on Quantum Game Theory

Quantum games and quaternionic strategies

It is well-known that the phenomenon of entanglement plays a fundamental role in quantum game theory. Occasionally, games constructed via maximally entangled initial states (MEIS) will have new Nash equilibria yielding to the players higher payoffs than the ones they receive in the classical version of the game. When examining these new games for Nash equilibrium payoffs, a fundamental question arises; does a suitable choice of an MEIS improve the lot of the players? In this paper, we show that the answer to this question is yes for at least the case of a variant of the well-known two player, two strategy game of Chicken. To that end, we generalize Landsburg's quaternionic representation of the payoff function of two player, two strategy maximally entangled states to games where the initial state is chosen arbitrarily from a circle of maximally entangled initial states and for the corresponding quantized games show the existence of superior Nash equilibrium payoffs when an MEIS is appropriately chosen.

Gaming the quantum

In the time since the merger of quantum mechanics and game theory was proposed formally in 1999, the two distinct perspectives apparent in this merger of applying quantum mechanics to game theory, referred to henceforth as the theory of ``quantized games'', and of applying game theory to quantum mechanics, referred to henceforth as ``gaming the quantum'', have become synonymous under the single ill-defined term ``quantum game''. Here, these two perspectives are delineated and a game-theoretically proper description of what makes a multiplayer, non-cooperative game quantum mechanical, is given. Within the context of this description, finding Nash equilibrium in a zero-sum quantum game is exhibited to be equivalent to finding a solution to a simultaneous distance minimization problem in the state space of quantum objects, thus setting up a framework for a game theory inspired study of ``equilibrium'' behavior of quantum physical systems such as those utilized in quantum information processing and computation.

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.