Quantum Game Research Articles

Application of the Eisert–Wilkens–Lewenstein quantum game scheme to decision problems with imperfect recall

We investigate implementations of the Eisert–Wilkens–Lewenstein (EWL) scheme of playing quantum games beyond strategic games. The scope of our research is decision problems, i.e. one-player extensive games. The research is based on the examination of their features when the decision problems are carried out via the EWL protocol. We prove that unitary operators can be adapted to play the role of strategies in decision problems with imperfect recall. Furthermore, we prove that unitary operators provide the decision maker with possibilities that are inaccessible for classical strategies.

Simulation of continuous variable quantum games without entanglement

A simulation scheme of quantum version of Cournot's duopoly is proposed, in which there is a new Nash equilibrium that may also be Pareto optimal without any entanglement involved. The unique property of this simulation scheme is decoherence-free against the symmetric photon loss. Furthermore, we analyze the effects of the asymmetric information on this simulation scheme and investigate the case of asymmetric game caused by asymmetric photon loss. A second-order phase transition-like behavior of the average profits of firms 1 and 2 in a Nash equilibrium can be observed with the change of the degree of asymmetry of the information or the degree of ‘virtual cooperation’. It is also found that asymmetric photon loss in this simulation scheme plays a similar role as that with the asymmetric entangled states in the quantum game.

Nash equilibria in quantum games

For any two-by-two game G \mathbf {G} , we define a new two-player game G Q \mathbf {G}^Q . The definition is motivated by a vision of players in game G \mathbf {G} communicating via quantum technology according to the protocol introduced by J. Eisert and M. Wilkins. In the game G Q \mathbf {G}^Q , each player’s (mixed) strategy set consists of the set of all probability distributions on the 3-sphere S 3 \mathbf {S}^3 . Nash equilibria in the game can be difficult to compute. Our main theorems classify all possible mixed-strategy equilibria. First, we show that up to a suitable definition of equivalence, any strategy that arises in equilibrium is supported on at most four points; then we show that those four points must lie in one of a small number of allowable geometric configurations.

About Our Authors

About Our Authors

New scheme on how to realize quantum NOT gate

Recently, some schemes on how to realize quantum NOT gate have been reported. In this paper, we theoretically propose a new scheme by means of quantum theory. Here we use a quantum coin as a quantum NOT gate and its logic function is realized by the quantum game theory. In our schemes, classical noise does not disturb the realization of quantum NOT gate.

Quantum Learning: Learn Without Learning

Quantum Education is the natural way to learn---motivating and exciting people to take responsibility for their own education.. The Montessori Model represents the closest example of Quantum Education, where the environment is prepared with didactic materials for the children to absorb at their own pace. Where children learn without formal pedagogical machinations, without consciously learning how to learn---by doing. Like Quantum Logic or Quantum Physics or Quantum Games, quantum thinking is an insightful, body/mind approach, attempting to connect our classical worldwhere objects or things have definite identitieswith our new quantum worldwhere things take on multiple realities simultaneously. The concept of mind, limiting our perceptual abilities is no longer confined to the brain or even the body---all organs are, in some ways, thinking organs, permitting restructuring of our cognitive educational commitment toward infinite choice and possibility. Quantum Education has been defined by the Canadian Quantum 2000 Group as the need for a quantum shift in what students are expected to learn in Alberta public schools starting Y2000.

A periodic point-based method for the analysis of Nash equilibria in 2 × 2 symmetric quantum games

We present a novel method of looking at Nash equilibria in 2 × 2 quantum games. Our method is based on a mathematical connection between the problem of identifying Nash equilibria in game theory, and the topological problem of the periodic points in nonlinear maps. To adapt our method to the original protocol designed by Eisert et al (1999 Phys. Rev. Lett. 83 3077–80) to study quantum games, we are forced to extend the space of strategies from the initial proposal. We apply our method to the extended strategy space version of the quantum Prisoner's dilemma and find that a new set of Nash equilibria emerge in a natural way. Nash equilibria in this set are optimal as Eisert's solution of the quantum Prisoner's dilemma and include this solution as a limit case.

Entangled Games Are Hard to Approximate

We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness $1-1/\,\mathrm{poly}$ even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless $\mathrm{P}=\mathrm{NP}$, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games.

Quantum Financial Economics - Risk and Returns

Financial volatility risk is addressed through a multiple round evolutionary quantum game equilibrium leading to Multifractal Self-Organized Criticality (MSOC) in the financial returns and in the risk dynamics. The model is simulated and the results are compared with financial volatility data.

Quantum walks: Decoherence and coin-flipping games

We investigate the global chirality distribution of the quantum walk on the line when decoherence is introduced either through simultaneous measurements of the chirality and particle position, or as a result of broken links. The first mechanism drives the system towards a classical diffusive behavior. This is used to build new quantum games, similar to the spin-flip game. The second mechanism involves two different possibilities: (a) All the quantum walk links have the same probability of being broken. (b) Only the quantum walk links on a half-line are affected by random breakage. In case (a) the decoherence drives the system to a classical Markov process, whose master equation is equivalent to the dynamical equation of the quantum density matrix. This is not the case in (b) where the asymptotic global chirality distribution unexpectedly maintains some dependence with the initial condition. Explicit analytical equations are obtained for all cases.

Information sharing using entangled states and its applications to quantum card tricks

Quantum superposition states, especially, quantum entangled states, are useful for several fields such as quantum computation, quantum cryptography, quantum game theory, and so on. In this paper, first, we propose methods to share entangled states without communication among players that may not know each other. Next, we introduce quantum cards and propose some quantum card tricks using this sharing method. For example, we propose tricks such that by sharing an entangled state between two spectators secretly, magicians make a spectator's card number and another spectator's card number the same.

Quantum game interpretation for a special case of Parrondo’s paradox

By using the discrete Markov chain method, Parrondo’s paradox is studied by means of theoretical analysis and computer simulation, built on the case of game AB played in alternation with modulus M=4. We find that such a case does not have a definite stationary probability distribution and that payoffs of the game depend on the parity of the initial capital. Besides, this paper reveals the phenomenon that “processing in order produces non-deterministic results, while a random process produces deterministic results”. The quantum game method is used in a further study. 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.

Strategic insights from playing quantum tic-tac-toe

In this paper, we perform a minimalistic quantization of the classical game of tic-tac-toe, by allowing superpositions of classical moves. In order for the quantum game to reduce properly to the classical game, we require legal quantum moves to be orthogonal to all previous moves. We also admit interference effects, by squaring the sum of amplitudes over all moves by a player to compute his or her occupation level of a given site. A player wins when the sum of occupations along any of the eight straight lines we can draw in the 3 × 3 grid is greater than or equal to 3. We play the quantum tic-tac-toe first randomly, and then deterministically, to explore the impact different opening moves, end games and different combinations of offensive and defensive strategies have on the outcome of the game. In contrast to the classical tic-tac-toe, the deterministic quantum game does not always end in a draw. In contrast also to most classical two-player games of no chance, it is possible for player 2 to win. More interestingly, we find that player 1 enjoys an overwhelming quantum advantage when he opens with a quantum move, but loses this advantage when he opens with a classical move. We also find the quantum blocking move, which consists of a weighted superposition of moves that the opponent could use to win the game, to be very effective in denying the opponent his or her victory. We then speculate what implications these results might have on quantum information transfer and portfolio optimization.

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.

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 Game Theory and Coordination in Intellectual Property

Abstract The inventive and creative works protected by intellectual property laws are typically characterized as public goods. As in the case of ordinary public goods, it is often difficult for inventors and creators to erect non-legal barriers to prevent the use of their works (non-excludability), and any such use generally does not deplete their works (non-rivalry). Thus, scholars have long used classical game theoretic models of public goods to describe the strategies of players in intellectual property (IP) games. These models contain two seemingly common-sense assumptions: one, that if there are a finite number of decisions a player can make for any single move, the player must make exactly one decision; and two, that if a player engages only in a single game, the player will break a pre-game commitment with another player to follow a mutually beneficial strategy if it is in that player’s self-interest. Abstract Recent extensions to classical game theory using the theory of quantum mechanics -- known as quantum game theory (QGT) -- have dispensed with these two assumptions, yielding radical new results for many types of games. For instance, in the classical version of the prisoner’s dilemma, two prisoners will fail to cooperate even though it is in their mutual interest to do so. However, in the quantum version, the prisoners’ decisions are often “entangled” in a mutually beneficial way that overcomes the ostensible barriers of classical self-interest, leading the players to cooperate. In this regard, quantum game theorists have suggested that if some exogenous quantum mechanical mechanism -- like a quantum computer -- could be used to entangle players in a public-goods game, doing so would diminish or eliminate sub-optimal free riding.Abstract This paper contends that there are endogenous effects analogous to those in quantum games -- specifically, quantum game-like results that emerge in the absence of external quantum computers -- in certain types of IP games that may act to reduce classically predicted free riding, duplicated development costs, and deadweight losses. In particular, instead of modeling underlying IP rights as classical entities, this paper follows the suggestions of several earlier scholars that legal rights are probabilistic and, at least metaphorically, quantum in nature. In so doing, it shows that rights, including IP rights, exhibit an inherent quantum structure that allows players to avoid making a single classical choice for each move. By allowing the government -- as a mechanism designer -- to engage in quantum strategies, ordinary players can exhibit forms of seemingly altruistic behavior and cooperation that are absent in classical models. The paper concludes by commenting briefly on how QGT might be applied more broadly to other areas of the law.

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.

Doves and hawks in economics revisited: An evolutionary quantum game theory based analysis of financial crises

The last financial and economic crisis demonstrated the dysfunctional long-term effects of aggressive behaviour in financial markets. Yet, evolutionary game theory predicts that under the condition of strategic dependence a certain degree of aggressive behaviour remains within a given population of agents. However, as a consequence of the financial crisis, it would be desirable to change the “rules of the game” in a way that prevents the occurrence of any aggressive behaviour and thereby also the danger of market crashes. The paper picks up this aspect. Through the extension of the well-known hawk–dove game by a quantum approach, we can show that dependent on entanglement, evolutionary stable strategies also can emerge, which are not predicted by the classical evolutionary game theory and where the total economic population uses a non-aggressive quantum strategy.

Noisy non-transitive quantum games

We study the effect of quantum noise in 3 × 3 entangled quantum games. By taking into account different noisy quantum channels, we analyze how a two-player, three-strategy Rock–Scissor–Paper game is influenced by the quantum noise. We consider the winning non-transitive strategies R, S and P such that R beats S, S beats P and P beats R. The game behaves as a noiseless game for the maximum value of the quantum noise. It is seen that Alice's payoff is heavily influenced by the depolarizing noise as compared to the amplitude damping noise. A depolarizing channel causes a monotonic decrease in players' payoffs as we increase the amount of quantum noise. In the case of the amplitude damping channel, Alice's payoff function reaches its minimum for α = 0.5 and is symmetrical. This means that larger values of quantum noise influence the game weakly. On the other hand, the phase damping channel does not influence the game. Furthermore, the Nash equilibrium and non-transitive character of the game are not affected under the influence of quantum noise.

Experimental implementation of a four-player quantum game

Game theory is central to the understanding of competitive interactions arising in many fields, from the social and physical sciences to economics. Recently, as the definition of information is generalized to include entangled quantum systems, quantum game theory has emerged as a framework for understanding the competitive flow of quantum information. Up till now, only two- and three-player quantum games have been demonstrated with restricted strategy sets. Here, we report the first experiment that implements a four-player quantum minority game over tunable four-partite entangled states encoded in the polarization of single photons. Experimental application of appropriate player strategies gives equilibrium payoff values well above those achievable in the classical game. These results are in excellent quantitative agreement with our theoretical analysis of the symmetric Pareto optimal strategies. Our results demonstrate for the first time how nontrivial equilibria can arise in a competitive situation involving quantum agents.