Probabilistic analysis of three-player symmetric quantum games played using the Einstein-Podolsky-Rosen-Bohm setting

This Letter extends our probabilistic framework for two-player quantum games to the multiplayer case, while giving a unified perspective for both classical and quantum games. Considering joint probabilities in the Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) setting for three observers, we use this setting in order to play general three-player noncooperative symmetric games. We analyze how the peculiar non-factorizable joint probabilities provided by the EPR-Bohm setting can change the outcome of a game, while requiring that the quantum game attains a classical interpretation for factorizable joint probabilities. In this framework, our analysis of the three-player generalized Prisoner's Dilemma (PD) shows that the players can indeed escape from the classical outcome of the game, because of non-factorizable joint probabilities that the EPR setting can provide. This surprising result for three-player PD contrasts strikingly with our earlier result for two-player PD, played in the same framework, in which even non-factorizable joint probabilities do not result in escaping from the classical consequence of the game.     

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.     

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.     

The equivalence of Bell's inequality and the Nash inequality in a quantum game-theoretic setting

The interaction of competing agents is described by classical game theory. It is now well known that this can be extended to the quantum domain, where agents obey the rules of quantum mechanics. This is of emerging interest for exploring quantum foundations, quantum protocols, quantum auctions, quantum cryptography, and the dynamics of quantum cryptocurrency, for example. In this paper, we investigate two-player games in which a strategy pair can exist as a Nash equilibrium when the games obey the rules of quantum mechanics. Using a generalized Einstein–Podolsky–Rosen (EPR) setting for two-player quantum games, and considering a particular strategy pair, we identify sets of games for which the pair can exist as a Nash equilibrium only when Bell's inequality is violated. We thus determine specific games for which the Nash inequality becomes equivalent to Bell's inequality for the considered strategy pair.     

Non-factorizable joint probabilities and evolutionarily stable strategies in the quantum prisoner's dilemma game

The well-known refinement of the Nash Equilibrium (NE) called an Evolutionarily Stable Strategy (ESS) is investigated in the quantum Prisoner's Dilemma (PD) game that is played using an Einstein-Podolsky-Rosen type setting. Earlier results report that in this scheme the classical NE remains intact as the unique solution of the quantum PD game. In contrast, we show here that interestingly in this scheme a non-classical solution for the ESS emerges for the quantum PD.     

Quantum games entropy

We propose the study of quantum games from the point of view of quantum information theory and statistical mechanics. Every game can be described by a density operator, the von Neumann entropy and the quantum replicator dynamics. There exists a strong relationship between game theories, information theories and statistical physics. The density operator and entropy are the bonds between these theories. The analysis we propose is based on the properties of entropy, the amount of information that a player can obtain about his opponent and a maximum or minimum entropy criterion. The natural trend of a physical system is to its maximum entropy state. The minimum entropy state is a characteristic of a manipulated system, i.e., externally controlled or imposed. There exist tacit rules inside a system that do not need to be specified or clarified and search the system equilibrium based on the collective welfare principle. The other rules are imposed over the system when one or many of its members violate this principle and maximize its individual welfare at the expense of the group.     

A discussion on the origin of quantum probabilities

We study the origin of quantum probabilities as arising from non-Boolean propositional-operational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorovian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).     

Stability of Mixed Nash Equilibria in Symmetric Quantum Games

In bimatrix games the Bishop-Cannings theorem of the classical evolutionary game theory does not permitpure evolutionarily stable strategies (ESSs) when a mixed ESS exists. We find the necessary form of two-qubit initialquantum states when a switch-over to a quantum version of the game also changes the evolutionary stability of a mixedsymmetric Nash equilibrium.     

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.     

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.     

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.     

Quantum Parrondo game based on a quantum ratchet effect

A Parrondo game is a counterintuitive game where two losing games can be combined to form a winning game. We construct a quantum version of a Parrondo game based on a quantum ratchet effect for a delta-kicked model, which can be realized in optical lattices. A game set is presented and a quantum anti-Parrondo game is also investigated.     

Quantum games via search algorithms

We build new quantum games, similar to the spin flip game, where as a novelty the players perform measurements on a quantum system associated to a continuous time search algorithm. The measurements collapse the wave function into one of the two possible states. These games are characterized by a continuous space of strategies and the selection of a particular strategy is determined by the moments when the players measure.     

Efficiency of Classical and Quantum Games Equilibria

Nash equilibria and correlated equilibria of classical and quantum games are investigated in the context of their Pareto efficiency. The examples of the prisoner’s dilemma, battle of the sexes and the game of chicken are studied. Correlated equilibria usually improve Nash equilibria of games but require a trusted correlation device susceptible to manipulation. The quantum extension of these games in the Eisert–Wilkens–Lewenstein formalism and the Frąckiewicz–Pykacz parameterization is analyzed. It is shown that the Nash equilibria of these games in quantum mixed Pauli strategies are closer to Pareto optimal results than their classical counter-parts. The relationship of mixed Pauli strategies equilibria and correlated equilibria is also studied.     

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.     

Tripartite Dynamic Zero-Sum Quantum Games

The Nash equilibrium plays a crucial role in game theory. Most of results are based on classical resources. Our goal in this paper is to explore multipartite zero-sum game with quantum settings. We find that in two different settings there is no strategy for a tripartite classical game being fair. Interestingly, this is resolved by providing dynamic zero-sum quantum games using single quantum state. Moreover, the gains of some players may be changed dynamically in terms of the committed state. Both quantum games are robust against the preparation noise and measurement errors.     

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.     

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.     

Entanglement and dynamic stability of Nash equilibria in a symmetric quantum game

We study the evolutionary stability of Nash equilibria (NE) in a symmetric quantum game played by the recently proposed scheme of applying ‘identity’ and ‘Pauli spin-flip’ operators on an initial state with classical probabilities. We show that in this symmetric game dynamic stability of a NE can be changed when the game changes its form, for example, from classical to quantum. It happens even when the NE remains intact in both forms.