Analyzing Three-Player Quantum Games in an EPR Type Setup

The -player quantum games are analyzed that use an Einstein-Podolsky-Rosen (EPR) experiment, as the underlying physical setup. In this setup, a player’s strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players’ strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for -qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the players’ payoffs and mixed Nash equilibrium are determined. Players’ payoff matrices are then defined using linear functions so that common two-player games can be easily extended to the -player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners’ Dilemma game for general . We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs. By dispensing with the standard unitary transformations on state vectors in Hilbert space and using instead rotors and multivectors, based on Clifford’s geometric algebra (GA), it is shown how the N-player case becomes tractable. The new mathematical approach presented here has wide implications in the areas of quantum information and quantum complexity, as it opens up a powerful way to tractably analyze N-partite qubit interactions.

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“…These general results thus subsume previous analyses for two-player and three-player games in an EPR setting [49], [50].…”

Section: Discussionsupporting

confidence: 83%

“…Following [49], [50], we can then form the bivectors , which are non-commuting for , with but if we have . We also have the trivector.finding and furthermore, that commutes with each vector , thus acting in a similar fashion to the unit imaginary .…”

Section: Introductionmentioning

confidence: 99%

N-Player Quantum Games in an EPR Setting

2012

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“…Classically defined games have been analyzed and it has been found that a quantum team has an advantage over any classical team [81]. Quantum games have also been analyzed by using geometric algebra [82]–[84], and they have been implemented using quantum computers [85]–[88]. For further background on quantum games, we refer to [89], [90].…”

Section: Introductionmentioning

confidence: 99%

Coevolution of Quantum and Classical Strategies on Evolving Random Networks

Iqbal

Perc

et al. 2013

PLoS ONE

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We study the coevolution of quantum and classical strategies on weighted and directed random networks in the realm of the prisoner’s dilemma game. During the evolution, agents can break and rewire their links with the aim of maximizing payoffs, and they can also adjust the weights to indicate preferences, either positive or negative, towards their neighbors. The network structure itself is thus also subject to evolution. Importantly, the directionality of links does not affect the accumulation of payoffs nor the strategy transfers, but serves only to designate the owner of each particular link and with it the right to adjust the link as needed. We show that quantum strategies outperform classical strategies, and that the critical temptation to defect at which cooperative behavior can be maintained rises, if the network structure is updated frequently. Punishing neighbors by reducing the weights of their links also plays an important role in maintaining cooperation under adverse conditions. We find that the self-organization of the initially random network structure, driven by the evolutionary competition between quantum and classical strategies, leads to the spontaneous emergence of small average path length and a large clustering coefficient.

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“…Later, Iqbal et al [26] studied evolutionarily stable strategies in quantum games and Kay et al [27] presented an evolutionary quantum game. Other related studies have also been performed [28][29][30][31][32]. For further background on quantum games, see [33,34].…”

Section: Introductionmentioning

confidence: 99%

Evolution of quantum and classical strategies on networks by group interactions

Iqbal

Chen

et al. 2012

New J. Phys.

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In this paper, quantum strategies are introduced within evolutionary games in order to investigate the evolution of quantum and classical strategies on networks in the public goods game. Comparing the results of evolution on a scale-free network and a square lattice, we find that a quantum strategy outperforms the classical strategies, regardless of the network. Moreover, a quantum strategy dominates the population earlier in group interactions than it does in pairwise interactions. In particular, if the hub node in a scale-free network is occupied by a cooperator initially, the strategy of cooperation will prevail in the population. However, in other situations, a quantum strategy can defeat the classical ones and finally becomes the dominant strategy in the population.

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Analyzing Three-Player Quantum Games in an EPR Type Setup

Cited by 18 publications

References 61 publications

N-Player Quantum Games in an EPR Setting

N-Player Quantum Games in an EPR Setting

Coevolution of Quantum and Classical Strategies on Evolving Random Networks

Evolution of quantum and classical strategies on networks by group interactions

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