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Analysis of TwoPlayer Quantum Games in an EPR Setting Using Clifford’s Geometric Algebra
, 2012
"... The framework for playing quantum games in an EinsteinPodolskyRosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players ’ strategy sets remain identical to the ones in the classical mixedstrategy v ..."
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The framework for playing quantum games in an EinsteinPodolskyRosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players ’ strategy sets remain identical to the ones in the classical mixedstrategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for twoplayer games, and as examples, we analyze the games of Prisoners ’ Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantummechanical analysis without the use of complex numbers or the Dirac Braket notation, and hence is more accessible to the nonphysicist.
NPlayer Quantum Games in an EPR Setting
, 2012
"... The Nplayer quantum games are analyzed that use an EinsteinPodolskyRosen (EPR) experiment, as the underlying physical setup. In this setup, a player’s strategies are not unitary transformations as in alternate quantum gametheoretic frameworks, but a classical choice between two directions along ..."
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The Nplayer quantum games are analyzed that use an EinsteinPodolskyRosen (EPR) experiment, as the underlying physical setup. In this setup, a player’s strategies are not unitary transformations as in alternate quantum gametheoretic 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 mixedstrategy 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 Nqubit GHZ and Wtype states, subject to general measurement directions, from which the expressions for the players ’ payoffs and mixed Nash equilibrium are determined. Players ’ NN payoff matrices are then defined using linear functions so that common twoplayer games can be easily extended to the Nplayer case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners ’ Dilemma game for general N§2. 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 Nplayer 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 Npartite qubit interactions.