N-Player Quantum Games in an EPR Setting

Abstract

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.

Maximum Likelihood Estimation of Stochastic Fractional Singular Models

“…Due to increasing the matrix dimension, computational complexity may limit some practical implementations. For some classes of problems with large matrices, it has been shown that use of geometric algebra can result in improvement in terms of tractabilty [21]. This problem together with the investigation of filter stability, continuous-time case studies, etc.…”

“…Equation (21) provides some additional data about the state x k together with the state vectors x i for i > k, which can be indicated as exogenous variables. Using the elementary row operations, the state vectors x i , i > k can be dropped out to eliminate the rest of the measurement.…”

Chris Cannings: A Life in Games

“…The concept of games, including evolutionary games, has been combined with quantum dynamics to give quantum games, [53] with potential applications to evolution at the molecular level. Evolutionary aspects have particularly been considered by Iqbal and colleagues for the two-player and multiplayer settings [45,[80][81][82]. See Orlin Grabbe [122] for an introduction to this area.…”

Evolution of quantum and classical strategies on networks by group interactions

“…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].…”