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Game Theoretic Interaction and Decision: A Quantum Analysis

Author

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  • Ulrich Faigle

    (Mathematisches Institut, Universität zu Köln)

  • Michel Grabisch

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris sciences et lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, IUF - Institut Universitaire de France - M.E.N.E.S.R. - Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche)

Abstract

An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a result, cooperation systems, decision systems and quantum systems all become visible as manifestations of special interaction systems. The treatment of the theory is purely mathematical and does not require any special knowledge of physics. It is shown how standard notions in cooperative game theory arise naturally in this context. In particular, states of general interaction systems are seen to arise as linear superpositions of pure quantum states and Fourier transformation to become meaningful. Moreover, quantum games fall into this framework. Finally, a theory of Markov evolution of interaction states is presented that generalizes classical homogeneous Markov chains to the present context.

Suggested Citation

  • Ulrich Faigle & Michel Grabisch, 2017. "Game Theoretic Interaction and Decision: A Quantum Analysis," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-03220813, HAL.
    • Handle: RePEc:hal:cesptp:halshs-03220813
    DOI: 10.3390/g8040048
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-03220813
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    References listed on IDEAS

    as
    1. Christophe Labreuche & Michel Grabisch, 2008. "A value for bi-cooperative games," Post-Print halshs-00308738, HAL.
    2. Grabisch, Michel & Rusinowska, Agnieszka, 2013. "A model of influence based on aggregation functions," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 316-330.
    3. Grabisch, Michel & Rusinowska, Agnieszka, 2011. "Influence functions, followers and command games," Games and Economic Behavior, Elsevier, vol. 72(1), pages 123-138, May.
    4. Hsiao, Chih-Ru & Raghavan, T E S, 1992. "Monotonicity and Dummy Free Property for Multi-choice Cooperative Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(3), pages 301-312.
    5. Marc Roubens & Michel Grabisch, 1999. "An axiomatic approach to the concept of interaction among players in cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(4), pages 547-565.
    6. Ulrich Faigle & Michel Grabisch, 2016. "Bases and linear transforms of TU-games and cooperation systems," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(4), pages 875-892, November.
    7. Ulrich Faigle & Michel Grabisch, 2012. "Values for Markovian coalition processes," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 51(3), pages 505-538, November.
    8. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
    9. Jean-Pierre Aubin, 1981. "Cooperative Fuzzy Games," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 1-13, February.
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    Cited by:

    1. Masih Fadaki & Babak Abbasi & Prem Chhetri, 2022. "Quantum game approach for capacity allocation decisions under strategic reasoning," Computational Management Science, Springer, vol. 19(3), pages 491-512, July.

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    More about this item

    Keywords

    cooperative game; decision system; evolution; Fourier transform; interaction system; measurement; quantum game;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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