IDEAS home Printed from https://ideas.repec.org/p/hal/journl/halshs-03220813.html
   My bibliography  Save this paper

Game Theoretic Interaction and Decision: A Quantum Analysis

Author

Listed:
  • 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," Post-Print halshs-03220813, HAL.
    • Handle: RePEc:hal:journl:halshs-03220813
    DOI: 10.3390/g8040048
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-03220813
    as

    Download full text from publisher

    File URL: https://shs.hal.science/halshs-03220813/document
    Download Restriction: no
    File URL: https://libkey.io/10.3390/g8040048?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Grabisch, Michel & Rusinowska, Agnieszka, 2011. "Influence functions, followers and command games," Games and Economic Behavior, Elsevier, vol. 72(1), pages 123-138, May.
    2. Christophe Labreuche & Michel Grabisch, 2008. "A value for bi-cooperative games," Post-Print halshs-00308738, HAL.
    3. Grabisch, Michel & Rusinowska, Agnieszka, 2013. "A model of influence based on aggregation functions," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 316-330.
    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. repec:hal:pseose:halshs-00749950 is not listed on IDEAS
    9. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
    10. repec:hal:pseose:halshs-00906367 is not listed on IDEAS
    11. Jean-Pierre Aubin, 1981. "Cooperative Fuzzy Games," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 1-13, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Fujimoto, Katsushige & Kojadinovic, Ivan & Marichal, Jean-Luc, 2006. "Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices," Games and Economic Behavior, Elsevier, vol. 55(1), pages 72-99, April.
    3. Brânzei, R. & Dimitrov, D.A. & Tijs, S.H., 2002. "Convex Fuzzy Games and Participation Monotonic Allocation Schemes," Discussion Paper 2002-13, Tilburg University, Center for Economic Research.
    4. Michel Grabisch & Agnieszka Rusinowska, 2010. "Different Approaches to Influence Based on Social Networks and Simple Games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00514850, HAL.
    5. Michel Grabisch & Fabien Lange, 2007. "Games on lattices, multichoice games and the shapley value: a new approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(1), pages 153-167, February.
    6. Förster, Manuel & Grabisch, Michel & Rusinowska, Agnieszka, 2013. "Anonymous social influence," Games and Economic Behavior, Elsevier, vol. 82(C), pages 621-635.
    7. Yan-An Hwang & Yu-Hsien Liao, 2010. "The unit-level-core for multi-choice games: the replicated core for TU games," Journal of Global Optimization, Springer, vol. 47(2), pages 161-171, June.
    8. Gerard van der Laan & René van den Brink, 2002. "A Banzhaf share function for cooperative games in coalition structure," Theory and Decision, Springer, vol. 53(1), pages 61-86, August.
    9. Michel Grabisch & Christophe Labreuche, 2010. "A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid," Annals of Operations Research, Springer, vol. 175(1), pages 247-286, March.
    10. Sébastien Courtin & Rodrigue Tido Takeng & Frédéric Chantreuil, 2020. "Decomposition of interaction indices: alternative interpretations of cardinal-probabilistic interaction indices ," Working Papers hal-02952516, HAL.
    11. Marichal, Jean-Luc & Mathonet, Pierre, 2011. "Weighted Banzhaf power and interaction indexes through weighted approximations of games," European Journal of Operational Research, Elsevier, vol. 211(2), pages 352-358, June.
    12. Molina, Elisenda & Tejada, Juan, 2004. "Linear production games with committee control: Limiting behaviour of the core," European Journal of Operational Research, Elsevier, vol. 154(3), pages 609-625, May.
    13. Ramón Flores & Elisenda Molina & Juan Tejada, 2014. "Pyramidal values," Annals of Operations Research, Springer, vol. 217(1), pages 233-252, June.
    14. Sascha Kurz, 2018. "Importance In Systems With Interval Decisions," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 21(06n07), pages 1-23, September.
    15. Conrado M. Manuel & Daniel Martín, 2020. "A Monotonic Weighted Shapley Value," Group Decision and Negotiation, Springer, vol. 29(4), pages 627-654, August.
    16. Molina, Elisenda & Tejada, Juan, 2013. "The Shapley group value," DES - Working Papers. Statistics and Econometrics. WS ws133430, Universidad Carlos III de Madrid. Departamento de Estadística.
    17. Ulrich Faigle & Michel Grabisch, 2014. "Bases and Linear Transforms of Cooperation Systems," Documents de travail du Centre d'Economie de la Sorbonne 14010r, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised May 2015.
    18. Calvo, Emilio & Santos, Juan Carlos, 2000. "A value for multichoice games," Mathematical Social Sciences, Elsevier, vol. 40(3), pages 341-354, November.
    19. Boonen, Tim J. & Tsanakas, Andreas & Wüthrich, Mario V., 2017. "Capital allocation for portfolios with non-linear risk aggregation," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 95-106.
    20. Conrado M. Manuel & Daniel Martín, 2021. "A Monotonic Weighted Banzhaf Value for Voting Games," Mathematics, MDPI, vol. 9(12), pages 1-23, June.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:journl:halshs-03220813. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.