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Quantum version of Aumann’s approach to common knowledge: Sufficient conditions of impossibility to agree on disagree

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  • Khrennikov, Andrei

Abstract

Aumann’s theorem states that if two agents with classical processing of information (and, in particular, the Bayesian update of probabilities) have the common priors, and their posteriors for a given event E are common knowledge, then their posteriors must be equal; agents with the same priors cannot agree to disagree. This theorem is of the fundamental value for theory of information and knowledge and it has numerous applications in economics and social science. Recently a quantum-like version of such theory was presented in Khrennikov and Basieva (2014b), where it was shown that, for agents with quantum information processing (and, in particular, the quantum update of probabilities), in general Aumann’s theorem is not valid. In this paper we present conditions on the inter-relations of the information representations of agents, their common prior state, and an event which imply validity of Aumann’s theorem. Thus we analyze conditions implying the impossibility to agree on disagree even for quantum-like agents. Here we generalize the original Aumann approach to common knowledge to the quantum case (in Khrennikov and Basieva (2014b) we used the iterative operator approach due to Brandenburger and Dekel and Monderer and Samet). Examples of applicability and non-applicability of the derived sufficient conditions for validity of Aumann’s theorem for quantum(-like) agents are presented.

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  • Khrennikov, Andrei, 2015. "Quantum version of Aumann’s approach to common knowledge: Sufficient conditions of impossibility to agree on disagree," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 89-104.
  • Handle: RePEc:eee:mateco:v:60:y:2015:i:c:p:89-104
    DOI: 10.1016/j.jmateco.2015.06.018
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    1. Diederik Aerts & Emmanuel Haven & Sandro Sozzo, 2018. "A proposal to extend expected utility in a quantum probabilistic framework," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 65(4), pages 1079-1109, June.
    2. Diederik Aerts & Emmanuel Haven & Sandro Sozzo, 2016. "A Proposal to Extend Expected Utility in a Quantum Probabilistic Framework," Papers 1612.08583, arXiv.org.
    3. Ismaël Rafaï & Sébastien Duchêne & Eric Guerci & Irina Basieva & Andrei Khrennikov, 2022. "The triple-store experiment: a first simultaneous test of classical and quantum probabilities in choice over menus," Theory and Decision, Springer, vol. 92(2), pages 387-406, March.
    4. Aerts, Diederik & Geriente, Suzette & Moreira, Catarina & Sozzo, Sandro, 2018. "Testing ambiguity and Machina preferences within a quantum-theoretic framework for decision-making," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 176-185.
    5. Basieva, Irina & Khrennikova, Polina & Pothos, Emmanuel M. & Asano, Masanari & Khrennikov, Andrei, 2018. "Quantum-like model of subjective expected utility," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 150-162.
    6. Ismaël Rafaï & Sébastien Duchêne & Eric Guerci & Ariane Lambert-Mogiliansky & Fabien Mathy, 2016. "A Dual Process in Memory: How to Make an Evaluation from Complex and Complete Information? An Experimental Study," GREDEG Working Papers 2016-23, Groupe de REcherche en Droit, Economie, Gestion (GREDEG CNRS), Université Côte d'Azur, France, revised Jan 2018.
    7. Patricia Contreras-Tejada & Giannicola Scarpa & Aleksander M. Kubicki & Adam Brandenburger & Pierfrancesco La Mura, 2021. "Observers of quantum systems cannot agree to disagree," Nature Communications, Nature, vol. 12(1), pages 1-7, December.
    8. Haven, Emmanuel & Khrennikova, Polina, 2018. "A quantum-probabilistic paradigm: Non-consequential reasoning and state dependence in investment choice," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 186-197.
    9. Khrennikova, Polina & Patra, Sudip, 2019. "Asset trading under non-classical ambiguity and heterogeneous beliefs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 562-577.

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