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On the non-emptiness of the fuzzy core

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  • Nizar Allouch
  • Arkadi Predtetchinski

Abstract

The seminal contribution of Debreu-Scarf (1963) connects the two concepts of core and competitive equilibrium in exchange economies. In effect, their core-equilibrium equivalence result states that, when the set of economic agents is replicated, the set of core allocations of the replica economy shrinks to the set of competitive allocations. Florenzano (1989) defines the fuzzy core as the set of allocations which cannot be blocked by any coalition with an arbitrary rate of participation and then shows the asymptotic limit of cores of replica economics coincides with the fuzzy core. In this note, we provide an elementary proof of the non-emptiness of the fuzzy core for an exchange economy. Unlike the classical Debreu-Scarf limit theorem and its numerous extensions our result does not require any asymptotic intersection -or limit- of the set of core allocations of replica economies.
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Suggested Citation

  • Nizar Allouch & Arkadi Predtetchinski, 2008. "On the non-emptiness of the fuzzy core," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(2), pages 203-210, June.
    • Handle: RePEc:spr:jogath:v:37:y:2008:i:2:p:203-210
    DOI: 10.1007/s00182-007-0105-2
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    Cited by:

    1. Nizar Allouch & Myrna Wooders, 2017. "On the nonemptiness of approximate cores of large games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(1), pages 191-209, January.
    2. Liu, Jiuqiang & Tian, Hai-Yan, 2014. "Existence of fuzzy cores and generalizations of the K–K–M–S theorem," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 148-152.
    3. Chiara Donnini & Marialaura Pesce, 2021. "Fairness and fuzzy coalitions," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(4), pages 1033-1052, December.
    4. Jiuqiang Liu & Xiaodong Liu, 2014. "Existence of Edgeworth and competitive equilibria and fuzzy cores in coalition production economies," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(4), pages 975-990, November.
    5. Liu, Jiuqiang & Liu, Xiaodong, 2013. "A necessary and sufficient condition for an NTU fuzzy game to have a non-empty fuzzy core," Journal of Mathematical Economics, Elsevier, vol. 49(2), pages 150-156.
    6. Maria Gabriella Graziano & Marialaura Pesce & Maria Romaniello, 2021. "Two characterizations of Cost Share Equilibria," CSEF Working Papers 628, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy, revised 24 May 2023.

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    More about this item

    Keywords

    Fuzzy core; Payoff-dependent balancedness; Exchange economies; D51; C71;
    All these keywords.

    JEL classification:

    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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