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Weak maximal elements and weak equilibria in ordinal games with applications to exchange economies

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  • Vincenzo Scalzo

    (University of Naples Federico II)

Abstract

We study binary relations (preferences) and ordinal games in the case where no continuity-like properties are assumed at all. We introduce generalizations of the maximal element and Nash equilibrium, called, respectively, the weak maximal element and weak equilibrium, and give existence results when binary relations satisfy only convexity conditions. The weak maximal element (weak equilibrium) is equivalent to the maximal element (Nash equilibrium) if and only if a generalization of continuity is given. Moreover, we obtain the existence of quasi-Pareto optimal allocations in exchange economies.

Suggested Citation

  • Vincenzo Scalzo, 2018. "Weak maximal elements and weak equilibria in ordinal games with applications to exchange economies," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 6(1), pages 29-39, April.
    • Handle: RePEc:spr:etbull:v:6:y:2018:i:1:d:10.1007_s40505-017-0121-8
    DOI: 10.1007/s40505-017-0121-8
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    References listed on IDEAS

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    1. Yannelis, Nicholas C. & Prabhakar, N. D., 1983. "Existence of maximal elements and equilibria in linear topological spaces," Journal of Mathematical Economics, Elsevier, vol. 12(3), pages 233-245, December.
    2. Pavlo Prokopovych, 2016. "Majorized correspondences and equilibrium existence in discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 541-552, March.
    3. Philip J. Reny, 2020. "Nash Equilibrium in Discontinuous Games," Annual Review of Economics, Annual Reviews, vol. 12(1), pages 439-470, August.
    4. Shafer, Wayne & Sonnenschein, Hugo, 1975. "Equilibrium in abstract economies without ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 345-348, December.
    5. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
    6. Guilherme Carmona & Konrad Podczeck, 2016. "Existence of Nash equilibrium in ordinal games with discontinuous preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 457-478, March.
    7. Vincenzo Scalzo, 2016. "Remarks on the existence and stability of some relaxed Nash equilibrium in strategic form games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 571-586, March.
    8. Michael R. Baye & Guoqiang Tian & Jianxin Zhou, 1993. "Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 60(4), pages 935-948.
    9. Pavlo Prokopovych, 2013. "The single deviation property in games with discontinuous payoffs," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 53(2), pages 383-402, June.
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    More about this item

    Keywords

    Binary relations; Preference relations; Weak maximal element; Ordinal games; Weak equilibrium; Exchange economies; Quasi-Pareto optimality;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies

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