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Unified treatment of the problem of existence of maximal elements in binary relations: a characterization

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  • Llinares, Juan-Vicente

Abstract

The aim of this paper is twofold. On the one hand to present by means of a unique statement an existence result which covers both ways (convexity and acyclicity) of analyzing the problem of existence of maximal elements of non transitive binary relations. And on the other hand, to introduce the concept of an abstract convexity structure, which we call mc-spaces, that generalizes the notion of usual convexity. It is presented as a powerful tool which allows many problems which have only been analyzed (previously) under convexity conditions to be solved.
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  • Llinares, Juan-Vicente, 1998. "Unified treatment of the problem of existence of maximal elements in binary relations: a characterization," Journal of Mathematical Economics, Elsevier, vol. 29(3), pages 285-302, April.
    • Handle: RePEc:eee:mateco:v:29:y:1998:i:3:p:285-302
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    1. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
    2. Bergstrom, Theodore C., 1975. "Maximal elements of acyclic relations on compact sets," Journal of Economic Theory, Elsevier, vol. 10(3), pages 403-404, June.
    3. Walker, Mark, 1977. "On the existence of maximal elements," Journal of Economic Theory, Elsevier, vol. 16(2), pages 470-474, December.
    4. 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.
    5. Guoqiang Tian, 1993. "Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 60(4), pages 949-958.
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    Cited by:

    1. Alcantud, J.C.R., 2008. "Mixed choice structures, with applications to binary and non-binary optimization," Journal of Mathematical Economics, Elsevier, vol. 44(3-4), pages 242-250, February.
    2. Francesco Ciardiello, 2007. "Convexity on Nash Equilibria without Linear Structure," Quaderni DSEMS 15-2007, Dipartimento di Scienze Economiche, Matematiche e Statistiche, Universita' di Foggia.
    3. M. Carmen Sánchez & Juan-Vicente Llinares & Begoña Subiza, 2003. "A KKM-result and an application for binary and non-binary choice functions," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 21(1), pages 185-193, January.
    4. Hougaard, Jens Leth & Tvede, Mich, 2001. "The existence of maximal elements: generalized lexicographic relations," Journal of Mathematical Economics, Elsevier, vol. 36(2), pages 111-115, November.
    5. Llinarès, Juan Vicente, 1998. "Existence of equilibrium in generalized games with non-convex strategy spaces," CEPREMAP Working Papers (Couverture Orange) 9801, CEPREMAP.
    6. J. V. Llinares, 2000. "Existence of Equilibrium in Generalized Games with Abstract Convexity Structure," Journal of Optimization Theory and Applications, Springer, vol. 105(1), pages 149-160, April.
    7. Llinarès, Juan Vicente, 1998. "Abstract convexity, some relations and applications," CEPREMAP Working Papers (Couverture Orange) 9803, CEPREMAP.
    8. Florian Brandl & Felix Brandt, 2020. "Arrovian Aggregation of Convex Preferences," Econometrica, Econometric Society, vol. 88(2), pages 799-844, March.

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