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Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations

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  • Guoqiang Tian

Abstract

This paper provides necessary and sufficient conditions for the existence of greatest and maximal elements of weak and strict preferences, and unifies two very different approaches used in the related literature (the convexity and acyclicity approaches). Conditions called transfer FS-convexity and transfer SS-convexity are shown to be necessary and, in conjunction with transfer closedness and transfer openness, sufficient for the existence of greatest and maximal elements of weak and strict preferences, respectively. The results require neither the continuity nor convexity of preferences, and are valid for both ordered and unordered binary relations. Thus, the results generalize almost all of the theorems on existence of maximal elements of preferences that appear in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:oup:restud:v:60:y:1993:i:4:p:949-958.
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    Citations

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    Cited by:

    1. Guilherme Carmona, 2011. "Understanding some recent existence results for discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(1), pages 31-45, September.
    2. Tian, Guoqiang, 2015. "On the existence of equilibria in games with arbitrary strategy spaces and preferences," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 9-16.
    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. Leandro Nascimento, 2011. "Remarks on the consumer problem under incomplete preferences," Theory and Decision, Springer, vol. 70(1), pages 95-110, January.
    5. Tian, Guoqiang, 2012. "A Full Characterization on Fixed-Point Theorem, Minimax Inequality, Saddle Point, and KKM Theorem," MPRA Paper 57929, University Library of Munich, Germany, revised Jul 2014.
    6. 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.
    7. Rabia Nessah & Guoqiang Tian, 2008. "The Existence of Equilibria in Discontinuous and Nonconvex Games," Working Papers 2008-ECO-14, IESEG School of Management, revised Mar 2010.
    8. Nessah, Rabia & Tazdaı¨t, Tarik, 2013. "Absolute optimal solution for a compact and convex game," European Journal of Operational Research, Elsevier, vol. 224(2), pages 353-361.
    9. 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.
    10. Tieying Huang & Jiuqiang Liu, 2022. "Fuzzy Strong Nash Equilibria in Generalized Fuzzy Games with Application in Urban Public-Sports Services," Mathematics, MDPI, vol. 10(20), pages 1-10, October.
    11. 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.
    12. Rabia Nessah & Raluca Parvulescu, 2017. "On the Existence of Pareto Efficient Nash Equilibria in Discontinuous Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-13, September.
    13. Vincenzo Scalzo, 2022. "Existence of alpha-core allocations in economies with non-ordered and discontinuous preferences," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(1), pages 1-12, May.
    14. Rabia Nessah & Guoqiang Tian, 2016. "On the existence of Nash equilibrium in discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 515-540, March.
    15. Tian, Guoqiang, 1994. "Generalized KKM theorem, minimax inequalities and their applications," MPRA Paper 41217, University Library of Munich, Germany.
    16. Guoqiang Tian, 2016. "On the existence of price equilibrium in economies with excess demand functions," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(1), pages 5-16, April.
    17. Nessah, Rabia & Tian, Guoqiang, 2008. "Existence of Equilibria in Discontinuous Games," MPRA Paper 41206, University Library of Munich, Germany, revised Mar 2010.
    18. Llinares, Juan-Vicente & Sanchez, M. Carmen, 1999. "Non-binary choice functions on non-compact sets," Economics Letters, Elsevier, vol. 63(1), pages 29-32, April.
    19. Nehring, Klaus, 1996. "Maximal elements of non-binary choice functions on compact sets," Economics Letters, Elsevier, vol. 50(3), pages 337-340, March.

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