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General conditions for the existence of maximal elements via the uncovered set

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  • Duggan, John

Abstract

This paper disentangles the topological assumptions of classical results (e.g.,Walker, 1977 on the existence of maximal elements from rationality conditions. It is known from the social choice literature that under the standard topological conditions—with no rationality assumptions on preferences—there is an element such that the upper section of strict preference at that element is minimal in terms of set inclusion, i.e., the uncovered set is nonempty. Assuming the finite subordination property, a condition that weakens known acyclicity and convexity assumptions, each such uncovered alternative is in fact maximal. Implications are a generalization of a result of Yannelis and Prabhakar (1983) on semi-convexity, an extension of Fan’s (1961) lemma on KKM correspondences, and the existence of fixed points for subordinate convex correspondences generalizing the work of Browder (1968).

Suggested Citation

  • Duggan, John, 2011. "General conditions for the existence of maximal elements via the uncovered set," Journal of Mathematical Economics, Elsevier, vol. 47(6), pages 755-759.
  • Handle: RePEc:eee:mateco:v:47:y:2011:i:6:p:755-759 DOI: 10.1016/j.jmateco.2011.09.008
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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. Schmeidler, David, 1969. "Competitive Equilibria in Markets with a Continuum of Traders and Incomplete Preferences," Econometrica, Econometric Society, vol. 37(4), pages 578-585, October.
    3. Shafer, Wayne & Sonnenschein, Hugo, 1975. "Equilibrium in abstract economies without ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 345-348, December.
    4. Alcantud, Jose C.R., 2006. "Maximality with or without binariness: Transfer-type characterizations," Mathematical Social Sciences, Elsevier, vol. 51(2), pages 182-191, March.
    5. Schofield, Norman, 1984. "Social equilibrium and cycles on compact sets," Journal of Economic Theory, Elsevier, pages 59-71.
    6. Tian, Guoqiang & Zhou, Jianxin, 1995. "Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 281-303.
    7. Banks, Jeffrey S. & Duggan, John & Le Breton, Michel, 2002. "Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections," Journal of Economic Theory, Elsevier, vol. 103(1), pages 88-105, March.
    8. Campbell, Donald E. & Walker, Mark, 1990. "Maximal elements of weakly continuous relations," Journal of Economic Theory, Elsevier, vol. 50(2), pages 459-464, April.
    9. Pavlo Prokopovych, 2011. "On equilibrium existence in payoff secure games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(1), pages 5-16, September.
    10. Banks, Jeffrey S. & Duggan, John & Le Breton, Michel, 2006. "Social choice and electoral competition in the general spatial model," Journal of Economic Theory, Elsevier, vol. 126(1), pages 194-234, January.
    11. Nehring, Klaus, 1996. "Maximal elements of non-binary choice functions on compact sets," Economics Letters, Elsevier, vol. 50(3), pages 337-340, March.
    12. Yannelis, Nicholas C., 1985. "Maximal elements over non-compact subsets of linear topological spaces," Economics Letters, Elsevier, vol. 17(1-2), pages 133-136.
    13. Bergstrom, Theodore C., 1975. "Maximal elements of acyclic relations on compact sets," Journal of Economic Theory, Elsevier, vol. 10(3), pages 403-404, June.
    14. Peris, Josep E. & Subiza, Begona, 1994. "Maximal elements of not necessarily acyclic binary relations," Economics Letters, Elsevier, vol. 44(4), pages 385-388, April.
    15. John Duggan, 2011. "Uncovered Sets," Wallis Working Papers WP63, University of Rochester - Wallis Institute of Political Economy.
    16. Horvath, Charles D. & Ciscar, Juan Vicente Llinares, 1996. "Maximal elements and fixed points for binary relations on topological ordered spaces," Journal of Mathematical Economics, Elsevier, vol. 25(3), pages 291-306.
    17. Mas-Colell, Andrew, 1974. "An equilibrium existence theorem without complete or transitive preferences," Journal of Mathematical Economics, Elsevier, vol. 1(3), pages 237-246, December.
    18. Walker, Mark, 1977. "On the existence of maximal elements," Journal of Economic Theory, Elsevier, vol. 16(2), pages 470-474, December.
    19. Donald J. Brown, 1973. "Acyclic Choice," Cowles Foundation Discussion Papers 360, Cowles Foundation for Research in Economics, Yale University.
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    Cited by:

    1. Barelli, Paulo & Duggan, John, 2015. "Extremal choice equilibrium with applications to large games, stochastic games, & endogenous institutions," Journal of Economic Theory, Elsevier, vol. 155(C), pages 95-130.

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