IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2006.16681.html
   My bibliography  Save this paper

The Yannelis-Prabhakar Theorem on Upper Semi-Continuous Selections in Paracompact Spaces: Extensions and Applications

Author

Listed:
  • M. Ali Khan
  • Metin Uyanik

Abstract

In a 1983 paper, Yannelis-Prabhakar rely on Michael's selection theorem to guarantee a continuous selection in the context of the existence of maximal elements and equilibria in abstract economies. In this tribute to Nicholas Yannelis, we root this paper in Chapter II of Yannelis' 1983 Rochester Ph.D. dissertation, and identify its pioneering application of the paracompactness condition to current and ongoing work of Yannelis and his co-authors, and to mathematical economics more generally. We move beyond the literature to provide a necessary and sufficient condition for upper semi-continuous local and global selections of correspondences, and to provide application to five domains of Yannelis' interests: Berge's maximum theorem, the Gale-Nikaido-Debreu lemma, the Gale-McKenzie survival assumption, Shafer's non-transitive setting, and the Anderson-Khan-Rashid approximate existence theorem. The last resonates with Chapter VI of the Yannelis' dissertation.

Suggested Citation

  • M. Ali Khan & Metin Uyanik, 2020. "The Yannelis-Prabhakar Theorem on Upper Semi-Continuous Selections in Paracompact Spaces: Extensions and Applications," Papers 2006.16681, arXiv.org.
    • Handle: RePEc:arx:papers:2006.16681
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2006.16681
    File Function: Latest version
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    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. Konard Podczeck, 1997. "Markets with infinitely many commodities and a continuum of agents with non-convex preferences (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(3), pages 385-426.
    3. 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.
    4. Prokopovych, Pavlo & Yannelis, Nicholas C., 2014. "On the existence of mixed strategy Nash equilibria," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 87-97.
    5. M. Ali Khan & Metin Uyanık, 2021. "Topological connectedness and behavioral assumptions on preferences: a two-way relationship," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(2), pages 411-460, March.
    6. Gale, D. & Mas-Colell, A., 1975. "An equilibrium existence theorem for a general model without ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 2(1), pages 9-15, March.
    7. Prokopovych, Pavlo & Yannelis, Nicholas C., 2017. "On strategic complementarities in discontinuous games with totally ordered strategies," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 147-153.
    8. Paulo Barelli & Idione Meneghel, 2013. "A Note on the Equilibrium Existence Problem in Discontinuous Games," Econometrica, Econometric Society, vol. 81(2), pages 813-824, March.
    9. Philip J. Reny, 2016. "Introduction to the symposium on discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 423-429, March.
    10. Cornet, Bernard, 1988. "General equilibrium theory and increasing returns : Presentation," Journal of Mathematical Economics, Elsevier, vol. 17(2-3), pages 103-118, April.
    11. Shafer, Wayne & Sonnenschein, Hugo, 1975. "Equilibrium in abstract economies without ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 345-348, December.
    12. 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.
    13. Majumdar, Mukul, 2009. "Equilibrium and optimality: Some imprints of David Gale," Games and Economic Behavior, Elsevier, vol. 66(2), pages 607-626, July.
    14. William Geller, 1986. "An Improved Bound for Approximate Equilibria," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(2), pages 307-308.
    15. Philip J. Reny, 2016. "Equilibrium in discontinuous games without complete or transitive preferences," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(1), pages 1-4, April.
    16. Morgan, Jacqueline & Scalzo, Vincenzo, 2007. "Pseudocontinuous functions and existence of Nash equilibria," Journal of Mathematical Economics, Elsevier, vol. 43(2), pages 174-183, February.
    17. Shafer, Wayne & Sonnenschein, Hugo, 1975. "Some theorems on the existence of competitive equilibrium," Journal of Economic Theory, Elsevier, vol. 11(1), pages 83-93, August.
    18. Scapparone, Paolo, 2015. "Existence of an upper hemi-continuous and convex-valued demand sub-correspondence," Mathematical Social Sciences, Elsevier, vol. 75(C), pages 123-129.
    19. Paul A. Samuelson, 2011. "The Collected Scientific Papers of Paul Samuelson," MIT Press Books, The MIT Press, edition 1, volume 7, number 0262015749 edited by Janice Murray, December.
    20. John Maynard Keynes, 2010. "Essays in Biography," Palgrave Macmillan Books, Palgrave Macmillan, number 978-1-349-59074-2.
    21. 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.
    22. M. Ali Khan & Metin Uyanik, 2019. "On an Extension of a Theorem of Eilenberg and a Characterization of Topological Connectedness," Papers 1912.12787, arXiv.org.
    23. McCabe, Peter J., 1981. "On two market equilibrium theorems," Journal of Mathematical Economics, Elsevier, vol. 8(2), pages 167-171, July.
    24. Mehta, Ghanshyam & Tarafdar, Enayet, 1987. "Infinite-dimensional Gale-Nikaido-Debreu theorem and a fixed-point theorem of Tarafdar," Journal of Economic Theory, Elsevier, vol. 41(2), pages 333-339, April.
    25. Toussaint, Sabine, 1984. "On the existence of equilibria in economies with infinitely many commodities and without ordered preferences," Journal of Economic Theory, Elsevier, vol. 33(1), pages 98-115, June.
    26. Schmeidler, David, 1971. "A Condition for the Completeness of Partial Preference Relations," Econometrica, Econometric Society, vol. 39(2), pages 403-404, March.
    27. 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.
    28. James C. Moore, 2007. "General Equilibrium and Welfare Economics," Springer Books, Springer, number 978-3-540-32223-8, December.
    29. Tarafdar, E., 1991. "A fixed point theorem and equilibrium point of an abstract economy," Journal of Mathematical Economics, Elsevier, vol. 20(2), pages 211-218.
    30. Aliprantis, Charalambos D. & Brown, Donald J., 1983. "Equilibria in markets with a Riesz space of commodities," Journal of Mathematical Economics, Elsevier, vol. 11(2), pages 189-207, April.
    31. Geistdoerfer-Florenzano, Monique, 1982. "The gale-nikaido-debreu lemma and the existence of transitive equilibrium with or without the free-disposal assumption," Journal of Mathematical Economics, Elsevier, vol. 9(1-2), pages 113-134, January.
    32. Oriol Carbonell-Nicolau & Richard P. McLean, 2019. "Nash and Bayes–Nash equilibria in strategic-form games with intransitivities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(4), pages 935-965, November.
    33. Philip J. Reny, 2020. "Nash Equilibrium in Discontinuous Games," Annual Review of Economics, Annual Reviews, vol. 12(1), pages 439-470, August.
    34. 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.
    35. Florenzano, Monigue & Le Van, Cuong, 1986. "A note on the Gale-Nikaido-Debreu lemma and the existence of general equilibrium," Economics Letters, Elsevier, vol. 22(2-3), pages 107-110.
    36. Oriol Carbonell-Nicolau & Richard McLean, 2013. "Approximation results for discontinuous games with an application to equilibrium refinement," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(1), pages 1-26, September.
    37. Tian, Guoqiang & Zhou, Jianxin, 1992. "Transfer Method for Characterizing the Existence of Maximal Elements of Binary Relations on Compact or Noncompact Sets," MPRA Paper 41227, University Library of Munich, Germany.
    38. Khan, M. Ali & Vohra, Rajiv, 1984. "Equilibrium in abstract economies without ordered preferences and with a measure space of agents," Journal of Mathematical Economics, Elsevier, vol. 13(2), pages 133-142, October.
    39. Nicholas Yannelis, 2009. "Debreu’s social equilibrium theorem with asymmetric information and a continuum of agents," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 38(2), pages 419-432, February.
    40. Eric Maskin & Kevin Roberts, 2008. "On the fundamental theorems of general equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 35(2), pages 233-240, May.
    41. McKenzie, Lionel W, 1981. "The Classical Theorem on Existence of Competitive Equilibrium," Econometrica, Econometric Society, vol. 49(4), pages 819-841, June.
    42. Anderson, Robert M, 1978. "An Elementary Core Equivalence Theorem," Econometrica, Econometric Society, vol. 46(6), pages 1483-1487, November.
    43. 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.
    44. Arrow, Kenneth J, 1994. "Methodological Individualism and Social Knowledge," American Economic Review, American Economic Association, vol. 84(2), pages 1-9, May.
    45. Wei He & Nicholas C. Yannelis, 2016. "Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 497-513, March.
    46. Florenzano Monique & Le Van Cuong, 1986. "Gale-nikaido-debreu lemma and the existence of general equilibrium (the)," CEPREMAP Working Papers (Couverture Orange) 8603, CEPREMAP.
    47. Gale, D. & Mas-Colell, A., 1979. "Corrections to an equilibrium existence theorem for a general model without ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 6(3), pages 297-298, December.
    48. Hidetoshi Komiya, 1997. "Inverse of the Berge maximum theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(2), pages 371-375.
    49. Borglin, Anders & Keiding, Hans, 1976. "Existence of equilibrium actions and of equilibrium : A note on the `new' existence theorems," Journal of Mathematical Economics, Elsevier, vol. 3(3), pages 313-316, December.
    50. Shafer, Wayne J., 1976. "Equilibrium in economies without ordered preferences or free disposal," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 135-137, July.
    51. Dutta, Prajit K. & Mitra, Tapan, 1989. "Maximum theorems for convex structures with an application to the theory of optimal intertemporal allocation," Journal of Mathematical Economics, Elsevier, vol. 18(1), pages 77-86, February.
    52. Ali Khan, M. & Rashid, Salim, 1982. "Approximate equilibria in markets with indivisible commodities," Journal of Economic Theory, Elsevier, vol. 28(1), pages 82-101, October.
    53. Galaabaatar, Tsogbadral & Khan, M. Ali & Uyanık, Metin, 2019. "Completeness and transitivity of preferences on mixture sets," Mathematical Social Sciences, Elsevier, vol. 99(C), pages 49-62.
    54. Vohra, Rajiv, 1992. "Marginal Cost Pricing under Bounded Marginal Returns," Econometrica, Econometric Society, vol. 60(4), pages 859-876, July.
    55. Shafer, Wayne J, 1974. "The Nontransitive Consumer," Econometrica, Econometric Society, vol. 42(5), pages 913-919, September.
    56. Robert M. Anderson & M. Ali Khan & Salim Rashid, 1982. "Approximate Equilibria with Bounds Independent of Preferences," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 49(3), pages 473-475.
    57. Guilherme Carmona, 2011. "Symposium on: Existence of Nash equilibria in discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(1), pages 1-4, September.
    58. Yannelis, Nicholas C., 1985. "Maximal elements over non-compact subsets of linear topological spaces," Economics Letters, Elsevier, vol. 17(1-2), pages 133-136.
    59. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June.
    60. Khan, M. Ali, 1975. "Some approximate equilibria," Journal of Mathematical Economics, Elsevier, vol. 2(1), pages 63-86, March.
    61. Walker, Mark, 1979. "A Generalization of the Maximum Theorem," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 20(1), pages 267-272, February.
    62. Yannelis, Nicholas C., 1987. "Equilibria in noncooperative models of competition," Journal of Economic Theory, Elsevier, vol. 41(1), pages 96-111, February.
    63. Lionel McKenzie, 1999. "Equilibrium, Trade, and Capital Accumulation," The Japanese Economic Review, Japanese Economic Association, vol. 50(4), pages 371-397, December.
    64. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, December.
    65. Uyanık, Metin & Khan, M. Ali, 2019. "On the consistency and the decisiveness of the double-minded decision-maker," Economics Letters, Elsevier, vol. 185(C).
    66. Hiroaki Osana, 1973. "On the Boundedness of an Economy with Externalities," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 40(3), pages 321-331.
    67. Unknown, 1986. "Letters," Choices: The Magazine of Food, Farm, and Resource Issues, Agricultural and Applied Economics Association, vol. 1(4), pages 1-9.
    68. Abdellaoui, Mohammed & Wakker, Peter P., 2020. "Savage for dummies and experts," Journal of Economic Theory, Elsevier, vol. 186(C).
    69. 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.
    70. BEWLEY, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," LIDAM Reprints CORE 122, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    71. Yannelis, Nicholas C., 1990. "On the upper and lower semicontinuity of the Aumann integral," Journal of Mathematical Economics, Elsevier, vol. 19(4), pages 373-389.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Prokopovych, Pavlo & Yannelis, Nicholas C., 2023. "On monotone pure-strategy Bayesian-Nash equilibria of a generalized contest," Games and Economic Behavior, Elsevier, vol. 140(C), pages 348-362.
    2. Pavlo Prokopovych & Nicholas C. Yannelis, 2022. "On nondegenerate equilibria of double auctions with several buyers and a price floor," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 73(2), pages 625-654, April.
    3. Uyanik, Metin & Khan, M. Ali, 2022. "The continuity postulate in economic theory: A deconstruction and an integration," Journal of Mathematical Economics, Elsevier, vol. 101(C).
    4. Adolfo Pimienta & Manuel Sanchis, 2023. "Continuous Selections and Extremally Disconnected Spaces," Mathematics, MDPI, vol. 11(4), pages 1-8, February.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Charalambos Aliprantis & Kim Border & Owen Burkinshaw, 1996. "Market economies with many commodities," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 19(1), pages 113-185, March.
    2. Wei He & Nicholas C. Yannelis, 2016. "Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 497-513, March.
    3. Rabia Nessah, 2022. "Weakly continuous security and nash equilibrium," Theory and Decision, Springer, vol. 93(4), pages 725-745, November.
    4. Basci, Erdem & Sertel, Murat R., 1996. "Prakash and Sertel's theory of non-cooperative equilibria in social systems -- twenty years later," Journal of Mathematical Economics, Elsevier, vol. 25(1), pages 1-18.
    5. M. Ali Khan & Metin Uyanık, 2021. "Topological connectedness and behavioral assumptions on preferences: a two-way relationship," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(2), pages 411-460, March.
    6. Hichem Ben-El-Mechaiekh & Philippe Bich & Monique Florenzano, 2009. "General equilibrium and fixed-point theory: a partial survey," PSE-Ecole d'économie de Paris (Postprint) hal-00755998, HAL.
    7. Anderson, Robert M. & Duanmu, Haosui & Khan, M. Ali & Uyanik, Metin, 2022. "On abstract economies with an arbitrary set of players and action sets in locally-convex topological vector spaces," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    8. Charalambos Aliprantis & Rabee Tourky, 2009. "Equilibria in incomplete assets economies with infinite dimensional spot markets," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 38(2), pages 221-262, February.
    9. Prokopovych, Pavlo & Yannelis, Nicholas C., 2019. "On monotone approximate and exact equilibria of an asymmetric first-price auction with affiliated private information," Journal of Economic Theory, Elsevier, vol. 184(C).
    10. Oriol Carbonell-Nicolau & Richard P. McLean, 2019. "Nash and Bayes–Nash equilibria in strategic-form games with intransitivities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(4), pages 935-965, November.
    11. Konrad Podczeck & Nicholas C. Yannelis, 2022. "Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences, without free disposal, and without compact consumption sets," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 73(2), pages 413-420, April.
    12. 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.
    13. Florenzano Monique, 1991. "Quasiequilibria in abstract economies application to the overlapping generations models," CEPREMAP Working Papers (Couverture Orange) 9117, CEPREMAP.
    14. 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.
    15. 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.
    16. Scalzo, Vincenzo, 2020. "Doubly Strong Equilibrium," MPRA Paper 99329, University Library of Munich, Germany.
    17. Uyanik, Metin & Khan, M. Ali, 2022. "The continuity postulate in economic theory: A deconstruction and an integration," Journal of Mathematical Economics, Elsevier, vol. 101(C).
    18. Filipe Martins-da-Rocha, V. & Topuzu, Mihaela, 2008. "Cournot-Nash equilibria in continuum games with non-ordered preferences," Journal of Economic Theory, Elsevier, vol. 140(1), pages 314-327, May.
    19. 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.
    20. Maxfield, Robert R., 1997. "General equilibrium and the theory of directed graphs," Journal of Mathematical Economics, Elsevier, vol. 27(1), pages 23-51, February.

    More about this item

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
    • D62 - Microeconomics - - Welfare Economics - - - Externalities

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2006.16681. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.