IDEAS home Printed from https://ideas.repec.org/a/spr/etbull/v13y2025i1d10.1007_s40505-025-00287-z.html
   My bibliography  Save this article

Alternative proofs of the GNKD and KKMS lemmas: a game-theoretic underpinning

Author

Listed:
  • M. Ali Khan

    (Johns Hopkins University)

  • Richard P. McLean

    (Rutgers University)

  • Metin Uyanik

    (University of Queensland)

Abstract

In this paper, we provide alternative proofs of fundamental results of Walrasian general equilibrium theory, and of cooperative game theory, by viewing them as results pertaining to a two-person qualitative or a one-person generalized game: in short, by giving them a game-theoretic underpinning. Under the first category, we have the Gale–Nikaido–Kuhn–Debreu (GNKD) lemma in mind, and under the second, the Knaster–Kuratowski–Mazurkiewicz–Shapley (KKMS) lemma.

Suggested Citation

  • M. Ali Khan & Richard P. McLean & Metin Uyanik, 2025. "Alternative proofs of the GNKD and KKMS lemmas: a game-theoretic underpinning," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 13(1), pages 1-20, April.
    • Handle: RePEc:spr:etbull:v:13:y:2025:i:1:d:10.1007_s40505-025-00287-z
    DOI: 10.1007/s40505-025-00287-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s40505-025-00287-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s40505-025-00287-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    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. 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.
    4. M. Ali Khan & Metin Uyanik, 2021. "The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 799-840, April.
    5. Florenzano, Monique, 2009. "Two lemmas that changed general equilibrium theory," Games and Economic Behavior, Elsevier, vol. 66(2), pages 603-605, July.
    6. Hervés-Beloso, Carlos & Moreno-García, Emma, 2009. "Large economies and two-player games," Journal of Mathematical Economics, Elsevier, vol. 45(9-10), pages 603-608, September.
    7. Khan, M. Ali & McLean, Richard P. & Uyanik, Metin, 2024. "On constrained generalized games with action sets in non-locally-convex and non-Hausdorff topological vector spaces," Journal of Mathematical Economics, Elsevier, vol. 111(C).
    8. Philip J. Reny, 2020. "Nash Equilibrium in Discontinuous Games," Annual Review of Economics, Annual Reviews, vol. 12(1), pages 439-470, August.
    9. 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.
    10. P. Jean-Jacques Herings, 1997. "An extremely simple proof of the K-K-M-S Theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(2), pages 361-367.
    11. 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.
    12. Philippe Bich & Bernard Cornet, 2004. "Fixed-point-like theorems on subspaces," Post-Print halshs-03330770, HAL.
    13. Reny, Philip J. & Holtz Wooders, Myrna, 1998. "An extension of the KKMS theorem," Journal of Mathematical Economics, Elsevier, vol. 29(2), pages 125-134, March.
    14. 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.
    15. 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.
    16. Yannelis, Nicholas C., 1985. "Maximal elements over non-compact subsets of linear topological spaces," Economics Letters, Elsevier, vol. 17(1-2), pages 133-136.
    17. M. Ali Khan, 2021. "On the Finding of an Equilibrium: Düppe–Weintraub and the Problem of Scientific Credit," Journal of Economic Literature, American Economic Association, vol. 59(2), pages 590-633, June.
    18. Shapley, Lloyd & Vohra, Rajiv, 1991. "On Kakutani's Fixed Point Theorem, the K-K-M-S Theorem and the Core of a Balanced Game," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(1), pages 108-116, January.
    19. McCabe, Peter J., 1981. "On two market equilibrium theorems," Journal of Mathematical Economics, Elsevier, vol. 8(2), pages 167-171, July.
    20. Philippe Bich & Bernard Cornet, 2004. "Fixed-point-like theorems on subspaces," Cahiers de la Maison des Sciences Economiques b04064, Université Panthéon-Sorbonne (Paris 1).
    21. Ichiishi, Tatsuro & Idzik, Adam, 1991. "Closed Covers of Compact Convex Polyhedra," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 161-169.
    22. Greenberg, Joseph & Shitovitz, Benyamin & Wieczorek, Andrzej, 1979. "Existence of equilibria in atomless production economies with price dependent preferences," Journal of Mathematical Economics, Elsevier, vol. 6(1), pages 31-41, March.
    23. 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.
    24. Bidard, Christian & Hosoda, Eiji, 1987. "On Consumption Baskets in a Generalized von Neumann Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 28(2), pages 509-519, June.
    25. Philippe Bich & Bernard Cornet, 2004. "Fixed-point-like theorems on subspaces," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-03330770, HAL.
    26. Komiya, Hidetoshi, 1994. "A Simple Proof of K-K-M-S Theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(3), pages 463-466, May.
    27. Zhou Lin, 1994. "A New Bargaining Set of an N-Person Game and Endogenous Coalition Formation," Games and Economic Behavior, Elsevier, vol. 6(3), pages 512-526, May.
    28. Motoki Otsuka, 2024. "The existence of Walrasian equilibrium: infinitely many commodities, measure space of agents, and discontinuous preferences," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 12(2), pages 119-140, December.
    29. Bergstrom, Theodore C. & Parks, Robert P. & Rader, Trout, 1976. "Preferences which have open graphs," Journal of Mathematical Economics, Elsevier, vol. 3(3), pages 265-268, December.
    Full references (including those not matched with items on IDEAS)

    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. M. Ali Khan & Metin Uyanik, 2021. "The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 799-840, April.
    2. 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.
    3. Bhowmik, Anuj & Yannelis, Nicholas C., 2024. "Equilibria in abstract economies with a continuum of agents with discontinuous and non-ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 115(C).
    4. Jean Guillaume Forand & Metin Uyanık, 2019. "Fixed-point approaches to the proof of the Bondareva–Shapley Theorem," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(1), pages 117-124, May.
    5. 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.
    6. Aouani, Zaier & Cornet, Bernard, 2009. "Existence of financial equilibria with restricted participation," Journal of Mathematical Economics, Elsevier, vol. 45(12), pages 772-786, December.
    7. Azrieli, Yaron & Shmaya, Eran, 2014. "Rental harmony with roommates," Journal of Economic Theory, Elsevier, vol. 153(C), pages 128-137.
    8. Bernard Cornet & Ramu Gopalan, 2010. "Arbitrage and equilibrium with portfolio constraints," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(1), pages 227-252, October.
    9. Philippe Bich & Bernard Cornet, 2009. "Existence of pseudo-equilibria in a financial economy," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00426399, HAL.
    10. Gourdel, Pascal & Le Van, Cuong & Pham, Ngoc-Sang & Tran Viet, Cuong, 2023. "Hartman-Stampacchia theorem, Gale-Nikaido-Debreu lemma, and Brouwer and Kakutani fixed-point theorems," MPRA Paper 116541, University Library of Munich, Germany.
    11. Khan, M. Ali & McLean, Richard P. & Uyanik, Metin, 2024. "On constrained generalized games with action sets in non-locally-convex and non-Hausdorff topological vector spaces," Journal of Mathematical Economics, Elsevier, vol. 111(C).
    12. Florenzano Monique, 1991. "Quasiequilibria in abstract economies application to the overlapping generations models," CEPREMAP Working Papers (Couverture Orange) 9117, CEPREMAP.
    13. Aouani, Zaier & Cornet, Bernard, 2011. "Reduced equivalent form of a financial structure," Journal of Mathematical Economics, Elsevier, vol. 47(3), pages 318-327.
    14. Andrikopoulos, Athanasios & Zacharias, Eleftherios, 2008. "General solutions for choice sets: The Generalized Optimal-Choice Axiom set," MPRA Paper 11645, University Library of Munich, Germany.
    15. Le, Thanh & Le Van, Cuong & Pham, Ngoc-Sang & Sağlam, Çağrı, 2020. "Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma," MPRA Paper 110933, University Library of Munich, Germany, revised 28 Oct 2020.
    16. Bagh, Adib, 1998. "Equilibrium in abstract economies without the lower semi-continuity of the constraint maps," Journal of Mathematical Economics, Elsevier, vol. 30(2), pages 175-185, September.
    17. 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.
    18. 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).
    19. 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.
    20. Liu, Jiuqiang & Tian, Hai-Yan, 2014. "Existence of fuzzy cores and generalizations of the K–K–M–S theorem," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 148-152.

    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:spr:etbull:v:13:y:2025:i:1:d:10.1007_s40505-025-00287-z. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.