IDEAS home Printed from https://ideas.repec.org/p/unl/unlfep/wp442.html
   My bibliography  Save this paper

Nash and limit equilibria of games with a continuum of players

Author

Listed:
  • Guilherme Carmona

Abstract

We show that a strategy is a Nash equilibrium in a game with a continuum of players if and only if there exists a sequence of finite games such that its restriction is an "n-equilibria, with "n converging to zero. In our characterization, the sequence of finite games approaches the continuum game in the sense that the set of players and the distribution of characteristics and actions in the finite games converge to those of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance. Also, they suggest defining a refinement of Nash equilibria for games with a continuum of agents as limit points of equilibria of finite games. This allows us to discard those Nash equilibria that are artifacts of the continuum model, making limit equilibrium a natural equilibrium concept for games with a continuum of players.

Suggested Citation

  • Guilherme Carmona, 2004. "Nash and limit equilibria of games with a continuum of players," Nova SBE Working Paper Series wp442, Universidade Nova de Lisboa, Nova School of Business and Economics.
    • Handle: RePEc:unl:unlfep:wp442
    as

    Download full text from publisher

    File URL: https://run.unl.pt/bitstream/10362/83488/1/WP442.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Drew Fudenberg & David Levine, 2008. "Limit Games and Limit Equilibria," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 2, pages 21-39, World Scientific Publishing Co. Pte. Ltd..
    2. Khan, M. Ali & Rath, Kali P. & Sun, Yeneng, 1997. "On the Existence of Pure Strategy Equilibria in Games with a Continuum of Players," Journal of Economic Theory, Elsevier, vol. 76(1), pages 13-46, September.
    3. Postlewaite, A & Schmeidler, David, 1978. "Approximate Efficiency of Non-Walrasian Nash Equilibria," Econometrica, Econometric Society, vol. 46(1), pages 127-135, January.
    4. Hart, Sergiu & Hildenbrand, Werner & Kohlberg, Elon, 1974. "On equilibrium allocations as distributions on the commodity space," Journal of Mathematical Economics, Elsevier, vol. 1(2), pages 159-166, August.
    5. Barlo, Mehmet & Carmona, Guilherme, 2015. "Strategic behavior in non-atomic games," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 134-144.
    6. Dubey, Pradeep & Mas-Colell, Andreau & Shubik, Martin, 1980. "Efficiency properties of strategies market games: An axiomatic approach," Journal of Economic Theory, Elsevier, vol. 22(2), pages 339-362, April.
    7. Khan, M. Ali & Sun, Yeneng, 2002. "Non-cooperative games with many players," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 46, pages 1761-1808, Elsevier.
    8. Mas-Colell, Andreu, 1984. "On a theorem of Schmeidler," Journal of Mathematical Economics, Elsevier, vol. 13(3), pages 201-206, December.
    9. Hildenbrand, W & Mertens, J F, 1972. "Upper Hemi-Continuity of the Equilibrium-Set Correspondence for Pure Exchange Economies," Econometrica, Econometric Society, vol. 40(1), pages 99-108, January.
    10. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, University Library of Munich, Germany.
    11. Green, Edward J., 1980. "Noncooperative price taking in large dynamic markets," Journal of Economic Theory, Elsevier, vol. 22(2), pages 155-182, April.
    12. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
    13. Carmona, Guilherme, 2004. "On the purification of Nash equilibria of large games," Economics Letters, Elsevier, vol. 85(2), pages 215-219, November.
    14. Novshek, William & Sonnenschein, Hugo, 1983. "Walrasian equilibria as limits of noncooperative equilibria. Part II: Pure strategies," Journal of Economic Theory, Elsevier, vol. 30(1), pages 171-187, June.
    15. HILDENBRAND, Werner, 1970. "On economies with many agents," LIDAM Reprints CORE 61, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    16. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
    17. Green, Edward J, 1984. "Continuum and Finite-Player Noncooperative Models of Competition," Econometrica, Econometric Society, vol. 52(4), pages 975-993, July.
    18. Bamon, Rodrigo & Fraysse, Jean, 1985. "Existence of Cournot Equilibrium in Large Markets," Econometrica, Econometric Society, vol. 53(3), pages 587-597, May.
    19. Wooders, Myrna & Selten, Reinhard & Cartwright, Edward, 2001. "Some First Results for Noncooperative Pregames: Social Conformity and Equilibrium in Pure Strategies," Economic Research Papers 269360, University of Warwick - Department of Economics.
    20. Rath, Kali P, 1992. "A Direct Proof of the Existence of Pure Strategy Equilibria in Games with a Continuum of Players," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(3), pages 427-433, July.
    21. Hildenbrand, Werner, 1970. "On economies with many agents," Journal of Economic Theory, Elsevier, vol. 2(2), pages 161-188, June.
    22. Mas-Colell, Andreu, 1983. "Walrasian equilibria as limits of noncooperative equilibria. Part I: Mixed strategies," Journal of Economic Theory, Elsevier, vol. 30(1), pages 153-170, June.
    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. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, University Library of Munich, Germany.
    2. Robin Nicole & Peter Sollich, 2018. "Dynamical selection of Nash equilibria using reinforcement learning: Emergence of heterogeneous mixed equilibria," PLOS ONE, Public Library of Science, vol. 13(7), pages 1-37, July.
    3. Robin Nicole & Peter Sollich, 2017. "Dynamical selection of Nash equilibria using Experience Weighted Attraction Learning: emergence of heterogeneous mixed equilibria," Papers 1706.09763, arXiv.org.
    4. Bodoh-Creed, Aaron, 2013. "Efficiency and information aggregation in large uniform-price auctions," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2436-2466.
    5. Hafalir, Isa E. & Hakimov, Rustamdjan & Kübler, Dorothea & Kurino, Morimitsu, 2018. "College admissions with entrance exams: Centralized versus decentralized," Journal of Economic Theory, Elsevier, vol. 176(C), pages 886-934.
    6. Daniel Lacker & Kavita Ramanan, 2019. "Rare Nash Equilibria and the Price of Anarchy in Large Static Games," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 400-422, May.
    7. Aaron Bodoh-Creed & Brent Hickman, 2016. "College Assignment as a Large Contest," Working Papers 2016-27, Becker Friedman Institute for Research In Economics.
    8. Guilherme Carmona, 2004. "On the existence of equilibrium bank runs in a Diamond-Dybvig environment," Nova SBE Working Paper Series wp448, Universidade Nova de Lisboa, Nova School of Business and Economics.
    9. Bodoh-Creed, Aaron L. & Hickman, Brent R., 2018. "College assignment as a large contest," Journal of Economic Theory, Elsevier, vol. 175(C), pages 88-126.
    10. Yang, Jian, 2011. "Asymptotic interpretations for equilibria of nonatomic games," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 491-499.
    11. Carmona, Guilherme, 2007. "Bank failures caused by Large withdrawals: An explanation based purely on liquidity," Journal of Mathematical Economics, Elsevier, vol. 43(7-8), pages 818-841, September.

    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. Guilherme Carmona, 2004. "Nash equilibria of games with a continuum of players," Nova SBE Working Paper Series wp466, Universidade Nova de Lisboa, Nova School of Business and Economics.
    2. Guilherme Carmona & Konrad Podczeck, 2022. "Approximation and characterization of Nash equilibria of large games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 73(2), pages 679-694, April.
    3. Carmona, Guilherme & Podczeck, Konrad, 2020. "Pure strategy Nash equilibria of large finite-player games and their relationship to non-atomic games," Journal of Economic Theory, Elsevier, vol. 187(C).
    4. Edward Cartwright & Myrna Wooders, 2009. "On equilibrium in pure strategies in games with many players," International Journal of Game Theory, Springer;Game Theory Society, vol. 38(1), pages 137-153, March.
    5. Barlo, Mehmet & Carmona, Guilherme, 2015. "Strategic behavior in non-atomic games," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 134-144.
    6. Jara-Moroni, Pedro, 2012. "Rationalizability in games with a continuum of players," Games and Economic Behavior, Elsevier, vol. 75(2), pages 668-684.
    7. Askoura, Y., 2017. "On the core of normal form games with a continuum of players," Mathematical Social Sciences, Elsevier, vol. 89(C), pages 32-42.
    8. Wooders, Myrna & Edward Cartwright & Selten, Reinhard, 2002. "Social Conformity And Equilibrium In Pure Strategies In Games With Many Players," The Warwick Economics Research Paper Series (TWERPS) 636, University of Warwick, Department of Economics.
    9. Khan, M. Ali & Sun, Yeneng, 1999. "Non-cooperative games on hyperfinite Loeb spaces1," Journal of Mathematical Economics, Elsevier, vol. 31(4), pages 455-492, May.
    10. Kalai, Ehud & Shmaya, Eran, 2018. "Large strategic dynamic interactions," Journal of Economic Theory, Elsevier, vol. 178(C), pages 59-81.
    11. Jian Yang, 2021. "Analysis of Markovian Competitive Situations Using Nonatomic Games," Dynamic Games and Applications, Springer, vol. 11(1), pages 184-216, March.
    12. Carmona, Guilherme, 2008. "Large games with countable characteristics," Journal of Mathematical Economics, Elsevier, vol. 44(3-4), pages 344-347, February.
    13. Camacho, Carmen & Kamihigashi, Takashi & Sağlam, Çağrı, 2018. "Robust comparative statics for non-monotone shocks in large aggregative games," Journal of Economic Theory, Elsevier, vol. 174(C), pages 288-299.
    14. Yang, Jian, 2011. "Asymptotic interpretations for equilibria of nonatomic games," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 491-499.
    15. Carmona, Guilherme & Podczeck, Konrad, 2009. "On the existence of pure-strategy equilibria in large games," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1300-1319, May.
    16. Lorenzo Rocco, 2007. "Anonymity in nonatomic games," International Review of Economics, Springer;Happiness Economics and Interpersonal Relations (HEIRS), vol. 54(2), pages 225-247, June.
    17. Cerreia-Vioglio, Simone & Maccheroni, Fabio & Schmeidler, David, 2022. "Equilibria of nonatomic anonymous games," Games and Economic Behavior, Elsevier, vol. 135(C), pages 110-131.
    18. Carmona, Guilherme & Podczeck, Konrad, 2014. "Existence of Nash equilibrium in games with a measure space of players and discontinuous payoff functions," Journal of Economic Theory, Elsevier, vol. 152(C), pages 130-178.
    19. Ennio Bilancini & Leonardo Boncinelli, 2016. "Strict Nash equilibria in non-atomic games with strict single crossing in players (or types) and actions," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(1), pages 95-109, April.
    20. Khan, M. Ali & Qiao, Lei & Rath, Kali P. & Sun, Yeneng, 2020. "Modeling large societies: Why countable additivity is necessary," Journal of Economic Theory, Elsevier, vol. 189(C).

    More about this item

    Keywords

    Nash equilibrium; limit equilibrium; noncooperative games; continuum of players;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:unl:unlfep:wp442. 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: Susana Lopes (email available below). General contact details of provider: https://edirc.repec.org/data/feunlpt.html .

    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.