IDEAS home Printed from
   My bibliography  Save this article

Large games and the law of large numbers


  • Al-Najjar, Nabil I.


This paper introduces discrete large games where the set of players is a countable dense 'grid' with a finitely additive distribution. In these games any function from player names to mixed actions is a legitimate strategy profile. No extraneous continuity or measurability conditions are assumed. Randomness can be modeled explicitly and an exact law of large numbers holds. Equilibria enjoy a strong purification property: every realization of every mixed strategy equilibrium is a pure strategy equilibrium almost surely. Every continuum-player game has a discrete large game representation that preserves the original payoffs, strategy profiles and equilibria. It is argued that strategy profiles in continuum-player games have an ambiguous meaning because measurability requirements force the smoothing out of individual variations. These variations have clear strategic meaning in finite-player games and can be expressed in discrete large games, but not when the set of players is the continuum.

Suggested Citation

  • Al-Najjar, Nabil I., 2008. "Large games and the law of large numbers," Games and Economic Behavior, Elsevier, vol. 64(1), pages 1-34, September.
  • Handle: RePEc:eee:gamebe:v:64:y:2008:i:1:p:1-34

    Download full text from publisher

    File URL:
    Download Restriction: Full text for ScienceDirect subscribers only

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

    References listed on IDEAS

    1. Nabil I. Al-Najjar & Luca Anderlini & Leonardo Felli, 2006. "Undescribable Events," Review of Economic Studies, Oxford University Press, vol. 73(4), pages 849-868.
    2. Judd, Kenneth L., 1985. "The law of large numbers with a continuum of IID random variables," Journal of Economic Theory, Elsevier, vol. 35(1), pages 19-25, February.
    3. Al-Najjar, Nabil I., 2004. "Aggregation and the law of large numbers in large economies," Games and Economic Behavior, Elsevier, vol. 47(1), pages 1-35, April.
    4. Harald Uhlig, 1996. "A law of large numbers for large economies (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 41-50.
    5. Ehud Kalai, 2004. "Large Robust Games," Econometrica, Econometric Society, vol. 72(6), pages 1631-1665, November.
    6. Dubey, Pradeep & Shapley, Lloyd S., 1994. "Noncooperative general exchange with a continuum of traders: Two models," Journal of Mathematical Economics, Elsevier, vol. 23(3), pages 253-293, May.
    7. Mas-Colell, Andreu, 1984. "On a theorem of Schmeidler," Journal of Mathematical Economics, Elsevier, vol. 13(3), pages 201-206, December.
    8. 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.
    9. Al-Najjar, Nabil Ibraheem, 1995. "Decomposition and Characterization of Risk with a Continuum of Random Variables," Econometrica, Econometric Society, vol. 63(5), pages 1195-1224, September.
    10. Feldman, Mark & Gilles, Christian, 1985. "An expository note on individual risk without aggregate uncertainty," Journal of Economic Theory, Elsevier, vol. 35(1), pages 26-32, February.
    11. Sun, Yeneng, 2006. "The exact law of large numbers via Fubini extension and characterization of insurable risks," Journal of Economic Theory, Elsevier, vol. 126(1), pages 31-69, January.
    Full references (including those not matched with items on IDEAS)


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

    Cited by:

    1. Anna Rubinchik & Roberto Samaniego, 2013. "Demand for contract enforcement in a barter environment," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(1), pages 73-97, June.
    2. Yang, Jian, 2011. "Asymptotic interpretations for equilibria of nonatomic games," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 491-499.
    3. 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.
    4. Tolvanen, Juha & Soultanis, Elefterios, 2012. "A correction to “Large games and the law of large numbers” [Games Econom. Behav. 64 (2008) 1–34]," Games and Economic Behavior, Elsevier, vol. 76(1), pages 340-343.
    5. Bodoh-Creed, Aaron, 2013. "Efficiency and information aggregation in large uniform-price auctions," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2436-2466.
    6. Tsitsiklis, John N. & Xu, Yunjian, 2015. "Pricing of fluctuations in electricity markets," European Journal of Operational Research, Elsevier, vol. 246(1), pages 199-208.
    7. Patrizia Berti & Michele Gori & Pietro Rigo, 2009. "A note on the law of large numbers in economics," Working Papers - Mathematical Economics 2009-10, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa, revised Nov 2010.
    8. Pablo Casas-Arce & F. Asís Martínez-Jerez, 2009. "Relative Performance Compensation, Contests, and Dynamic Incentives," Management Science, INFORMS, vol. 55(8), pages 1306-1320, August.
    9. Hannu Salonen, 2010. "On the existence of Nash equilibria in large games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(3), pages 351-357, July.
    10. repec:eee:proeco:v:191:y:2017:i:c:p:178-193 is not listed on IDEAS
    11. repec:spr:jogath:v:46:y:2017:i:2:d:10.1007_s00182-016-0539-5 is not listed on IDEAS
    12. Jian Yang, 2015. "A Link between Sequential Semi-anonymous Nonatomic Games and their Large Finite Counterparts," Papers 1510.06809,, revised Jun 2016.
    13. Jian Yang, 2015. "Analysis of Markovian Competitive Situations using Nonatomic Games," Papers 1510.06813,, revised Apr 2017.

    More about this item


    Access and download statistics


    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:eee:gamebe:v:64:y:2008:i:1:p:1-34. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu). General contact details of provider: .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.