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Large games and the law of large numbers

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  • Al-Najjar, Nabil I.
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    Abstract

    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Games and Economic Behavior.

    Volume (Year): 64 (2008)
    Issue (Month): 1 (September)
    Pages: 1-34

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    Handle: RePEc:eee:gamebe:v:64:y:2008:i:1:p:1-34

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    Web page: http://www.elsevier.com/locate/inca/622836

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    References

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    1. Ehud Kalai, 2002. "Large Robust Games," Discussion Papers 1350, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. 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.
    3. Mas-Colell, Andreu, 1984. "On a theorem of Schmeidler," Journal of Mathematical Economics, Elsevier, vol. 13(3), pages 201-206, December.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Nabil I. Al-Najjar & Luca Anderlini & Leonardo Felli, 2006. "Undescribable Events," Review of Economic Studies, Oxford University Press, vol. 73(4), pages 849-868.
    10. 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.
    11. Harald Uhlig, 1996. "A law of large numbers for large economies (*)," Economic Theory, Springer, vol. 8(1), pages 41-50.
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    Cited by:
    1. Anna Rubinchik & Roberto Samaniego, 2013. "Demand for contract enforcement in a barter environment," Social Choice and Welfare, Springer, vol. 41(1), pages 73-97, June.
    2. Carmona, Guilherme & Podczeckz, Konrad, 2008. "On the Existence of Pure-Strategy Equilibria in Large Games," FEUNL Working Paper Series wp531, Universidade Nova de Lisboa, Faculdade de Economia.
    3. Patrizia Berti & Michele Gori & Pietro Rigo, 2010. "A note on the law of large numbers in economics," Quaderni di Dipartimento 131, University of Pavia, Department of Economics and Quantitative Methods.
    4. Hannu Salonen, 2010. "On the existence of Nash equilibria in large games," International Journal of Game Theory, Springer, vol. 39(3), pages 351-357, July.
    5. 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.
    6. Bodoh-Creed, Aaron, 2013. "Efficiency and information aggregation in large uniform-price auctions," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2436-2466.
    7. Yang, Jian, 2011. "Asymptotic interpretations for equilibria of nonatomic games," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 491-499.

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