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Nash Equilibria of Games with a Continuum of Players

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  • Guilherme Carmona

    (Universidade Nova de Lisboa)

Abstract

We characterize Nash equilibria of games with a continuum of players (Mas-Colell (1984)) in terms of approximate equilibria of large finite games. For the concept of $(\epsilon,\epsilon)$ - equilibrium --- in which the fraction of players not $\epsilon$ - optimizing is less than $\epsilon$ --- 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 $(\epsilon_n,\epsilon_n)$ - equilibria, with $\epsilon_n$ converging to zero. The same holds for $\epsilon$ - equilibrium --- in which almost all players are $\epsilon$ - optimizing --- provided that either players' payoff functions are equicontinuous or players' action space is finite. Furthermore, we give conditions under which the above results hold for all approximating sequences of games. In our characterizations, a sequence of finite games approaches the continuum game in the sense that the number of players converges to infinity and the distribution of characteristics and actions in the finite games converges to that of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance.

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Bibliographic Info

Paper provided by EconWPA in its series Game Theory and Information with number 0412009.

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Length: 31 pages
Date of creation: 21 Dec 2004
Date of revision:
Handle: RePEc:wpa:wuwpga:0412009

Note: Type of Document - pdf; pages: 31. This is a revised version of part of my paper ''Nash and Limit Equilibria of Games with a Continuum of Players''
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Web page: http://128.118.178.162

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Keywords: Nash equilibrium; Games with a continuum of players; Games with finitely many players; approximate equilibria.;

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References

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  1. Carmona, Guilherme, 2004. "On the Existence of Pure Strategy Nash Equilibria in Large Games," FEUNL Working Paper Series wp465, Universidade Nova de Lisboa, Faculdade de Economia.
  2. 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.
  3. Drew Fudenberg & David K. Levine, 1986. "Limit Games and Limit Equilibria," Levine's Working Paper Archive 220, David K. Levine.
  4. 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.
  5. Barlo, Mehmet & Carmona, Guilherme, 2011. "Strategic behavior in non-atomic games," MPRA Paper 35549, University Library of Munich, Germany.
  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. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
  8. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, EconWPA.
  9. Wooders, M. & Selten, R. & Cartwright, E., 2001. "Some First Results for Noncooperative Pregames : Social Conformity and Equilibrium in Pure Strategies," The Warwick Economics Research Paper Series (TWERPS) 589, University of Warwick, Department of Economics.
  10. 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.
  11. 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.
  12. Green, Edward J., 1982. "Continuum and Finite-Player Noncooperative Models of Competition," Working Papers 418, California Institute of Technology, Division of the Humanities and Social Sciences.
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Cited by:
  1. Carmona, Guilherme, 2006. "On the Existence of Pure Strategy Nash Equilibria in Large Games," FEUNL Working Paper Series wp487, Universidade Nova de Lisboa, Faculdade de Economia.
  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. Carmona, Guilherme, 2007. "Intermediate Preferences and Behavioral Conformity in Large Games," FEUNL Working Paper Series wp523, Universidade Nova de Lisboa, Faculdade de Economia.
  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. 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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