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Nash and Limit Equilibria of Games with a Continuum of Players

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  • 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 $\varepsilon_n$-equilibria, with $\varepsilon_n$ 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.

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File URL: http://128.118.178.162/eps/game/papers/0311/0311004.pdf
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Bibliographic Info

Paper provided by EconWPA in its series Game Theory and Information with number 0311004.

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Length: 39 pages
Date of creation: 18 Nov 2003
Date of revision:
Handle: RePEc:wpa:wuwpga:0311004

Note: Type of Document - pdf; prepared on win xp; to print on general; pages: 39; figures: 0. none
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Web page: http://128.118.178.162

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Keywords: Nash equilibrium; limit equilibrium; games with a continuum of players;

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References

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  1. Green, Edward J, 1984. "Continuum and Finite-Player Noncooperative Models of Competition," Econometrica, Econometric Society, vol. 52(4), pages 975-93, July.
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  4. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
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  6. 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.
  7. HILDENBRAND, Werner & MERTENS, Jean-François, . "Upper hemi-continuity of the equilibrium set correspondence for pure exchange economies," CORE Discussion Papers RP -109, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  8. Green, Edward J., 1980. "Noncooperative price taking in large dynamic markets," Journal of Economic Theory, Elsevier, vol. 22(2), pages 155-182, April.
  9. Drew Fudenberg & David Levine, 1983. "Limit Games and Limit Equilibria," UCLA Economics Working Papers 289, UCLA Department of Economics.
  10. Bamon, Rodrigo & Fraysse, Jean, 1985. "Existence of Cournot Equilibrium in Large Markets," Econometrica, Econometric Society, vol. 53(3), pages 587-97, May.
  11. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, EconWPA.
  12. Barlo, Mehmet & Carmona, Guilherme, 2011. "Strategic behavior in non-atomic games," MPRA Paper 35549, University Library of Munich, Germany.
  13. 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.
  14. Rath, Kali P, 1992. "A Direct Proof of the Existence of Pure Strategy Equilibria in Games with a Continuum of Players," Economic Theory, Springer, vol. 2(3), pages 427-33, July.
  15. Guilherme Carmona, 2003. "On the Purification of Nash Equilibria of Large Games," Game Theory and Information 0311007, EconWPA.
  16. 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.
  17. 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.
  18. Postlewaite, A & Schmeidler, David, 1978. "Approximate Efficiency of Non-Walrasian Nash Equilibria," Econometrica, Econometric Society, vol. 46(1), pages 127-35, January.
  19. Hildenbrand, Werner, 1970. "On economies with many agents," Journal of Economic Theory, Elsevier, vol. 2(2), pages 161-188, June.
  20. 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.
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Citations

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Cited by:
  1. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, EconWPA.
  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, 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.
  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. Guilherme Carmona, 2004. "On the Existence of Equilibrium Bank Runs in a Diamond-Dybvig Environment," Finance 0404009, EconWPA.

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