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Nash and Limit Equilibria of Games with a Continuum of Players

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  • Carmona, Guilherme

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.

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Bibliographic Info

Paper provided by Universidade Nova de Lisboa, Faculdade de Economia in its series FEUNL Working Paper Series with number wp442.

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Length: 39 pages
Date of creation: 2004
Date of revision:
Handle: RePEc:unl:unlfep:wp442

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Keywords: Nash equilibrium; limit equilibrium; noncooperative games; continuum of players;

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References

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  1. 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.
  2. Green, Edward J, 1984. "Continuum and Finite-Player Noncooperative Models of Competition," Econometrica, Econometric Society, vol. 52(4), pages 975-93, July.
  3. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, EconWPA.
  4. M Ali Khan & Yeneng Sun, 1996. "Non-Cooperative Games with Many Players," Economics Working Paper Archive 382, The Johns Hopkins University,Department of Economics.
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  6. 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.
  7. 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.
  8. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
  9. Fudenberg, Drew & Levine, David, 1986. "Limit Games and Limit Equilibria," Scholarly Articles 3350443, Harvard University Department of Economics.
  10. 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.
  11. HART, Sergiu & HILDENBRAND, Werner & KOHLBERG, Elon, . "On equilibrium allocations as distributions on the commodity space," CORE Discussion Papers RP -183, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  12. 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.
  13. Carmona, Guilherme, 2003. "On the Purification of Nash Equilibria of Large Games," FEUNL Working Paper Series wp436, Universidade Nova de Lisboa, Faculdade de Economia.
  14. Postlewaite, A & Schmeidler, David, 1978. "Approximate Efficiency of Non-Walrasian Nash Equilibria," Econometrica, Econometric Society, vol. 46(1), pages 127-35, January.
  15. 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.
  16. 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).
  17. Barlo, Mehmet & Carmona, Guilherme, 2011. "Strategic behavior in non-atomic games," MPRA Paper 35549, University Library of Munich, Germany.
  18. Bamon, Rodrigo & Fraysse, Jean, 1985. "Existence of Cournot Equilibrium in Large Markets," Econometrica, Econometric Society, vol. 53(3), pages 587-97, May.
  19. Green, Edward J., 1980. "Noncooperative price taking in large dynamic markets," Journal of Economic Theory, Elsevier, vol. 22(2), pages 155-182, April.
  20. 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.
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Citations

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Cited by:
  1. Bodoh-Creed, Aaron, 2013. "Efficiency and information aggregation in large uniform-price auctions," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2436-2466.
  2. Carmona, Guilherme, 2004. "Symmetric Approximate Equilibrium Distributions with Finite Support," FEUNL Working Paper Series wp441, Universidade Nova de Lisboa, Faculdade de Economia.
  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. Carmona, Guilherme, 2004. "On the Existence of Equilibrium Bank Runs in a Diamond-Dybvig Environment," FEUNL Working Paper Series wp448, Universidade Nova de Lisboa, Faculdade de Economia.
  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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