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Nash and Limit Equilibria of Games with a Continuum of Players

  • Guilherme Carmona

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://econwpa.repec.org/eps/game/papers/0311/0311004.pdf
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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
Contact details of provider: Web page: http://econwpa.repec.org

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  1. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, EconWPA.
  2. Fudenberg, Drew & Levine, David, 1986. "Limit games and limit equilibria," Journal of Economic Theory, Elsevier, vol. 38(2), pages 261-279, April.
  3. 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).
  4. M Ali Khan & Yeneng Sun, 2002. "Non-Cooperative Games with Many Players," Economics Working Paper Archive 482, The Johns Hopkins University,Department of Economics.
  5. Guilherme Carmona, 2003. "On the Purification of Nash Equilibria of Large Games," Game Theory and Information 0311007, EconWPA.
  6. 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).
  7. 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.
  8. 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.
  9. Hildenbrand, Werner, 1970. "On economies with many agents," Journal of Economic Theory, Elsevier, vol. 2(2), pages 161-188, June.
  10. Green, Edward J., 1980. "Noncooperative price taking in large dynamic markets," Journal of Economic Theory, Elsevier, vol. 22(2), pages 155-182, April.
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  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. 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.
  14. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
  15. Barlo, Mehmet & Carmona, Guilherme, 2015. "Strategic behavior in non-atomic games," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 134-144.
  16. Postlewaite, A & Schmeidler, David, 1978. "Approximate Efficiency of Non-Walrasian Nash Equilibria," Econometrica, Econometric Society, vol. 46(1), pages 127-35, January.
  17. 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.
  18. Bamon, Rodrigo & Fraysse, Jean, 1985. "Existence of Cournot Equilibrium in Large Markets," Econometrica, Econometric Society, vol. 53(3), pages 587-97, May.
  19. Rath, Kali P, 1992. "A Direct Proof of the Existence of Pure Strategy Equilibria in Games with a Continuum of Players," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(3), pages 427-33, July.
  20. 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.
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