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On Equilibrium in Pure Strategies in Games with Many Players

  • Edward Cartwright

    ()

    (Department of Economics, Keynes College, University of Kent)

  • Myrna Wooders

    ()

    (Department of Economics, Vanderbilt University)

Treating games of incomplete information with countable sets of actions and types and finite but large player sets we demonstrate that for every mixed strategy profile there is a pure strategy profile that is 'epsilon-equivalent'. Our framework introduces and exploits a distinction between crowding attributes of players (their external effects on others) and their taste attributes (their payoff functions and any other attributes that are not directly relevant to other players). The main assumption is a 'large game' property, dictating that the actions of relatively small subsets of players cannot have large effects on the payoffs of others Since it is well known that, even allowing mixed strategies, with a countable set of actions a Nash equilibrium may not exist, we provide an existence of equilibrium theorem. The proof of existence relies on a relationship between the 'better reply security' property of Reny (1999) and a stronger version of the large game property. Our purification theorem are based on a new mathematical result, of independent interest, applicable to countable strategy spaces.

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File URL: http://www.accessecon.com/pubs/VUECON/vu05-w11.pdf
File Function: First version, 2005
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Paper provided by Vanderbilt University Department of Economics in its series Vanderbilt University Department of Economics Working Papers with number 0511.

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Date of creation: Apr 2005
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Handle: RePEc:van:wpaper:0511
Contact details of provider: Web page: http://www.vanderbilt.edu/econ/wparchive/index.html

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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. Edward Cartwright & Myrna Wooders, 2009. "On equilibrium in pure strategies in games with many players," International Journal of Game Theory, Springer;Game Theory Society, vol. 38(1), pages 137-153, March.
  3. Conley, John P. & Wooders, Myrna H., 2001. "Tiebout Economies with Differential Genetic Types and Endogenously Chosen Crowding Characteristics," Journal of Economic Theory, Elsevier, vol. 98(2), pages 261-294, June.
  4. Kovalenkov, Alexander & Wooders, Myrna Holtz, 1999. "Approximate Cores Of Games And Economies With Clubs," The Warwick Economics Research Paper Series (TWERPS) 535, University of Warwick, Department of Economics.
  5. Wooders, Myrna & Edward Cartwright & Selten, Reinhard, 2002. "Social Conformity And Equilibrium In Pure Strategies In Games With Many Players," The Warwick Economics Research Paper Series (TWERPS) 636, University of Warwick, Department of Economics.
  6. Myrna Wooders & Edward Cartwright & Reinhard Selten, 2005. "Behavioral Conformity in Games with Many Players," Vanderbilt University Department of Economics Working Papers 0513, Vanderbilt University Department of Economics.
  7. Mario Rui Pascoa, 1998. "Nash equilibrium and the law of large numbers," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(1), pages 83-92.
  8. Wooders, Myrna Holtz, 1994. "Equivalence of Games and Markets," Econometrica, Econometric Society, vol. 62(5), pages 1141-60, September.
  9. Green, Edward J, 1984. "Continuum and Finite-Player Noncooperative Models of Competition," Econometrica, Econometric Society, vol. 52(4), pages 975-93, July.
  10. Edward Cartwright & Myrna Wooders, 2009. "On purification of equilibrium in Bayesian games and expost Nash equilibrium," International Journal of Game Theory, Springer;Game Theory Society, vol. 38(1), pages 127-136, March.
  11. Anderson, Robert M., 1992. "The core in perfectly competitive economies," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 14, pages 413-457 Elsevier.
  12. Edward Cartwright & Myrna Wooders, 2003. "Conformity and Bounded Rationality in Games with Many Players," Working Papers 2003.123, Fondazione Eni Enrico Mattei.
  13. Mas-Colell, Andreu, 1984. "On a theorem of Schmeidler," Journal of Mathematical Economics, Elsevier, vol. 13(3), pages 201-206, December.
  14. 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.
  15. 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.
  16. Theo Offerman & Jan Potters & Joep Sonnemans, 2002. "Imitation and Belief Learning in an Oligopoly Experiment," Review of Economic Studies, Oxford University Press, vol. 69(4), pages 973-997.
  17. Rath, Kali P. & Yeneng Sun & Shinji Yamashige, 1995. "The nonexistence of symmetric equilibria in anonymous games with compact action spaces," Journal of Mathematical Economics, Elsevier, vol. 24(4), pages 331-346.
  18. Rui Pascoa, Mario, 1993. "Approximate equilibrium in pure strategies for non-atomic games," Journal of Mathematical Economics, Elsevier, vol. 22(3), pages 223-241.
  19. M Ali Khan & Kali P Rath & Yeneng Sun, 1994. "On the Existence of Pure Strategy Equilibria in Games with a Continuum of Players," Economics Working Paper Archive 381, The Johns Hopkins University,Department of Economics, revised Feb 1997.
  20. Pascoa Mario Rui, 1993. "Noncooperative Equilibrium and Chamberlinian Monopolistic Competition," Journal of Economic Theory, Elsevier, vol. 60(2), pages 335-353, August.
  21. Guilherme Carmona, 2003. "On the Purification of Nash Equilibria of Large Games," Game Theory and Information 0311007, EconWPA.
  22. Martin W. Cripps & Godfrey Keller & Sven Rady, 2002. "Strategic Experimentation: The Case of Poisson Bandits," CESifo Working Paper Series 737, CESifo Group Munich.
  23. Friedman, Daniel, 1996. "Equilibrium in Evolutionary Games: Some Experimental Results," Economic Journal, Royal Economic Society, vol. 106(434), pages 1-25, January.
  24. Mark Walker & John Wooders, 2001. "Minimax Play at Wimbledon," American Economic Review, American Economic Association, vol. 91(5), pages 1521-1538, December.
  25. Ehud Kalai, 2001. "Ex-Post Stability in Large Games," Discussion Papers 1351, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  26. Conley, John P. & Wooders, Myrna, 1996. "Taste-homogeneity of optimal jurisdictions in a Tiebout economy with crowding types and endogenous educational investment choices," Ricerche Economiche, Elsevier, vol. 50(4), pages 367-387, December.
  27. Conley, John P. & Wooders, Myrna H., 1997. "Equivalence of the Core and Competitive Equilibrium in a Tiebout Economy with Crowding Types," Journal of Urban Economics, Elsevier, vol. 41(3), pages 421-440, May.
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