Pure-Strategy Nash Equilibrium Points in Non-Anonymous Games

  • M. Ali Khan
  • Kali P. Rath
  • Yeneng Sun
Part of the Theory and Decision Library book series (TDLC, volume 18)


We present an example of a nonaiomic game without pure Nash equilibria. In the example, the set of players is modelled on the Lebesgue unit interval with an equicontinuous family of payoff functions, and an identical action set given by [–1,1]. This example is sharper than that recently presented by Rath-Sun-Yamashige in that the relationship between societal responses and individual payoffs is linear. We also present a theorem on the existence of pure strategy Nash equilibria in nonatomic games in which the set of players is modelled on a nonatomic Loeb measure space.


Nash Equilibrium Payoff Function Pure Strategy Pure Nash Equilibrium Societal Response 
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  1. 1.
    Anderson, R. M. (1991) . “Non-st andard Methods in Economics,” in W. Hildenbrand and H. Sonnenschein (eds.) Handbook of Mathematical Economics. Amsterdam: North Holland Publishing Company.Google Scholar
  2. 2.
    Anderson, R. M. (1992), “The Core in Perfectly Competitive Economies,” in R. J. Aumann and S. Hart (eds.) Handbook of Game Theory, Volume 2. Amsterdam: NorthHolland Publishing Company.Google Scholar
  3. 3.
    Artstein, Z.(1983). “Distributions of random sets and random selections.” Israel Journal of Mathematics 46, 313–324.CrossRefGoogle Scholar
  4. 4.
    Berge, C. (1959) Topological Spaces. London: Oliver & Boyd.Google Scholar
  5. 5.
    Billingsley, P. (1968). Convergence of Probability Measures. New York: John Wiley.Google Scholar
  6. 6.
    Castaing, C., and Valadier, M. (1977). Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics no. 580, Berlin and New York: Springer-Verlag, 1977.CrossRefGoogle Scholar
  7. 7.
    Fan, K. (1952) . “Fixed Points and Minimax Theorems in Locally Convex Linear Spaces.” Proc. Nat. Acad. Sci. U.S.A 38, 121–126.Google Scholar
  8. 8.
    Fudenberg, D., and J. Tirole (1991). Game Theory. Cambridge: MIT Press.Google Scholar
  9. 9.
    Glicksberg, I. (1952). “A Further Generalization of Kakutani’s Fixed Point Theorem with Application to N ash Equilibrium Points.” Proc. A mer. Math. Soc. 38, 170–172.Google Scholar
  10. 10.
    Hart, S. and E. Kohlberg (1974), Equally Distributed Correspondences. Jour. Math. Econ. 1, 167–174.CrossRefGoogle Scholar
  11. 11.
    Hart, S., W. Hildenbrand and E. Kohlberg (1974), “On Equilibrium Allocations as Distributions on the Commodity Space”, Jour. Math. Econ. 1, 159–166.CrossRefGoogle Scholar
  12. 12.
    Khan, M. Ali (1986). “Equilibrium Points of Nonatomic Games over a Banach Space.” Trans. Amer. Math. Soc. 293, 737–749.CrossRefGoogle Scholar
  13. 13.
    Khan, M. Ali, Rath, K. P., and Sun, Y. N. (1994) “On Games with a Continuum of Players and Infinitely Many Pure Strategies.” Johns Hopkins Working Paper No. 322. Jour. Ec. Theory forthcoming.Google Scholar
  14. 14.
    Khan, M. Ali, Rath, K. P., and Sun, Y. N. (1995) “On Private Information Games without Pure Strategy Equilibria.” Johns Hopkins Working Paper No. 352.Google Scholar
  15. 15.
    Khan, M. Ali and Sun, Y. N. (1995a). “Pure Strategies in Games with Private Information.” J. Math. Econ. 24, 633–653.CrossRefGoogle Scholar
  16. 16.
    Khan, M. Ali and Sun, Y. N. (1995b) “Non-Cooperative Games on Hyperfinite Loeb Spaces.” Johns Hopkins Working Paper No. 359.Google Scholar
  17. 17.
    Loeb, P. A. (1975). “Conversion from Nonstandard to Standard Measure Spaces and Applications in Probability Theory.” Trans. Amer. Math. Soc. 211, 113–122.CrossRefGoogle Scholar
  18. 18.
    Loeb, P. A., and Rashid, S. (1987). “Non-standard Analysis,” in J. Eatwell et al. (eds.) The New Palgrave. London: The MacMillan Publishing Co.Google Scholar
  19. 19.
    Nash, J. F. (1950). “Equilibrium Points in N-person Games.” Proc. Natl. Acad. Sci. U.S.A. 36, 48–49.CrossRefGoogle Scholar
  20. 20.
    Nash, J. F. (1951), “Noncooperative Games.” Ann. Math. 54, 286–295.CrossRefGoogle Scholar
  21. 21.
    Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. New York: Academic Press.Google Scholar
  22. 22.
    Pascoa, M. R. (1993) . “Approximate Equilibrium in Pure Strategies for Non-atomic games.” J. Math. Econ. 22, 223–241.CrossRefGoogle Scholar
  23. 23.
    Rashid, S. (1987). Economies with Many Agents. Baltimore: The Johns Hopkins University Press.Google Scholar
  24. 24.
    Rath, K. (1992) . “A Direct Proof of the Existence of Pure Strategy Equilibria in Games with a Continuum of Players.” Ec. Theory 2, 427–433.CrossRefGoogle Scholar
  25. 25.
    Rath, K., Sun, Y., and Yamashige, S. (1995). “The Nonexistence of Symmetric Equilibria in Anonymous Games with Compact Action Spaces.” J. Math. Econ. 24, 331–346.CrossRefGoogle Scholar
  26. 26.
    Rudin, W. (1974). Real and Complex Analysis. New York: McGraw Hill.Google Scholar
  27. 27.
    Sun, Y. N. (1993a) “Distributional Properties of Correspondences on Loeb Spaces.” J. Func. Anal. 139, 68–93.CrossRefGoogle Scholar
  28. 28.
    Sun, Y. N. (1993b) “Integration of Correspondences on Loeb Spaces.” Trans. Amer. Math. Soc. forthcoming.Google Scholar
  29. 29.
    Schmeidler, D. (1973) “Equilibrium Points of Nonatomic Games.” J. Stat. Phys. 7, 295–300.CrossRefGoogle Scholar
  30. 30.
    Von Neumann, J. (1932). “Einige Sätze über Messbare Abbildungen.” Ann. Math. 33, 574–586.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • M. Ali Khan
    • 1
  • Kali P. Rath
    • 2
  • Yeneng Sun
    • 3
    • 4
  1. 1.Department of EconomicsThe Johns Hopkins UniversityBaltimoreUSA
  2. 2.Department of EconomicsUniversity of Notre DameNotre DameFrance
  3. 3.Department of MathematicsNational University of SingaporeSingapore
  4. 4.Cowles FoundationYale UniversityNew HavenUSA

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