Economic Theory

, Volume 58, Issue 1, pp 161–182 | Cite as

Pure-strategy Nash equilibria in nonatomic games with infinite-dimensional action spaces

  • Xiang Sun
  • Yongchao Zhang
Research Article


This paper studies the existence of pure-strategy Nash equilibria for nonatomic games where players take actions in infinite-dimensional Banach spaces. For any infinite-dimensional Banach space, if the player space is modeled by the Lebesgue unit interval, we construct a nonatomic game which has no pure-strategy Nash equilibrium. But if the player space is modeled by a saturated probability space, there is a pure-strategy Nash equilibrium in every nonatomic game. Finally, if every game with a fixed nonatomic player space and a fixed infinite-dimensional action space has a pure-strategy Nash equilibrium, the underlying player space must be saturated.


Infinite-dimensional action space Nonatomic game  Pure-strategy Nash equilibrium Saturated probability space 

JEL Classification

C62 C72 


  1. Bewley, T.F.: Existence of equilibria in economies with infinitely many commodities. J. Econ. Theory 4, 514–540 (1972)CrossRefGoogle Scholar
  2. Carmona, G., Podczeck, K.: On the existence of pure-strategy equilibria in large games. J. Econ. Theory 144, 1300–1319 (2009)CrossRefGoogle Scholar
  3. Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Rhode Island (1977)CrossRefGoogle Scholar
  4. Dvoretsky, A., Wald, A., Wolfowitz, J.: Relation among certain ranges of vector measures. Pac. J. Math. 1, 59–74 (1951)CrossRefGoogle Scholar
  5. Dvoretsky, A., Wald, A., Wolfowitz, J.: Elimination of randomization in certain statistical decision procedures and zero-sum two-person games. Ann. Math. Stat. 22, 1–21 (1951)CrossRefGoogle Scholar
  6. Fajardo, S., Keisler, H.J.: Model Theory of Stochastic Processes. A.K. Ltd, Massachusetts (2002)Google Scholar
  7. Fan, K.: Fixed points and minimax theorems in locally convex linear spaces. Proc. Natl. Acad. Sci. USA 38, 121–126 (1952)CrossRefGoogle Scholar
  8. Fremlin, D.H.: Measure algebra. In: Monk, J.D., Bonnet, R. (eds.) The Handbook of Boolean Algebra (vol. 3), pp. 877–980. North Holland, Amsterdam (1989)Google Scholar
  9. Fu, H.: From large games to bayesian games: a unified approach on pure strategy equilibria, working paper. National University of Singapore (2007)Google Scholar
  10. Glicksberg, I.L.: A further generalization of Kakutani’s fixed point theorem with application to Nash equilibrium points. Proc. Am. Math. Soc. 3, 170–174 (1952)Google Scholar
  11. Hoover, D., Keisler, H.J.: Adapted probability distributions. Trans. Am. Math. Soc. 286, 159–201 (1984)CrossRefGoogle Scholar
  12. Kakutani, S.: Construction of a non-separable extension of the Lebesque measure space. Proc. Acad. Tokyo 20, 115–119 (1944)CrossRefGoogle Scholar
  13. Keisler, H.J., Sun, Y.: Why saturated probability spaces are necessary. Adv. Math. 221, 1584–1607 (2009)CrossRefGoogle Scholar
  14. Khan, M.A., Zhang, Y.: On sufficiently diffused information and finite-player games with private information, working paper, Johns Hopkins University, Baltimore (2012)Google Scholar
  15. Khan, M.A.: Equilibrium points of nonatomic games over a Banach space. Trans. Am. Math. Soc. 293, 737–749 (1986)CrossRefGoogle Scholar
  16. Khan, M.A., Majumdar, M.: Weak sequential convergence in \(L_1(\mu, X)\) and an approximate version of Fatou’s lemma. J. Anal. Appl. 114, 569–573 (1986)CrossRefGoogle Scholar
  17. Khan, M.A., Rath, K.P., Sun, Y.: On the existence of pure-strategy equilibria in games with a continuum of players. J. Econ. Theory 76, 13–46 (1997)CrossRefGoogle Scholar
  18. Khan, M.A., Rath, K.P., Sun, Y.: On a private information game without pure strategy equilibria. J. Math. Econ. 31, 341–359 (1999)CrossRefGoogle Scholar
  19. Khan, M.A., Rath, K.P., Sun, Y.: The Dvoretzky-Wald-Wolfowitz Theorem and purification in atomless finite-action games. Int. J. Game Theory 34, 91–104 (2006)CrossRefGoogle Scholar
  20. Khan, M.A., Sun, Y.: Pure strategies in games with private information. J. Math. Econ. 24, 633–653 (1995)CrossRefGoogle Scholar
  21. Khan, M.A., Sun, Y.: Non-cooperative games on hyperfinite Loeb spaces. J. Math. Econ. 31, 455–492 (1999)CrossRefGoogle Scholar
  22. Khan, M.A., Yannelis, N.C.: Equilibrium Theory in Infinite Dimensional Space. Springer, Berlin (1991)CrossRefGoogle Scholar
  23. Khan, M.A., Zhang, Y.: Set-valued functions, Lebesgue extensions and saturated probability spaces. Adv. Math. 229, 1080–1103 (2012)CrossRefGoogle Scholar
  24. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces (I). Springer, Berlin (1977)CrossRefGoogle Scholar
  25. Loeb, P.A., Sun, Y.: Purification and saturation. Proc. Am. Math. Soc. 137, 2719–2724 (2009)CrossRefGoogle Scholar
  26. Maharam, D.: On homogeneous measure algebras. Proc. Natl. Acad. Sci. USA 28, 108–111 (1942)CrossRefGoogle Scholar
  27. Noguchi, M.: Existence of Nash equilibria in large games. J. Math. Econ. 45, 168–184 (2009)CrossRefGoogle Scholar
  28. Podczeck, K.: On the convexity and compactness of the integral of a Banach space valued correspondence. J. Math. Econ. 44, 836–852 (2008)CrossRefGoogle Scholar
  29. Rath, K.P.: A direct proof of the existence of pure strategy equilibria in games with a continuum of players. Econ. Theory 2, 427–433 (1992)CrossRefGoogle Scholar
  30. Rustichini, A., Yannelis, N.C.: What is perfect competition. In: Khan, M.A., Yannelis, N.C. (eds.) Equilibrium Theory in Infinite Dimensional Spaces, pp. 249–265. Springer, Berlin (1991)CrossRefGoogle Scholar
  31. Stokey, N.L., Lucas, R.E., Prescott, E.C.: Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge (1989)Google Scholar
  32. Sun, Y.: Integration of correspondences on Loeb spaces. Trans. Am. Math. Soc. 349, 129–153 (1997)CrossRefGoogle Scholar
  33. Sun, Y., Yannelis, N.C.: Saturation and the integration of Banach valued correspondences. J. Math. Econ. 44, 861–865 (2008)CrossRefGoogle Scholar
  34. Sun, Y., Zhang, Y.: Individual risk and Lebesgue extension without aggregate uncertainty. J. Econ. Theory 144, 432–443 (2009)CrossRefGoogle Scholar
  35. Topkis, D.M.: Equilibrium points in nonzero-sum \(n\)-person submodular games. SIAM J. Control Optim. 17, 773–787 (1979)CrossRefGoogle Scholar
  36. Walsh, J.L.: A closed set of normal orthogonal functions. Am. J. Math. 45, 5–24 (1923)CrossRefGoogle Scholar
  37. Yannelis, N.C.: On the upper and lower semicontinuity of the Aumann integral. J. Math. Econ. 19, 373–389 (1990)CrossRefGoogle Scholar
  38. Yannelis, N.C.: Debreu’s social equilibrium theorem with symmetric information and a continuum of agents. Econ. Theory 38, 419–432 (2009)CrossRefGoogle Scholar
  39. Yu, H.: Rationalizability in large games. Econ. Theory (2013). doi: 10.1007/s00199-013-0756-0
  40. Yu, H., Zhu, W.: Large games with transformed summary statistics. Econ. Theory 26, 237–241 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Economics and Management SchoolWuhan UniversityWuhanChina
  2. 2.School of EconomicsShanghai University of Finance and EconomicsShanghaiChina
  3. 3.Key Laboratory of Mathematical Economics (SUFE)Ministry of EducationShanghaiChina

Personalised recommendations