Conditional expectation of correspondences and economic applications

Research Article


We characterize the properties of convexity, compactness and preservation of upper hemicontinuity for conditional expectation of correspondences via the condition of “nowhere equivalence,” and hence extend the classical results on integration of correspondences. To illustrate the economic applications of those properties, we present new results on large games, abstract economies with asymmetric information and stochastic games.


Conditional expectation Correspondences Nowhere equivalence Convexity Compactness Preservation of upper hemicontinuity Large games Abstract economies Stochastic games 

JEL Classification

C60 C62 C72 


  1. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Springer, Berlin (2009)CrossRefGoogle Scholar
  2. Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)CrossRefGoogle Scholar
  3. Balbus, Ł., Dziewulski, P., Reffett, K., Woźny, Ł.: Differential information in large games with strategic complementarities. Econ. Theory 59, 201–243 (2015). doi: 10.1007/s00199-014-0827-x CrossRefGoogle Scholar
  4. Balder, E.J.: A unifying approach to existence of Nash equilibria. Int. J. Game Theory 24, 79–94 (1995)CrossRefGoogle Scholar
  5. Barelli, P., Duggan, J.: Extremal choice equilibrium with applications to large games, stochastic games, & endogenous institutions. J. Econ. Theory 155, 95–130 (2015)CrossRefGoogle Scholar
  6. Bilancini, E., Boncinelli, L.: Strict Nash equilibria in non-atomic games with strict single crossing in players (or types) and actions. Econ. Theory Bull. 4, 95–109 (2016)CrossRefGoogle Scholar
  7. Blackwell, D.: Discounted dynamic programming. Ann. Math. Stat. 36, 226–235 (1965)CrossRefGoogle Scholar
  8. Bogachev, V.I.: Measure Theory, vol. 1. Springer, Berlin (2007)CrossRefGoogle Scholar
  9. Carmona, G., Podczeck, K.: Existence of Nash equilibrium in games with a measure space of players and discontinuous payoff functions. J. Econ. Theory 152, 130–178 (2014)CrossRefGoogle Scholar
  10. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics No. 580. Springer, Berlin (1977)CrossRefGoogle Scholar
  11. Chow, Y.S., Teicher, H.: Probability Theory: Independence, Interchangeability, Martingales. Springer, New York (2012)Google Scholar
  12. Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 38, 803–886 (1952)CrossRefGoogle Scholar
  13. Duffie, D., Geanakoplos, J., Mas-Colell, A., McLennan, A.: Stationary Markov equilibria. Econometrica 62, 745–781 (1994)CrossRefGoogle Scholar
  14. Duggan, J.: Noisy stochastic games. Econometrica 80, 2017–2045 (2012)CrossRefGoogle Scholar
  15. Dynkin, E.B., Evstigneev, I.V.: Regular conditional expectations of correspondences. Theory Probab. Appl. 21, 325–338 (1977)CrossRefGoogle Scholar
  16. Greinecker, M., Podczeck, K.: Edgeworth’s conjecture and the number of agents and commodities. Econ. Theory 62, 93–130 (2016a). doi: 10.1007/s00199-015-0866-y CrossRefGoogle Scholar
  17. Greinecker, M., Podczeck, K.: Core equivalence with differentiated commodities. Working paper (2016b)Google Scholar
  18. Hanen, A., Neveu, J.: Atomes conditionnels d’un espace de probabilité. Acta Mathematica Hungarica 17, 443–449 (1966)CrossRefGoogle Scholar
  19. He, W., Sun, Y.: Stationary Markov perfect equilibria in discounted stochastic games. J. Econ. Theory 169, 35–61 (2017)CrossRefGoogle Scholar
  20. He, W., Sun, X., Sun, Y.: Modeling infinitely many agents. Theor. Econ. 12, 771–815 (2017)CrossRefGoogle Scholar
  21. Hildenbrand, W.: Core and Equilibria of a Large Economy. Princeton University Press, Princeton (1974)Google Scholar
  22. Jacobs, K.: Measure and Integral. Academic Press, London (1978)Google Scholar
  23. Khan, M.A., Sun, Y.: Non-cooperative games with many players. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory, Chapter 46, vol. 3, pp. 1761–1808. North-Holland, Amsterdam (2002)Google Scholar
  24. Khan, M.A., Rath, K.P., Sun, Y., Yu, H.: Large games with a bio-social typology. J. Econ. Theory 148, 1122–1149 (2013)CrossRefGoogle Scholar
  25. Khan, M.A., Rath, K.P., Sun, Y., Yu, H.: Strategic uncertainty and the ex-post Nash property in large games. Theor. Econ. 10, 103–129 (2015)CrossRefGoogle Scholar
  26. Klein, E., Thompson, A.C.: Theory of Correspondences. Wiley, New York (1984)Google Scholar
  27. Loeb, P.A.: Real Analysis. Birkhäuser, Boston (2016)CrossRefGoogle Scholar
  28. Ma, T.-W.: On sets with convex sections. J. Math. Anal. Appl. 27, 413–416 (1969)CrossRefGoogle Scholar
  29. Maharam, D.: On homogeneous measure algebras. Proc. Natl. Acad. Sci. 28, 108–111 (1942)CrossRefGoogle Scholar
  30. Matheron, G.: Random Sets and Integral Geometry. Wiley, London (1975)Google Scholar
  31. Martins-da-Rocha, V.F., Topuzu, M.: Cournot-Nash equilibria in continuum games with non-ordered preference. J. Econ. Theory 140, 314–327 (2008)CrossRefGoogle Scholar
  32. Neveu, J.: Atomes conditionnels d’espaces de probalite et theorie de l’information. In: Symposium on Probability Methods in Analysis. Springer, Berlin, pp. 256–271 (1967)Google Scholar
  33. Nowak, A.S., Raghavan, T.E.S.: Existence of stationary correlated equilibria with symmetric information for discounted stochastic games. Math. Oper. Res. 17, 519–527 (1992)CrossRefGoogle Scholar
  34. 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
  35. Qiao, L., Sun, Y., Zhang, Z.: Conditional exact law of large numbers and asymmetric information economies with aggregate uncertainty. Econ. Theory 62, 43–64 (2016). doi: 10.1007/s00199-014-0855-6.pdf
  36. 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). doi: 10.1007/BF01211424 CrossRefGoogle Scholar
  37. Rauh, M.T.: Non-cooperative games with a continuum of players whose payoffs depend on summary statistics. Econ. Theory 21, 901–906 (2003). doi: 10.1007/s00199-001-0252-9 CrossRefGoogle Scholar
  38. Royden, H.L., Fitzpatrick, P.M.: Real Analysis, 4th edn. Prentice Hall, Boston (2010)Google Scholar
  39. 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
  40. Schmeidler, D.: Equilibrium points of nonatomic games. J. Stat. Phys. 7, 295–300 (1973)CrossRefGoogle Scholar
  41. Shapley, L.: Stochastic games. Proc. Natl. Acad. Sci. USA 39, 1095–1100 (1953)CrossRefGoogle Scholar
  42. Soler, J.-L.: Notion de liberté en statistique mathématique. Université Joseph-Fourier - Grenoble I, Modélisation et simulation (1970). (in French)Google Scholar
  43. Sun, Y., Yannelis, N.C.: Perfect competition in asymmetric information economies: compatibility of efficiency and incentives. J. Econ. Theory 134, 175–194 (2007a)CrossRefGoogle Scholar
  44. Sun, Y., Yannelis, N.C.: Core, equilibria and incentives in large asymmetric information economies. Games Econ. Behav. 61, 131–155 (2007b)CrossRefGoogle Scholar
  45. Sun, X., Zhang, Y.: Pure-strategy Nash equilibria in nonatomic games with infinite-dimensional action spaces. Econ. Theory 58, 161–182 (2015). doi: 10.1007/s00199-013-0795-6 CrossRefGoogle Scholar
  46. Sun, X., Sun, Y., Wu, L., Yannelis, N.C.: Equilibria and incentives in private information economies. J. Econ. Theory 169, 474–488 (2017)CrossRefGoogle Scholar
  47. Tourky, R., Yannelis, N.C.: Markets with many more agents than commodities: Aumann’s “Hidden” assumption. J. Econ. Theory 101, 189–221 (2001)Google Scholar
  48. Yannelis, N.C.: Integration of Banach-valued correspondences. In: Khan, M.A., Yannelis, N.C. (eds.) Equilibrium Theory in Infinite Dimensional Spaces. Springer, Berlin (1991)Google Scholar
  49. Yannelis, N.C.: Debreu’s social equilibrium theorem with asymmetric information and a continuum of agents. Econ. Theory 38, 419–432 (2009). doi: 10.1007/s00199-007-0246-3 CrossRefGoogle Scholar
  50. Yu, H.: Rationalizability in large games. Econ. Theory 55, 457–479 (2014). doi: 10.1007/s00199-013-0756-0 CrossRefGoogle Scholar
  51. Yu, H., Zhu, W.: Large games with transformed summary statistics. Econ. Theory 26, 237–241 (2005). doi: 10.1007/s00199-004-0516-2 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of EconomicsThe Chinese University of Hong KongShatinHong Kong
  2. 2.Departments of Economics and MathematicsNational University of SingaporeSingaporeSingapore

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