Equilibrium Points of Non-Cooperative Random and Bayesian Games

  • Nicholas C. Yannelis
  • Aldo Rustichini
Part of the Studies in Economic Theory book series (ECON.THEORY, volume 2)

Abstract

We provide random equilibrium existence theorems for non-cooperative random games with a countable number of players. Our results yield as corollaries generalized random versions of the ordinary equilibrium existence result of J. Nash [22]. Moreover, they can be used to obtain equilibrium existence results for games with incomplete information, and in particular Bayesian games. In view of recent work on applications of Bayesian games and Bayesian equilibria, the latter results seem to be quite useful since they delineate conditions under which such equilibria exist.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C.D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York & London, 1985.Google Scholar
  2. 2.
    K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica 22 (1954), 265–290.CrossRefGoogle Scholar
  3. 3.
    R. J. Aumann, Correlated equilibrium as an expression of Bayesian rationality, Econometrica 55 (1987), 1–18.CrossRefGoogle Scholar
  4. 4.
    R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1–12.au]CrossRefGoogle Scholar
  5. 5.
    R. J. Aumann, Y. Katznelson, R. Radner, R. W. Rosenthal, and B. Weiss, Approximate purification of mixed strategies, Math. Oper. Res. 8 (1983), 327–341.CrossRefGoogle Scholar
  6. 6.
    E. J. Balder, Generalized equilibrium results for games with incomplete information, Math. Oper. Res. 13 (1988), 265–276.au]CrossRefGoogle Scholar
  7. 7.
    K. C. Border, Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, Cambridge, 1985.CrossRefGoogle Scholar
  8. 8.
    F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283–301.CrossRefGoogle Scholar
  9. 9.
    C. Castaing and M. Valadier, Convex Analysis and Measurable Multifonctions, Lecture Notes in Math. #580, Springer-Verlag, Berlin-New York, 1977.Google Scholar
  10. 10.
    G. Debreu, A social equilibrium existence theorem, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 886–893.CrossRefGoogle Scholar
  11. 11.
    K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121–126.CrossRefGoogle Scholar
  12. 12.
    K. Fan, Extensions of two fixed-point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240.CrossRefGoogle Scholar
  13. 13.
    K. Fan, Applications of a theorem concerning sets with convex sections, Math. Ann. 163 (1966), 189–203.au]CrossRefGoogle Scholar
  14. 14.
    J. C. Harsanyi, Game with incomplete information played by Bayesian players, Management Sci. 14 (1967–8); Part I, 159–182; Part II, 320-334; Part III, 486-502.CrossRefGoogle Scholar
  15. 15.
    C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72.Google Scholar
  16. 16.
    T. Kim, K. Prikry, and N. C. Yannelis, Carathéodory-type selections and random fixed point theorems, J. Math. Anal. Appl. 122 (1987), 383–407.CrossRefGoogle Scholar
  17. 17.
    T. Kim, K. Prikry, and N. C. Yannelis, On a Carathéodory-type selection theorem, J. Math. Anal. Appl. 135 (1988), 664–670.CrossRefGoogle Scholar
  18. 18.
    A. Mas-Colell, On a theorem of Schmeidler, J. Math. Econom. 13 (1984), 201–206.au]CrossRefGoogle Scholar
  19. 19.
    E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 361–382.CrossRefGoogle Scholar
  20. 20.
    P. Milgrom and R. Weber, Distributional strategies for games with incomplete information, Math. Oper. Res. 10 (1985), 619–632.CrossRefGoogle Scholar
  21. 21.
    R. Myerson, Bayesian equilibrium and incentive compatibility: An introduction, in: L. Hurwicz, D. Schmeidler, and H. Sonnenschein eds., Social Goals and Social Organization: Essays in Memory of Elisha Pazner (Cambridge University Press, Cambridge, 1985), Chapter 8, pp. 229–259.Google Scholar
  22. 22.
    J. Nash, Non-cooperative games, Ann. of Math. 54 (1951), 286–295.CrossRefGoogle Scholar
  23. 23.
    T. Palfrey and S. Srivastava, Private information in large economies, J. Econom. Theory 39 (1986), 34–58.CrossRefGoogle Scholar
  24. 24.
    T. Palfrey and S. Srivastava, On Bayesian implementable allocations, Rev. Econom. Stud. 54 (1987), 193–208.CrossRefGoogle Scholar
  25. 25.
    J. Peck and K. Shell, Market uncertainty: Correlated equilibrium and sunspot equilibrium in imperfect competitive economies, Mimeo, Department of Economics, Cornell University, 1987.Google Scholar
  26. 26.
    A. Postlewaite and D. Schmeidler, Implementation in differential information economies, J. Econom. Theory 39 (1986), 14–33.CrossRefGoogle Scholar
  27. 27.
    R. Radner and R. Rosenthal, Private information and pure-strategy equilibria, Math. Oper. Res. 7 (1982), 401–409.CrossRefGoogle Scholar
  28. 28.
    S. Rashid, The approximate purification of mixed strategies with finite observation sets, Econom. Lett. 19 (1985), 133–135.CrossRefGoogle Scholar
  29. 29.
    A. Rustichini, A counterexample and an exact version of Fatou’s lemma in infinite dimensions, Arch. Math. (Basel) 52 (1989), 357–362.CrossRefGoogle Scholar
  30. 30.
    A. Rustichini and N. C. Yannelis, What is perfect competition?, in: M. Ali Khan and N. C. Yannelis eds., Equilibrium Theory with Infinitely Many Commodities (Springer-Verlag, Berlin & New York, 1991), pp. 249–265.Google Scholar
  31. 31.
    W. J. Shafer and H. F. Sonnenschein, Equilibrium in abstract economies without ordered preferences, J. Math. Econom. 2 (1975), 345–348.CrossRefGoogle Scholar
  32. 32.
    N. C. Yannelis and N. D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom. 12 (1983), 233–245.CrossRefGoogle Scholar
  33. 33.
    N. C. Yannelis, Equilibria in non-cooperative models of competition, J. Econom. Theory 41 (1987), 96–111.CrossRefGoogle Scholar
  34. 34.
    N. C. Yannelis, On the upper and lower semicontinuity of the Aumann integral, J. Math. Econom. 19 (1990), 373–389.CrossRefGoogle Scholar
  35. 35.
    N. C. Yannelis, Fatou’s lemma in infinite dimensional spaces, Proc. Amer. Math. Soc. 102 (1988), 303–310.Google Scholar
  36. 36.
    N. C. Yannelis, Integration of Banach-valued correspondences, in: M. Ali Khan and N. C. Yannelis eds., Equilibrium Theory with Infinitely Many Commodities (Springer-Verlag, Berlin & New York, 1991), pp. 1–35.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Nicholas C. Yannelis
    • 1
  • Aldo Rustichini
    • 2
  1. 1.Department of EconomicsUniversity of Illinois at Urbana-ChampaignChampaignUSA
  2. 2.Department of EconomicsNorthwestern UniversityEvanstonUSA

Personalised recommendations