Correlated Equilibrium of Bertrand Competition

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5385)

Abstract

This paper explores the relation between equilibrium coarsenings and equilibrium refinements via Bertrand competition example and similar situations, it shows that the typical equilibrium coarsening -— a unique correlated equilibrium -— is equivalent to the unique Nash equilibrium itself, is also equivalent to the equilibrium refinement, for the standard n-firms Bertrand competition model with linear demand and symmetric, linear costs in the most special and simplest case, and compares some wonderful and remarkable differences of the existence, uniqueness, stability, connectivity, and strategic property of Nash equilibrium and correlated equilibrium between Cournot and Bertrand model. We also propose some open questions.

Keywords

Equilibrium coarsenings equilibrium refinements strategic correlation principle positive correlated equilibrium negative correlated equilibrium duality gap 

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References

  1. 1.
    Abreu, D., Milgrom, P., Pearce, D.G.: Information and Timing in Repeated Partnerships. Econometrica 59(6), 1713–1733 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aoyagi, M.: Collusion in dynamic Bertrand oligopoly with correlated private signals and communication. Journal of Economic Theory 102, 229–248 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aumann, R.J.: Subjectivity and correlation in randomized strategies. Journal of Mathematic Economics 1, 67–96 (1974)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Aumann, R.J., Dreze, J.H.: Rational Expectations in Games. American Economic Review 98(1), 72–86 (2008); See also: When All is Said and Done, How Should You Play and What Should You Expect? Discussion paper no.387, Center for the Study of Rationality of The Hebrew University of Jerusalem (2005)CrossRefGoogle Scholar
  5. 5.
    Aumann, R.J., Serrano, R.: An economic Index of Riskiness. Discussion paper no.446, Center for the Study of Rationality of The Hebrew University of Jerusalem (2007)Google Scholar
  6. 6.
    Bergemann, D., Morris, S.: Belief Free Incomplete Information Games. In: Papadimitriou, Zhang (eds.) Proceeding of Workshop on Internet and Network Economics. Springer, Heidelberg (2008)Google Scholar
  7. 7.
    Bernheim, D.: Rationalizable strategic behavior. Econometrica 52, 1007–1028 (1984)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brandenburger, A., Dekel, E.: Rationalizability and correlated equilibrium. Econometrica 55, 1391–1402 (1987)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bulow, J., Geanakoplos, J.D., Klemperer, P.D.: Multimarket oligopoly: Strategic Substitutes and Complements. Journal of Political Economy 93(3), 488–511 (1985)CrossRefGoogle Scholar
  10. 10.
    Calvó-Armengol, A.: The Set of Correlated Equilibria of 2*2 Games, http://selene.uab.es/acalvo
  11. 11.
    Chwe, M.S.-Y.: Incentive Compatibility Implies Signed Covariance (2006), www.chwe.net/michael/i.pdf
  12. 12.
    Foster, D.P., Hart, S.: An Operational Measure of Riskiness, Discussion paper no.454, Center for the Study of Rationality of The Hebrew University of Jerusalem (2007)Google Scholar
  13. 13.
    Foster, D.P., Hart, S.: A Reserve-based Axiomatization Of the Measure of Riskiness, Discussion paper, Center for the Study of Rationality of The Hebrew University of Jerusalem (2008)Google Scholar
  14. 14.
    Fudenberg, D., Levine, D.: Repeated Games with Frequent Signals. Quarterly Journal of Economics (to appear, 2008)Google Scholar
  15. 15.
    Gul, F.: A Comment on Aumann’s Bayesian View. Econometrica 66, 923–927 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hart, S.: Five Questions on Game Theory. In: Hendricks, V.F., Hansen, P.G. (eds.) Game Theory: 5 Questions, pp. 97–107. Automatic Press (2007)Google Scholar
  17. 17.
    Jackson, M.O.: The Economics of Social Networks. In: Blundell, R., Newey, W., Persson, T. (eds.) Advances in Economics and Econometrics, Theory and Applications. Cambridge University Press, Cambridge (2006)Google Scholar
  18. 18.
    Jackson, M.O., Wolinsky, A.: A strategic Model of Social and Economic networks. Journal of Economic Theory 71, 44–74 (1996)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Liu, L.: Correlated equilibrium of Cournot oligopoly competition. Journal of Economic Theory 68, 544–548 (1996)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mahdian, M., McAfee, R.P., Pennock, D.: The Secretary Problem with a Hazard Rate Condition. In: Papadimitriou, Zhang (eds.) Proceeding of Workshop on Internet and Network Economics. Springer, Heidelberg (2008)Google Scholar
  21. 21.
    Maskin, E.S.: Introduction to Recent Developments in Game Theory. Edward Elgar Publishing (1999)Google Scholar
  22. 22.
    Milgrom, P., Roberts, J.: Rationalizability, learning and equilibrium in games with strategic complementarities. Econometrica 58, 1255–1278 (1990)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Myerson, R.B.: Dual Reduction and Elementary Games. Games and Economic Behavior 21, 183–202 (1997)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Nau, R.F., Gomez Canovas, S., Hansen, P.: On the Geometry of Nash Equilibria and Correlated Equilibria. International Journal of Game Theory 32, 443–453 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Neyman, A.: Correlated Equilibrium and Potential Games. International Journal of Game Theory 26, 223–227 (1997)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pearce, D.G.: Rationalizable Strategic Behavior and the Problem of Perfection. Econometrica 52(4), 1029–1050 (1984)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Qin, C.-Z., Stuart, C.: Are Cournot and Bertrand equilibria evolutionary stable strategies? Journal Evolutionary Economics 7, 41–47 (1997)CrossRefGoogle Scholar
  28. 28.
    Sekiguchi, T.: Uniqueness of equilibrium payoffs in finitely repeated game with imperfect monitoring. The Japanese Economic Review 56(3), 317–331 (2005)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Van Damme, E.: Strategic Equilibrium. In: Aumann, R., Hart, S. (eds.) Handbook of Game Theory, ch. 41, vol. III. North Holland, Amsterdam (2002)Google Scholar
  30. 30.
    Van Damme, E.: On the State of the Art in Game Theory: An Interview with Robert Aumann. Games and Economic Behavior 24, 181–210 (1998)CrossRefGoogle Scholar
  31. 31.
    Viossat, Y.: Openness of the set of games with a unique correlated equilibrium. cahier du laboratoire d’éeconoméetrie 2005-28, Ecole polytechnique, France (revised in, 2006)Google Scholar
  32. 32.
    Ui, T.: Correlated Equilibrium and Concave Games. International Journal of Game Theory 37(1), 1–13 (2008)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Yi, S.: On the Existence of a Unique Correlated Equilibrium in Cournot Oligopoly. Economics Letters 54, 235–239 (1997)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Young, H.P.: The Possible and the Impossible in Multi-Agent Learning. Artificial Intelligence 171, 429–433 (2007)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Zheng, B.: Approximate efficiency in repeated games with correlated private signal. Games and Economic Behavior 63(1), 406–416 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • John Wu
    • 1
  1. 1.School of BusinessEast China Normal UniversityShanghaiChina

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