Theory and Decision

, Volume 40, Issue 1, pp 1–27 | Cite as

On games under expected utility with rank dependent probabilities

  • Klaus Ritzberger


Expected utility with rank dependent probabilities is a generalization of expected utility. If such preference representations are used for the payoffs in the mixed extension of a finite game, Nash equilibrium may fail to exist. Set-valued solutions, however, do exist even for those more general utility functions. But some set-valued solutions may have certain conceptual shortcomings. The paper thus proposes a new set-valued solution concept, called fixed sets under the best reply correspondence. All set-valued solution concepts are robust to perturbations of the expected utility hypothesis.

Key words

Non-expected utility non-cooperative normal-form games setvalued solutions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Allais, M.: 1953, ‘Le Comportement de l'Homme Rationnel devant le Risque’, Econometrica 21, 503–546.Google Scholar
  2. Basu, K. and Weibull, J.W.: 1991, ‘Strategy Subsets Closed Under Rational Behavior’, Economics Letters 36, 141–146.Google Scholar
  3. Berge, C.: 1963, Topological Spaces, Macmillan, New York.Google Scholar
  4. Bernheim, B.D.: 1984, ‘Rationalizable Strategic Behavior’, Econometrica 52, 1007–1028.Google Scholar
  5. Chew, S.H., Karni, E., and Safra, Z.: 1987, ‘Risk Aversion in the Theory of Expected Utility with Rank Dependent Probabilities’, Journal of Economic Theory 42, 370–381.Google Scholar
  6. Crawford, V.P.: 1990, ‘Equilibrium without Independence’, Journal of Economic Theory 50, 127–154.Google Scholar
  7. Debreu, G.: 1952, ‘A Social Equilibrium Existence Theorem’, Proceedings of the National Academy of Sciences 38, 886–893.Google Scholar
  8. Dekel, E., Safra, Z., and Segal, U.: 1991, ‘Existence and Dynamic Consistency of Nash Equilibrium with Non-expected Utility Preferences’, Journal of Economic Theory 55, 229–246.Google Scholar
  9. Dow, J. and Werlang, S.R.d.C.: 1994, ‘Nash Equilibrium under Knightian Uncertainty: Breaking Down Backward Induction’, Journal of Economic Theory 64, 305–324.Google Scholar
  10. Dresher, M.: 1970, ‘Probability of a Pure Equilibrium Point in n-Person Games’, Journal of Combinatorial Theory 8, 134–145.Google Scholar
  11. Eichberger, J. and Kelsey, D.: 1993, ‘Non-Additive Beliefs and Game Theory’, mimeo.Google Scholar
  12. Ellsberg, D.: 1961, ‘Risk, Ambiguity and the Savage Axioms’, Quarterly Journal of Economics 75, 643–669.Google Scholar
  13. Fan, K.: 1952, ‘Fixed Point and Minimax Theorems in Locally Convex Topological Linear Spaces’, Proceedings of the National Academy of Sciences 38, 121–126.Google Scholar
  14. Gilboa, I.: 1987, ‘Expected Utility Theory with Purely Subjective Non-Additive Probabilities’, Journal of Mathematical Economics 16, 65–88.Google Scholar
  15. Glicksberg, I.L.: 1952, ‘A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilibrium Points’, Proceedings of the American Mathematical Society 38, 170–174.Google Scholar
  16. Green, J.: 1987, ‘“Making Book Against Oneself” The Independence Axiom and Nonlinear Utility Theory’, Quarterly Journal of Economics 102, 785–796.Google Scholar
  17. Harsanyi, J.C.: 1973, ‘Games with Randomly Disturbed Payoffs: A New Rational for Mixed-Strategy Equilibrium Points’, International Journal of Game Theory 2, 1–23.Google Scholar
  18. Harsanyi, J.C. and Selten, R.: 1988, A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge, Mass.Google Scholar
  19. Hendon, E., Jacobsen, H.J., Sloth, B., and Tranæs, T.: 1994, ‘Game Theory with Lower Probabilities’, mimeo.Google Scholar
  20. Kahneman, D. and Tversky, A.: 1979, ‘Prospect Theory: An Analysis of Decision under Risk’, Econometrica 47, 263–291.Google Scholar
  21. Karni, E. and Safra, Z.: 1988, ‘Vickrey Auctions in the Theory of Expected Utility with Rank Dependent Probabilities’, Economic Letters 20, 15–18.Google Scholar
  22. Klibanoff, P.: 1993, ‘Uncertainty, Decision, and Normal Form Games’, mimeo.Google Scholar
  23. Kohlberg, E. and Mertens, J.-F: 1986, ‘On the Strategic Stability of Equilibria’, Econometrica 54, 1003–1037.Google Scholar
  24. Machina, M.J.: 1982, ‘Expected Utility Analysis without the Independence Axiom’, Econometrica 50, 277–323.Google Scholar
  25. Nash, J.: 1950, ‘Non-Cooperative Games’, Dissertation at Princeton Univ.Google Scholar
  26. Pearce, D.G.: 1984, ‘Rationalizable Strategic Behavior and the Problem of Perfection’, Econometrica 52, 1029–1050.Google Scholar
  27. Quiggin, J.: 1982, ‘A Theory of Anticipated Utility’, Journal of Economic Behavior and Organization 3, 225–243.Google Scholar
  28. Ritzberger, K. and Weibull, J.W.: 1993, ‘Evolutionary Selection in Normal Form Games’, Stockholm Univ. Working Paper 5 WE.Google Scholar
  29. Sarin, R. and Wakker, P.: 1992, ‘A Simple Axiomatization of Nonadditive Expected Utility’, Econometrica 60, 1255–1272.Google Scholar
  30. Schmeidler, D.: 1989, ‘Subjective Probability and Expected Utility without Additivity’, Econometrica 57, 571–587.Google Scholar
  31. Selten, R.: 1975, ‘Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games’, International Journal of Game Theory 4, 25–55.Google Scholar
  32. van Damme, E.: 1987, ‘Stability and Perfection of Nash Equilibria’, Springer Verlag, Berlin-Heidelberg-New York.Google Scholar
  33. Wakker, P.: 1994, ‘Separating Marginal Utility and Probabilistic Risk Aversion’, Theory and Decision 36, 1–44.Google Scholar
  34. Yaari, M.E.: 1987, ‘The Dual Theory of Choice Under Risk’, Econometrica 55, 95–115.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Klaus Ritzberger
    • 1
  1. 1.Dept. of EconomicsInstitute for Advanced StudiesViennaAustria

Personalised recommendations