Advertisement

Randomisation, Mixed Strategies and the Reduction Axiom

  • Michele Bernasconi

Abstract

In the last twenty years an accumulating body of evidence has been produced against the Independence axiom of the Expected Utility theory (EU, henceforth) that preferences over random prospects are linear in the probabilities of the final outcomes. Such evidence has stimulated the development of several alternative nonlinear preference models. The main objective of this lecture is to present one specific class of results from that literature: we will discuss the experimental evidence and the theoretical models which have rejected the Independence axiom, in so far as it prescribes indifference toward randomisation of equally good alternatives. This restriction, known in the literature as the Betweenness axiom (Chew, 1983; and Dekel, 1986), is widely used in generalisations of expected utility and it is fundamental for their applications to game theory, because players whose preferences violate Betweenness may be unwilling to randomise as the mixed strategy Nash equilibrium requires. A central part of the lecture will indeed be concerned with the modifications which the rejection of the Betweenness axiom implies for the notion of (equilibrium in) mixed strategies in game theory.

Keywords

Nash Equilibrium Pure Strategy Indifference Curve Expected Utility Theory Certainty Equivalent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allais, M. (1953). “Le comportement de l’homme rationnel devant le risque: critique des postulates et axiomes de l’école americaine”, Econometrica, 21, pp.503-556.Google Scholar
  2. Bernasconi, M. (1994). “Nonlinear Preferences and Two-stage Lotteries: Theories and Evidence”, Economic Journal, 104, pp. 54–70.CrossRefGoogle Scholar
  3. Bernasconi, M. and G. Loomes, (1992) “Failure of the Reduction Principle in an Ellsberg-type problem”, Theory and Decision, 32, pp. 77–100.CrossRefGoogle Scholar
  4. Camerer, C. (1989). “An experimental Test of Several Generalized Utility Theories”, Journal of Risk and Uncertainty, 2, pp. 61–104.CrossRefGoogle Scholar
  5. Camerer, C. (1992). “Recent Tests of Generalizations of EU Theories”, in Utility: Theories, Measurement, and Applications, in W. Edwards (ed.), Dordrecht; Kluwer.Google Scholar
  6. Camerer, C.F., and T.H. Ho (1994). “Violations of the Betweenness axiom and Nonlinearity in Probability”, Journal of Risk and Uncertainty, 8, pp. 167–196.CrossRefGoogle Scholar
  7. Chew, S.H., and K.R. MacCrimmon (1979), “Alpha-nu Theory: an Axiomatization of Expected Utility”, University of British Columbia Working Paper No. 669.Google Scholar
  8. Chew, S.H. (1983). “A Generalization of the Quasilinear Mean with Applications to the Measurement of the Income Inequality and Decision Theory Resolving the Allais Paradox”, Econometrica, 53, pp. 1065–1092.Google Scholar
  9. Chew, S.H., Epstein, L.G. and U. Segal (1991). “Mixture Symmetry and Quadratic utility”, Econometrica, 59, pp. 139–163.CrossRefGoogle Scholar
  10. Conlisk, J. (1989). “Three Variants on the Allais Example”, American Economic Review, 79, pp. 392–407.Google Scholar
  11. Crawford, P. (1990). “Equilibrium without Independence”, Journal of Economic Theory, 50, pp. 127–154.CrossRefGoogle Scholar
  12. Dekel, E. (1986). “An axiomatic Characterisation of Preference under Uncertainty: Weakening the Independence Axiom”, Journal of Economic Theory, 40, pp. 304–318.CrossRefGoogle Scholar
  13. Dekel, E., Z. Safra, and U. Segal (1991). “Existence and Dynamic Consistency of Nash Equilibrium with Non-expected Utility Preferences”, Journal of Economic Theory, 55, pp. 229–246.CrossRefGoogle Scholar
  14. Dow, J., and S.R. da Costa Werlang (1994), “Nash Equilibrium under Knightian Uncertainty: Breaking down Backward Induction”, Journal of Economic Theory, 64, pp. 305–324.CrossRefGoogle Scholar
  15. Fishburn, P.C. (1983), “Transitive Measurable Utility”, Journal of Economic Theory, 31, pp. 293–317.CrossRefGoogle Scholar
  16. Fishburn P.C. and R. Rosenthal (1986), “Non-cooperative Games and Nontransitive Preferences”, Mathematical Social Science, 12, pp. 1–7.CrossRefGoogle Scholar
  17. Gul, F. (1991). “A Theory of Disappointment Aversion”, Econometrica, 59, pp. 667–687.CrossRefGoogle Scholar
  18. Harless, D.W., and C.F. Camerer (1994), “The Predictive Utility of Generalized Expected Utility Theories”, Econometrica, 65, pp. 1251–1289.CrossRefGoogle Scholar
  19. Hey, J.D. and, C. Orme, (1994), “Investing generalizations of expected utility theory using experimental data”, Econometrica, 62, pp. 1291–1326.CrossRefGoogle Scholar
  20. Kahneman, D. and A. Tversky, (1979). “Prospect Theory: an Analysis of Decision under Risk”, Econometrica, 47, pp. 263–291.CrossRefGoogle Scholar
  21. Kahneman, D. and A. Tversky (1984). “Choices, values and frames”, American Psychologist, 39, pp. 341–350.CrossRefGoogle Scholar
  22. Kami, E. and Z. Safra (1989a), “Ascending Bid Auctions with Behaviorally Consistent Bidders”, Annals of Operation Research, 19, pp. 435–444.CrossRefGoogle Scholar
  23. Kami, E. and Z. Safra (1989b), “Dynamic Consistency, Revelation in Auctions, and the Structure of Preferences”, Review of Economic Studies, 56, pp. 421–434.CrossRefGoogle Scholar
  24. Loomes, G., and R. Sugden (1986). “Disappointment and dynamic consistency in choice under uncertainty”, Review of Economics Studies, 53, pp. 271–282.CrossRefGoogle Scholar
  25. Machina, M.J. (1982). Expected Utility” Analysis without the Independence Axiom”, Econometrica, 50, pp. 277–323.CrossRefGoogle Scholar
  26. Machina, M.J. (1989). “Dynamic” Consistency and Non-Expected Utility Models of Choice Under Uncertainty”, Journal of Economic Literature, 27, pp. 1622–1668.Google Scholar
  27. Marschak, J. (1950), “rational behaviour, uncertain prospects, and Expected Utility”, Econometrica, 18, pp. 111–141.CrossRefGoogle Scholar
  28. Quiggin, J. (1982). “A theory of Anticipated Utility”, Journal of Economic Behaviour and Organization, 3, pp. 323–343.CrossRefGoogle Scholar
  29. Segal, U. (1990), “Two-stage Lotteries without the Reduction axiom”, Econometrica, 58, pp. 349–377.CrossRefGoogle Scholar
  30. Skala, H.J. (1989), “Nonstandard Utilities and the Foundations of Game Theory”, International Journal of Game Theory, 1989, pp. 67–81.Google Scholar
  31. Yaari, M.E. (1987), “The Dual Theory of Choice under Risk”, Econometrica, 55, pp. 95–115.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Michele Bernasconi

There are no affiliations available

Personalised recommendations