Skip to main content
Log in

The Generalized Means Model (GMM) for non-deterministic decision making: Its normative and descriptive power, including sketch of the representation theorem

Theory and Decision Aims and scope Submit manuscript

Abstract

Two schools of thought have been arguing during the last thirty years about the foundations of the theory of choice under uncertainty, namely: The neo-Bernoullian or American school defending an Expected Utility Model (EUM); and the Allais or French school proposing a model based on the moments of the probability distribution over psychological values (MM).

In this paper we present a unified theory: the Generalized Means Model (GMM). By using the well known concept of the generalized mean it is possible to derive both contesting models from the same core of axioms, that — surprisingly enough — includes an extended form of the hotly debated Substitution Principle.

It can be seen that the differences between the two models occur at the beginning and at the end of their axiomatic derivation, as follows: the EUM starts from a probability distribution function over consequences whereas the MM begins with a probability distribution over psychological values. The EUM finishes with an early introduction of a behavioural axiom on the existance of utility, whereas the MM uses first the properties of the distribution function, and then introduces the behavioural assumptions.

The simplest MM consistent with all axioms of the GMM is the model proposed by the author some years ago. It is suggested that a reduced version, the Three Moments Model (TMM), is sufficient for practical applications.

The second part of the paper demonstrates how the TMM solves in a very natural way the Allais Paradox, the certainty effect, the reflection effect, and several other behavioural observations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • Allais, M. (1953), ‘Le Comportement de l'Homme Rationnel Devant le Risque: Critique des Postulats et Axiomes de l'Ecole Américaine’, Econometrica, 21 (1953) 503–546.

    Google Scholar 

  • Allais, M. (1979), ‘The Foundations of a Positive Theory of Choice involving Risk and a Criticism of the Postulates and Axioms of the American School’, 27–145 and ‘The So-called Allais Paradox and Rational Decisions Under Uncertainty’, 437–681 in M. Allais and O. Hagen (Eds.) Expected Utility Hypotheses and the Allais Paradox, D. Reidel Publishing Co, Dordrecht, Holland (1979) 714 pp.

  • Amihud, Y. (1979), ‘Critical Examination of the New Foundation of Utility’ 149–160 in M. Allais and O. Hagen (Eds.). Expected Utility Hypotheses and the Allais Paradox, D. Reidel Publishing Co, Dordrecht, Holland (1979) 714 pp.

    Google Scholar 

  • Bell, D. (1982), ‘Regret in Decision Making under Uncertainty’, Operations Research, 30, No. 5 (Sept–Oct 1982) 961–981.

    Google Scholar 

  • Chew, S.H. (1983), ‘A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox’, Econometrica, 51, No. 4 (July 1983) 1065–1092.

    Google Scholar 

  • Coombs, C.H., T.G. Bezembinder and F.M. Goode (1967), ‘Testing Expectation Theories of Decision Making without Measuring Utility or Subjective Probability’, Journal of Mathematical Psychology, 4 (1967) 72–103.

    Google Scholar 

  • Coombs, C.H. and L. Huang, (1970), ‘Tests of a Portfolio Theory of Risk Preference’, Journal of Experimental Psychology, 85, No. 1 (1970) 23–29.

    Google Scholar 

  • de Finetti, B. (1931), ‘Sul Concetto di Media’, Giornale dell'Istituto Italiano degli Attuari, 2 (1931) 369–396.

    Google Scholar 

  • Hardy, G.H., J.E. Littlewoodand G. Polya (1934), Inequalities, Cambridge University Press, Cambridge, U.K., second edition (1952) 324 pp.

    Google Scholar 

  • Kahneman, D. and A. Tversky (1979), ‘Prospect Theory: An Analysis of Decision Under Risk’, Econometrica, 47, No. 2 (March, 1979) 263–291.

    Google Scholar 

  • Keeney, R.L. and H. Raiffa (1976), Decisions with Multiple Objectives: Preferences and Value Tradeoffs, John Wiley, New York (1976) 569 pp.

    Google Scholar 

  • Kolmogoroff, A. (1930), ‘Sur la Notion de la Moyenne’, Atti della Reale Accademia Nazionale dei Lincei, Rendiconti, Serie 6, 12 (1930) 388–391.

    Google Scholar 

  • Ledersnaider, D. (1981), ‘Comparacion de Algunos Modelos Teoricos para Analisis de Decisiones de Inversion Bajo Incertidumbre’, B.Sc. dissertation, Department of Industrial Engineering, University of the Andes, Bogota, Colombia (June 1981) 121 pp + 2 App.

    Google Scholar 

  • Luce, R. D. and H. Raiffa (1958), Games and Decisions, second printing, John Wiley, New York (1958) 509 pp.

    Google Scholar 

  • McCord, M. and R. de Neufville (1983), ‘Fundamental Deficiency of Expected Utility Decision Analysis’, 279–305, in S. French, L.C. Thomas, R. Hartley and D.J. White (Eds.), Multi-Objective Decision Making, Academic,Press, London (1983) 325 pp.

    Google Scholar 

  • Munera, H.A. (1978), ‘Modeling of Individual Risk Attitudes in Decision Making under Uncertainty: An Application to Nuclear Power’, Ph.D. dissertation, Department of Engineering, University of California, Berkeley, California, USA (September 1978) 266 pp.

    Google Scholar 

  • Munera, H.A., and R. de Neufville (1983), ‘A Decision Analysis Model when the Substitution Principle is not Acceptable’, 247–262, in B.P. Stigum and F. Wenstop (Eds.), Foundations of Utility and Risk Theory with Applications, D. Reidel Publishing Co., Dordrecht, Holland (1983).

    Google Scholar 

  • Royden, H.L., P. Suppes and K. Walsh (1959), ‘A Model for the Experimental Measurement of the Utility of Gambling’, Behavioural Science, 4, No. 1 (Jan. 1959) 11–18.

    Google Scholar 

  • Savage, L. J. (1972), The Foundations of Statistics, 2nd Revised edition, Dover Publications, New York (1972) 310pp.

    Google Scholar 

  • Shohat, J.A. and J.D. Tamarkin (1943), ‘The Problem of Moments’, Mathematical Surveys, No. 1, The American Mathematical Society, New York (1943) 144 pp.

    Google Scholar 

  • Slovic, P. and A. Tversky (1974), ‘Who Accepts Savage's Axiom?’, Behavioural Science, 19 (1974) 368–373.

    Google Scholar 

  • Tintner, G. (1941), ‘The Theory of Choice under Subjective Risk and Uncertainty’, Econometrica, 9 (1941) 298–304.

    Google Scholar 

  • Tversky, A. (1967), ‘Additivity, Utility and Subjective Probability’, Journal of Mathematical Psychology, 4 (1967) 175–201.

    Google Scholar 

  • von Neumann, J. and O. Morgenstern (1944), Theory of Games and Economic Behavior, Princeton University Press, 3rd. edition, Science Editions, John Wiley & Sons, New York (1964) 640 pp.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Munera, H.A. The Generalized Means Model (GMM) for non-deterministic decision making: Its normative and descriptive power, including sketch of the representation theorem. Theor Decis 18, 173–202 (1985). https://doi.org/10.1007/BF00134073

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00134073

Keywords

Navigation