A General Criterion of Choice, with Discussion of Borch Paradox

Author(s)
Benito V. Frosini

ABSTRACT

The author resumes a proposal by Frosini of a criterion of choice between probability prospects, which realizes a suggestion by Allais of taking account, beside the expected utility of the dispersion or variability of utilities. The suggested criterion is unidimensional, and is increasing with expected utility, and decreasing, for most people, who are risk averse, with the absolute deviation of utilities; a parameter multiplying this dispersion measure allows for risk-averse or risk-prone behaviour, according to its sign, and also for more or less departure from a certain prospect. This composite criterion shares practically all desirable conditions of rationality, and allows explaining all popular paradoxes in the literature about utility theory. Then the author deals with an apparent, but really false paradox, raised by Borch in connection with the representation of probability prospects in a Markowitz-type plot. This kind of analysis is modified from the traditional reference to points of type (mean, standard deviation) to the reference which replaces the standard deviation with the mean absolute deviation; no essential change is involved. The paper closes with some numerical examples which show the correctness of the suggested criterion, as compared to unaccettable conclusions of the expected utility approach.

The author resumes a proposal by Frosini of a criterion of choice between probability prospects, which realizes a suggestion by Allais of taking account, beside the expected utility of the dispersion or variability of utilities. The suggested criterion is unidimensional, and is increasing with expected utility, and decreasing, for most people, who are risk averse, with the absolute deviation of utilities; a parameter multiplying this dispersion measure allows for risk-averse or risk-prone behaviour, according to its sign, and also for more or less departure from a certain prospect. This composite criterion shares practically all desirable conditions of rationality, and allows explaining all popular paradoxes in the literature about utility theory. Then the author deals with an apparent, but really false paradox, raised by Borch in connection with the representation of probability prospects in a Markowitz-type plot. This kind of analysis is modified from the traditional reference to points of type (mean, standard deviation) to the reference which replaces the standard deviation with the mean absolute deviation; no essential change is involved. The paper closes with some numerical examples which show the correctness of the suggested criterion, as compared to unaccettable conclusions of the expected utility approach.

Cite this paper

Frosini, B. (2014) A General Criterion of Choice, with Discussion of Borch Paradox.*Theoretical Economics Letters*, **4**, 691-696. doi: 10.4236/tel.2014.48087.

Frosini, B. (2014) A General Criterion of Choice, with Discussion of Borch Paradox.

References

[1] Allais, M. (1953) Le Comportement de l’Homme Rationnel Devant le Risque: Critique des Postulats et Axioms de l’Ecole Américaine. Econometrica, 21, 503-546. http://dx.doi.org/10.2307/1907921

[2] Ellsberg, D. (1961) Risk, Ambiguity, and the Savage Axioms. Quarterly Journal of Economics, 75, 643-669. http://dx.doi.org/10.2307/1884324

[3] Kahneman, D. and Tversky, A. (1979) Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47, 263-291. http://dx.doi.org/10.2307/1914185

[4] Frosini, B.V. (2012) Realistic Utility versus Game Utility: A Proposal for Dealing with the Spread of Uncertain Prospects. Statistica, 72, 3-22.

[5] Von Neumann, J. and Morgenstern, O. (1953) Theory of Games and Economic Behavior. 3rd Edition, Princeton University Press, Princeton.

[6] Lindley, D.V. (1985) Making Decisions. 2nd Edition, Wiley, London.

[7] Markowitz, H. (1952) The Utility of Wealth. Journal of Political Economy, 60, 151-158.

http://dx.doi.org/10.1086/257177

[8] Markowitz, H. (1959) Portfolio Selection. Wiley, New York.

[9] Liu, L. (2004) A New Foundation for the Mean-Variance Analysis. European Journal of Operational Research, 158, 229-242.

[10] Yaari, M.E. (1987) The Dual Theory of Choice Under Risk. Econometrica, 55, 95-115.

http://dx.doi.org/10.2307/1911158

[11] Frosini, B.V. (1997) The Evaluation of Risk Attitudes: A New Proposal. Statistica Applicata, 9, 435-458.

[12] Frosini, B.V. (2010) Realistic Utility versus Game Utility: A Proposal for Dealing with the Spread of Uncertain Prospects. Dipartimento di Scienze statistiche, Università Cattolica del Sacro Cuore, Serie E.P.N. 140. http://dipartimenti.unicatt.it/scienze statistiche_statistiche_2183.html

[13] Quiggin, J. (1982) A Theory of Anticipated Utility. Journal of Economic Behavior and Organization, 3, 323-343. http://dx.doi.org/10.1016/0167-2681(82)90008-7

[14] Quiggin, J. (1993) Generalized Expected Utility Theory. The Rank-Dependent Model, Kluwer, Boston. http://dx.doi.org/10.1007/978-94-011-2182-8

[15] Borch, K. (1968) Indifference Curves and Uncertainty. The Swedish Journal of Economics, 70, 19-24. http://dx.doi.org/10.2307/3438982

[16] Johnstone, D. and Lindley, D. (2012) Mean-Variance and Expected Utility: The Borch Paradox. Statistical Science, 28, 223-237. http://dx.doi.org/10.1214/12-STS408

[17] Levy, H. and Sarnat, M. (1969) A Note on Indifference Curves and Uncertainty. The Swedish Journal of Economics, 71, 206-208. http://dx.doi.org/10.2307/3439370

[1] Allais, M. (1953) Le Comportement de l’Homme Rationnel Devant le Risque: Critique des Postulats et Axioms de l’Ecole Américaine. Econometrica, 21, 503-546. http://dx.doi.org/10.2307/1907921

[2] Ellsberg, D. (1961) Risk, Ambiguity, and the Savage Axioms. Quarterly Journal of Economics, 75, 643-669. http://dx.doi.org/10.2307/1884324

[3] Kahneman, D. and Tversky, A. (1979) Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47, 263-291. http://dx.doi.org/10.2307/1914185

[4] Frosini, B.V. (2012) Realistic Utility versus Game Utility: A Proposal for Dealing with the Spread of Uncertain Prospects. Statistica, 72, 3-22.

[5] Von Neumann, J. and Morgenstern, O. (1953) Theory of Games and Economic Behavior. 3rd Edition, Princeton University Press, Princeton.

[6] Lindley, D.V. (1985) Making Decisions. 2nd Edition, Wiley, London.

[7] Markowitz, H. (1952) The Utility of Wealth. Journal of Political Economy, 60, 151-158.

http://dx.doi.org/10.1086/257177

[8] Markowitz, H. (1959) Portfolio Selection. Wiley, New York.

[9] Liu, L. (2004) A New Foundation for the Mean-Variance Analysis. European Journal of Operational Research, 158, 229-242.

[10] Yaari, M.E. (1987) The Dual Theory of Choice Under Risk. Econometrica, 55, 95-115.

http://dx.doi.org/10.2307/1911158

[11] Frosini, B.V. (1997) The Evaluation of Risk Attitudes: A New Proposal. Statistica Applicata, 9, 435-458.

[12] Frosini, B.V. (2010) Realistic Utility versus Game Utility: A Proposal for Dealing with the Spread of Uncertain Prospects. Dipartimento di Scienze statistiche, Università Cattolica del Sacro Cuore, Serie E.P.N. 140. http://dipartimenti.unicatt.it/scienze statistiche_statistiche_2183.html

[13] Quiggin, J. (1982) A Theory of Anticipated Utility. Journal of Economic Behavior and Organization, 3, 323-343. http://dx.doi.org/10.1016/0167-2681(82)90008-7

[14] Quiggin, J. (1993) Generalized Expected Utility Theory. The Rank-Dependent Model, Kluwer, Boston. http://dx.doi.org/10.1007/978-94-011-2182-8

[15] Borch, K. (1968) Indifference Curves and Uncertainty. The Swedish Journal of Economics, 70, 19-24. http://dx.doi.org/10.2307/3438982

[16] Johnstone, D. and Lindley, D. (2012) Mean-Variance and Expected Utility: The Borch Paradox. Statistical Science, 28, 223-237. http://dx.doi.org/10.1214/12-STS408

[17] Levy, H. and Sarnat, M. (1969) A Note on Indifference Curves and Uncertainty. The Swedish Journal of Economics, 71, 206-208. http://dx.doi.org/10.2307/3439370