# Further Analysis of the Markowitz Model of Utility with a small degree of probability distortion

## DOI:

https://doi.org/10.5750/jgbe.v2i3.537## Keywords:

Markowitz Utility Function, Expo-Value Utility Function, Probability Distortion, Gambling.## Abstract

Explanation of the Allais paradox and the preference of many for multiple prize lottery tickets provide a rationale for why a model of agent’s choice under uncertainty should embody the assumption that they distort probabilities. However the degree of probability distortion required to explain gambling on long shots in Cumulative Prospect Theory appears problematic since it implies subjective expected rates of return are dramatically higher than objective returns. Here we show that a Markowitz model of expected utility, supplemented by a small degree of probability distortion, has qualitatively similar predictions as Cumulative Prospect Theory for numerous experimental outcomes as well as the indifference curves between expected return and objective probabilities for a given stake gamble. In addition we show how a small degree of probability distortion can lead to a preference for a multiple prize lottery which has a rather different prize structure and associated probabilities than the optimally chosen one prize lottery even though the utility gain is small.## References

Abdellaoui, B. M, Bleichrodt, H. and Paraschiv, C. (2007)” Loss Aversion Under

Prospect Theory: A Parameter-Free Measurement”. Management Science, 1659-1674.

Battalion, R.C, Kagel, J.C. Jiranyakul.K. (1990). “Testing Between Alternative Models of Choice Under Uncertainty: Some Initial Results,” Journal of Risk and Uncertainty, 3: 25-50.

Cain, M., D. Law and D.A. Peel (2003) ‘The Favourite-Longshot bias and the Gabriel and Marsden Anomaly: An Explanation Based on Utility Theory’, in L. Vaughan Williams (ed.), The Economics of Gambling (London: Routledge) pp. 2- 13

Cain, M., Law. D and Peel, D.A. (2008)” "Bounded cumulative prospect theory: some implications for gambling outcomes," Applied Economics. 40(1), 5-15

Camerer, C.F. (2000) Prospect Theory in the Wild: Evidence from the field. D. Kahneman, A, Tversky Eds. Choices, Values and Frames. Cambridge University Press, New York, 288-300.

Cain, M., Law. D and Peel, D.A. (2002).” Skewness as an explanation of gambling by locally risk averse agents.”. Applied Economics Letters,15,1025-1028.

Friedman, M. and Savage, L.J. (1948), “The Utility Analysis of Choices Involving Risk,” Journal of Political Economy, LV1, 279-304.

Kahneman, D. and Tversky, A. (1979), “Prospect Theory: An Analysis of Decision under Risk,” Econometrica, 2, 263-91.

Markowitz, H. (1952) “The Utility of Wealth”. The Journal of Political Economy, 60, 151-158.

Paya, I.A. Peel. D.A. Law, D. and Peirson, J. (2005) Testing for Market Efficiency in Gambling Markets: Some observations and New Statistical Tests based on Bootstrap Method.pp346-365. (In Information Efficiency in Financial and betting Markets (ed. L.Vaughan Williams. Cambridge University Press.

Peel, D.A. and Law, D. (2008) “A More General Non-Expected Utility Model as an Explanation of Gambling Outcomes for Individuals and Markets* Forthcoming Economica

Peel, D.A., Zhang, J. and Law, D. (2008). "The Markowitz model of utility supplemented with a small degree of probability distortion as an explanation of outcomes of Allais experiments over large and small payoffs and gambling on unlikely Outcomes " Applied Economics, vol. 40(1), 17-26.

Prelec, D. (1998), “The Probability Weighting Function,” Econometrica, 66, 497-527.

Prelec, D. (2000), “Compound Invariant Weighting Functions in Prospect Theory”, pp. 67-92 in Choices, Values, and Frames, (eds.) Daniel Kahneman and Amos Tversky , Cambridge University Press

Quiggin, J. (1991), 'On the optimal design of lotteries', Economica 58(1), 1-16.

Rabin, M (2000), "Risk Aversion and Expected-Utility Theory: A Calibration Theorem", Econometrica, 68, 5, 1281-1292.

Rieger, M, O and Wang, M (2006) “Cumulative Prospect Theory and the St. Petersburg paradox”. Economic Theory, 28.665-679.

Saha, A. (1993), “Expo-Power Utility: A Flexible Form for Absolute and

Relative Risk Aversion,” American Journal of Agricultural Economics, (75), 905–913.

Samuelson, P.A. (1963), "Risk and Uncertainty: A Fallacy of Large Numbers", Scientia, 153-158.

Tversky, A. and Kahneman, D. (1992), “Advances in Prospect Theory: Cumulative Representation of Uncertainty,” Journal of Risk and Uncertainty, 5:4, 297-323.

Vaughan Williams, L. (1999), “Information Efficiency in Betting Markets: a Survey”', Bulletin of Economic Research, 53, 1-30.