By T. Goodall
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Additional resources for Adequate Decision Rules for Portfolio Choice Problems
This is the case for the class of normal distributions and the class of log-normal distributions. 50 Up to the present day, any discussion of Markowitz’s µ–σ2 rule includes this conclusion on its ‘rationality’. It seems worth emphasising that this entire argument is based on declaring the EU principle the only ‘rational’ one. It is true that there has been a time when the von Neumann–Morgenstern deﬁnition of rationality was irrefutable; and most works on the relationship of the EU principle and Markowitz’s µ–σ2 rule stem from this time.
The necessity to distinguish between ‘utility’ and ‘risk attitude’ has led Shoemaker (1982), as well as Sugden (1986), to call u(r) a ‘utility function 23 chapter three A N A LY S I S O F P R O M I N E N T DECISION RULES A D E Q U AT E D E C I S I O N R U L E S F O R PORTFOLIO CHOICE PROBLEMS under certainty’, and ω(r) a ‘utility function under risk’. But this choice of wording necessitates further deﬁnitions, as well as a discussion of the relation between the two functions. Furthermore, it does not remedy the confusing use of the word ‘utility’.
They must be obeyed if preference indices are to express preference relations. Only under Debreu’s axioms can utility functions mirror preference relations among results, or decision rules express preference relations among gambles. Thus, if the expected gain rule’s recommendation is based on the law of large numbers, it can only be called ‘suitable’ or ‘rational’ for decision situations in which the gamble is repeated inﬁnitely often. It is then inapplicable to situations in which the gamble is played only once.
Adequate Decision Rules for Portfolio Choice Problems by T. Goodall