By T. Goodall
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Additional info for Adequate Decision Rules for Portfolio Choice Problems
These axioms provide for the existence of a general function that indicates preferences among unspeciﬁed elements 24 of a general set. The axioms designed for the normative justiﬁcation of the EU principle provide both for the existence of the risk attitude function ϕ(u) and for the mathematical form of the decision principle itself. 28 The combination of these eight axioms with the infamous axiom of ‘independent evaluation of chances’, or ‘independence axiom’ for short, leads to the EU principle.
Again, this is not to say that the EU principle may not be claimed as ‘rational’ in other decision situations. ‘Rationality’ is a matter of opinion and cannot be disputed. What is argued here is that the predominance of expected values in decision theory, and especially in decision rules for portfolio choice problems, can be explained by decision theory’s history of thought. The ﬁrst decision rules were expected values, applied to games of chance in settings of inﬁnitely many repeats, and supported by a notion that was to become the law of large numbers.
But their logical content and interpretation are different, and a distinction must thus be made. Only under very special assumptions are the two identical in form and spirit. 26 But this means depriving the EU principle of much of its appeal and generality. The other such assumption is that all individuals again evaluate all possible results at face value, that is, omit the function u(r) altogether by setting u(r) = r, which amounts to setting ω(r) = ϕ(u). But the subjective character of all decision situations of Knightian risk calls for allowing individual evaluations of results.