By T. Goodall
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Additional info for Adequate Decision Rules for Portfolio Choice Problems
For gambles repeated ﬁnitely often, there is no reason to believe that the accumulated net gain ﬂuctuates around zero. 18 For the St Petersburg game, no ﬁnite expected value exists, and the law of large numbers is inapplicable. It is thus impossible to derive the ‘fair’ entrance fee to the St Petersburg game from the law of large numbers, on which the expected gain rule rests. 19 This would induce an entirely different decision situation, of course, and so does not remedy the expected gain rule’s inapplicability to the St Petersburg game.
The unfortunate lack of distinction between ϕ(u) and u(r) by von Neumann and Morgenstern, and their equally unfortunate choice of wording, has led to two different kinds of misinterpretation. First, it has led to claims that the EU principle was identical to Bernoulli’s ‘moral expectation’ rule. 25 Of course, the two are not identical at all, unless ω(r) is deﬁned being equal to ln(r). Second, it has led to claims that the EU principle was identical to Bayes’s rule. This may be correct with regard to their mathematical form, since u(r) and ω(r) are mathematically indistinguishable.
Telser (1955/56) and Kataoka (1963) proposed two more safety ﬁrst rules for portfolio choice settings. Because of their emphasis on disastrous occurrences, safety ﬁrst rules have always lent themselves to appeals to intuition and introspection. Their application to problems of decision under Knightian risk was thus almost always justiﬁed on rather behavioural reasoning. It might thus be argued that they should only be treated as descriptive decision rules. As such they have indeed been tested empirically, mainly in studies where behaviour under potential crisis results was analysed.