By T. Goodall
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Today, there is nothing paradoxical about the St Petersburg game. As Feller (1968) points out, modern probability theory has shown the word ‘fair’ to be misleading, if it is applied to gambles with inﬁnite second moments. All that can be said about such gambles rests on the law of large numbers. All that the law of large numbers asserts is that the accumulated net gain or loss of a ‘fair’ gamble is likely to be of smaller magnitude than the number of times the game is played. Nothing more can be said.
It should indeed be called a decision principle, rather than a decision rule. Until the function ϕ(u) is speciﬁed, the EU principle may recommend or describe vastly different decision behaviour. The EU principle’s widespread acceptance is due to the axiomatic embedding it has received, which goes beyond the axioms of linearity, reﬂexivity and transitivity. 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.
Because of their inﬂuence on recent developments, it will prove illuminating to discuss their original versions as well. When exchanging arguments for and against decision rules used in portfolio choice theory, it will prove helpful to start with an analysis of the archetype of all decision rules, the ‘expected gain rule’. 6 But this alone is no good reason for discussing the expected gain rule. What the discussion will bring forth are arguments for and against all decision rules that make use of expected values.
Adequate Decision Rules for Portfolio Choice Problems by T. Goodall