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THE AXIOMS OF UTILITY: In 1989, Peter Wakker, a leading Dutch Bayesian economist on the faculty of Erasmus University in Rotterdam published his magnum opus Additive Representations of Preferences: A New Foundation of Decision Analysis. Peter for example analysed ‘Choquet expected utility’ and other models using a general trade off technique for analysing cardinal utility and based on a continuum of outcomes. In his later book Prospect Theory for Risk and Ambiguity he analyses Tversky and Kahneman’s cumulative prospect theory as yet another valid alternative to Savage’s expected utility. In 2013, Peter was awarded the prestigious Frank P. Ramsey medal by the INFORMS Decision Analysis Society for his high quality endeavours.
    Peter has recently advised me (personal communication) that modifications to Savage’s expected utility which put positive premiums on the positive components of a random monetary reward which are regarded as certain to occur, and negative premiums on those negative components which are thought to be certain to occur, are currently regarded as the state of the art e.g. as it relates to Portfolio Analysis. For an easy introduction to these ideas see Ch. 4 of my 1999 book [15].
    In an important special case, which my Statistics 775: Bayesian Decision and Control students, including the Catalonian statistician Josep Ginebra Molins and the economist Jean Deichmann, validated by a small empirical study in Madison, Wisconsin during the 1980s, a positive premium ε can be shown to satisfy ε =2φ-1 whenever a betting probability φ, which should be elicited from the investor, exceeds 0.5.
    The preceding easy-to-understand approach, which contrasts with the ramifications of Prospect theory, was axiomatized in 2007 by Alain Chateauneuf, Jürgen Eichberger and Simon Grant in their article ‘Choice under uncertainty with the best and the worst in mind: New additive capacities’ which appeared in the Journal of Economic Theory.
    While enormously complex, the new axiom system is nevertheless the most theoretically convincing alternative that I know of to the highly normative Savage axioms. Their seven mathematically expressed axioms, which refer to a preference relation on the space of monetary returns, may be referred to under the following headings:
Axiom 0: Non-trivial preferences
Axiom 1: Ordering (i.e. the preference relation is complete, reflexive and transitive)
Axiom 2: Continuity
Axiom 3: Eventwise monotonicity
Axiom 4: Binary comonotonic independence
Axiom 5: Extreme events sensitivity
Axiom 6: Null event consistency
    If you are able to understand all these axioms and wish to comply with them, then that puts you under some sort of obligation to concur with either the Expected Utility Hypothesis, or the simple modification suggested in Ch.4 of [15] or obvious extensions of this idea. Alternatively, you could just replace expected utility by whatever modification or alternative best suits your practical situation. Note that, if your alternative criterion is sensibly formulated, then it might, at least in principle, be possible to devise a similarly complex axiom system that justifies it, if you really wanted to. In many such cases, the cart has been put before the proverbial horse. It’s rather like a highly complex spiritual biblical prophecy being formulated after the event to be prophesied has actually occurred. Maybe the Isaiahs of decision theoretic axiomatics should follow in the footsteps of Maurice Allais and focus a bit more on empirical validations, and the scientific and socio-economic implications of the methodology which they are striving to self-justify---.

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