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Schmidt, Ulrich, Axiomatic Utility Theory under Risk: Non-Archimedean representations and applications to insurance economics, Springer-Verlag Berlin, Heidelberg 1998,PP201+xv

The lengthy title of this book neatly summarises its contents. The book is divided into three distinct parts. The first part is a survey of axiomatic utility under risk, beginning with expected utility and proceeding to cover the generalisations of expected utility developed over the past twenty years. The second part is a discussion of non-Archimedean representations of preferences, incorporating a range of boundary effects. The third part is a discussion of a range of problems in insurance economics from the perspective of generalized expected utility models incorporating first-order risk-aversion.

The objectives of the first part of the book are modest -- to present an updated version of the numerous surveys already available and to lay the basis for subsequent sections. The basic approach is closest in spirit to that of Chew and Epstein, classifying the main generalizatons of expected utility as either rank-dependent or betweenness-preserving and then considering various unifying models.

Discussion of the empirical evidence is brief and somewhat casual, but is used to draw the conclusion that the bulk of the observed violations of EU may be attributed to certainty effects and, more generally, boundary effects arising when the probability of some outcome falls to zero. This leads the author to the consideration of 'non-Archimedean' models in which the assumption of continuity is dropped, but which are consistent with EU at 'interior' points (roughly speaking, on the interior of any face of the unit simplex).

Schmidt briefly discusses arguments for and against the imposition of a continuity argument. One sort of argument relies on appeals to introspection of the form 'Consider the worst possible consequence, and some trivial positive consequence. Would you take any risk of the worst consequence, however small, in order to receive the positive consequence with probability close to 1'. After one looks at a couple of examples (risking execution for a few dollars, risking road death to drive to the video store), it seems that not much more can be said, so Schmidt is right to be brief on this point.

On the other hand, I think more attention should be paid to the argument that continuity is a mere mathematical convenience, with no observable outcomes. To put this argument in context, consider that the original and strongest counterexamples to EU, the original Allais paradox and the coincidence of lottery gambling and insurance, involve large positive or negative outcomes occurring with probabilities of 1 per cent or less. By contrast, few recent experiments have involved such small probabilities and large payoffs, partly because protocols involving some gambles being played 'for real' cannot be credibly implemented if some payoffs exceed the experimenters' budget.

The question of continuity relevant to the current debate may be rephrased as 'do violations of EU occur near boundaries or exactly on boundaries'. For example, would it affect the sales of lottery tickets if the set of prizes remained the same, but the odds of winning were reduced. A model based on boundary effects would suggest not, whereas a probability weighting model would suggest that demand for lottery tickets should be quite sensitive to chances of winning.

Although this is issue downplayed in his initial discussion of continuity, Schmidt's analysis of non-Archimedean models brings him back to a closely related point in the end. After a brief survey of the early literature on lexicographic preferences, Schmidt analyses a model of expected utility with certainty preferences (EUCP), in which an expected utility representation holds for non-degenerate lotteries, while a separate valuation function is applied to degenerate lotteries. By virtue of the expected utility representation each non-degenerate lottery has a well-defined conditional certainty equivalent that may be used in 'folding back' decision trees and other EU operations. Assuming the valuation functions are separately continuous, each non-degenerate lottery also has an unconditional certainty equivalent. Schmidt characterises certainty preference by the requirement that the unconditional certainty equivalent should be greater than the conditional certainty equivalent. This representation works neatly enough in accommodating the certainty effect and in allowing for a characterisation of risk-aversion, but it is easy to see that it must involve violations of first-order stochastic dominance near the boundary.

The main interest in EUCP, therefore, is to motivate a general consideration of representations referred to by Schmidt as expected utility on restricted sets (EURS). The basic idea is that a representation may be obtained by covering the choice set with subsets, on each of which preferences satisfy expected utility, then patching together the preferences on the subsets to obtain a complete representation. EUCP is one example and betweenness provides another. The theory is also shown to include the Cohen-Jaffray model of the certainty effect as a special case.

Since EURS incorporates the betweenness-conforming theories (of which implicit expected utility IEU is the most general) as a special case, but intersects with rank-dependent utility only for EU preferences, Schmidt argues that 'EURS in conjunction with RDU provides a better classification of non-expected utility than IEU in conjunction with RDU'. It should be noted, however, that to reach this conclusion Schmidt must exclude rank-dependent representations in which the probability weighting function is discontinuous at zero and one.

Returning the problem of stochastic dominance violations, Schmidt offers a notion of e-certainty equivalence in which certainty effects arise when the probability of some outcome rises to 1-e for some suitably small e. This seems to fit the limited data available on gambles with small probabilities reasonably well, and preserves continuity and stochastic dominance. Thus, we return to our starting point. It is clear enough that most violations of EU occur near the boundary of the relevant simplex. Should we interpret this as meaning that violations are associated with small probabilities, or that violations are associated specifically with the difference between zero probabilities and non-zero probabilities?

The third part of the book deals with applications to insurance economics and particularly with the implications of first-order risk-aversion. Topics covered include co-insurance, bilateral risk sharing and agency theory. The results extend the basic observation of Segal and Spivak that an agent with first-order risk-aversion may demand full insurance even at actuarially unfair odds. As an example, in a risk-sharing problem where both parties have striclty risk-averse dual expected utility preferences as in Yaari (), the less risk-averse agent will bear all the risk, contrary to the standard EU result given by the Borch sharing rule. Although there is nothing remarkably surprising in this section, the high standard of rigor and accuracy that characterises the rest of the book is maintained here.

In summary, this book provides a useful new view of the debate on certainty and boundary effects. The idea of expected utility on restricted sets is an interesting and different way of weakening the independence axiom. The question of whether continuity is a merely mathematical assumption or a point of substance remains unresolved, but Schmidt's work has advanced the debate.

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