A unified derivation of classical subjective expected utility models through cardinal utility

Abstract

Classical foundations of expected utility were provided by Ramsey, de Finetti, von Neumann and Morgenstern, Anscombe and Aumann, and others. These foundations describe preference conditions to capture the empirical content of expected utility. The assumed preference conditions, however, vary among the models and a unifying idea is not readily transparent. Providing such a unifying idea is the purpose of this paper. The mentioned derivations have in common that a cardinal utility index for outcomes, independent of the states and probabilities, can be derived. Characterizing that feature provides the unifying idea of the mentioned models.

Introduction

A characteristic property of expected utility is the separation of probabilities, describing the uncertainty of a decision maker regarding a state space, and utilities, describing the value of outcomes. That separation is reflected in existing preference axiomatizations, that can be classified into two groups accordingly. First, there are the likelihood-oriented axiomatizations, in which the central property of subjective expected utility is probabilistic sophistication (i.e., uncertainty is expressed in terms of probabilities). A rich structure is imposed on the relevant uncertainties and preference axioms are based on that structure. This approach can be used in decision under risk, where probabilities are already given (von Neumann and Morgenstern, 1944). Also, the approach of Savage (1954), which does not assume probabilities given, is likelihood oriented. His postulate P4 allows the derivation of probabilities. Utilities are then derived similarly to von Neumann and Morgenstern (1944).

The second group of axiomatizations is utility-oriented and is the subject of this paper. Here a rich structure is imposed on the outcome set and preference axioms are based on that structure. The central property of subjective expected utility (SEU) now is cardinality of utility, i.e., a meaningful ordering of utility differences (Vickrey, 1945). The approaches of Ramsey (1931), de Finetti, 1931, de Finetti, 1937, and Anscombe and Aumann (1963) can be classified in this group. We will show that all these models can be derived from one unifying principle, ensuring the existence of cardinal utility, invariant across different states of nature or context. Probability then results from the utility-exchange rate between different states.

In a formal manner, the unifying principle can also be introduced in the likelihood-oriented axiomatization of von Neumann and Morgenstern (1944). A corresponding derivation of their model will thus be provided. The remaining classical axiomatization (Savage, 1954) can be derived in a similar fashion if the results of qualitative probability theory (Fishburn, 1986, Wakker, 1989b, Wakker, 1993) can be used; this idea is not elaborated here. Hence Savage's axiomatization is not covered in this paper. Derivations of SEU that were directly based on the unifying principle have been presented by Wakker (1984), Wakker (1989b, 1993) and Wakker and Tversky (1993). The principle was used in experimental measurements of utility by Bouzit and Gleyses (1996), Wakker and Deneffe (1996), Fennema and van Assen (1997), Abdellaoui (1998), Bleichrodt and Luis Pinto (1998). It can already be recognized in the `standard sequence invariance' of Krantz et al. (1971).

Nonexpected utility models can be characterized by appropriate weakenings of the unifying principle. For example, Wakker (1989a); Wakker (1989b, Chapter VI) used a `comonotonic' weakening to characterize Choquet expected utility (Schmeidler, 1989). Wakker (1994) used a similar comonotonic weakening for risk to characterize rank-dependent utility (Quiggin, 1981). In these models, there is no complete separation of decision weight and utility because the decision weight of an event depends on the rank-ordering of the associated outcome. The invariant ordering of utility-differences is therefore not generally applicable but only under special circumstances (`comonotonicity'). Nonexpected utility models will not be discussed in this paper.

Section 2presents the basic method for deriving expected utility from utility tradeoffs, invariant across states of nature. In subsequent sections, it is demonstrated that the utility tradeoffs can be recognized in the classical axiomatizations of SEU, i.e., by de Finetti (1937) in Section 3, Anscombe and Aumann (1963) in Section 4, von Neumann and Morgenstern (1944) in Section 5, and Ramsey (1931) in Section 6. Appendix Abriefly demonstrates that utility tradeoffs can also be recognized in recent SEU axiomatizations. Appendix Bpresents proofs not given in the main text.

For each SEU axiomatization discussed, it is first shown how the unifying principle can be recognized in the axioms used and then how the principle can serve to provide alternative derivations.