The Present State of Utility Theory

Summary and Conclusion

The present state of the utility theory can be summarized by four propositions:

• 1. The utility function u (x) derived from the vNM (von Neumann & Morgenstern) axiomatics is not the cardinal utility of the certain outcome x ; this last utility is a different function v (x).

• 2. The relation between u and v expresses the utility of uncertainty. If u = v, as assumed in /14/, the expected utility, a preference functional among uncertain prospects, expresses a risk neutral behavior.

• 3. The axioms of vNM shall be dropped. In particular, single stage and multiple equivalent bets are not indifferent, i. e. of equal utility. This propos ition, which requires the abandonment of the independance axiom as expressed by axiom 3:C:b of vNM and by the formulations of Samuelson /2/ and Malinvaud /3/, is necessary in order to be able to establish a general utility function of uncertain prospects such that it covers the intrinsic utility of uncertainty.

• 4. Such a general utility function is non-linear in probalities.

At present three schools are offered. The first, represented among others by the three anthors quoted in Section 2 of this paper, tries to formulate and empirically verify the points 1 and 2 of this conclusion. The second, of, among others, the three authors quoted in Section 3 of this paper, tries to prove conclusions 3 and 4.

My proposal of 1966 – 1974 /19 to 25/ assumes all four conclusions. But it lacks formal proof and is also critized as it can lead to violations of the principle of dominance. It is shown in /22/ and recalled here that such violations are in fact fruitful explanations of possible actual behaviors and can be considered as rational.

Our younger followers should seek a synthesis of these schools.