THE ORDINAL UTILITY UNDER UNCERTAINTY AND THE MEASURE OF RISK AVERSION IN TERMS OF PREFERENCES 1. INTRODUCTION A choice is said to be rational if it follows a criterion, which is usually introduced as a preference model. The von Neumann-Morgenstern theory 1 not only assumes such a rationality but also other axioms with which the neo-Bernoullian utility is connected.
Since a preference model can be represented by ordinal utility also in the yon Neumann-Morgenstern case, 2 a first analysis concerns the determination in this case of the relationships between the different utility indices of the actions on the one hand and the utilities of their consequences and probabilities on the other.
A second analysis concerns the measure of risk aversion, usually given by the Arrow-Pratt index, which is referred to the neo-Bernoullian utility. But a more general measure is necessary if we accept that a preference model can be considered without assuming, for instance, the independence axiom. A new index of risk aversion is proposed in this paper. It requires only the existence of a certainty equivalent for each action. This index turns out to be zero when the von Neumann-Morgenstern axioms hold and its derivative to be proportional to the Arrow-Pratt index.
It is also shown that the measure of risk aversion can be positive if the yon Neumann-Morgenstern axioms are not all assumed.
2. THE ORDINAL UTILITY FUNCTION OF UNCERTAIN ACTIONS 3
Every action is represented by a probability distribution of consequences. Then, by indicating the set of the consequences with C (where C is any set, not necessarily a Euclidean one), the actions are probability distributions on C, i.e., functions a: C ~1, where I is the real unitary interval, with a(c)>~ 0 and No,ca(c) = 1. A set A of actions is considered as well as a preference