On the Pollatsek-Tversky Theorem on Risk

This paper offers remarks about, and a generalization of, the Pollatsek and Tversky theorem on a measure of risk. As in the Pollatsek-Tversky paper, attention is limited to criteria that are additive with respect to convolution. The objective is to weaken the scalar monotonicity axiom or to replace it with another one to arrive at the more flexible form of an additive criterion as a finite linear combination of cumulants of higher orders. The new criterion can be taken in two different ways. The first consists of a direct generalization of the scalar monotonicity axiom. The second appeals to a continuity condition that is stronger than the usual one. This paper also discusses properties of the additive criteria and the independence axiom.