On the relationship between comparisons of risk aversion of different orders

We show conditions which ensure that the comparisons between risk aversion of different orders of two decision makers are related. In particular, we derive a condition ensuring that greater downside risk aversion implies greater risk aversion and a different condition ensuring that the opposite implication holds. We then generalize these results to higher order greater risk aversion, obtaining conditions which make it possible to infer the direction of the comparison for risk aversion of a given order from the knowledge of the direction for\r\na different order.