A large strand of research has identified conditions on preferences under which (i) a single risk is undesirable and (ii) two independent risks aggravate each other. We extend this line of inquiry by
establishing conditions such that (iii) the degree of mutual aggravation is greater for more severe risks. Here, the severity of risks is characterized by means of general stochastic dominance shifts
and also via moment-preserving stochastic transformations. Greater mutual aggravation is implied by all commonly used utility functions and may thus be regarded a typical property of expected utility preferences. We show that greater mutual aggravation determines the comparative statics of risk changes in several risk management problems, including precautionary saving, intertemporal
risk-taking, and self-protection. Greater mutual aggravation further explains recent experimental findings on higher-order risk preferences. Finally, it offers a new, simple, and efficient method to
elicit risk preferences up to “very” high orders.

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