Bringing order to rankings of utility functions by strong increases in nth order aversion to risk

Rankings of utility functions generated by simple nth order risk-averse transformations are not partial orders, and therefore, do not yield reliable comparative statics predictions, except at the second order. Restrictions have been identified that rectify this deficiency at the third order concerning downside risk aversion: the strong order and the Schwarzian. We show that these concepts and their characterizations generalize to all higher orders of risk preference, the latter in two ways, pathwise (parametric) infinitesimal increases and n-monotone increases in aversion to risk, and we provide applications to intertemporal choice problems for self-protection and saving.