Four notions of mean-preserving increase in risk, risk attitudes and applications to the rank-dependent expected utility model

This article presents various notions of risk generated by the intuitively appealing single-crossing operations between distribution functions. These stochastic orders, Bickel and Lehmann dispersion or (its equal-mean version) Quiggin's monotone mean-preserving increase in risk and Jewitt's location-independent risk, have proved to be useful in the study of Pareto allocations, ordering of insurance premia and other applications in the expected utility (EU) setup. These notions of risk are also relevant to the Quiggin-Yaari rank-dependent expected utility (RDEU) model of choice among lotteries. Risk aversion is modeled in the vNM expected utility model by Rothschild and Stiglitz's mean-preserving increase in risk (MPIR). Realizing that in the broader rank-dependent setup this order is too weak to classify choice, Quiggin developed the stronger monotone MPIR for this purpose. This paper reviews four notions of mean-preserving increase in risk-MPIR, monotone MPIR and two versions of location-independent risk (renamed here left- and right-monotone MPIR)-and shows which choice questions are consistently modeled by each of these four orders.