New methods in the classical economics of uncertainty: comparing risks

Second-order stochastic dominance answers the question “Under what conditions will all risk-averse agents prefer $$\tilde{x}_2$$ x ~ 2 to $$\tilde{x}_1$$ x ~ 1 ?” Consider the following related question: “Under what conditions will all risk-averse agents who prefer lottery $$\tilde{x}_1$$ x ~ 1 to a reference lottery $$\tilde{\omega }$$ ω ~ also prefer lottery $$\tilde{x}_2$$ x ~ 2 to that reference lottery?” Each of these two questions is an example of a broad category of questions of great relevance for the economics of risk. The second question is an example of a contingent risk comparison, while the question behind second-order stochastic dominance is an example of a non-contingent risk comparison. The stochastic order arising from a contingent risk comparison is obviously weaker than that arising from the corresponding non-contingent risk comparison, but we show that the two stochastic orders are closely related, so that the answer to a non-contingent risk comparison problem always provides the answer to the corresponding contingent risk comparison problem. In addition to showing the connection between parallel contingent and non-contingent risk comparison problems, we articulate a method for solving both kinds of problems using the “basis” approach. The basis approach has often been used implicitly, but we argue that there is value in making its use explicit, particularly in indicating which new, previously unsolved problems can readily be solved by the basis approach and which cannot.