This chapter lays down the foundations of stochastic dominance rules from a rigorous and general setting. We choose to present the theory from …rst principles.
In Chapter 1 we obtained quantitative presentations for preference relations of individuals in terms of their utility functions. However, in practice, it is hard to specify utility functions of individuals. Thus, it seems natural to ask whether we could go a step further to represent the total order de…ned by utility functions by other stochastic orders which depend only on distributions of random variables involved, but not on utility functions. As we will see, this is possible only locally, i.e., when we consider a coarsening of risk attitudes of decision-makers and address the problem in each category of risk attitude. As explained in Chapter 1, each type of risk attitude is characterized by the shape of utility functions. Thus, the derivation of a quantitative representation of preference relations for an individual belonging to a given class of risk attitude depends on the shape of utility functions of individuals in that class. Moreover, the quantitative representations obtained lose the power of a total order, in other words, they are only partial orders. The practical aspect of this “…ner”representation of preference relations is that, within each class of risk attitude, decisions could be based only on distributions of random variables (e.g., future returns on investment prospects) which,
will discuss reliable estimation based on such data. In order to derive presentations of preferences, in terms of distribution
functions, in each category of risk attitude, we need some results on approximations of appropriate functions as well as a lemma involving the Stieltjes integral.