Third-Degree Stochastic Dominance

Abstract

Here F(x) and G(x) are less-than cumulative probability distributionis where x is a continuous or discrete random variable representing the outcome of a prospect. The closed interval [a, b] is the sample space of both prospects. The integral shown in Rule 2 and those shown throughout the paper are Stieltjes integrals. Recall that the Stieltjes integral fb f(x)dg(x) exists if one of the functions f and g is continuous and the other has finite variation in [a, b]. Let D1, D2, and D3 be three sets of utility functions ?(x). D1 is the set containing all utility functions with 4(x) and +1(x) continuous, and 41(x) >0 for all xE[a, b]. D2 is the set with ?(x), ?1(x), ?2(x) continuous, and q$j(x)>0, 02(x)?O for all xC[a, b]. D3 is the set with ?(x), ?1(x), ?2(X), ?3(X) continuous, and +1(x) > 04 2(x) O O for all xC[a, b]. Here +1(x) denotes the ith derivative of +(x). Hadar and Russell proved that Rule 1 is valid for all ,CD1 and Rutle 2 is valid for all ED2. The authors point out that the set of probability distributions that can be ordered by means of second-degree stochastic dominance is, in general, larger than that which can be ordered by means of first-degree stochastic dominance. Note that in Rule 2, they assume that +(x) is not only an increasing function of x but also exhibits weak global risk aversion, a condition guaranteed by requiring the second derivative of ?(x) to be nonpositive. In this paper, a condition which will be called third-degree stochastic dominance is considered. It is based on the following assumption about the form of the utility function ?(x). From a normative point of view, one expects the risk premium associated with an uncertain prospect to become smaller the greater is the individual's wealth. The plausibility and implications of this assumption h'ave been explored by John Pratt, as well as others. The risk premium of an uncertain prospect is that amount by which the certainty equivalent of the prospect differs from its expected value. In mathematical terms, given the prospect F(x) with expected value A, the corresponding risk premium -t is obtained by solving the following equation. rb

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