STOCHASTIC

andom variables when the utility functions are quadratic or cubic. In
Exercise ME-5 another type of dominance relation of relevance to convex
costs rather than concave utilities, is considered. Exercise ME-11 examines
dominance relations for concave utility functions which are not assumed to be
nondecreasing. Exercise ME-7 illustrates the caution which must be exercised
when dealing with dominance relations, in that the addition of a random
variable to each side of the relation may render it invalid. It should be noted
that all such dominance relations are relevant only to the problem of choosing
one random variable from a given set of random variables. In the portfolio
problem, the possibility of investing partially in each of several assets will
generally introduce all of the random assets into the optimal solution, even
if one of them strictly dominates the others. This diversification behavior is
illustrated in the extreme in the paradoxical Exercise CR-12. Stochastic
dominance relations between different total portfolios are what matters in the
portfolio allocation problem. Exercise ME-22 utilizes stochastic dominance
results in the analysis of a reinsurance problem.
The Brumelle and Vickson paper examines stochastic ordering relations
from a unified view, based on recent work in the field of measure theory.
The emphasis is on simple tools which permit better understanding of the
underlying structure of such dominance relations, and which allow some new
dominance relations to be discovered in a straightforward manner. The
"technical" level of the paper is not unduly high, but the general tone is
perhaps somewhat abstract. The paper emphasizes the fact that most of the
interesting classes of utility functions are convex sets in a function space, in
the sense that they are closed under addition and multiplication by nonnegative
constants. If the sets are represented as closed convex hulls of their
extreme points (just as for convex sets in ordinary Euclidean space), then
necessary and sufficient conditions for dominance are obtained by restricting
one's attention to the extreme points. Since the extreme points can often be
determined by inspection, the stochastic dominance tests drop out immediately.
The paper reexamines first- and second-degree dominance from this viewpoint,
and then treats the interesting case of third-degree dominance. This is the
ordering relation relevant to the class of concave nondecreasing utility functions
having nonnegative third derivative (i.e., having convex first derivative).
The interest in this class arises from the fact that it includes many of the
economically relevant utility functions for wealth. This ordering relation was
introduced by Whitmore (1970). In Exercise ME-6 it is shown that thirddegree
dominance is equivalent to an ordering of double integrals of cumulative
distributions. Exercise ME-20 extends the theory to nth-degree stochastic
dominance.
The paper also discusses an alternative formulation of second-degree
dominance which emphasizes the underlying probabilistic content of the
ordering. The ordering is shown to be equivalent to the existence of a bivariate
INTRODUCTION
83