STOCHASTIC

A UNIFIED APPROACH TO STOCHASTIC DOMINANCE
II. Examples of Stochastic Dominance Relations
In the following examples, "increasing" should be interpreted in the weak
sense, i.e., as "nondecreasing," and is denoted by the symbol /. An analogous
convention is for "decreasing". The operators A and v are used for
"minimum" and "maximum," respectively. For example, min[a,b~\ = a/\b.
EXAMPLE 1
The first example is referred to as first-order stochastic dominance by Hadar
and Russell [4] and appeared in the work of Lehmann [6] (also see Hanoch
and Levy [5]). This example shows that an investor with an increasing utility
function prefers X to Y if P[Y^ w] ^ P[X^ w~] for all values of w, i.e.,
1 — G(w) ^ 1 — F(w) for each w.
Define the function /[xgw] to be 1 if x ^ w and 0 if x < w. Then P[Y}t w] =
£/[Y^] and P[X^w] = £/tXgW]- Given a function/, the function / + is
called the positive part of/and is defined by/ + = max[/,0]. Similarly,
/" = min[/,0]. Note that/=/ + +/".
Theorem 2.1 Suppose that / is an increasing function defined on R =
(—00,00) with v(/~)>— oo. Let G = {/[.§w], w e R} and suppose that
v(g) ^ n(g) for each g e G. Then v(/) ^ n(f). Note that v(g) — oo, it follows by the dominated convergence
theorem [7, p. 125] that -co < v(/") = lim„v(/„-) ^ lim^/T) = A* CD-
Since/=/ + +/ , v(/) ^ A