Northwestern University
Marciano Siniscalchi
Fall 2009
Econ 331-0
STOCHASTIC DOMINANCE
1.
Introduction
Our analysis so far builds upon expected-utility theory. An unpleasant implication of this ap-
proach is that, without knowledge of an individual’s utility function, there is little we can say about
her preferences.
True, if we know that she is risk-averse, we may be able to determine whether she prefers a
certain amount of money to a random variable describing uncertain or risky terminal wealth; we
are able to construct the eﬃcient frontier if we consider normally distributed returns; etc. But note
that, in both cases, we are restricting attention to
speciﬁc
comparisons—e.g. comparing certain
sums of money with random variables, comparing normally distributed portfolios, etc. However,
for
general
random variables
X,Y
, we cannot say whether
X
±
Y
,
Y
±
X
, or
X
∼
Y
.
It is clear that we cannot completely remedy this situation—otherwise, utility theory would
be useless! However, we
can
make some progress on the sole basis of economically meaningful
assumptions such as “more money is better” and risk aversion. This leads to the notion of
stochastic
dominance
.
2.
First-Order Stochastic Dominance
With this as backdrop, consider two random variables
X
and
Y
. The following seems a plausible
deﬁnition of the assertion that
X
is more likely to yield better outcomes than
Y
:
Deﬁnition 1.
For any two random variables
X,Y
, say that
X
ﬁrst-order stochastically dom-
inates
Y
(or
X
FOSD
Y
) iﬀ, for all
m
∈
R
,
Pr[
X > m
]
≥
Pr[
Y > m
]
.
The important point is that the above inequality is required to hold for
all
real numbers
m
. In
a way,
X
is “unequivocally better” than
Y
. Note also that FOSD may be seen as establishing a
transitive, but
partial
(as opposed to complete) order on random variables. On the other hand, EU
yields a complete order.
For instance, suppose