Leigh Tesfatsion. Discussion Paper No. ... Leigh Tesfatsion. In attempting to ... article [2J by Josef Hadar and William Russell is marred by such re- strictions.
.1
STOCHASTIC DOMINANCE AND THE MAXIMIZATION OF EXPECTED UTILITY by
Leigh Tesfatsion Discussion Paper No. 74 - 37,
January 1974
Center for Economic Research to
Department of Economics Universi ty of Minnesota Minneapolis, Minnesota 55455
STOCHASTIC DOMINANCE AND THE MAXIMIZATION OF EXPECTED UTILITY'"
by Leigh Tesfatsion In attempting to construct a general framework for the analysis of choice under uncertainty, researchers have long sought to establish reasonable criteria for the selection of one prospect over another. Among current researchers the concept of stochastic dominance 1 has attracted considerable attention.
This paper attempts to clarify and
generalize certain basic relationships between stochastic dominance and the maximization of expected utility. The paper begins with a critique of an article by Giora Hanoch and Haim Levy [lJ. ,I
.,
;
Although Hanoch and Levy propose a series of inter-
esting theorems relating stochastic dominance to the maximization of expected utility, errors appear in the statement and proof of these theorems which prevent (or should prevent) the researcher from using them directly.
The necessary modifications are given in Part I below.
An undesirable feature of many articles in the area of stochastic dominance are the regularity conditions imposed on the utility functions (e.g., bounded, differentiable) and the random variables (e.g., absolutely continuous distribution function, nonnegative).
The important 1971
article [2J by Josef Hadar and William Russell is marred by such restrictions.
" ,I
Using this paper as a base, a series of relatively general
Research underlying this paper was supported by National Science Foundation Grant GS-3317 to the University of Minnesota lFirst degree stochastic dominance is said to hold between two distribution functions F and G when F(x) ::;; G(x) VX E R; and second degree stochastic dominance when [F(t) - G(t) J dt ::;; 0 \fx
IX-co
E
R
2
results have been obtained which appear especially interesting in their relation to the modified theorems of Part I.
These results are presented
below in Part II. Notation
1.
F
and
G
for arbitrary right continuous
distribution functions. 2.
U~'< = [u:R
-+
R
Iu
nondecreasing and
continuous} . 3.
U~~(F, G)
[u
I J udF
U·/(
E
J udG
-
well
defined} • 4.
U~b'
co
f~-
f N+ [1 0-
ud F
+
f
N-;·
[1 - F] du
0-
SN+[l -
~
>
~ u(O) lim N-PJ
[1 - F(N+)]
0-
~
Using
f
N+
F du
-
0-
~
f 0- ud
F
lim [-u N-PJ
F] du] = 0 - [-u(O)[l - F(O-)]
0-
- F] du ~
~
f 0- [1
- F] du
is finite, and
u(O)[l - F(O-)] + f~ [1 - F] du • 0-
fo-
7
I 0- [1 00
Suppose
JN+_ [1
- F] du
Then"
+
~
- F] du
is finite.
[1 - F(N )][u(N) - u(M)]
M
~ 0 "'if N > M ~
I
N+
I M [1 00
00
>
- F] du == lim N-[F x,
- GJ IB(t) dt} .
and, since
it is clear that
By definition of
[F - GJ (t) IB (t)
of
~
CX)
y
IA (t) dt
,
. Lx [G
T
and
- FJ I A (t) dt I ~ 2 I x - y I
T(x) = inf [x': Then
FJ
x
[x
T ~
and positivity
x'J
~
x' [F - GJ (t) IB (t) dt ~L [F - GJ (t) IB (t) dtJ (X)
00
[T(x) ~ T(x') J •
function of
x,
Hence
T
is a monotone nondecreasing
continuous and differentiable
its essential domain
D
=
[x: T(x) > - a:>} •
a.e.
over
/
Step II ~b'
Since
G(T(t)) - F(T(t)) = G( - co) - F( - co) = 0
[F(t) - G(t) J IB (t) = 0
t < t', a.e.
holds
,'d:
C
over
Step I)
D
=
ov,~r
a. e.
[t: T(t)
at any p'oint
t
a.e.
[t: t < t ' } ,
= - co}. where
DC
holds
,'d:
U D = R,
a.e.
over
[t: t
=
< t'} •
for a11 and hence
Differentiating
T' (t)
IA (T(t)) [G(T(t» - F(T(t)) J T '(t) Hence
over
for a11
(see
-k
exists, one gets
IB (t) [F(t) - G(t) J .
> -
D = [t: T(x)
the proof is complete.
oo}.
Since
/
Step III For any
u
E Uid:
SX
(F, G),
[G(t) - F(t) J duet) ~ 0 'if
X
-(X)
J udF - J udG
hence Proof:
Let
x
:c;
L
(X)
+
Gdu
u
-(X)
For any
L: I
E U,b':
(F, G).
Fdu
- [J_oo udG + J-oo udFJ + constant
x
J
~ 0
o
x,
G - F
0
I du
0
[G - FJ dt ~ 0 V x ~ 3: x' E R s.t.
-00
G(x') > F(x')
~
(by rt. continuity of
exists an interval Define 1'0
u(x) = 13
IG-
F
-co
I
[x if
dt F(x)
0 -co
.:md
0 -co
I 13
Then
well defined
xdG
I
.
xdF
are
,
[G - F J dt =
Jx13,
[G - F J dt
/
[J x [G -00
~.
G) there
and by Lemma 1>"
-co
,
I-oo
xdG
and
if x < 13 •
u(x) = x
13,
I
F
//
- FJ dt ~ 0 V x, G 1= FJ
18
H-L
D.
Theorem 3
Let
F
and
G
for some
x' x',
,
fx -0)
J
x
"
I dt
,
- GJ (x) u' (x) dx
G 3:
u' ([x* + I3J/2)) •
are sufficiently near zero, and
approaches zero sufficiently rapidly as that the sign of
U'
and
I
it is clear GJ (x) u' (x) dx
(x) dx
can be made
-co
negative. It is also clear that one may require so rapidly for
Ixl --- +
co
that
u(x)
=
lu' (x) I
Lx
!Xl U '
(t) dt
to approach zero is a bounded
21 function of
x.
In this case
u E D , l
J udG
and are finite, and by Lemma 1 ~'( Hence
[G (x*) > F (x*)
for some
x*J
-
J udF and J udG both exist J udF = S [F - G J (x) u' (x) dx
~ not
< 0 •
II
[GDF wrt D1J
H-R Corollary 1 If
then all the odd moments of
G O. The inequality appearing in the proof (top of page 333) is wrong for aM ~ 1 . A direct counterexample to the proof is obtained by setting the expected return per dollar invested in the risky asset equal to the return of the safe asset. In this case aM = 1 and daM/dwo = 0 independently of R'. ]
25 Y
to complement one's current prospects
EY
a fluctuating income and
~
0. 6
X,
where
X may represent
All of the eight theorems presented
below [except Theorem I', part iJ are proved without the density and nonnegativity as.sumptions. Finally, the conclusions of many of the theorems below are stated in terms of the stochastic dominance of one (set of) distribution function(s) by another.
It is clear from Theorems
l~"
and 2"kpresented
in Part II that such conclusions can be directly translated in terms of the maximization of expected utility over broad classes of utility functions. In Theorem I ' the decision maker is confronted with the choice of transforming his current portfolio containing a random prospect into a diversified portfolio containing a sure prospect and a specified amount of the original random prospect.
Part ii), in particular, gives a
necessary and sufficient condition for the second degree stochastic dominance of one portfolio over the other, assuming the diversified portfolio contains a positive amount of the random prospect. Theorem I ' Let
Y
a
X be a random variable with finite mean
+ bX.
Let
Y respectively. i)
and define
C be the distribution functions of X and
F and Then:
If
and ii)
x
If
F(x) = 0 b
~
I
b ~ 0,
for
x < 0,
then
[G ~ FJ
then
a
~
0
[a + bx ~ xJ ~ [G «
FJ •.
6For an approach covering this latter problem where both its coefficient are left unrestricted in sign, see [4J.
Y and
26 Proof of i) By definition of
= F([y
- aJ/b)
if
Y and right continuity of ~
y
a
[y - a J/b ~ y V Y ~ a
G(y)
~
and
= F([y
G(y)
=0
G(y)
if
F,
G(y)
Y< a
Since
- aJ/b)
~
F(y) V Y
a + x
x
~
~
a
~
Hence
F(y) V Y •
Proof of ii) Sufficiency Case I:
1
b
G(y)
= F(y - a) V y
F(y)
~
E
G(y) V y,
Case II:
~
b
R.
But
[G «
and
~
a
~
o.
Hence
FJ •
1. 0
~
F(y) = F( [y . s
-
y = a + bx
By hypothesis
a/I - b •
x = Joo
y -
Hence by Lemma 1*
~
y ~ [y - a J/b ,
for
a J/b) = G(y)
-00
~
i. e.
x
x
for finite.
xdG
0 .
~
x ~ y~'(,
For any 00
Ix
I
[F - GJ dt
by the above.
toox [F - GJ dt
=
[F - GJ dt +
Ix
00
I
00
-00
And similarly, for
00 -00
[F - GJ dt -
[G - FJ dt ~ 0
x < y~'(
,I x [F
- GJ dt ~ 0 •
-00
Hence
[G < < F J •
Case III:
b
In this case trivially.
=
0 Y
= a.
Assume
F
If
t
G.
F
=G
the corollary follows
The hypotheses guarantee that
27
SO
\G(t) - F(t) \dt
'
0, [G«
l'
and
But
- X.
Then
Y = a
+
bX
Hence by Theorem 2'
F(z) = 1 - F(-z) :o;;F(z) V z.
Hence
[G «
FJ.
Corollary B Let
X
be a random variable with distribution function
finite mean
0
G(x)
¢,
X=
0 = - b,
+ oEX = a + bx ;;:: EX = - x
a
FJ.
Define
[G «
and let
x
x < v,
for
v
and
be the value of a sure venture.
=1
G(x)
for
x;;:: v.
Then
F
and Let
[v;;::
x]
F] •
Proof Let
o
b
in Theorem 1',
ii).
/
The next theorem extends and makes precise the often cited result that if and
X
Y,
is "preferred" to
then
X
Y,
and
W is independent of both
X
+ W will be "preferred" to Y + W •
Theorem 2'7 Let
X,
functions of
X
aX
+ bW
a ;;:: O.
Y,
F,
and
G,
Y.
and
aY
and
W denote random variables with distribution
and
H .respectively.
Assume that
W is independent
Let the distribution functions of the random variables
+ bW be denoted by F and 6 respectively,
where
Then: i)
[G < F J .~
[G
> 0
a.a.z
for
5.
f]
and
[6 < FJ
if
dH (z)
7This theorem corresponds to Theorem 5 in [2J. However, the hypotheses given for Theorem 5 are insufficient to guarantee the conclusions presented. Moreover, in Theorem 5 it is assumed that W has a density and b;;:: 0 These restrictions are removed in Theorem 2' above.
I
29
SIX)
ii)
I < IX)
IF(w) - G(w)
and
[G«
FJ
-IX)
dH (z) > 0
Proof of
~
a.a.z
i)
If a
for
a
= 0, G - F
and
=0
dH
a.e.
and the Theorem holds.
0 . will be assumed below. Suppose
b
= o.
Then
F(z) - G(z)
inequality holding for some
z,
so
[G
=
F
(~)
G(~) ~
-
0 V z,
strict
< FJ
Suppose
(z - t) dH
;-. IX) F J_ oo
a
\
[G < FJ
Since
by assumption, the integrand is A
everywhere nonnegative, and positive over some interval. and
[6
0
. and
W is
Hence the proof goes through as above.
Suppose
a
FJ
Then
ii) i),
A
:::;
a.a.z.
.
As in the proof of
;;:: 0 V x,
for
Hence
~
0
will be assumed below.
b
s tric t inequali ty ho lding for some
x.
Hence
[F < < GJ
30
SX [F(z) A
b > O.
Suppose
Then
A
=
- G(z)] dz
-00
J':oo
[L: [F (z :
[J:
t) - G (z : t)1dH
[F (z : t)
(~)]dZ
G (z : t)] dzJ
Loooo
=
dH
(~)
where the interchange of integra-
oo
S-0000
tion is· justified since Since
[F«
dH(w) > 0
G] for
The case
! F(w) - G(w) ! dw < 00
by hypothesis, a.a.w,
b < 0
[F
«
and
e]
by hypothesis.
[F <
G2 (t).
•
Then
~
(~), G 2
= O[f(-t)J
+ ItIJ/[l -
~
f
(say)
= sup
G (t) 2
"
B V t, G (t) 2
Jltlf(t) dt < ~
[[Ixl
f
[II + 1 2 J dt
0 < kl < k2 < 1, G (t) 2
Hemce
"0
(~ : ~,) + F (~ ,)
L:
k'
=
such that
(k')[k - k*J
and
G l
k*
and
F
k E[k l ,k J. 2
and
Gl (t)
=
= sup
Since
It/ ~ + ~ over,
k
i)] / at
(~ : ~ ,)t~ : ~ ')2J] dt
(~ : ~)], a k '
F
k E[k 1 ,k 2 J
are measurable and
= O[f(t)J
Gl (t) as
BVt
It I ~ + ~.
as More-
It follows, since
by hypothesis, that k J 2
2
G
2
functions, independent of Convergence Theorem,
lim k-tk* k E [k , k J 2 1
(t)
k.
are Lebesgue integrable Hence by Lebesgue's Dominated
[eM/OkJ (k*)
=
lim
6 M(k)
k-it* k E [k l ,k2 J
[[F (~) F(~ : ~) F( k~ ) F(~ : ~* ) ] /[k
- k*
JJ dt
41 L(Xl(Xl wf (w) [F
For
k
co =J
(X -
[k , k ], 2 1
E
(1 k:
k~'()W)
define
(~
- F
~::W)] dw
-
D
(k~")
(say).
D(k) = [D(k) - D(k~'() ]/k - k*
t:,.
wf (w) [[S(k, w) - S(k*, w)]/k - k*] dw,
S(k, w) -
=
[F (x - (i - k)W) _ F (~ = ~w)].
where
Differentiation
under the integral sign can now be justified using the same technique as used above for
cM/~,
with the hypothesis of
a finite variance replacing that of a finite mean. (Xl result obtained is wf(w)
The
J
_(Xl
f
f
(x - ( k~'( k*) w) [w _ x ] dw 1 -
(~ = ~:::w)
Letting
[x - w] dw
1 (1 - kk) 3
first integral becomes f (u) [x - uJ du.
1
(1 _ k~'()2
u = [x - (1 - k*)w/k*]
Jco [x -
uk~'(
]
f
-(Xl
the
(x - Uk~'() 1 -
k~',
Combining this with the second integral,
one obtains: 1 (1 -
k~")
3
J~
J_coco f
1 (1 -
few) f
(~ = ~::w)[[x
k~'()3
(w) f
(x - k~' 0 3: N s. t < /':,.
Thus
•
if n
I_:
x
x
II_co Hn,k (z) dz - I_co ~(z) dz
>N
~ (z) dz,
I
being the limit of convex
functions, is itself convex.
Combining Steps I and III, (0, 1)
and symmetric over
M( • , x)
is convex in
(0, 1)
about the poi.nt
hence i t attains a minimum over this is true for any (0,1),
using
ifXER
x
E R
[H ~
:S;:S;
o
I_ooF(X) dx
G(x')
and
~ ~ ~G'
Suppose
for some
x
,
Then
(~F'
OF)
= G(y)
F(x)
= [y -
~GJ/6G
[FDG wrt U')'
[1\,
«
~
x.
kv + (1 - k) X •
1\] .
x
and
Let
1\
Then
v
the
be the
49 Theorem 5'
Let (
II.
r
G
F
and
, u~ 2) G
x"
(~F'
have finite means and variances
. 1 y. respectLve
( - co,
E
G
x' I\. 0 J
1.
F(x) ~ G(x)
2.
Sx"
x
3: x'
Suppose
( - co,
E
+ 00
J
oi) and
such that the following hold: x ~ x'
for all
[G - F J dt ..,. '- 0
for all
x
E
[x", x
'J,
and x
,
,
Sx"
SX" x
[1 - GJ(t) dt
t,
3.
Either
x'
= 0,
or
F(x' -) ::; G(x' -); F(x ") x
4.
S
G(x")
[1 - FJ(t) dt
> 0 F(x')
either
and
if
F x"
x'
= G(x')
and
x"
or
= 0
F(x"-) ~ G(x"-)
[G - F J (t) dt ~ 0 'if x ::; x"
-00
strict inequality holding if strict in-
Then
equality holds for some 5'
Corollary Let
F
x
in
1, 2, or 3, or
f.1i
>
f.1~
.
A and
G
be the distribution functions of two non-
negative random variables with a common mean and finite variances,
6~
respectively.
Then
[F < < G J
50
5'
Corollary
and
Let
F
(IJ.G'
52) G
x
~
0
B
F
~
and
G
for
x
~
F
and
x
F
5~) ,
G intersect at ~
~
G
for
~
x
x
strict inequality holding
then
F 'Ie G •
i::: 5'
If
respectively.
with
(IJ. , F
have finite means and variances
G
Corollary If
F
C
and
G
such that F(x)
have finite means and variances
~
G(x) "If x
~ X'k
F(x) ::;; G(x) "If x
