JOURNAL
OF ECONOMIC
THEORY
3522317 (1992)
56,
Stochastic Dominance
under
Bayesian Learning*
SUSHILBIKHCHANDANI
Anderson Graduate
University
of California,
School of Management,
Los Angeles, California
90024
UZI SEGAL
Department
of Economics, University of Toronto,
Toronto. Canada M55 IA1
AND
SLTNILSHARMA
Department
of Economics,
University
of California,
Los Angeles, California
90024
Received January 22, 1990; revised June 11, 1991
The concept of first-order stochastic dominance defined on distributions is inadequate in models with learning. We extend this concept to the space of distributions
on distributions. We discuss conditions under which for all common observations
one person’s beliefs (over a set of probability distributions) dominate another
person’s beliefs by first-order stochastic dominance. We obtain sufficient conditions
for this partial order and show that the sufticient conditions are necessary, provided
that the underlying distributions satisfy an additional assumption. These conditions
can be verified without taking any observations. Applications are discussed. Journal
of Economic Literature
Classification Numbers: Cll, DSO, D81, D83. 0 1992
Academic
Press, Inc.
1. INTRODUCTION
The concept of first-order stochastic dominance is usually defined on
probability distributions over final outcomes (simple lotteries). Although
this definition has been widely applied in economics, there are situations
where it is inadequate. In this paper we analyze the concept of first-order
* We are grateful to Ken Burdett, Seonghwan Oh, Lars Olson, Joe Ostroy, Zvi Safra,
Lloyd Shapley, and two anonymous referees for helpful comments.
352
0022-0531192 $3.00
Copyright
All rights
0 1992 by Academic Press, Inc.
of reproduction
in any form reserved.
STOCHASTIC
DOMINANCE
UNDER
LEARNING
353
stochastic
dominance for probability distributions
over probability distributions
(compound lotteries) under Bayesian learning.
e discuss
conditions under which one person’s beliefs dominate another person’s
beliefs by first-order stochastic dominance regardless of what they observe
in common. We provide sufficient conditions on prior beliefs under whit
this is true. These conditions can be easily verified without taking any
observations.
One may claim that it is unnecessary to define stochastic dominance
relationships between updated beliefs. For instance, one may multipfy
probabilities in the compound lotteries and apply the usual definition of
stochastic dominance to the resulting actuarially equivalent simple lotteries.
However, the partial order on compound lotteries obtained by ~ornpar~n~
actuarially equivalent simple lotteries may not be very useful from a normative standpoint. When a decision-maker faces a series of decisions, with
some resolution of uncertainty between decisi
valuable information
ution on distributions.
may be lost by multiplying
probabilities in a di
In any setting where there is learning, the usual concept of first-order
stochastic dominance for distributions
may be inadequate.
This is
illustrated by the following example, adapted from ~ikhchanda~i
and
Sharma [2].
Consider a risk-neutral decision-maker searching
uentially (with or
m distribution on
without recall) for the lowest price. Let F, be a un
CO, I] and let G, , G, be uniform distributions on CO,i) and [i, I], respectively. F is the compound lottery which is degenerate at F, and G is the
compound lottery which yields the simple lottery Gi with probability 1,
i = 1,2. The decision-maker may either take price samples from F only, or
from G only. The cost of each sample is i. It is well kn
searching from a known distribution (i.e., a simple lottery)
to stop as soon as one observes a price less than some reservation price
(see, for example, Lippman and McCall [lo]). The reservation price, r, is
obtained by solving
c=srH(x)dx,
0
(1.1)
where c is the cost of each sample. Moreover, the minimum expected cost
is equal to r. (The cost includes the price paid for the good as well as the
sampling cost.) Thus, when searching from F, the minimum expected cost
and the reservation price are both equal to l/d.
On the other hand, when
searching from G, the decision-maker knows after exactly one observation
whether he is searching from G, or G,. Using (1.1) it is easy to check that
it is optimal to stop after the first observation from 6, and the minimum
expected cost is 0.75. Clearly, both
and G have the same actuarially
F
354
BIKHCHANDANI,
SEGAL,
AND
SHARMA
equivalent simple lottery, and yet the expected cost is lower under F. The
usual definition of first-order stochastic dominance, when applied to
actuarially equivalent simple lotteries, is inadequate.’
Our concept of stochastic dominance under Bayesian learning, what we
call Bayes’first-order
stochastic dominance, states that a compound lottery
P dominates another compound lottery G if after any sequence of
(common) observations, the actuarially equivalent simple lottery of the
posterior distribution
of p dominates the actuarially equivalent simple
lottery of the posterior distribution of G, by ordinary first-order stochastic
dominance. In the search example above, after one observation the
posterior distribution
of F remains unchanged whereas the posterior
distribution of G is either degenerate at G, or degenerate at G,. Thus F
and G cannot be compared by Bayes’ first-order stochastic dominance. It
can be shown that if a compound lottery $ dominates another compound
lottery 6 by Bayes’ first-order stochastic dominance, then the expected
total cost (price of good plus sampling cost) is lower under G.
Bayes’ first-order stochastic dominance is not a new concept. This and
other related concepts have been used in the literature. For instance, Bayes’
first-order stochastic dominance is the same as Berry and Fristedt’s [ 1 ]
concept of “strongly to the right,” which is useful in deriving comparative
static results for the bandit problem. For the case when the simple lotteries
have two final outcomes, Berry and Fristedt provide an equivalent condition for Bayes’ first-order stochastic dominance. However, in order to
check this condition all possible sequences of observations have to be
considered. The contribution
of our paper is that it provides easily
verifiable conditions for Bayes’ first-order stochastic dominance between
compound lotteries with finitely many final outcomes. More importantly,
our conditions can be checked without taking any observations.2
i This example shows that from a normative
viewpoint,
the partial order obtained by comparing actuarially
equivalent
simple lotteries may be inadequate
when there is learning. From
a descriptive
viewpoint
this partial order may be inappropriate,
even in situations
where there
is no learning, since decision-makers
may violate the reduction
of compound
lotteries axiom.
That is, they may not be indifferent
between a compound
lottery and its actuarially
equivalent
simple lottery (see Segal 1181 and the references cited there). If we modify the above example
so that the decision-maker
can search exactly once from either For G, then an individual
who
does not subscribe
to the reduction
axiom would not be indifferent
between
F and G.
Bikhchandani,
Segal, and Sharma [3] obtain necessary and sufficient conditions
for stochastic
dominance
under Bayesian learning for such decision-makers.
‘The question
of ordering
distributions
on distributions
has been addressed
before by
Bohnenblust,
Shapley,
and Sherman
[5], and Blackwell
141. Bohnenblust,
Shapley,
and
Sherman
obtain
a partial
order
on information
systems,
that is, on distributions
on
distributions,
based on their value to the decision-maker.
Blackwell
shows that this partial
order is the same as the one obtained
from the statistical
notion of sufficiency.
This notion
compares
two information
systems
based on the criterion
of second-order
stochastic
dominance
of the expected posterior
distributions
after one observation.
STOCHASTIC DOMINANCE UNDER LEAIRNING
355
A related question has been addressed by Whitt [I93 and aileron
[14]. They are interested in conditions under which the posterior distribution of a compound lottery updated after an observation dominates
another posterior distribution of the same compound lottery updated after
a less favorable observation, by first-order stochastic dominance. We
discuss the relationship
between this concept and
ayes’ first-order
stochastic dominance in Section 3.
This paper assumes the existence of probabilities of probabilities. Some
authors argue against this concept (for a discussion see ~arschak [127,
and in particular de Finetti [6]). We presume that they mean that when
there is no learning, probabilities
of probabilities
are equivalent to
ies, but when there is learning, the two are different. That, at
is Totrep’s view in Kreps’ [9, p. 150] splendid drama concerning
Our results are as follows. We show that when each underlying simple
lottery is over the same two final outcomes, a compound lottery F
dominates another compound lottery G by ayes’ first-order stochastic
dominance if and only if F is a convex transformation
of G (in a sense
made precise later), if and only if the expected final outcome from any
updated version of F is greater than the expected final outcome from an
updated version of G using the same observations. This generalizes to the:
case where the underlying simple lotteries yieid a finite number of final
outcomes provided that the simple lotteries can be completely ordered
by ordinary first-order stochastic dominance and satisfy an a~d~tio~a~
assumption
The paper is organized as follows. We give necessary and sufhcient
conditions for Bayes’ first-order stochastic domina~cc in Section 2.
first analyze the case when there are two final outcomes and then the
when there are finitely many final outcomes. Ap~~~cat~o~s are discusse
Section 3. All proofs are in the Appendix.
2. BAYES’
FIRST-ORDER
STOCHASTIC
Let L, s {(PI, P,, .... P,):Pi>Q, CPi=l)
be the s ace of probability
distributions
on a finite set (x1, x2, .... x,>, x1 <x2< ... <xM. An
element of L,, Xi= (PiI, Pi2, .... PiM) represents a simple lottery yielding
outcome xi with probability P,.
Let I,,= ((X,, a,; X,, a,; .... X,,afg): Cti38, xUi=l,
XfEL,)
be the
space of probability distributions with finite support in L,. Elements of L,,
called compound
lotteries, are denoted by 17, G, etc. The Lottery
’ Trade-off talking rational economic person.
356
BIKHCHANDANI,
SEGAL,
AND
SHARMA
F= (I,, a,; X2, a,; ... . X,, aN) yields the simple lottery Xi with probability
aj. When comparing two compound lotteries F and G we write, without
loss of generality, F= (a,, Q, .,., a,) and G = (pi, p2, .... PN), where some
of the ai and pi may be zero. By multiplying the probabilities under F, say,
we obtain its actuarially equivalent simple lottery E[F] EL,, where
E[F]
= (C-i” criPil, C: aiPia, .... Cy cxiPiM)<
An observation from a compound lottery F is an outcome, Xi. We will
sometimes refer to xi as the final outcome. The simple lottery Xi which
gives this final outcome is not observed. Successive observations are
independent draws from the same simple lottery Xi. After observing ti
realizations of xi, t, realizations of x2, and so on, the decision-maker uses
Bayes’ rule to obtain a posterior compound lottery in L, which is denoted
by F(t,, t,, .... tM), or F(T), where T=(tl,
t,, .... tM), Let a,(T) be the
posterior probability given T that Xj is the true simple lottery. By Bayes’
rule,
Since we assume that P, > 0 for all i, j, ai( T) is well defined for all i and
T.4 Thus F(T)= (q(T), a,(T), .,., a,(T)).
DEFINITION 1. Let Xi, X, E L,. Xi dominates
X2 by first-order
stochastic dominance, denoted by X, FOSD X,, if and only if for all
increasing functions U: R -+ R, c,E 1 P, U(xj) > c,E 1 P, U(xj).
It is well known that X1 FOSD X, if and only if
Ql<M.
(2.2)
Since for every F, GEL,, E[fl, E[G] EL,, the partial order FOSD
induces a stochastic dominance relation on L, in an obvious way. This
definition of stochastic dominance is useful when decision-makers subscribe
to the reduction of compound lotteries axiom (i.e., they are interested only
in the probabilities of final outcomes), and there is no learning. However,
the search example in the Introduction
shows that comparing ECF] and
E[G] by FOSD is inadequate when there is learning. The following
definition
requires that stochastic dominance
be maintained
under
Bayesian learning.
4 We assume
and j.
P,>O
for simplicity.
Our
results
can be extended
to allow
P,=O
for some i
STOCHASTIC
DOMINANCE
UNDER
357
LEARNING
DEFINITION
2. Let F = (ul, Q, .... CY~), G = (PI, p2, -.., fiN) E L,. F
dominates G by Bayes’first-order
stochastic dominance, denoted by F+= 6,
if and only if VT, E[F( T)] FQSD E[G( T)].
Our definition of Bayes’ first-order stochastic dolminance is identical
the concept of strongly to the right in Berry and Fristedt [I].’
to
2.1. Two Final Outcomes
We first present the case when the simple lotteries Xi, i= 1,2, .... IV,
yield two final outcomes, x1 and x2, x1 <x2. Without loss of generality
let Pll>PZl>
.,. b-PNl, where Xi = (Pi,, Pi2). Thus X,,, FOSD X,_ 1 I ..
The expected final outcome under the posterior distribution
E[xl F(T)]
= i
j=l
F(T) is
t x,cc,(T)P,.
i=l
E[x ( G(T)] is similarly defined. Berry and Fristedt [l]
of two final outcomes that F+ G if and only if
ECx IF(T)1 2 E[x I G(T)],
proved for the case
(2.3)
VT.
However, (2.3) is an impractical condition for applications, since it requires
that the posterior distributions F(T) and G(T) be computed for all T. It is
our aim here to find easily computable conditions equivalent to P”+ 6.
Consider the following
conditions
on F = (or,, Q, .... a,), G =
(aI, pz, .... fiN) E L,. There exist a 2 1, b < N such that
Moreover,
F= (0, .... 0, a,, a,+ I, . ... EN),
aj>O, Viaa
(2.4)
G = (PI, Pz, ...> Pb, 0, .... 01,
p,>o,
(2.5)
Vi6b.
if a < b then
cli
%+ai+l
Pi
a<i<b
(2.6)
‘Bi+Pi+l’
5 A stronger definition would be to require that stochastic dominance be maintained after
any sequences of observations T from F and T’ from G. Thus, corresponding to 3 we may
define F+* G iff E[F(T)] FOSD E[G(T’)],
VT, T’. Let P= (A’,, pi; X,, p2; .... X,, p,),
p,>O,
and G= (Y,, ql; Yz, q2;.... Yr, qL), ql>O. It is easily verified that F>* G if and only
if Xi FOSD Y,, Vi, 1. First note that for all T the support of I;tT) is (X,, X,, .... X,), and for
all T’ the support of G( T’) is (Y,, Y,, .... YL). Therefore a sufficient condition for F+* G is
that Xi FOSD YI, Vi, 1. From (2.1) it is clear that for any i and 1 one can find T and T’ such
that F(T) places most of its mass on X,, and G(T’) on YI. Thus a necessary condition for
F+* G is that X, FOSD Y,, Vi, 1.
358
BIKHCHANDANI,
SEGAL,
AND
SHARMA
or, equivalently,
-- cli
@-,+1
Pi
a<i<b.
%+ly
Conditions (2.4) and (2.5) imply that if for some i, cli = 0 and fii > 0, then
ak = 0 Vk < i. Also, if for some i, cli > 0, pi = 0, then Pk = 0 Vk > i. Condition
(2.6) implies that if ai, a,+i, /Ii, /Ii+ i > 0 for some i then the conditional
distribution of I; on Xi, X,+r dominates the conditional distribution of G
on Xi, Xi+ 1 by FOSD.
The main result of this section is stated below. The proof is omitted since
Theorem 1 is a special case of Theorem 3 below.
THEOREM
1. Let X1, X,, .... X~E L, be simple lotteries with two final
outcomes x1 and x2. Let F= (a,, q, .... ~1~) and G= (PI, p2, .... PN) be
compound lotteries with outcomes in the ordered set (X,, X,, .... X,). The
following statements are equivalent:
(i) F+G.
(ii) E[x/i(T)]
>E[xjG(T)],
VT;
(iii) F, G satisfy (2.4), (2.5), and (2.6);
(iv) F, G satisfy (2.4) and (2.5), and there exists a convex function, f,
such that Fi = f(Gi) Vi E {a, a + 1, .... b), where a and b are as defined
in (2.4) and (2.5), and Fi=zf=,
a,, Gi= CL=, /3r are the cumulative
distribution functions of F and G, respectively.
When F and G have the same support, that is, when F= (a,, CQ,.... a,),
G = (PI, Pz, .... /IN), and cli > 0, pi > 0 Vi, condition (iv) of Theorem 1 states
that the cumulative distribution function of F is a convex transformation of
the cumulative distribution function of G. Conditions (iii) and (iv) can be
checked easily, since they do not require any observations. Conditions (i)
and (ii) cannot be checked in general, since they are conditions on the
updated distributions F(T) and G(T) for all T.
2.2. Many Final Outcomes
In this section we consider the case when each simple lottery yields
finitely many outcomes. We restrict ourselves to the case where
X, FOSD .XNmI . . . FOSD X1. We first show that (2.4), (2.5), and (2.6) are
sufficient for F+ G.
THEOREM
2. Suppose that X, FOSD X,,- I . .. FOSD Xi. Let F =
al,
m2,
.
.
.
.
aN)
and G = Ml,
P2, . . . . BN) be compound lotteries with outcomes
(
in the ordered set (X1, X,, .... X,). If F and G satisfy (2.4), (2.5), and (2.6)
then F> G.
STOCHASTIC DOMINANCE UNDER LEARNING
359
It can be shown that even when X,, X,, .... X, are ordered by F
(2.4), (2.5), and (2.6) are not necessary for F+ G. However, these condil
tions are necessary when X1, X,, .... X,,, are of a special type defined below.
DEFINITION 3. The simple lotteries Xi= (PiI, Pi2, .~.,PiM), i= 1, 2, .,., Iv
are of type 1 if Xi+ I is a convex transformation of X,, i= 1, 2, ...) W- 1.
That is, there exist convex functionsfi, i= 1, 2, .... W- 1 such that
j,
Pjil,i=f(
i
j=
Pg],
It can be shown (see Lemma
are of type 1 if and only if
pi
----<
j+l
pi+I,j+l
pij
Vl<M,
A.2 in the Appendix)
Qj<M,
pi+l,j
Qi=1,2
,..., IV-I.
(2.7)
1
that
Qi= 1, 2, .... N-
1.
G3)
’
Condition (2.8) is equivalent to the monotone likelihood ratio property
(see Ferguson [S]). This assumption is used in a number of models in
auction theory, principal agent problems, etc. (see Milgrom 1141).
DEFINITION
4. The simple lotteries Xi= (PiI, Pip, ..D,PiM), i= 1,2, .... N2
are of type 2 if
p,
DEFINITION
2
pi+
1, j9
Vj’j< M, Qi= I, 2, ..~)N-
1.
(2.9)
5. The simple lotteries Xi = (Pi,, Pi2, .... PiM), i= 1, 2, ~..)Ar
are of type 3 if
pijGpi+l.j,
Qj> 1, Qi= 1, 2, .. .. N-
1.
(2.10)
In the two final outcomes case considered in Section 2.1 any collection
of simple lotteries are of types 1, 2, and 3. To see this, label the simple lotteries over two final outcomes X,, X,, .... X, so that P,, > Pzl > . . >
Clearly X,, X,, .... X, satisfy (2.8), (2.9), and (2.10). For the case of
final outcomes, types 1, 2, and 3 are illustrated in Figs. 1, 2, and 3, re
tively. These triangle diagrams show simple lotteries in the P, - P, s
Choose any Xi and plot it on a triangle diagram as shown in Fig.
points (PI, Pz, P3) below the line AE satisfy PJP, 3 Pi2/Pil,
a
points above the line BD satisfy P,/P2 3 Pix/Pi2. Thus (2.8) implies that if
Xl, x2, .... X, are of type 1 then Xi+ 1 must lie in the region AC
Xi- 1 must lie in BCE.
6 Such diagrams were reintroduced into the literature by Mark Machina. He attributes
these diagrams to Jacob Marschak (see Machina [ll]).
360
BIKHCHANDANI,
SEGAL,
AND
SHARMA
p3=
Prob(x3)
P, = Prob(x,)
FIGURE
1
A similar argument establishes that if Xi, X,, .... 1, are of type 2 and Xi
is as shown in Fig. 2, then Xi+ i must lie in the region AFCD, and X,-i
must lie in HCE. Also, if Xi, X,, .... X,,, are of type 3 and Xi is as in Fig. 3,
then Xj+l is in FCD and Xiel is in HCEB. These figures also show that
the three types are not identical.
In each of Figs. 1, 2, and 3, all simple lotteries that dominate Xi
(by FOSD) are northwest of Xi, and Xi dominates all simple lotteries
to its southeast. Thus, at least for three outcomes, if X,, X,, .... X, are of
1
p3’
Prob(xJ
P, = Probdx,)
FIGURE
2
STOCHASTICDOMINANCE
UNDER
361
LEARNING
P, =
Prob(x,)
P, = Prob(xJ
FIGURE
type 1, 2, or 3, then X, FOSD X,establishes this in general.
LEMMA 1. rf X, , X,, .... X,
X, FOSD X, p I . . . FOSD X1.
3
1 . ~. FOSD X, ~ The following lemma
are
of
type
1,
2,
or
3,
then
As is evident from Figs. 1, 2, and 3, the converse of Lemma 1 is not true.
Also, there exist Xi, X,, .... X, which are of types 1, 2, and 3. For example,
we can choose X,, X,, . ... X, which satisfy (2.8) with P,= Pi+l.i,
9=2,3 , ....M-1.
i=l,2 ,..., N- 1. In the case of three outcomes, if
Xi, X2, ...) X, lie on the line DH in Fig. 2 then they are of ty
If the underlying simple lotteries are of type I, 2, or 3, then for any i such
ahat ai, Mi+ 1, Bi7 Pi+ 1 ) 0, it is possible to find a sequence of observations
T such that (i) F(T) and G(T) assign most of their mass to Xj and Xi+ i,
and (ii) the relative mass assigned to Xi and Xi+ 1 by F(T) and G( 7-g
can be made arbitrarily close to the relative mass assigned to them by
F and 6, respectively. Therefore, E[F((T)] FOSD E[G(T)]
only if
ai(T)/aj+ ,(T) d P,(WB,+,(T)
only if c+~+ 16 Pi/Pi+ 1. Thus (2.6) is
necessary for F>i 6. The main result of this section generalizes Theorem 1.
THEOREM3. Let X,, X,, .... XNg L, be of type 1, 2, OY 3. Let
F= (a,, a2, ..*, ~~1 and G = (P1, ,%, .... PN) be compound lotteries wit
outcomes in the ordered set (X,, X2, .... X,). The following statements are
equivalent:
642/56/2-9
362
BIKHCHANDANI,
(i)
SEGAL,
AND
SHARMA
F+G;
(ii) E[xIF(T)I3E[xIG(T)I,
VT;
(iii) F, G satisfy (2.4), (2.5), and (2.6);
(iv) F, G satisfy (2.4) and (2.5), and there exists a convex function, f,
such that Fj = f(Gi), Vi E {a, a + 1, .... b}, where a and b are defined in (2.4)
and (2.5) and Fi=Cfzla,.,
Gi = CF= 1 p, are the cumulative distribution
functions of F and G, respectively.
Theorem 2 established that if XN FOSD X,- 1 . . . FOSD X, , then (2.4),
(2.5), and (2.6) are sufficient for F+ G. However, (2.4), (2.5), and (2.6) are
not necessary for F+ G under these assumptions. This is shown in
Bikhchandani, Segal, and Sharma [ 31.’
So far we have assumed that the supports of the simple lotteries and the
compound lotteries are finite sets. The sufficient conditions in Theorem 2
can be generalized to the case where the support of the compound lotteries
is an infinite set (and the support of the simple lotteries is a finite set). That
is, if 9 is an infinite set of simple lotteries ordered by FOSD, and F and
G are compound lotteries with support 9, then if F is a convex transformation of G then F& G (see Bikhchandani, Segal, and Sharma [3]).
Although we do not have any results for the case where the supports of
the simple lotteries and the compound lotteries are infinite sets, we close
this section with an example in this setting. Let E; be normally distributed
with unknown mean M, and precision 1, and G be normally distributed
with unknown mean Mg and precision 1. Further M, and Mg are each
normally distributed with means pf and p respectively, and precision r. It
can be verified that if ,M$ pLgthen F&G. b’
3.
APPLICATIONS
Bayes’ first-order stochastic dominance is useful in sequential decision
models where there is uncertainty about probabilities. There are two kinds
of questions that arise in applications of this concept. When do one
person’s beliefs dominate another person’s beliefs regardless of what they
observe in common? This question comes up in the first example below, in
which we point to a duality between comparative risk aversion and Bayes’
7 In this example,
Xi = (0.5, 0.3, 0.2), X, = (0.5, 0.25, 0.25), X’s = (0.25,
(0.25,0.45,0.3)
are simple lotteries over three final outcomes. Clearly, X,
FOSD Xi and Xi, X,, X-,, X, are not of any type I, 2, or 3. Let F=
G = (0.75,0.1,0.15,0).
F and G violate
(2.6), since aJ(clg + s) > &/(&
+
s Suppose
that one observes k samples with mean X from F[G].
distribution
of M/
[M,]
is a normal
distribution
with
mean
[(rpc, + mZ)/(r + m)] and precision z + VI. Clearly,
the expected posterior
F dominates
the expected posterior
distribution
under G by FOSD.
0.5,0.25),
and X, =
FOSD X, FOSD X,
(0, 0.1, 0.1,0.8)
and
8s). However,
F+ G.
Then the posterior
(r~~+mx)/(z+m)
distribution
under
STOCHASTIC
DOMINANCE
UNDER
LEARNING
363
first-order stochastic dominance. The second question we address is which
compound lottery would a decision-maker choose today knowing that he
will behave optimally in the future? This comes up in the other examples
in this section. We first discuss the relationship between Bayes’ first-order
stochastic dominance and the papers of Whitt [19] and Milgrom [14].
Let F be a distribution on distributions (i.e., a compound lottery) and let
F(yr, yZr .... y,) be the posterior distribution
(updated by Bayes’ rule),
after final outcomes yl, y2, .... y, are observed. To be consistent with our
earlier notation, each yj~ (x1, x2, .... x~M). The foclilowing ass~rn~t~~~ is
often made in economic models.
H)EFINITION
6. A compound
lottery F is increasing if
EEF(Y*, Y2, ...>Yjt ‘..) Y~)I FOSD ELF(y,y Yz, “‘3 y.lT ..‘> ym)I,
V(Yt,
Y2, -,
Ymh
VYJG
Yj,
Vm.
A compound lottery, F, is increasing if higher observations in the past
imply that future observations are more likely to be higher.’ It follows from
Definition 2 that a compound lottery F is increasirrg if and only if
F(Y,, Y2, .... Yj, --A>Y,~)>J’(;(YI, Y2v ..-gJ;‘, .~.>~,n),
WYI,
Y2, ..‘> y,),
vyj<
yj,
Vm.
Let F be a compound lottery which yields simple !otteries X, , X,, .... X,.,.
Milgrom
[ 141 has shown that if X,, X2, ..~,X, have the monotone
likelihood ratio property”’ (which, as proved in Lemma A.2, is equivalent
to assuming that the simple lotteries are of type I), then P is i~cr~as~~~
(see also Whitt [19]). This can also be proved from Theorem 3 (see
ikhchandani, Segal, and Sharma [3]).
Comparative risk aversion and Bayes’first-order stochastic dominance.
say that one decision-maker is less risk averse than another if he is w~~~i~g
to pay more than the other one for every lottery. Let two expected utility
maximizers 4 and II have the same utility function U. Let both face the random variable (x, S; y, 1 S), x > y, yielding x if S happens and y if S does
not happen. Assume, without loss of generality, that U(X) = 1 and u(y) = 0.
The two individuals have the opportunity
of jointly observing a series
of identical experiments under which S may or may not hap
9 Related
assumptions
are affiliation
(see Milgrom
and Weber
1151) and condirionrrl
szochastic dominance (see Riley [ 161).
lo The monotone
likelihood
ratio property
is often measured
in models in auction theory,
principai
agent problems,
etc.
‘I Think of the lottery as a slot machine that pays a positive amount if S happens and zero
otherwise.
364
BIKHCHANDANI,
SEGAL,
AND
SHARMA
Initially,
decision-makers I and II have beliefs F= (Q , CI~, .... rxN) and
G = (PI, 82, .... PN), respectively, over the possible values of the probability
of the event S, pl, p2, .... pN. The prices P, and P, that decision-makers I
and II are willing to pay for this lottery are given by P,= u-‘(C aipi),
P,= u-‘(C /lipi). It follows from Theorem 1 that I is willing to pay more
than 11 for this lottery for all possible Bayesian updating (based on
common observations of realizations of the event S or 1 S) if and only if
F+ G if and only if F is a convex transformation
of G.
There is a striking duality between the definition of risk aversion in
expected utility theory and Bayes’ first-order stochastic dominance. In
expected utility theory, one decision-maker is less risk averse than another
if and only if his utility function is a convex transformation of the other’s
utility function. In the above story, where there is uncertainty about the
probability, the equivalent condition is that one decision-maker’s beliefs
over possible values of the probability of success is a convex transformation
of the other%.
A screening problem. Consider an employer who cannot observe the
ability of potential employees, Output in each period is a random function
of the worker’s ability, which may take the values a,, a2, . ... aN, and the
effort (or investment, or some other input), 8 2 0, made by the employer in
that period. Specifically, let X1, X2, .... X, be a set of simple lotteries over
the employee’s input levels (.x1, x2, .... x,}, where 0 < x1 <x2 < .. . < xM.
If the worker’s ability is ai then his input level in any period is an independent draw from Xi. We assume that the simple lotteries (X,, X2, .... X,)
are of type 1; that is, they are ordered by the monotone likelihood ratio
property. Under this assumption, higher input levels from a worker imply
that he is more likely to be of higher ability. If the employer’s input is 8 in
any period, and the worker’s input is xi, the output (in dollars) isf(8, xj),
where fs, fX > 0, fee Q 0, and feX > 0. The employer incurs a cost c(0) for
providing the input in each period, where c’( .) > 0 and c”( .) 3 0.
Suppose that the employer has to choose between two potential
employees, F= (cI~, a2, .... aN) and G= (pr, pZ, .... PN). That is, the
employer’s beliefs that worker F is of ability ai is cli etc. If his objective is
to maximize expected gross profit over T< cc periods, which one should
he choose?12 His expected gross profit if he selects F is
nF=
i
k=l
dkB[mdy
fE[f(eky
Yk)
1 F(YIF
. . . . Yk-dl-C(~k)}IF1,
I2 Since there is no moral hazard
in this example,
the question
of designing an optimal
incentive
scheme for the worker
does not arise. Therefore,
we assume that the workers
are
paid a constant wage which is exogenously
determined.
STOCHASTIC
DOMINANCE
UNDER
LEARNING
345
where 6 is a discount factor and F( y,, .... yk- r) E P for k = 1. In each term
of the summation the outer expectation is over y,, ..‘, y,- 1 and the inner
expectation is over yk. The expected gross profit from selecting G, ‘, is
similarly defined.
Consider the following example. Let f(e, x) = Bx, c(6) = @, T= 2, and
6= I. Further, M=2
and N= 3 with x1 =O, x2= 1, and X, = (1,0),
X, = (3, f), X, = (0, 1). Thus the input of a worker with ability level
and
is always zero, etc. Let F= (aI, CQ,ax) = (0.45,0.05,0.5)
a1
G = (jl, pZ, b3) = (0.5,0,0.5). Not only is the expected input level under F
greater than that under G (i.e., E[x 1F] > E[x 1G]), but also, when F and
G are considered as simple lotteries over ability 1eveHs (a,, a*, Q), %;
dominates G by ordinary first-order stochastic dominance (i.e., 01~< PI ar
01~+ c(~< /I1 + b2). However, direct calculation shows that the employer
better off choosing G. Specifically 0.1808 = ZIF < IiTc = 0.1875.
As the next proposition shows, if F> G, then thle expected gross pr
is greater under F. Thus, if F is a convex transformation of G then
employer should choose F.
PROPOSITION 1. If F+ G then RF3 IIf’.
A sampling problem. An employer wishes to hire one of two workers F
and 6, each of whom can produce zero or one unit per period.
Each worker’s output is an independent draw in every period, but the
probability of success for each is unknown. An employer has a distribution
over the probabilities of success of each of the two workers. Specifically, Pet
O<p, <pz< ... <pN< 1, and let F= (tq, cc*, .‘., 01~) and 6=(pr, pZ9..., ,!Iw)
be such that C cli = 2 /Ii = 1. The employer believes that there is probability
cli [pi] that the probability of success for worker F [G] is pi, i = I, 2, .#.,pd.
Qf course, some of the ai and pi may be zero.
Although the employer wants to hire only one of the two workers, be
may hire both workers during an initial probationary period and make a
final decision later. If he hires both, he can observe the total quantity
produced in each period, but he cannot tell how many units (zero or one)
each worker produced.r3 Every period the employer updates his beliefs
about the workers by using Bayes’ rule. Even if initially pi dominates
by ordinary
first-order
stochastic dominance
(in the sense that
cf=, CZ~<C~= I pi, VI) this may change once the employer gets more information about their performance. For example, let pl = 0, p2 = 0.5, p3 = 1,
I3 Alternatively,
the employer
may wish to hire both the workers
ability worker
(the one with a higher expected probability
of success)
job. In order to gather more information
about the workers’
abilities
their joint output for m periods.
and place the higher
in a more demanding
he decides to observe
366
BIKHCHANDANI,
SEGAL,
AND
SHARMA
and let F= (0,0.5,0.5),
G= (0.5,0,0.5). Obviously F dominates G by
ordinary first-order stochastic dominance. If in the first period two units of
output are observed, the updated G dominates the updated F by ordinary
first-order stochastic dominance. Therefore, the employer may wish to hire
both workers initially, and make a final decision after m periods. However,
as we prove in Proposition 2, if F+ G then after an initial period of observation the updated F dominates the updated G by ordinary first-order
stochastic dominance, regardless of the sequence of observed outputs. Thus
he can avoid the cost of hiring both initially and select F to begin with.
PROPOSITION
2. Suppose that F& G and the employer hires both workers
initially. Let F* = (a:, a$, .... a$) and G* = (p:, /?z, .... jQ,) be his updated
distributions after m periods of observation. Then cf= 1 a* < cf= 1 p*, Vl.
Sequential search. Consider a risk-neutral individual who searches for the
lowest price at which to buy a good. He can elicit price quotations from
different sellers at the rate of one price quotation per time period. He can
search for at most L time periods and, for simplicity, the cost of obtaining
each quotation is zero. Once he decides to stop searching, he buys the good
at the lowest price quotation obtained so far. Thus, an optimal strategy is
to obtain L price samples and select the lowest price. The possible prices
-.. <x,. The individual does not know the
are {x1, x2, .... x,}, Xl<XZ<
exact distribution over prices he searches from. Let X1, X,, .... X, be the set
of possible distributions of prices. Let F and G be two distributions on the
set (Xi, X,, .... XN}. Suppose that the individual can search for the lowest
price either from F or from G. Further, once he chooses one of them he
cannot switch to the other at a later stage. (The interpretation is that F and
G represent his beliefs on the distributions
over prices in two widely
separated shopping areas. If he goes to one shopping area, then he does not
have the time to go to the other.) Are there conditions on F and G such
that, without actually computing the expected minimum prices, one can
determine which one of the two distributions
on distributions will be
preferred by the individual ?
We show that FOSD cannot be used to choose between F and G. Let
X, = (1, 0), X, = (4, 4) and X3 = (0, 1) be simple lotteries over two final
outcomes x1 and x2, x1 <x2. That is, X, yields x1 with probability 1, etc.
Let F= (0, LO), G = (3, 0, f), and F’ = (f, 0, 4) be distributions on the set
(Xi, X,, X3}. First, consider the case when the individual is allowed to
take two price samples from either F or G. Let y, E (x1, x2}, k= 1, 2
denote the kth sample observation. Since
E[min( yr , y2) I F] = :x1 + $x2 < 5.~~+ $x2 = E[min( y,, y2) I G];
STOCHASTIC
DOMINANCE
UNDER
LEARNING
367
E; is preferred to G. If, instead, the choice is between F’ and G, and again
only two price samples are allowed, then
E[min(JJi,
y2) IF’] = ix1 + +x2 > $x1 + $x2 = E[min(y,,
y2) j G]
implies that G is preferred to F’. Since ELF] = I?[%;‘] = (f, ;f and
E[G] = ($, i), it follows that ELF] FOSD E[G] and ELF’] FOSDE[
Hence, FOSD is inadequate in this setting. In Bikhcha~dani,
Segal, an
Sharma [3] it is shown that if F>, G and the underlying simple lotteries
are of type 1,14 then when searching with recall over a finite horizon the
expected minimum price under G is less than the expected minimum price
under F (see also Bikhchandani and Sharma [Z]).
Bandit problems. Berry and Fristedt [ 1] use ayes’ first-order stochastic
dominance to compare optimal strategies (under different p
problems. Consider the two armed bandit with independent
in which one arm, say the second, has a known distribution.
ith arm yields an amount qi > 0 with probability xi, i= 1, 2, and yields
with probability 1 - rci. The player knows n2 and has a prior distrib~ti~
on or. IIis objective is to maximize the expected discounted reward over an
infinite horizon. Berry and Fristedt show that if a prior F on nil
first-order stochastically dominates another prior G, then the value
bandit (i.e., the supremum of the expected reward over all strategies) under
the F-prior is greater than the value under the G-prior.
Rothschild [17] uses the bandit problem to analyze the p
good when the seller does not know the demand curve (see also
L-131). othschild emphasizes that an individual will eventually settle on
one arm and play it forever. However, the chosen arm will not necessarily
be the “correct” one. That is, learning may be incomplete and an individual
may play the less attractive arm forever. For any given experience on t
unknown arm, if a prior F leads to incomplete Iearni
then for all priors
which are Bayes’ first-order stochastically dominate
by F, we obtain
incomplete learning. Whether similar results go through for
correlated arms (see Easley and Kiefer [7]) is an interesting open question.
Berry and Fristedt also show that for the case of two final outcomes,
F+ G is equivalent to condition (ii) of Theorem 1. owever, as pointed
out earlier, this is an impractical condition, since it requires that the
posterior distributions F( 2”) and G(T) be computed for all T. The sufhcient
conditions we obtain for F$ G are easily computable.
l4 If the simple lotteries are of type 1 then higher
higher future price realizations
are more likely.
price realizations
in the past imply
that
368
BIKHCHANDANI,
SEGAL,
AND
SHARMA
APPENDIX
We need the following result to prove Theorem 2.
LEMMA
A.l.
Let F and G be as in (2.4) and (2.5), respectively. Then (2.6)
implies
Vl<N.
(A.11
Proof of Lemma A.l. If b <a, (A.l) follows trivially. Suppose that
N} then again (A.l) follows
b > a. If 1E (1, 2, ...) a-l}
or Z~{b,b+l,...,
trivially. Suppose that there exists ZE (a, a + 1, .... b - l} such that
Without loss of generality we may assume that CT= i ai 6 CT= i fii, vs < 1.
Therefore, ar>P,. From (2.6) we know that a,/yi,<ai+l/Bi+l,
a< i< b.
Thus ai > pi, I< id b. This, together with (A.2), implies Cp= i a,>
w h’ic h contradicts the fact that Cp=i cl,< 1. Thus (A.l) must be
Cf=lpi=l,
true. 1
Proof of Theorem 2. From Lemma A.1 we know that (2.4), (2.5), and
(2.6) imply (A.l). Since X, FOSD X,_ i ... FOSD X,, it is easily checked
that (A.l) implies E[F] FOSD E[G]. Thus it is sufficient to show that
after taking one sample the updated distributions satisfy (2.4), (2.5), and
(2.6), and the lemma follows by repeated application.
Let Tj be an
M-vector with a 1 in the jth place and 0 everywhere else. Thus Tj
represents a sample of size one in which xi was observed. We will show
that F(Tj) and G(T’) satisfy (2.4), (2.5), and (2.6).
Since a,(Tj) > 0 if and only if ai> 0, (2.4) is automatically
satisfied by
F(Tj). Similarly, (2.5) is satisfied by G(T,). Suppose that b > a. Choose i
such that a < i < b. Since F, G satisfy (2.6), we know that clJc~,+ 1 </Ii//Ii+ 1.
Also, since ~i(Tj)/ai+l(Tj)=(P,ja,)l(Pi+l,jai+l),
and Pi(Tj>/P,+l(Tj)=
(f’,Bi)I(Pi+
l,jPi+ 1) we have
ai I < Pi(?)
ai+ l(Tj)
Pi+ I(?)’
(A.3)
But (A.3) implies that F( Tj) and G( Tj) satisfy (2.6).
1
In the discussion on type 1 we claimed
equivalent. A proof is provided below.
(2.7) and (2.8) are
that
STOCHASTIC
LEMMA
DOMINANCE
UNDER
The simple lotteries
A.2.
LEARNING
X1, Xz: .... X,
are
of type
1 $ and
only if
pi j+l
--<
pij
Pi+!,j+l
pi+l,j
Vj-cM,
Vi= 4, 2, .... N-
Proof of Lemma A.2. Suppose (2.8) is true. Take
pI = cj= i P,, and q, z cj= 1Pi+ I,j. Define fj as follows:
any i< N. Let
1= 1, 2, ...) M;
fi(PJ = 4r7
fz is continuous
1.
’
and between each pI and pI+ I it is linear. For all I< M - 2,
fi(PI+l)-fi(PI)
PI+l-PI
jfi(Pi+Z)-L(PII)
’
P1+2-Plfl
41+
1-
4i<
4r+2
-
4r+
1
PI+1-P11Pr+2-Plti
pi+I,I+1cpi+I,l+2
*
pi,/+
1
’
pg+2
The last inequality is implied by (2.8). Thus fi is convex an
are of type I.
Next suppose that X,, X,, .... X, are of type 1. Take any i < N. Let p,
and qr be as defined earlier. For some constants cr > 0, c2 > 0, dr , d2 define
pP = cl pI+ dI and 4.7= c2ql+ d,, I= 1, 2, .... M Clearly it is possible to
choose ci, di such that pjYI=qF-l=O
and pT+r=@+r=l.
Let
h*(x) z c2fj((x -d,)/c,) + d,. Obviously q? = fi*(pT), I= I, 2, .... hf. Since
fj is convex, and cr, c2 > 0, j-j* is convex. Therefore,
Also,
Similarly,
Pa/(Pu + Pi,,+,) = p:.
Thus (A.4) implies that Pi+l,l/
) <1 Pil/(Pil+Pi,r+l)y
which implies that Pi+l,~+l/P,+,,~3
(pi+-l,l+pi+l,I+l
Pi,,+l/Pil. Since this is true for all 1, i, we have established (2.8).
Proof of Lemma 1. The proof is immediate when X,, X2, .. .. X, are
either of type 2 or type 3. Suppose that X,, X2, .... X, are of type I. We can
rewrite (2.8) as
BIKHCHANDANI,
370
SEGAL, AND SHARMA
Define
JjSAL
j = 1, 2, ...) hf.
pi+i,j’
By assumption, 1,3 1, > . . . 3 1,. Also, 1, < 1 and A, > 1, otherwise
either cjE, P, # 1 or J$E 1 Pi, I, j # 1. Therefore there exists I* such that
&*>l>,&+,.
For all 161*,
i
Pij=
j=t
Similarly,
Inequalities
i
i
Ajpi+,3j>L/
i
pi+I,j3
j=l
j=l
(A.9
pi+l,j.
j=l
for all I> I*,
(AS) and (A.6) imply Xi+l
FOSD Xi.
1
Lemmas A.3 and A.4 below are required to prove Theorem
some notation. For 0,~ [0, 11, C 0,= 1, define
Cki(f)li
02,
. ..>
0,)
ckv4,
e2,
. . . . e,)
Cite,,
e2,
. . . . e,)’
3. First,
1~ i, k 6 N.
=
(A.81
We can rewrite (2.1) as
LEMMA A.3. Suppose that X1, X2, .... X, are of type 1, 2, or 3. For any
k < i there exist Oki
1 , 6’f> .... 0% such that
cri(ey,
eg .... eg) = I,
if
rE {k, i}
-=I1,
if
rE (1, 2, .... k-
l> u (i+ 1, .... N).
Proof of Lemma A.3. Suppose that X1, X,, .... X, are of type 1. Let
Sil = c$= 1 P,, VI < M. Then,
M-l
In ci(e,,
.... e,) = 1
Blln(S,-
S,,- i) +
ln(Si,
- Sj,,-
I= 1
a In Cj
~~=si~-si,I--I-si,i+l-sii
asit
e1
eI+ 1
=---0,
pil
4+l
Pi3l+l’
1)
STOCHASTIC
DOMINANCE
UNDER
LEARNING
371
Thus 8 ln Ci/aS, 3 0, QI < M if and only if
-- 81
pii
e It1
(k.10)
Vl<M.
‘pi,i+l’
By Lemma 1, S,,d Su, Vr > i, VI< M. Thus,
satisfying the inequalities (A.lO) then
if we choose IS,, .... 0,)
V’r>i.
C,(@l, a.., 0,) 6 Ci(O, >*.., O,),
Similarly,
(All)
8 In Ck/dSkl 6 0, if and only if
(Al?)
Thus for any (0,, .... 0,) satisfying (A.12), we have
c,(dl,
..., 8,)
d
ck(@,
‘drck.
> ..*> @Ml,
(A.13)
Since Xi, X2, .... X, are of type 1, (2.8) implies that there exist (0,, t)?, ...) 0,)
which satisfy (A.lO) and (A.12) with at least one strict inequality.
Cki(Pk,, Pkz, .. .. PkM) > 1 and cki(Pjl, Pjz5 .... PLM) < 1,
Moreover, Since
there exists a point (Sfi, el;j, .... 0:) on the chord joining X, and Xi such
that
cki(qi,
(A.14)
ey, .... e$, = 1.
Since (Oti, 19l;j,.... 0:) satisfy (A.lO) and (A.12), Eqs. (A.11) (A.13), and
(A.14) imply the lemma when X,, Xz, .. .. X, are of type 1.
The proof for type 2 or 3 is similar. If X,, X,, ..~,X.V are of type 2 t
d In CJaP, 3 0, Vj < M, if and only if
3->
&I’
and 8 ln
ck/dPkj
pij
1 -c:;i
j=l,2
P,’
>... .
- 1:
(A.15)
G 0 if and only if
Since X,, X,, .... X, are of type 2, it is possible to find 8, ) 6,, ...) GM whit
satisfy (A.15) and (A.16) and the rest of the proof is similar to the proof
for type 1.
When X,, X,, .... X, are of type 3, then one can fmd ol, 8,, ....
that a In c,Iap, d 0, Qj > 1 and a In
> 0, Qj > I, and the rest of the
proof is symmetric.
1
ckjapkj
372
BIKHCHANDANI,
SEGAL,
AND
SHARMA
LEMMA
A.4 Suppose that X,, X,, . ... X, are of type 1, 2, or 3. Zf F+ G
then F and G satisfy (2.4), (2.5), and (2.6).
Proof of Lemma A.4. Suppose that (2.4) does not hold. Therefore there
exists i, and k < i such that aj = 0, ak > 0, and pi > 0. (We disregard
the possibility that ai = pi= 0 because then, after dropping Xi, (2.4)
can be satisfied.) Without loss of generality we assume that 01,=0,
Vr E (k + 1, k + 2, .... i>. From the proof of Lemma A.3 it is clear that if
Xl, x2, *.., X, are of type 1 then (OF, .... 6:) and (0?, .... f3$), r > k, s > i,
when considered as simple lotteries,
are also of type t with
(ey, ...) 6;) FOSD (8?, .... SE). Thus we can find (6:, .... 8%) such that
(B~,~.YO~),
(0:, . ... &),
and (Oii+‘, .... 8$+“)
are of type t with
(0;’ , .... 8gf’) FOSD (e:, .... Q$) FOSD (0: .... (I:), and
Cri(6F, .... 6%) < 1,
Vr E (1, 2, .... k-
c,k(e:,
VrE{1,2 ,..., k-l}u{i+l,...,
. . . . 6t.f;)
<
l,
1) u (i+ 1, .... Nj
N}.
(A.17)
We can choose (&Jf , .... 0%) to be rational and T= (tI, t2, .... tM) such that
tj/c tj= 67, Vj. Define H, to be the compound lottery which yields X, with
probability 1. Substituting (A.17) in (A.9), we see that
lim F(zt,, .... zt,) = Hk
z-m
lim G(zt,, .... zt,)=
z-m
1
Of,,
r=k+l
where a,>O, C:=,+, a,= 1. Thus for large enough z, E[F(zt,, .... zt,)]
does not dominate E[G(zt,, .... ztM)] by FOSD and hence F% G.
A similar proof establishes that (2.5) is necessary for F+ G.
Next, suppose that b > a. Choose an increasing sequence of observations
T’= (t:, .... t’,) such that (t:E tj, t’Jc tJ!, .... t’,E tj)) converges to
(6;ii+1 ) ...) eg+ l ), where a<i< b and (Q:‘+‘, .... 19$+‘) is as defined in
Lemma A.3. Thus Lemma A.3 and (A.9) imply that
0,
lim clk( T’) =
I-m
ak
OEj+C$+l
0,
lim Bk( T’) =
I+co
Bk
Pi+Bi+l’
if
k$(i,i+l}
if
kE{i,i+l)
if
k#(i,i+l)
if
kE{i,i+l).
STOCHASTIC
DOMINANCE
UNDER
Also, since ai, a,+,, fli,fli+l>O,
lim,,,
defined compound lotteries. Therefore,
E[ lim F(T’)]
I-cc
3’33
LEARNING
F(T’) and iim,,,
G(T’) are well-
FOSD E[ lim
1-30
iff
Q'k<M
iff
ai
aj+ai+f
Pi
%+A+1.
Thus, for large enough 1, E[F( T’)] FOSD E[G( T’)] only if (2.6) ho1
Proof of Theorem 3. (i) o (iii). By L emma 8.4, (i) implies (iii) and by
Lemma 1 and Theorem 2, (iii) implies (i).
(i)+(ii).
If F+=G then E[F(T)] FOSD E[G(T)],
VT, which implies
E[x/F(T)]>E[xIG(T)],
VT.
(ii) s (i). Suppose that 8’2 G. Then (iii) implies that either (2.4) or (2.5)
or (2.6) is violated. Consider first the case when (2.6) is violated. Thus,
there exist i, i+ 1 such that ai, ai+:, pi, /3i+l > 0 and cc,/(ai+ ai+ 1) >
p,/(pi+ /Ii+ 1). Choose T’= (ti, t:, .... t’,) such that C I: -+ co and
lim -cj”=,
Iem
t’
i,i+
t,!=H’
1
’
vi
where f?,i+ ’ is as defined in Lemma A.3. For large enough I, P;(T’) an
G(T’) &ace most of their mass on Xi and Xi+ 1, and
Since, for arbitrarily small, positive E there exists I such that ai
ai+ ,(T”) 2 1 -E, and fii(T’) + fii+ ,(T’) >, 1 -E, we have E[.x j G(T’)]
E[x I F( T’)] for large enough I.
i
>
314
BIKHCHANDANI,
SEGAL,
AND
SHARMA
The other possibility, when F% G, is that either (2.4) or (2.5) is violated.
Assume that gi + pi > 0, Vi, otherwise we can drop Xi. Thus there exists
i, i + 1 such that either ai > 0, mi+ I =O, and pi+r>O,
or ai>O, Bi=O, and
pi+ 1 > 0. Using Lemma A.3 once again, we can find T* such that F( T*)
puts most of its mass on Xi, and G(T*)
on Xi+ 1, and thus
E[xl G(T*)]
> E[x 1F( T*)].
(iii)=>(iv).
Let Fi=~~=,
a,, Gj=CIS1
f(Gi) = Fig
p,, and definefas
i = a, a + 1, . ... b;
f is continuous, and between Gi and Gi+ r, i = a, a + 1, .... b - 1, it is linear.
The rest of the proof is similar to that of the first part of Lemma A.2.
(iv)*(iii).
Let iE (a, a+ 1, .... b-l},
and let F, and Gi be as defined
above. For some constants c1 > 0, c2 >O, dl, d, define GF = c,G,+ d,,
Fi*=cc,Fi+d2,
i=l,2 ,..., N. The rest of the proof is similar to that of the
second part of Lemma A.2. 1
Proof of Proposition 1. We give the proof for T< co. The T= CC
case follows by a simple continuity argument. Let yr, yz, .... ykP 1 E
{ Xl, x2, ---, x,} be the history of employee input levels in the first k- 1
periods. (By observing the output level, the employer can infer
yr, y2, .... yk- 1) since he knows 6,) 02, .... ok- 1.) Then, under G his
optimal choice in period k is
where the expectation
iS over yk. Given our assumptions on f( .) and c( .),
a unique 0: exists for each yr, yz, .... yk ~ 1. The expected gross profit under
G in period k is
19: and II:
are similarly
FOSDJTG(Y,,
y2,...,
defined. Since F+ G implies EII;(yl,
yk--l)l,
E[f(@,",
we
have
yk)(G(yl,...,yk-1)1-C(8kG)
<'[f(fl,",
yk)iFt;(yl,...,
ykpl)]
de,")
and, therefore,
WgYl,
y2r
..., Yk-d~n,F(Yl,
Y,,
...Y Yk-1).
y,, .... ykP r)]
STOCHASTIC
DOMINANCE
UNDER
LEARNING
375
Moreover, since X,, X,, .... X, are of type 1, F and G are increasing (in the
sense of Definition 6). Therefore, nr( .) and n,“( .) are increasing fu~~ti~~s
of their arguments. Thus, since E[F] FOSD E[G], we have
k=l
Proof of Proposition 2. In each period the employer observes either
two units of output (and he knows that each worker produced one unit),
or zero units (and he knows that each worker produced zero), or one unit.
Suppose that he observed mj periods of i units of output, i = 0, 1, 2, an
C mi = m. Since the exact sequence of these observations is not important,
we assume that in the first m, + m2 periods he observes zero or two uni
in each period, and in the final m, periods he observes one unit per perio
Let p= (a,, &, .... oi,,,) and G = (fll, B2, .... flN) be the updated distributions
after m,+m_,
periods. Since p= F(m,, m,), G= G(m,, mz), and F+Gh;:
clearly iQ= G.
Consider the remaining m, periods. For 0 < j d m
qj that in j of the m, periods worker F produced a u
not and that in the other m, -j periods worker F
worker G did. Let @ be the updated probability at the end of the r~
periods that the probability of success for worker F is pi, and let a: j be the
updated probability at the end of the m periods that the probability sf success for worker F is pi, given that in exactly j of ,the m, periods in whit
only one unit was produced, worker F was the one to produce it. Similarly,
define /I,+ and /3J;,. Of course, a* =‘j$& qjliaGj and /3* =cJTo qml-jfiJ,.
Finally, let F* = (a:, CY:, .... c(s) and G* = (p:, 82, .... /?$). We show that
F* FOSD G*. That is,
Since E[P] FOSD E[d] it follows that for j> mJ2, qj> qrn,- j’
Let ~i=~~!~q~ol$~,
d = c,q;>4, ..*, qh,), where qj=q&j=&(qj+qm,-i).
/I,! = cJ?O qi/3i;: j, and F’ = (a;, a;, .... ah), G’ = (pi, /I;, .... lj;).
It is well known that if I;, FOSD G1 then for I >a> b>O,
aF, + (1 -a)G, FOSD bF, + (1 -b)G1.
Moreover, if F1 FOSD G1 and
F2 FOSD G2 then VaE [0, 11, UP, + (1 -a)F, FOSD aG, + (I -a)G,.
376
BIKHCHANDANI,
SEGAL,
AND
SHARMA
Also, it is easily verified that if j> ml/2 then a$j FOSD aJrnlPj, where
(UT, cl;/ ...2aXi ). These facts imply that F* FOSD F’ and
G’ FOSD G*. Since &= 6, it follows that for every j, a~j FOSD p$j. We
therefore obtain the sequence of dominations, if (1) ml is an even number,
CC:j=
j,
j,
F*=
j
2 qjcx;j
FOSD
z q;alyj
j=O
j-0
WI2
42
=,goq~(~~j+~~rnI-j)
FOSD1 qi(B.Tj+P.Tnz-j)
j=O
= f q;&
z q,,- jB;j=
FOSD
G*
j=O
j=O
and, if (2) ml is an odd number,
F* = ffJ qjCt:j
FOSD
j=O
z qJCr!j
j=O
(ml- 1x2
=
C
j=O
9~(a~j+a.*I,,~j)+4;ml+l),2a~(m*+1)/2
FOSD
(ml- 1l/2
1 qj.(B-Tj+B~nrl-j)+q;M,+1)/2P~(m,+1)/2
j=O
= 2 q;p$j
FOSD
j=O
2 qm,- j&=
G”.
m
j=O
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