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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
IMPACT OF HIGHER-ORDER UNCERTAINTY
Jonathan Weinstein
Muhamet Yildiz
Working Paper 03-1
March 2003
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This paper can be downloaded without charge from the
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MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
MAY
3
2003
LIBRARIES
IMPACT OF HIGHER-ORDER UNCERTAINTY
JONATHAN WE1NSTEIN AND MUHAMET YILDIZ
Abstract.
large,
some games, the impact
of higher-order uncertainty
implying that present economic theories
theories
first
In
may be
very
misleading as these
assume common knowledge of the type structure
or the second orders of beliefs. Focusing
is
after specifying the
on normal-form games
the players' strategy spaces are compact metric spaces,
we show
in
which
that our
key condition, called "global stability under uncertainty," implies a variety
of results to the effect that the impact of higher-order uncertainty
Our
central result states that, under global stability, the
is
small.
maximum change
in
equilibrium strategies due to changes in players' beliefs at orders higher than
k
is
exponentially decreasing in
we can approximate
k.
Therefore, given any need for precision,
equilibrium strategies by specifying only finitely
many
orders of beliefs.
Key words: higher-order
uncertainty, stability, incomplete information,
equilibrium.
JEL Numbers: C72, C73.
1.
Most economic
Introduction
theories are based
on equilibrium analysis of models in which
the players' types (following Harsanyi (1967)) are simply taken as their beliefs
about some underlying uncertainty, such as the marginal cost of a firm or the
value of an object for a buyer, and rarely include a player's beliefs about the
other players' beliefs about the underlying uncertainty. Using such a type structure implicitly assumes that, conditional on the first-order beliefs about
some
Date: First Version: October, 2002; This version: March, 2003.
We
thank Daron Acemoglu, Glenn Ellison, Sergei Izmalkov, Paul Milgrom, Stephen Mor-
Robert Wilson and the seminar participants
ris,
Marchiano
We
are especially grateful to
Siniscalchi,
Drew Fudenberg, whose
dominance-solvability.
1
at
MIT and UPenn.
inquiries led us to our result
about
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
2
payoff-relevant uncertainty,
knowledge.
Even the
1
all
literature
There
(in
is
a finite-dimensional space of payoff uncertainty)
now an
as large
As Rubinstein
is
up to some
finite
order
economic theories
beliefs
it
is
may be
in
can be profoundly
mutually
this information is
misleading.
3
beliefs
have large impact, the present
This large impact
is
also disturbing
hard to believe that we would ever know a player's high-order
with any precision. Without such knowledge,
predictions
game
— no matter how many orders we consider.
Most importantly, when the higher-order
because
common knowledge
from the equilibria of games in which
only
some
and Morris (1998) and
(1989) illustrates, the equilibria of a
which a particular piece of information
known
this as-
an impact on equilibrium behavior
as lower-order uncertainty (see Rubinstein (1989), Kajii
different
makes
2
extensive literature, however, that emphasizes that in
games higher-order uncertainty has
Morris (2002)).
Damme
on global games (Carlsson and van
(1993)) and on forecasting others' forecasts (Townsend (1983))
sumption
common
of a player's higher-order beliefs are
when
we cannot make
the impact of higher-order uncertainty
assuming that higher-order
beliefs
large impact implies that the
is
large.
accurate
Moreover,
correspond to higher-order reasoning, such a
bounds of rationality are at
least as
important as
the basic incentives. This would necessitate a change of paradigm for analyzing
these problems. Therefore,
it is
of fundamental importance to classify
which high-order uncertainty has
In this paper,
little
we provide a broad
high-order uncertainty has
little
games
in
impact.
set of sufficient conditions
impact.
Our main
under which
sufficient condition is called
"global stability under uncertainty." It states that the variation in each player's
best response
is
always
less
than the variation in his beliefs about the others'
Here we use the standard terminology: a player's
first-order beliefs are his beliefs
about
the underlying uncertainty; his second-order beliefs are roughly his beliefs about the other
players' first-order beliefs,
2
For an illustration of
and so
how
on.
a model with such an assumption can be deceptive regarding
the impact of higher-order uncertainty, see Section 2.3.
For example, the Coase conjecture
as
shown by Feinberg and Skrzypacz
may
fail
(2002).
when we introduce second-order
uncertainty
HIGHER-ORDER UNCERTAINTY
actions (according to the
stant b that
less
is
than
embedding metric defined
1.
3
later), multiplied
by a con-
Under certain continuity assumptions, we show that
global stability under uncertainty
is
closely related to the standard concept of
global stability of best-response correspondence (under certainty).
For games
with one-dimensional strategy spaces, we further provide a simple second-order
condition that guarantees global stability under uncertainty.
We
consider finite-person
metric spaces and there
is
games
some
in
which the strategy spaces are compact
payoff-relevant source of uncertainty that
comes
We work in universal type space,
where
from a complete, separable metric space.
the players' types are their entire hierarchy of beliefs about the underlying uncertainty, allowing players to entertain
that,
our
when the
game
rium
is
(e.g.,
beliefs.
We show
best responses are always unique, global stability implies that
dominance-solvable.
4
In that case, whenever there exists an equilib-
under the conditions of Vives (1990)),
strategy profile. This
much
any coherent set of
it is
the unique rationalizable
important because rationalizability
is
is
considered to have
stronger epistemic support than equilibrium (see Bernheim (1984), Pearce
Aumann and Brandenburger
(1984),
will refer to
(1995)
and Dekel and Gul
(1997)).
We
equilibrium throughout the paper because our results also apply to
games without unique best responses, which might not be dominance-solvable.
We
a (Bayesian) Nash equilibrium of this game.
fix
Note
that, since every
type space can be embedded in universal type space, this corresponds to fixing
an equilibrium
beliefs
up
to
stability, the
all
for all
type spaces simultaneously.
a certain order
k.
maximum variation
his higher-order beliefs, is at
we want
(e.g., in
Our main
Let us also
result states that,
fix
a player's
assuming global
in the player's equilibrium strategy, as
most
b
k
we vary
times a constant. That means that,
if
to determine the equilibrium behavior within a certain margin of error
order to check the validity of a certain theoretical prediction),
need to specify
orders k*
is
finitely
many
orders of beliefs, where the required
we only
number
of
a logarithmic function of the desired precision. In particular, the
impact of an erroneous
common knowledge assumption
at orders higher
See Milgrom and Roberts (1990) for a related result in supermodular games.
than k*
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
4
will
be
less
than the specified bound. This
is
by Wilson (1987) of "successive reductions
a contribution to the goal set out
in the base of
common knowledge
required to conduct useful analyses of practical problems."
We
have so
far-
maximum
focused on the
change in a player's equilibrium
strategy due to any change in his higher-order beliefs.
We
also investigate
the relationship of the change in strategy to the size of the change in beliefs.
Towards
this goal, firstly,
we
define an "embedding metric"
on
beliefs at
each
order (as well as on beliefs about the other players' actions). This metric has
the crucial property of preserving the distances in lower-order beliefs
are
embedded
in the space of higher-order beliefs as point masses, allowing
us to sensibly compare variations at different orders.
player's strategy varies as
other beliefs fixed.
his beliefs,
beliefs,
we vary
his belief at
Now we
is
satisfied in traditional
the different orders of
We
by the
show
size of this
that,
is
at
most
b
k
change
under global
and the independence assumption, the marginal impact
of changes
times a constant. This formalizes our notion
that, under global stability, the marginal
impact of higher-order
beliefs decreases
exponentially. In that case, precision in lower-order beliefs will
important than the precision in higher-order
if
all his
can define the marginal impact of a change in fcth-order
measured by our embedding metric.
in fcth-order beliefs
It
how much a
"independent private value"
beliefs as the variation in equilibrium strategies divided
stability
ask
(To be able to do this without violating the coherency of
an assumption that
in beliefs as
We
some order k and keep
we need an independence assumption about
environments.)
when they
beliefs in
approximating a problem.
also follows that the players' equilibrium behavior
they formed erroneous higher-order
beliefs.
very natural; we should emphasize that they
be much more
would not change much
These assertions
may
all
sound
may easily fail when global stability
does not hold. In particular, with linear best-responses, the marginal impact of
fcth-order beliefs actually increases exponentially in k
whenever global
stability
does not hold.
It also
follows from our assumptions that equilibrium behavior
is
continu-
ous with respect to the product topology on type space that comes from the
HIGHER-ORDER UNCERTAINTY
5
embedding metric; when the best responses are always unique, the equilibrium
correspondence
be continuous.
will
Although there
is
a sizeable literature on the impact of higher-order uncer-
most studies has been
tainty following Rubinstein (1989), the focus of
ation of
common knowledge and
lower semi-continuity of equilibrium in the
worst-case scenarios, such as approximating
p-beliefs
relax-
common knowledge with common
(Monderer and Samet (1989)), robustness of equilibrium against (pos-
sibly substantial) payoff uncertainty with small probability (Fudenberg, Kreps,
and Levine (1988) and Kajii and Morris (1997)), and strong topologies under
which equilibrium
is
lower semi-continuous uniformly over
all
games (Monderer
and Samet (1997) and Kajii and Morris (1998)). Most closely related to our
work, Morris (2002) analyzes the impact of higher-order uncertainty within a
model with
linear best responses, reaching the conclusion that
order beliefs can be arbitrarily large
games. Our focus
two ways.
differs in
order uncertainty within a single
if
we
impact of higher-
require a uniform
Firstly,
bound
we measure the impact
game (dropping
over
all
of higher-
the uniformity requirement).
Second, while our sufficient condition implies continuity of best response, most
of these papers analyze matrix
on the mixed
The
strategies,
when the best response
outline of the paper
relation
between
games with
games and naturally use the supremum metric
stability
is
is
generically discontinuous.
as follows. In the next section,
we
illustrate the
and dampening impact
of higher-order beliefs using
we
present our basic model with
linear best responses. In Section 3,
independence assumption and introduce the embedding metric; we introduce
global stability in Section 4 and provide sufficient conditions
it
in Section
5.
Our major
results are presented in Section 6
assumption, and our main result
7.
Section 8 concludes.
2.
Some
is
and examples
for
with independence
extended beyond this assumption in Section
proofs are relegated to the Appendix.
Examples with Linear Best Responses
We will now show how dampening impact of higher-order uncertainty is equivalent to stability in
games with
linear best-response functions, such as the linear
.
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
6
Cournot duopoly. This
we
cepts which
illustrates the close relationship
between these two con-
a broader context in the later sections.
will establish in
Cournot Duopoly. Consider a Cournot duopoly where
2.1.
function
the inverse-demand
given by
is
P=l-Q
P
where
i
E
N=
function
{1,2}.
The marginal
cost of firm
Each firm knows
=
(Qi, 12)
i
The inverse-demand and payoff
is
i
where
qo
qt is
the supply of firm
denoted by q, so that
qt
-
-
<7i
functions are
own marginal
its
(1
cost. If
92
-
its
payoff
(k)
common
knowledge.
we assumed that the marginal
common knowledge, then we would have the classical
We
case.
costs
complete-information
could also allow incomplete information by assuming that (c\,C2)
drawn from a commonly known
Cj
+
q\
is
u
were
Q=
good and
the price of a
is
own
conditional on its
cost
is
distribution, representing the beliefs of j about
If
Cj.
we
further
assumed that
and
C\
C2
were
independently distributed, then this would correspond to the assumption that
the firms' beliefs about the other firms' cost are
paper,
if
Cj,
on
all levels
representing
ijj
its beliefs
representing
type
A
is
the entire
Firm
of uncertainty.
i's
list
about
beliefs
probability distribution i^
i's
t,
on
=
t
about
k
,~ l
,
Firm
Cj.
function of
of type
tj
its
type,
is
i
an equilibrium
.
:
about q. In general, firm
U
h->
q*{U) specifies firm
z's
[q,
(l-
ql
-
q* (tj)
-
c,)}
has
Firm
supply as a
q*(U) maximizes the expected payoff
iff
That
is,
equilibrium strategy
maximize the expected payoff
t
i.
i
.).
given the strategy q* of the other firm.
E
ij
has also a probability distribution
representing fcth-order beliefs of firm
{ci,t\,tf,
to allow
j has a probability distribution
j's beliefs
strategy profile (gj,^)? where q*
q* will
knowledge. In this
we do not make such strong informational assumptions; we want
variations in
on
common
=q
{
(l
-
q,
-
E
t
[q* (tj)]
-
a)
,
.
HIGHER-ORDER UNCERTAINTY
where expectation Ei
as q*
depends on the entire type
(tj)
(2-1)
Of
be determined by
will
course,
we
=
^
q*
=
^
-^te(*i)]-
- \e3
M
^= ~^- -^
1
(2-3)
1
^
Here
[cj]
n*
Ql
1
-
E
-
l
depends only on
depends only on
and EiEjEi
-a
[g*]
~
—2
[
t},
Cj ]
l
-
EjEj
about the
i
depends on the third and
Cl
l
~
Ei
4
-
+ \e,EM}-
the beliefs of
the beliefs of
i|,
ft)]
would yield
further substitution of (2.1) in (2.3)
l
[q*
we can obtain
Substituting (2.2) in (2.1),
_
at all levels,
.)
.
have
also
(2-2)
A
£L
its beliefs {tj,tf,
This implies that
tj.
8f
7
^+
4- -
i
- EiE
g
all
N
l
,
r
about the cost of
i
j,
E Ej
t
[ci\
about the cost of
beliefs of j
i,
higher-order beliefs. In general,
fcl
4+
1
F F F
T^ EjEjEj
•EF
^*1
i
[ qj \
k times
when k
odd; the last term
is
is
EEE
l
J
Ej[q*]/2
i
rium, each firm's supply will always be in
last
term
is
at
most l/2
fc
.
That
is, if
we
[0, 1];
k
when
k
is
even. In equilib-
hence the absolute value of the
fix the beliefs
up
the equilibrium strategy q* up to an error of at most l/2
to fcth order,
we know
fc
.
This also implies that we can write the equilibrium strategy as a convergent
series
*
Ql
where the
1
=
~
-c,
2
_
1
-E
z
[cj]
1
- EjEj
4
coefficient of the
[cj]
_
is
l/2
- EiEjEi
fc
.
The
significance of this formula
that the coefficients of expectations decrease exponentially as
order expectations.
[c-j]
16
8
kih term
1
we go
is
to higher-
.
JONATHAN WEINSTEIN AND MUHAMET
8
2.2.
be
YILDIZ
General Case with Linear Best Responses. The
with linear best-response functions
easily generalized to the case
BRi =
where a
t
is
s*
(2.4)
a,
+ bE
z
[sj]
the underlying parameter for player
the (unknown) action of player
=
+
a,
;
bEi
[a,-]
+
b
i
(such as
Now, the equilibrium
j.
2
analysis above can
EE
t
3
[a,]
+
—
(1
c*)
and
/2)
Sj is
strategies satisfy
+ ^E^E,
E
t
\s*]
v
k times
when k
is
odd.
The absolute
value of the coefficients will decrease exponentially,
resulting in a convergent infinite series as above,
•
Note that
When
and only
if \b\
<
1.
this corresponds precisely to the stability of the equilibrium of
the complete information
•
if
the equilibrium
is
game under the best-response correspondence.
unstable, the impact of higher-order beliefs
in equilibrium is actually higher
than that of lower-order
one must know the higher-order
beliefs to
beliefs,
an impossibly high
and
level of
precision in order to predict behavior.
•
Our
derivation in this section relies only on the formation of higher-order
expectations
— not on the particular type space used.
Hence
applies
it
to any type structure.
•
We
are only able to use the substitution trick here to derive a simple
formula because of the linearity of the best-response function.
In the
general case a player's best response depends on the details of the entire
distribution (as noted
by Morris (2002)) and there
is
no
direct relation-
ship between a player's best responses under certainty and uncertainty,
rendering such elementary analysis impossible and requiring the more
sophisticated tools of the following sections.
Note also that Morris and Shin (2003) and Morris (2002) obtain
examples with linear best responses similar to ours in
on
different issues;
this section.
specific
They
focus
Morris and Shin (2003) focus on the role of public information
while Morris (2002) focus on the large impact of higher-order expectations in
the worst-case scenario (when the slope of the best response approaches
1)
HIGHER-ORDER UNCERTAINTY
A
2.3.
each player
and
(0, 1)
E [a + bsj\xi] —
BRi =
The above
Check
We have ex ante a ~ AT (0,1), and
= a + £; where e ~ N (0, (1 — v) /v)
Structure.
gets a private signal x,
i
some v G
for
Type
Traditional
is all
+ 6 is
common
t
£2 are all independent.
[sj|xi] for
some
b
>
0,
For each
where
di
assume
i,
= E [a|x,] =
vx;.
knowledge.
^
whenever bv
that,
and
a, e\,
a,
9
we have a Bayesian Nash equilibrium
1,
s*
with
VXi
(2.5)
l-6u
1
When
<
bv
When
6u
>
1,
negative and hence s*
is
decreasing in x l
equilibrium seems intuitive.
1,
tuitively the coefficient of x^
is
however, counterin-
Now,
.
write s* as a series of higher order expectation as in (2.4). Since the fcth-order
expectation of a
s*
is
=
EiEjEi
vx
+
t
.
.
2
bv x z
.
Ej
[a]
=
+ &Vx
;
v
k
Xi,
we have
h b
-f
k
EiEjEi
E
{
\s*}
.
V
fc
Firstly, notice
that
when
bv
>
1,
times
higher-order terms increase exponentially,
yielding a divergent series. This explosively large impact of higher-order uncertainty,
ond,
however, does not appear
when
formula in
bv
<
1
<
6,
in
the directly computed formula in
we have a convergent
(2.5), despite
(2.5). Sec-
series yielding seemingly intuitive
the fact that marginal contributions of higher-order ex-
pectations increase exponentially. This
is
only because our single-dimensional
type space forced the variations in higher-order expectations to decrease exponentially,
5
compensating the increases in marginal contributions.
approximated
real-life situation,
doubts about this model.
But
in the
the players will probably have higher-order
In that case, their higher-order expectations
may
vary significantly, leading to dramatically different behavior (under the equilibrium of
more accurate model). In that
case, the model's predictions
about
the behavior will be misleading, and considerations about higher-order beliefs
within the model will yield a false sense of robustness.
J
This
is
a general
phenomenon
(see
Samet
(1998).)
>•
jonathan weinstein and muhamet yildiz
10
Model with independence
3.
We
ing uncertainty
Polish space
on
i
game among
consider a
A.
set
:
x
S
(In the
K
.
.
.
,
Xi—\,
Si,
where
Xi-\-i,
which
which at
.
.
.
,
Xn )
xz
Xn
X\,...,
list
j\—i,
anu
=
G
TV
and
open
least contains all
d_i (s_ n
We now define the players'
ter a. We confine ourself to
independent from
(i)
Ms own
v-^li
•
Each
=
player
Ylj^i %}, x -i
Ei—li ^ij 2-i+l
j
£ N, we define /_,
Yj, j
when we
-
:
•
i
%n)
X-i
we
—
for
fixed <r-algebra
use product spaces,
=
on
will
write di for the metric on Si for each
5_ by
t
s'_.)
= maxdj
{sj,s')
.
3
other orders.
We
do
this
because we want
vary a player's /cth-order beliefs without worrying about the
measure the impact of
(ii)
its
this
change on equilib-
impact through the changes
(The independence assumption
will
in the
be dropped
result.)
define the beliefs (or type) of a player
A =A
fc
.)
a metric
is
the belief structures where a player's beliefs are
His fcth-order beliefs (about i^T
e
X-i
i
i
inductively. His first order beliefs
(about a) are represented by a probability distribution
t\
[0, 1]
compact
d) is a
hierarchy of beliefs about the underlying parame-
player's beliefs at other orders.
We
€
X, suppressing the
rium strategies without worrying about
main
of sets, write
Xj —>
:
We
beliefs at
coherency of his beliefs and
in our
source of underly-
where (A,
(01,02)
\Xi, X—i)
sets;
define the metric eLj on
to be able to
A
The
Given any metric space (X,d), write A(X)
(fj {xj))-, v
always use the product cr-algebra.
i
n}.
,
.
S = Hi #•
the space of probability distributions on
X
.
a compact metric space, and utility function
is
Likewise, for any family of functions fj
Y-i by /_j (x-i)
=
Cournot example above a
Notation. Given any
\X\,
.
a complete and separable metric space), where d
(i.e.,
-»•
N = {1, 2,
a payoff-relevant parameter a €
is
has action space
m A
players
(AJJlJ)
1
)
.
=
e Ai
= A (A)
on A.
are represented by a probability distribution
on A£I*. The type of a player
t
t\
Ul
t
2
3
f
\
%
is
the
list
.
.
HIGHER-ORDER UNCERTAINTY
We write T
T = Yli^i f° r
of all these probability distributions.
types
t.
of player
ti
We
i.
also write
11
for the set of all possible
t
the se ^ OI an type profiles
"
His beliefs are represented by the product measure
beliefs (t},tf,tf,
.
.)
at each order; that
x
if
G
£_,-)
rifclo-^fc}
*
by changing
t\
to t\ in
of his
•
x T_;, the
s FlfcLi ^f POfc-i)-
We
we have used the independence assumption.)
for the belief structure obtained
x
if
C ^
given any rifclo^fc
is,
probability that he assigns to the event {(a,
(Here, of course,
x
ij
write t\tf
t\vL i and ti\tf are
i;
defined similarly.
Example
1.
(^Independent private value environment ) Take any incomplete-
information game with payoffs Ui
for each
(s; #,)
where each
i
pendently distributed with some probability distribution
by player
i,
and
this is
framework, by taking
and £
:
6-i i— >
common
A=U
H,^^., and
= 5e
t]
i;
r
knowledge.
taking t\
strategy of a player
i is
it
=
s, (i,)
Bayesian Nash equilibrium
=
s*
maximizes the expected value
i*
due to Vives (1990)
i\
embedded in our
be
= Uj^Pj
where P^i
k
>
where 5 X de-
2,
on {x}.
1
S{ (ti)
i
that determines which action
result,
= if = P_jo£
= o^-i /or eac/i
inde-
and privately known
game can
if
is
a measurable mapping
ti
the probability distribution
x
_1
notes the measure that puts probability
A
This
P
G Oj
9i
x
Oj,
he would choose given
(s\, s^
E
G
s*),
,
1
x
if
(see also
x
-
•
We fix
type U.
which must be such that
\ui (a,Si,s*_ l (i-i))
if
his
at each
|ij]
i,
of Ui (a,
and
for
s* (ti)
sl f ) under
Si,
each
i.
The next
Milgrom and Roberts (1990)), presents
conditions under which there exists an equilibrium.
(This result
is
proven by
the devise of considering the "agent-normal form game" in which each type
taken as a
omitted.
new
The
player.)
The
S and
is
conditions that are already true in our model are
stated conditions in this result will not be assumed in our paper.
Existence Theorem (Vives (1990)). For each
compact
a
lattice subset
of a Euclidean space, and Ui
upper semi- continuous on
S
z
.
i
is
G N, assume that
S, is a
bounded, supermodular on
Then, there exists an equilibrium
s*
.
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
12
Embedding
metric. Throughout the paper, we
tances between probability distributions.
we
metric, which
Given any
fi,
(3.1)
/J
G
A
MiA
will call
A (X),
=
,/
write
first
A (X
X) Imarg^ =
x
the marginal distribution on the
is
embedding metric don
A (X)
(3.2)
d(/j,,n')=
where
Ev
member
sense:
of
if \i
z'th
//,
marg2 */ =
X
x X.
copy of X.
and
inf
Ev
[d(x 1 ,x 2 )}
point masses at x and
// are
/i
and
Now we
where
[/,',
define our
by setting
easy to verify that this
It is
//}
with marginals
,
the expectation operator with respect to v and
is
dis-
therefore introduce the following
for the set of all joint probability distributions
margi
need a measure of the
embedding metric. Let (X, d) be any metric space.
we
{u G
We
will
is
(x\,
x2 )
is
a generic
an extension in the following
x', respectively,
then d(u,
=
An
fjf)
d(x,x') -- thus the notational convenience of using d for both metrics.
equivalent definition
given by
is
d(^//)=
(3.3)
inf
E[d{YX))
Y~ii,Y'~ij.
where
distributions
and
E
is
\i
and
and Y' of X-valued random variables with
fjf,
respectively,
and coming from the same probability space,
all
pairs
the expectation operator on this space.
The embedding
tinuity; the
proof
distribution of
Lemma
Y
taken over
inf is
1.
metric has the following property of preserving Lipschitz conis
in the Appendix. Notice in the
/ (Y)
for
a random variable
Y~
lemma
that
o
/i
f~
l
is
the
[i.
Let (X,dx) and (Z,dz) be two metric spaces, and f
:
X
— Z
»
be
such that
dz (f(x),f(x'))<Xdx (x,x')
(Vx,x')
for some X. Let also dx and dz be the embedding metrics on
respectively.
A (X)
Then,
dz
(/i
o
f-\u'
o
/- 1 ) < \dx
(ji, /i')
(V/i, a')
and
A (Z),
higher-order uncertainty
4.
13
and Higher-order Uncertainty
Stability, Rationalizability,
We are now ready to present our sufficient condition for the dampening impact
of higher order uncertainty:
function.
The
is
usually defined by the condition
less
than the variation in the other
global stability of equilibrium
that the variation in the best response
players' strategies
under the best-response
stability of equilibrium
under certainty.
6
is
We will
response function under uncertainty, which
first
is
extend this notion to the best
not directly related to the best
response function under certainty.
Best Responses. Given any player
A
x S-i, we write BRj
i
and any probability distribution
for the best
(tt)
response of player
when
i
on
it
his beliefs
about the underlying uncertainty a and the other players' actions s_ are rep2
resented by
Notice that
tt.
we
are taking the best response to be a function
rather than a correspondence. Under certain conditions
egy spaces are convex and
utilities are strictly
when the
(e.g.,
quasi-concave in
own
strat-
strategy),
the best-response correspondence will indeed be singleton. In general, however,
there
may be
multiple best responses. In those cases
we
assume that the
will
equilibrium uses a single consistent selection from the best-response correspondence. In the former case, the global stability defined below will be a property
of the game, while in the latter case,
we
the independence assumption,
and
/.i
When
G A(S'_
it
Z
).
In that case,
it
will
we
will
be a property of equilibrium. Under
have
tt
will write
does not lead to any confusion, we
BRi
when
—
t\
BR^
will
(e.g.,
BRt
denoting the best response of player
(a, s_,; t l ) 1
a and
s_j are
random
({i)
The
iff
there exists b G
we could
(t},/J,)
is fixed)
t\
instead of
i
or write
when
[0, 1)
We
G
A (A)
BR4
{-k).
it
in the
type
his
form of
is £;,
where
say that global stability under un-
such that, given any
usual definition appears to be different. For instance, in
need that the product of
some
for
variables.
Global Stability under uncertainty.
certainty holds
t\
/i
sometimes suppress some of
the arguments
write
x
maximum
rescale our metrics
variations
is
less
than
1.
Of
on each strategy space so that our
i
G N,
t\
G
A (^4),
two player games we only
course, under this condition,
definition
is
also satisfied.
,
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
14
and any
//
/z,
G
A (SLj),
d (BRi
%
where gLj
The
is
,
BR
r
the embedding metric on
(//))
<
BRi with
A (S-i))
6d_i (m,
A*')
A (5_j).
required condition for global stability
schitz continuity (of each
on
(ji)
is
the standard condition for Lip-
respect to the
embedding metric defined
with the additional requirement that the constant
which can be
b,
thought of as an upper bound on the absolute value of the slope, be
Of
1.
course, this
satisfying the
Our
first
is
the same as saying that for each
above condition, since we can take
b
entire type space to
Proposition
game, a strategy of a player
.
.
.
our game
i is
a
is
6,
G
than
[0,
1)
b n }.
,
is
dominance-
a function from his
5,-.
Assume
1.
there
= max {6j,
result states that global stability implies that
solvable. Notice that in our
i
less
that each player has single-valued best response corre-
spondence, and assume global stability under uncertainty. Then, there exists at
most one rationalizable strategy
s*
,
it is
We
profile. If in
addition there exists an equilibrium
the unique rationalizable strategy profile.
prove this proposition in the appendix for the general model developed
in Section
7.
Our proof
essentially
shows that the diameter of the space of
surviving strategies, measured as the
tions to any given type of
maximum
distance
among
any player, decreases by a factor of
b at
available ac-
each round.
Therefore, in the limit there can be at most one strategy profile. If there
equilibrium
isfied),
it
(e.g.,
will
when the
is
an
conditions in the existence theorem above are sat-
never be eliminated and hence will be the unique rationalizable
strategy profile.
The requirement
that there
is
always a unique best response
is
not superfluous. For example, for the second-price auction with private values,
there are multiple equilibria, and hence multiple rationalizable strategies, but
the dominant-strategy equilibrium
is
globally stable
and
satisfies all of
our other
results.
By a well-known
will
result of
Milgrom and Roberts (1990), a supermodular game
be dominance-solvable whenever there
is
a unique Nash equilibrium.
When
.
.
HIGHER-ORDER UNCERTAINTY
15
we can
there are two players, this requires a single-crossing property, which
guarantee only by a global stability condition on best responses under certainty
result extends this result
beyond supermodular games, by imposing our global
on the best responses under uncertainty.
stability condition directly
Global stability
sufficient to
is
guarantee that the impact of higher-order
beliefs
on equilibrium
result.
Consider a change in a player
so that
is
This
diminishing.
according to some mapping
The next
<f>.
maximum
—
1st order beliefs,
(s*
Assume
2.
Then, given any
k
t,
>
1
(u\t* o 0- )
and
Proof. Let
/j,
spectively.
Clearly,
same
—
from
to if
1st order beliefs
=
tfo0
_1
,
have changed
result states that, in that case, the
can be at most
i
t*
times the expected
b
change in the other players' equilibrium strategies due to the change
Proposition
dz
formally expressed in the next
fcth-order beliefs
i's
change in equilibrium strategy of player
in their k
is
believes that the other players' k
i
Our
each type of each player.)
(in the region of all possible intersections for
fjf
,
global stability
i
and any measurable function
< bE
s * (u))
and
beliefs of
o
s*_ z
[cL,
are
i.
under uncertainty for some
(
5 *_ 2
be the distributions of
s*_ l
state space T_j
1,
under the original
(u\<P
s*_ i
1
(t*! ))
under
two random
,
s%
and
tt
fi',
(t_0) \u]
ti\t% o
<j)~
respectively,
1
= dipRifafiiBRi^yl))
<
<
inf
bi
E[d-
i
{s- i ,s'_ i
)]
bE[d-i{sU,sU°<t>)\ti\
= bE
[<L«
(
s
v
(t_A0 (^T
1
))
,
under
Therefore,
^(tAt^- ),^))
;
re-
coming from the
variables
and have the distributions n and
[0, 1).
A£~* —» A£~-J
:
(f>
€
b
,
sU (u))
\u]
t{.
..
jonathan weinstein and muhamet yildiz
16
Sufficient conditions for stability
5.
we
In this section
present two sets of sufficient conditions for global stability
under uncertainty. Both
under
certainty.
We
under uncertainty
class
is
first
is
closely related to global stability
la. Best-response function of player
BR
(5.1)
A
x S-i
— X
>
is
i
and dx (g
t
i.e.,
s_
{a,
)
2
the vector of
and
<&
l
t
Q
i
This
takes the form of
xu 6
A (A
x 5_,); fi'.X-^Si and
,
g
r
(a, s'_,))
in (5.1)
all
<
such that
P> i
/^cL* (s_ 2
satisfied
is
,
di
(/,-
,
some
/; (x'))
u, is analytical
(The Taylor expansion
E
[gi\
moments.) The more substantial part of this assumption
are Lipschitz continuous.
Under
<
s'_j)
whenever
interior solution.
(x)
would imply such a functional form, where
for the first order condition
fi
certainty.
(a, S - i )])
there exist a, and
and the optimization problem has an
that
under
are two Lipschitz continuous functions defined through
Note that the functional form
is
(E[gi
taken with respect to
Banach space (X, dx)\
OLidx {x, x')
=f
1
(t i ,fJ,)
i
where expectation
:
present a general class of games where global stability
characterized by Assumption la.
Assumption
<7,
sets of conditions are closely related to global stability
certainty,
Assumption la
is
yields
a best response function
BRj
Our
(a, s-i)
=
fi
[g t (a,
s_
?
))
=
hi (a, s-i)
equilibrium would be stable under the best response correspondence
di [hi (a, S-i)
,
latter condition
Assumption
Proposition
< hd-i
hi (a, si,))
is
slightly
at each a for
some
&,-
<
1.
The
weaker than the following assumption.
lb. For each
3.
(s_i,s'_j)
if
i
G N, we have
6j
Assumptions la and lb imply
=
a,/^
<
1.
global stability
under uncertainty.
Proof. In the Appendix.
That
is,
under Assumption
the existence of
/ or g
is
cti's
and
/3/s
la, global stability
under uncertainty
is
implied by
that satisfy Assumption lb. Moreover, whenever
the identity, global stability under certainty and uncertainty will be
HIGHER-ORDER UNCERTAINTY
equivalent. Hence, there
Assumption
1
17
a close link between these two concepts. Although
is
might not be easy to check in general, our next example presents
a general class of games where these conditions can be easily checked.
Example
2.
For each
G N,_ take
i
Ui (a,
where
gi
:
Ax S-i — R
for each j
j3 i
=
s-i)
R
G
j3 t
>
differentiable functions with $[
gi is
(s^ gx
fa
(a,
i,i£l
for some
s_ 2 )
-d
(sj)
and
,
z
and for some
i
[x, x]
a continuously differentiate function with \dg /dsj\
is
*
^
Si,
=
Si
0,
and
7
<
<f>[
Lipschitz continuous with parameter
0,
and
<p {
d >
i
are twice continuously
Cj
and
0,
<
d[
>
Note that
0.
with respect to the changes in s_j.
f3 {
Check that
BR
i
where f
t
lution
x
<d
i
i
(x),
x
if z
to the first order condition
d
(x)
(z) is
x
(tlix)=f (E[g
if
z
function theorem,
fi
(x)
/<fr'
>
(a,s^)])
c(x)
/</>'
/<f/ (x),
=
(x)
where 7^ = min^^] {^ x 1$ x )) >
whenever b = max l6 jv Pi/li < 1{
)
the unique so-
By
the inverse-
parameter aj
Therefore, global stability
0.
i
it is
z otherwise.
also Lipschitz continuous with
is
and
=
l/j
{
is satisfied
Focusing on games where the agents' strategy spaces are one-dimensional, our
next result presents a simple sufficient condition for global
for
dampening impact
stability,
and hence
of higher order uncertainty, in terms of second derivatives
of the utility functions.
Proposition
4.
ferentiable, Ui (a,
C
R,
Ui (a,
S-i) is strictly concave,
2
2
For each
-,
i,
assume
Si
d Ui/ds
is
•)
is
twice- continuously dif-
bounded away from
zero,
and
5.2
6
;
<^
=max>
a
j-£
max s
\d
2
u
x
(a, s)
u
z
/dsidsj\
/
'
mm
.
s
\d
2
(a, s)
<
Then, we have global stability under uncertainty whenever
the interior of Si for all
Proof. In the Appendix.
t\
x
/j,,
or
(ii)
1.
/dsf\
Si is convex.
(i)
BF^
{t\,n)
is
in
—
,
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
18
Example
3.
and arbitrary
parameter
Consider Cournot duopoly with linear inverse- demand function
cost function
a.)
Check that
with
Ci
c'/
>
(where both
=
2
\d Ui/dSidsj\
\P'\
and
\d
P
2
and
Ui/ds
may depend on
ct
=
2
\
P
2\P'\
+
c" , so
that
= max ——
\P'\
bi
2\P'\
1
+ d!
<
- -2'
yielding global stability.
6.
Equilibrium Impact with Independence
In this section, using the
embedding metric defined above, we
will
put a
natural metric on the type space, which will allow us to compare variations in
different orders of the type space.
We will show that,
under the previously stated
conditions, variations in higher-order beliefs have a lower impact on equilibrium
behavior than comparable variations in lower-order
6.1.
Embedding metric on
We now
beliefs.
beliefs.
apply the embedding-metric
construction inductively to define our embedding metric on beliefs of each player
at each order. First, for k
=
1,
we extend d
d{t},t$)=
inf
a~t\
at each t},t] G
Ai and
to
A" -1 by
to
A (A)
At =
by setting
E[d(a,a')]
,a' ~t\
setting
d^P^m&xd^J.})
at each
l
t _ il
i\ i
£ A"
-1
.
For any k
d{tlt^)=
where
Y
and Y' take values
in
>
1,
we extend d
inf
to A*,
A£lJ (whose generic member
d(t k_ u ik_ i )=m^xd(t';,t^
G
A" -1
.
setting
E[d(Y,Y')l,
by setting
at each i'L^i'Li
by
is
t^'
1
),
and to
A
fc
HIGHER-ORDER UNCERTAINTY
6.2.
Dampening impact
stability,
we
will
now
find
19
of higher-order uncertainty. Assuming global
an upper bound
egy caused by a change in any fcth-order
changes (according to d) at
beliefs.
When we
bound
orders k, this
all
the change in equilibrium strat-
for
will
consider comparable
be decreasing exponen-
tially in k.
Proposition
uniformly on
d
(6.1)
for some
b.
t
5.
Assume
that, for
each
i
N
E
BR4
,
Lipschitz continuous
is
(•, /i)
i.e.,
\i,
{BPH $;ti),BRi
aGl.
Assume
(£;/x))
<
ad,
under uncertainty for parameter
also global stability
Then, in the model with independence, for any
d
(6.2)
i
{
S *(ti ),s*{ti
The conclusion can be
some order k while
\%))<abk
-1
strategy due to this change in the beliefs
at
is
k
U, k, and any i ,
i,
k
k
d(t ,i ).
Change the
spelled out as follows:
the other beliefs are fixed.
all
(V/z.^tl)
$,£)
beliefs of
The change
a player at
in the equilibrium
most an exponentially decreasing
function of k times the change in the beliefs according to our embedding metric.
bound
In other words, the
of the rate of change in equilibrium strategy as a
function of fcth-order belief
=
Proof. Firstly, for k
some k
— 1,
i.e.,
(6.3)
For any fixed
t
is
(6.2) is just (6.1).
k~ x
for all j E TV, i, and t
1,
J
d3
(s*
and
i
(t,-)
E N,
E A£~j. Fix U
d (f
Then, by
k.
Now assume
that (6.2) holds at
,
s*
,
let
/
at each t_~
exponentially decreasing in
(it-
Lemma
1
)
J (t^
1, for
d(t
(WH)
us define /
(£ri)
=
1
))
/,,
~2
A£lJ
:
= BR_
7
d
(i
—
(Atii
>
r\t k -i)
k
.
S-i by setting
1
)
so that our induction hypothesis (6.3)
< ab k
any ik_1
k
< «b k
~2
k
k
d{i -\i -')
k
k
(yi _- \t _-^
,
of-\t k of-i)<ab k ^d(t k ,t k ).
t
.
becomes
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
20
Notice that
k
t
o
f~
x
and
t
k
o
/
_1
are the distributions of
s*_i
under U and U\ik
.
Therefore, by global stability,
di {s;(t i ),s*(tA^))<bd^or\^or')<abb
In case
we only know a
player's beliefs
up
k- 2
d{tt^)<abk
-1
d{ttit)
to kih order and have no knowledge
of his beliefs at higher orders, the following result tells us the accuracy with
which we can predict his equilibrium behavior. This
is
important because, as
argued in the Introduction, modelers would prefer not to have to specify the
This result might be thought to be a corollary to
players' higher-order beliefs.
Proposition
k
1st
4-
will
and
can be obtained simply by adding the
5; it
all
higher orders.
The
form of this proposition and only assumes global
the strategy space.
7
9,
changes at
summation, however,
validity of this infinite
be established only when we prove Proposition
effects of
which
stability
is
a more general
and boundedness of
Notice also that our present result does not refer to any
topology on the type space
—although we used our embedding metric to reach
this result.
Proposition
Da = max aja
k
>
1.
6.
/
e/i
Under
d
and
the assumptions
(a, a').
Let
tt
,
U
be
the notation of Proposition 5,
such that
t\
= ^
for
all
I
<
let
k for some
Then, in the model with independence.
^( S *(t
2
),s*(tO)
<b k aD A /(l-b).
In certain cases, a modeler might want to predict the equilibrium behavior
within a certain margin of error. For example, checking the validity of certain
qualitative predictions of his theories
may
only require the knowledge of equi-
librium strategies within a certain margin of error. Proposition 6
This infinite summation would give us a proof only
for
/
any sequence {i^]}^ of types such that
orders, limj^oo d
corollary.
s
(t
[I]
,t)
t[l]
=0. Proposition 9
is
if
we had
identical to
tells
us
how
continuity at infinity,
some
fixed type
t
at the
i.e.,
first
implies the latter statement as an immediate
HIGHER-ORDER UNCERTAINTY
many
orders of uncertainty he needs to specify.
and any
t
G T,
if
we know
,
fRA
(6-4)
k
t
It
21
implies that, given any
>
e
up to the order
—
\og(e)-\og(aD A /(l-b))
>
777
,
log (6)
maximum
then we can compute the equilibrium strategies up to a
Notice that the expression on the right-hand side
e.
decreasing in
6.3.
is
error of
and
increasing in b
e.
Continuity in product topology. Many authors emphasized that
librium strategy
is
equi-
not continuous with respect to the product topology on type
space and introduced stronger topologies, such as the topology of uniform convergence, in order to
make
the equilibrium strategies continuous (see Monderer
and Samet (1996) and Kajii and Morris
dence
fails
to be lower semi-continuous.)
vergence over
e-mail
all
These authors require uniform con-
games, in essence focusing on the worst-case games, such as
game which has high dependence on
consider the
(The equilibrium correspon-
(1998)).
games with
higher order beliefs. Moreover, they
discrete strategy-spaces,
where the best-response
respondence cannot usually have any continuous selection, which
global stability. Here
of this
sult
game
we
fix
is
cor-
needed
for
a game, and ask whether the equilibrium strategies
are continuous with respect to a product topology.
Our next
re-
answers this question in the affirmative for games satisfying global stability
under uncertainty and
embedding metric on
Note that
for the
beliefs.
this topology
embedding metric. That
to this topology
iff,
for
[t m
product topology on type space generated by the
the topology of pointwise convergence under the
is
is,
equilibrium strategy s*
any sequence
-£
where convergence of beliefs
Vfc
{t z>m }
e N] =»
meN
[a* fc,
is
continuous with respect
of types,
m
)
-a
;
(*)]
at each order is according to the
Also, because the space of beliefs
is
,
embedding metric.
compact under the embedding metric,
this
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
22
topology
metrized by the metric <4 (called a Frechet metric) defined by
is
oo
db
any number in
(tJ
where
b is
bility,
the equilibrium strategy
metric,
and hence
Proposition
the
it is
)=Y
1
(0, 1).
is
J
fc=i
i is
k-
Our next
l
,
d{ttil),
result states that,
sta-
continuous in the product topology.
Under the assumptions and
7.
under global
Lipschitz continuous with respect to a Frechet
model with independence, for each
player
b
i
G
the notation of Proposition 5, in
N
,
the equilibrium strategy s*
Lipschitz continuous with respect to d^. In that case, s*
of
continuous
is
with respect to the product topology on type space generated by the embedding
metric on
any two types
Proof. Fix
U,k
each order.
beliefs at
and U of player
tr
i.
For each k G N, define the type
by setting
i
k
-7
i
I.
We
otherwise
t)
I
at each order
if/<fc,
t\
— J
have
oo
oo
d
t
(s* (t,),s*
<
(U))
]T> (s* (t
i!k ),s* (t iifc _!))
=Y,di
fc=i
(*J (U,k),s* (t iik \ii))
fc=i
oo
fc=l
where q
equality, note that
we can change U
Hence, by Proposition
a\ (s*
e.
(U)
Since
tion; the
,
To
as defined in Proposition 5, proving the result.
is
s* (t,))
e is
<
6,
for
J2[ =1 d
x
each
(5* (t
e
a
)
i,
>
0,
s*
(ti, fc
,
1. Let S* be the set of
sistent selection
5,
t* to t\
-i))+e
and the
< Er=i
The next
/
such that
dt (s* (t hk )
equality
last equality is
Bayesian Nash Equilibria
by
first in-
one at a time.
there exists an integer
arbitrary, this yields the inequality.
next inequality Proposition
Corollary
by changing
to
see the
is
,
s* (t ilfc _i))+
by
defini-
definition.
s* that
use a con-
from best-response correspondence, and define E* by setting
HIGHER-ORDER UNCERTAINTY
E*
(t)
sition
=
{s*
E*
5,
G 5*}
(t) \s*
is
at each t
G T. Then, under
23
the assumptions of Propo-
lower semi- continuous with respect to the product topology on
type space generated by the embedding metric on beliefs at each order.
Proof.
Take any
t,
any
s* (t)
By
in the topology above.
equilibrium s* at
G E*
(t),
and sequence
definition s*
(t)
Then, by Proposition
t.
is
(n) that converges to
t
Nash
the value of a Bayesian
7, s* (t (n))
G E*
t
(n)) converges to
(t
s* (i).
7.
Without independence
We will now define the universal type space without imposing independence.
We will show that our main result, namely Proposition 6, generalizes to this
structure.
General model (without independence). The independence assumption
was
built into our previous
model
to allow for simpler notation
and a
clearer
consideration of the effects of changing beliefs at a single order. In order to allow
for the general case,
we
will
need to consider the usual (and more complicated)
construction of the universal type space by Brandenburger and Dekel (1993), a
variant of an earlier construction by Mertens and Zamir (1985).
of /cth-order beliefs in our two models are not parallel, as in the
/cth-order belief will contain information
about
define our types using the auxiliary sequence
by Xq
=A
and Xk
=
[A
n
(X)~-i)}
x
X
all
{Xk} of
A
is
the
We
will
sets defined inductively
>
with the weak topology and the cr-algebra generated by
a standard separable Borel space as
new model
lower orders as well.
for each k
k -i
The meanings
0.
We endow
each
Xk
this topology, yielding
a Polish space.
A
player
z's first or-
der beliefs are represented by a probability distribution r\ on Xo, second order
beliefs (about all players' first order beliefs
and the underlying uncertainty) are
represented by a probability distribution r? on X\, etc. Therefore, a type r of
z
a player
i
is
a
member
of I7fcli
^ P^fc-i)-
Since a player's Mh-order beliefs
now
contain information about his lower order beliefs, we need the usual coherence
requirements.
We
common knowledge
write
T
for the subset of (Ofc^=i
A PGt-i))"
that the players' beliefs are coherent,
i.e.,
i
n which
the players
it is
know
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
24
their
own
beliefs
and their marginals from
the variables t, f G
3
is
T
different orders agree.
The
as generic type profiles.
rest of the
We
model
will use
in Section
unchanged.
Dropping the independence assumption causes two complications.
First, since
a player's higher-order beliefs contain information about his lower-order
we can no
I
>
beliefs,
longer vary a player's belief at order k and keep his beliefs at order
k constant
—as
in Proposition 5
ments. Instead, we allow
in Proposition 6).
all
— without violating the coherency require-
the beliefs at
orders higher than k to vary (as
all
now
(possibly)
we need
to extend
Second, since the other players' actions are
perceived to be correlated with the underlying uncertainty,
our definition of global stability to allow such correlation. To do
any metric on
A
this, let d-i
be
x S^ such that
t
d-i ((a,s-i), (a,s'_,))
(7.1)
=
d_ (s-^s'^)
?:
A and s_,, s'_ G 5-,;, so that the metric cL; on <SL, is preserved when
S-i is embedded in A x 5L,-. Extend also d-i to A (A x S-i) using the embedding
metric as before. We are able to leave the metric d-i only partially specified
for each a
G
r
since global stability
is
related only to responsiveness of the best response with
Thus we
respect to the changes in the other players' strategies.
prove that as long as the inequality below
satified for
is
some
will
be able to
d-i our results
will hold.
Global Stability under uncertainty in general model.
stability
under uncertainty holds
ding metric d-i on
and any
n,
tt'
G
A (A
(BR
(tt)
,
some
there exist
x S-i) satifying
x S-i) with
d,
(7.2)
The next
A (A
iff
(7.1)
b
G
[0,
(tt'))
<
say that global
and some embed-
and such that, given any
marg^ = marg^', we
BRt
1)
We
bd.
t
(tt, tt')
i
.
can also check that Example 2 of Section 5 remains valid under the new
we
will
whenever max,
\P'\
have global stability under the
<
2 min„ \P'\.
new
N
have
proposition extends Propositions 3 and 4 to the present set up.
tion, while
G
definition in
One
defini-
Example
3
,
HIGHER-ORDER UNCERTAINTY
Proposition
8.
certainty,
For each
(b)
25
Assumptions la and lb imply global
(a)
assume S
i,
t
C
K, U{
entiable, Ui(a, -,s_j) is strictly concave,
(a,
•)
d 2 Ui/ds 2
under un-
stability
is twice- continuously differis
bounded away from zero,
and
_ v-^
7 3)
(
bi
-
27ien
we
7
=
2-J
2_J
max S)a \d 2 Ui (a, s) /dsidsj\
L
31
i3
2
min^ld
^ (a,s)/ds 2 <
/iaue global stability
interior of Si for all
or
tt,
(ii)
3lw3r
\
under uncertainty whenever
(i)
BRi
(tt)
in the
is
Si is convex.
Proof. In the Appendix.
We
are
now ready
main
to prove our
which extends Proposition 6 to
result,
our general model.
Proposition
D$ = maxi e jy swpvw e &, Ay s ^ di (BRi (tt) BRi (tt')) G M.
such that r\ = f\ for all < k for some k > 0. Assume
9. Let
Let also r,f €
global stability
T
,
be
.
,
I
under uncertainty for parameter
b.
Then, in the general model,
ditfWtsKftf^tfDs.
(7-4)
Notice that our result assumes only global stability and boundedness of the
strategy space.
Under these two assumptions we reach the conclusion
know the beliefs up
to a certain order k,
a bound of error that
maximum impact
all
as a
bound on the
k,
any topology on the type space.
chosen
variations in equilibrium outcomes. If there are other
known
then we can replace
Ds
prove our proposition.
2. Let (X,
We
start
with the following technical lemma.
Ex), (Y,Ey)
;
(Z,T, Z ) be separable standard Borel spaces,
X xY x Z
with the a-algebras generated
by the corresponding product topologies. Let probability measures
x
Y
and
X
restrictions)
with these bounds. In the remainder of the section we
and endow XxY,YxZ,XxZ, and
X
Ds
Our
is
Finally,
bounds on the equilibrium outcomes (perhaps due to some support
Lemma
we
bounding the
the higher-order beliefs can have on equilibrium.
result does not refer to
if
we can know the equilibrium play within
an exponentially decreasing function of
is
that,
x Z, respectively, be such that marg x P
= marg x P'.
P
and P' on
Then, there
,
JONATHAN WEINSTEIN AND MUHAMET
26
exists a probability
marg XxZ P
=
measure
P
on
X
x
Y
x
Z
such that marg XxY P
= P
and
P'.
Proof. In the
Appendix.
Proof of Proposition
9.
Define
= A x T to be the universal state space. This
n
Q = A x (Jlfcli A (Xk-i)) in which coherency
Q,
is
the subset of the larger space
is
common knowledge. By Brandenburger and
for every
i
E N, there
exists
Dekel (1993),
and
yielding a standard separable Borel space,
and
YILDIZ
for every
t
Q, is
=
a Polish space,
(ti,
a probability distribution kt
.
G
.
.
.
A
,
rn ) G
T
(Q) such
that
maigXki KTi =T^
(7.5)
and k Ti
(fi)
=
1.
(Vfc),
Let
:
(a,r) h^ (a,
s!_,
(t_,))
,
and write
tt Ti
for
k Tj o /T 1 6
A (.4
x 5_
t
)
the joint distribution of the underlying uncertainty and the other players'
actions induced by
We
k
=
>
0,
will
r,.
Notice that s*
use induction on
and assume that the
hypothesis.
We
fc.
For
result
is
= BRi (7r r J.
(t$)
=
A;
0,
this
true for k
true by definition.
is
—
Fix any
Take any r and r as in the
1.
have
4(s:(Ti)>Sifo))
= diiBRii-K^^RiiTVr,))
< 6d_ -(7r Tt.,7r^)
t
=
(7.6)
6
inf
^t^TTx 'T=.T
Ev
[d_ { ((a,
s_0
A
^-
,
(a',
aL,))]
-
I
where the inequality
The
is
rest of the proof is
due to global
]
stability
and
7Tt
devoted to constructing ai/6
A
1Tt
the induction hypothesis,
(7.7)
Ev
[(!_,
((o,s_0
,
(a'.s^).)]
<
1
ft*"
/^-
is
)7r
.
defined by (3.1).
such that, under
.
HIGHER-ORDER UNCERTAINTY
Combining
We will
(7.6)
and
we obtain
(7.7),
decompose ClasCl
(7.4).
AxLxH where
=
k— 1
oo
L = 11 (A (AVj))
(7.8)
71
H = \{ (A {X .
and
l
1=1
that
L
is
a singleton
and
set,
(7.5),
we have
beliefs.
=A
Xk-i
raargx,^^ =
=
r\
H
n Ti and n^ on
i
= maigXki K fi
t\
there exists
2,
A
the marginals of a on the cross product of
of
we use the convention
1,
in the following
such that
Cl
,
by our hypothesis. Since we have separable stan-
is
Lemma
dard Borel spaces, by
=
x L.
probability distributions
where the second equality
For k
G L can simply be ignored
I
analysis for that case. Note that
By
n
l ))
l=k
and higher-order
are the spaces of lower
27
fc
A (Xk-i
a G
_ x with the
Hx
x
first
H) such
that
and second copies
are
=
marg 12 o"
marg 13 (j
and
n Ti
=
Kf
t
,
respectively.
=
Now, consider u
7
(7.9)
a
:
o
(a,
7
I,
-1
A
G
h u h2 )
[(A x S-i)
.-> (/3 (a,
Notice that the marginal of u on the
margjZ/
=
margj (a o 7
and similarly marg2f
=
We now prove (7.7).
/.
By
(7.9),
'K-i
.
a',
)
But by
(7.9), s_,
and f
=
(l,h 2 ),
=
s!_,
(f _,)
,
/? (a, /, /i 2 ))
copy of
first
(marg 12 a) o
= 7 (A
fc
-i
x
and hence by
d-i {(a, s_ ? )
(7.10)
=
M
A
=
(3~ l
and
,
(a', si,-))
s'_j
=
H
is
z
k Ti o
[3~ 1
j7T
-
x H) and take any
=
7r Ti
,
.
((a, s_,-)
,
(a', s'_ 2 ))
(7.1),
=
rf_, (*_-, s'_,)
.
sl f (t_j) for some type profiles f
which agree up to the order k
—
induction hypothesis,
(7.11)
x S-
Therefore, by definition, v G A„. T
Write I
we have a =
_1
/,
where
)
d^^s'^Kb^Ds.
1
by
(7.8).
=
(I,
hi)
Then, by the
G
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
28
Combining
(7.10)
and
(7.11),
d_
(7.12)
Since suppi/
C
t
we obtain
((a,
s_
)
2
,
(a', s'_ 2 ))
< b^Ds.
/ (by construction), (7.12) implies (7.7).
Conclusion
8.
Present economic theories are mostly based on equilibrium analysis of models
on only a few low orders of uncertainty,
all
higher-order
We
know, how-
some games higher-order uncertainty has a profoundly
large impact
in which, conditional
unwarrantedly assumed to be
beliefs are
ever, that in
in equilibrium. In this
stability
paper we presented a
knowledge.
sufficient condition,
is
low. Using the universal type space, in
any coherent
set of beliefs,
specify the players' beliefs
up
we have shown under
to
some order
we
k,
which players can enter-
this
will
assumption that
know
behavior within a bound that decreases exponentially in k
That
k
(e)
orders of beliefs, where k
Proposition
(e) is
a logarithmic function of
by specifying
e.
Under a
independence assumption we also formalize our notion that, under
marginal impact of higher-order uncertainty
order
Propositions 2 and
(cf.
9).
8
5).
That
is,
is
when
may grow
further
(exponentially) decreasing in the
the problem must be approximated
stability does not hold, as the
exponentially. In the latter case,
first
stability, the
using lower-order uncertainty rather than higher-order uncertainty; this
reversed
we
their equilibrium
(cf.
of error, then the researcher can validate his theory
e
if
a theoretical prediction requires knowledge of the strategies within a
is, if
margin
namely global
under uncertainty, which guarantees that the impact of higher-order
uncertainty
tain
common
may be
impact of higher-order uncertainty
we
believe that accurate prediction
using traditional analysis will be impossible.
When
the best responses are always unique,
game, and hence our analysis would not change
we have a dominance-solvable
if
we considered refinements
of
equilibrium or non-equilibrium concepts, such as rationalizability. Nevertheless,
in general,
our use of normal-form representation and the solution concept of
°It also follows
from these assumptions that the equilibrium strategy
player's type with respect to a product topology
(cf.
Proposition
7).
is
continuous in
HIGHER-ORDER UNCERTAINTY
29
Nash equilibrium does impose an important
(unrestricted) Bayesian
which requires further research.
Many
limitation
on extensive-form rep-
theories are based
resentations and use refinements, such as sequential rationality (Selten (1974),
Kreps and Wilson (1982)). Their predictions are often driven by these
ments when equilibrium
It is
itself
refine-
does not have any predictive power in their games.
then crucial to extend our analysis to such a framework, using extensive-
form constructions, such as Battigalli and Siniscalchi (1999).
Appendix A. Omitted Proofs
A.l.
Proof of
dx
definition of
Lemma
(li, Li'),
there exists u €
A^'
dx
)]
Define /
J"
1
:
X2 — Z
2
*
by /
(£1,2:2)
=
(/
(
A (A'),
G
//
fj,,
Ev [d x (x 1 ,x 2 <
(A.l)
i/o
Take any
1.
and
any
fix
>
e
By
0.
such that
(fjt,
fjf)
+
e.
Then, by
x i) / ( x 2))>
definition,
6 A^ of -iy of -i. Hence,
dz
{^of-
Jor
l
< Euof -
l
)
i
[d z (z 1
,z 2 )}
= Ev [dz (f(x 1 ),f(x 2 ))}
< Eu
[Xd x {xi,
< Xdx
since
e is
A. 2.
+
is
by change
last inequality is
plies that our
elimination
by
game
is
1.
dominance
We
will
now show
is
the case whenever
vex.) For each
i,
let
U{ is
M, C
by
by the hypothesis,
solvable. Beforehand,
i,
we formally
define our
in the proof.
his best response correspon-
by the best response function BRt. (Recall that
continuous and strictly quasi-concave, and Si
5,
the
(3.2);
that global stability im-
method and develop some notation needed
single- valued, given
is
is
(A.l).]
Rationalizability. Assume that, for each player
is
(xi,x 2 )]
Ae;
of variables, the next inequality
Proof of Proposition
dence
= \E„ [d x
arbitrary, the result follows. [Here, the first inequality
next equality
and the
{Li,Lt)
x 2 )]
'
be the set of all measurable functions
s;
:
is
this
con-
%—
>
Si,
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
30
the set of
i.e.,
all
allowable strategies for player
iteratively as follows. Set
be the
=
Sf
Sj\ For each k
set of all possible beliefs of player
that are not eliminated in the
beliefs of
i
A
on
x S-i by his type
and
all
set S*
best responses of
=
Yl T €T &i
Proof of Proposition
(
The
r 0-
Dk =
We
will
actions
and
=
s^ (t^)
d
E*"
1
.
G S\
=
and metric
d_,,
and take any
=
BR
(7r r
S; (t^)
A (M_j)
G
belief c_i
.,
and
_.)
!<T
G
{.B.Ri (tiy^,,) |<t_,
about
1
E*:" } for
E^
k,
s'
t
each
and any
(t ? )
is
i
.
i
in the limit
The second
part simply
hence
k,
assume global
0,
,
there cannot be any two distinct
first part.
for
1
k, define
(s l (T i ),s'l (T l ))
showing the
.,
induced
for the
i
Tj against all his beliefs in
0; therefore,
Towards showing that lim^oo D^
definition,
l
.
.
D M_;)
set of all rationalizable strategies for
follows from the observation that s\
b
7r Tii<7 _
available to any type t x of any player
of the process of elimination,
meter
=
(tj)
sup
show that lim^oo Dk
Si (t,)
and his
r^
0, 1,
1
players' allowable strategies
For each non-negative integer
1.
E^ = A (S^
rounds. Write
with type
i
>
=
k
,
1
0, let
on other
i
1
Write Sf
the other players' strategies.
the set of
—
k
first
Define sets 5f
i.
5°°.
1
some para-
stability for
G N,Ti G %,Si,
i
= BR
G
s*
some o^i
a >_.) for
(tt Ti
G S*.
s^
>
1
By
o'_ i
G
Hence,
(A.2)
di {Si
On
ML
a;
(r 2 ) ,5- (r,))
=
di
(BRi
(ir TttCr _
the other hand, for each
—
^o x
define also
butions on
tively.
and
z/
° p^
\°~'-i
/i
A
w
=
1
))
//
given any
u
fi'
:
)
,BR(n T ^ a <_U <
(a,r), define fi^
s_, i—> s_^ (r_,)
Notice that
.
fj,
and
u
bd-
=
/j^
dn Ti
J/i w
(•)
u =
(a, r),
(to)
G
_
i
A
and any
= f
ffT
a_
w () dnTi (u),
fj,
>7T
,
Ti ,<T-i,
x
(cr_ z o
is
,
(a, s'_j
n Tita>_. )
.
p" 1 ) and
the point mass at
7r T
.
ct_j
>(7 /_.
and
= /
Moreover, since
.
((a, s_; (r_,))
\J
are the probability distri-
z
vt t?i(7
t
Sa
and 5 a
x S- conditional on u, induced by behefs
Notice also that
=
=
u>
where pw
w x
t
a'_ {
,
respec-
u (-)dKTi (u),
fjf
G
cr_j,cr'_,
(t-0)) e SU PPA*W we
>
E^
fi
1
,
ave
HIGHER-ORDER UNCERTAINTY
d- ((a,s- (r_,))
z
{
(A.3)
,
J_, fflV i ,(7_ i>
By combining,
=
(a,s'_ { (r_i)))
(A. 2)
< £„
7rT il <r'_J
and
d-i (s_* (r_j)
taking the
fc
-i, yielding
D k -i-
<
Ejx [J_ t (xi,x 2 )]
we obtain
(A.3),
<
di(Si{Ti) ,S;( r i))
By
=
(xi,x 2 )]
[<J_i
< D
(t_;))
s'_,
,
31
supremum on both
^fc-i-
we obtain
sides,
D k <bDk -i.
A.3.
random
and
BR,
showing that
Proof of Proposition
A (S-i).
S-i
D k < b k Do,
<
Therefore,
Take any
3.
=
Recall that d_, (n,fj?)
(t},n)
with distributions
=
fi
(E
[
and
\i
t|
dkiBMtiiBRiW)) =
< o (£
2
inequality
first
is
and the
by
will
(jm)
Proof of Proposition
assume
convex,
a,-ftE [d_i (s_,,
last inequalities are
(BR,
i?/?,
(^,m) an d
we can take BRi
(t,-,
,
4.
##i
/x)
BRi
{E[gi
(
(a,s'_ i
s'_
E
t]
m
£?/?, (£j, //)
We
(/i,
A (A)
£
variables
1,
.
we have
Hence,
)]))
a 5 -z))])
>
[d-
t
{
fj!
(a, s'_,)])
[&
=b E
sf_i)]
< M-i
(U,n') are
the variations in unconstrained optima.)
i
£
,
)]
and
in that case the variations in the constrained
U
z
sL f )])
(a,
due to Assumption
(//))
Take any
and
(£ [#
fi,
Take any
(s_i,s'_j)l.
By Assumption
[d x (gi (a, s_f) jft
triangle inequality. Since s_j
dt
A.4.
s_
and any
and any two random
(a, S - i ))),fi
\g t (a,
A (A),
E [d_i
=£
(tj, //)
D
0.
£
t}
//, respectively.
a,dx (E
<
E N,
i
A (/I),
di (fi (E[gi
<
where the
€
and BR,
9l (a, s-i)])
Dk =
infs_ t-~/z,3^_.~//
variable a with distribution
s'_ l
lirrifc_ 00
1
t
,
(s_i, s'_ 2 )]
,
and the second
are arbitrary, this yields
//)
•
and
/*,//
the interior of
G
Si.
A (5-,). We
(When S
t
is
as the unconstrained optima, as
optima are
write
(s;t})= Jui(a,s)dt}(a)
if
anything
less
than
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
32
and write
U
l
and U^ for the
U\, U^,
with respect to
Firstly, since
BRi
for optimization
and the
s i;
(fi)
first
and second order
cross partial with respect
and BRi
(//)
problems with
and
Sj
are in the interior, the
and
\i
partial derivatives of
first
Sj, respectively.
order conditions
// yield
EptiBIk^)^^)] =
(A.4)
and
E[Ui(BR
(A.5)
i
(»'), S '_ i )]=0,
respectively. Let
= E[ir {BR
J
be the value of the derivative at BRi
We
will
now
(??). First
find
we
\E[U\
< £
will yield
[BR
equality
following inequality
(p)
sU) - Ul (BRi
,
(li),^.)
-
V\
(BR
(/i)
(
M)
,
s_
?:
)]
|
,s_0|]
^max^^^lcU-^,^)
^maxl^.^;^)^^;^,^)].
=
inequality,
and these bounds
for \J\,
//.
= lEptiBRiMis'-i)]]
< E[\Ut {BR
first
problem with
for the optimization
upper and lower bounds
=
Here the
(/.*)
M),J_ i )]
find an upper bound:
\J\
(A.6)
i (f
i
is
is is
we write
U-
by
definition, the
by the
(BR^
second equality
triangle inequality.
(jj,)
,
—
s'_,)
U- (BR,,
To
(/.i)
,
is
by (A.4), and the
derive the penultimate
s_j) as the
some
of the
changes that we would get by changing each coordinate in turn, and apply the
mean
value theorem to each, obtaining
|i£ (Bifc
M y_<) -
0?
(BifcM,s_0
1
<
^max|t£.(s;tj)|| 5i -sj|
<
^maxl^.fatJJId-^S-i.slO
HIGHER-ORDER UNCERTAINTY
where the
last inequality is
33
by our definition of the metric
To
cLj.
find our lower
bound, we write
= \E[Ui{BBd {p),s'_
\J\
= \E[Ui{BR
i
= E[\U\ {BR,
(A.7)
Here the
The
first
sign.
The
{p)
)-Ux {BR
)
s'_ %
,
-
U\ {BR,
is
crucial;
{^-BR
equalities are
we have
and hence U\ {BRi
equality here because U\
—
(p) ,s'_ t )
is
U\ {BRi
{/j,,
=
p')
we obtain
\BRi (p)
-BRi{n)\<
d_ 4
(a*, A'
Emax
(s;
/ds 2
\
t,-)|
< Jmax s
dt] (a)
its
the
&
[^-»
(
s -«i 5 -t)]
irrw g f ni
|^(s;ti)|.
s
and min s
U; (a, s) fdsidsj] dt\ (a)
|
C/^ (5; t})\
Therefore,
.
max
^ J |mm
Ulj (s; t\)\
s
2
|<9
s'_A is strictly
a constant. Combining
is
i'n$s- 1 ~n,s'_ ~^'
^ min
)
(•;
respectively.
never changes
(//) ,s'_ ? )
j¥«
Check that max s \U^
) \]
mean value theorem, and
again by the
are arbitrary,
2
s'_ t
by definition and (A. 5),
s'_j
\d Ui (a, s)
,
_ i )]\
i { f,')\.
because the term inside the expectation
and that s_j and
{pi)
rmn'\Uii {s;t})\\BRi
inequality in the next line
.
i { fj,'),s
=
and (A.7) and observing that cLj
f min s
f
l
E mmpt^tfjWBRi^-BRi^)
and the second
last equality is
(A. 6)
{p),s'_ i
)]\
>
third equality
decreasing,
i
2
|9
s
I
^>
min,|££(s;tj)|
v->
~
/"
^J
s
max,
«t
(a, s)
|^
/ds ds ] dt
(a, s)
|<s>V (a, 5)
mm
l
Jds* dtj (a)
|
/ds ds J
l
2
s
\d Ul {a, S )/ds
EmaXj \d Ui
2
mm
(a)
|
,
\
"<
2
W
\
(a, s) /<9sidsj|
l
s
\o Ui (a.s) /osf\
completing the proof.
A. 5.
Proof of Proposition
proof of Proposition 4 above.
8.
We
The proof
will
of part (b)
prove part
(a).
is
very similar to the
>
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
34
Under Assumptions la and
gi (a, S-i)
marg A 7r
BFU
= gi (a, s'_^) =
= marg A we
yielding dx
(tt'),
{BR,
Now assume that /3 >
exists Mi >
such that
Define a metric c^, on
onAx
define d_j
A
Z
X
,
Since
d x (g l {a,s_ t ),gl
(A.8)
and
{tt)
0.
{
BR
BR
have
7r',
each
(a) for
g~i
any
lb, take
=
(tt')) =
f {E%
Firstly, if
))
< M_*
<M
d^
= f (En
(g z (a)))
z
{
x £Lj
/4
any
for
(tt, tt')
=
(a, a')
then
0,
tt, it'
with
(g { (a)))
,
=
(L*.
compact, there
is
(V^s^a',^)
t
=
/3 i
hence, for each
continuous and
<& is
by setting
Af.
(a, s_;, s'_,),
{tt)
(a',s'_ l
6
t
.
M;//3j at each distinct
a, a',
S-i by
cL* ((a, s_
?:
)
(a,
,
s'_
?;
Now, take any two random
= d^
))
(a, a')
variables (a,S-i)
~
from the same probability space and write p
+
7r
for
d_
(s_i,
7;
and
s'_,-)
(a',s'__j)
.
~
come
that
7r'
^
the probability that a
a'.
Note that
E
(A.9)
[d_, ((a,
s_0
,
= P M /f3 + E
s^-))]
(a',
r
l
[d_, (s_ i; s'_.)]
.
Moreover, we have
diiBRiWtBIUW)) < a E[dx
i
(gi (a, S - i ),gi (a,s'_ i ))]
otiE [d x
[gi (a,
+a E
z
<
< OipMj + <*&£
where the
first
equality
is
by
nuity of
gi,
inequality
)
,&
(a, s'_,))
[d_; (s_i, s'_.)
[d_i (s_,-,
=
b,E [d-i((a, s^), (a'.sLj)]
is
i
i
i
a
^ a']
a
:
=
by definition of
are arbitrary, this shows that
is
6,
(s-i ,^_i )])
,
rfj
(A.9). Since (a, S-i)
(•£>!?; (7r)
,
the next
3,
by (A.8) and the Lipschitz
by the non-negativity of
and
BR,
a']
a]
derived as in the proof of Proposition
additivity, the next equality is
=
a
:
sLj]
bi
{pM /P + E[(L
:
(a, s'_,))
,#
=
the next inequality
equalities are
z
[d x (gi (a, s_,-)
+ Qi^S
ttjpMi
s_
{tt'))
<
bz
d_j,
and the
~ and (a!
E [d_,- (tt, tt')]
tt
,
conti-
last
.s'_
7
'
.
)
two
~
tt'
.
,
HIGHER-ORDER UNCERTAINTY
A. 6.
Proof of
Lemma
P =
Let
2.
35
= marg x P'.
marg x P
Since
we have
separable standard Borel spaces, there exists conditional probability P(-||-)
(Ex x y) x {X x Y)
ply write
P (B\x)
—
>
[0, 1]
P (X
for
with respect to the
cr-field T,
x x {Y}, and we
x B\\ {x,y)) where y can be chosen
define P' (C|x) similarly for each
probability distributions on (Y,
C
E Hz- Notice that
Ey) and
P (-|x)
(Z, Ez), respectively.
9
sim-
arbitrarily.
and P'
We
are
(-|x)
For each
:
i6X,
let
Px =
be the product measure of
measure
P
P(.|x)xP(-|x)
P (|x)
and P'
at each measurable set
FCX
Fx =
A
P{AxBxC)=
J
Y
x
{(y,z)
Notice that, for any rectangle
x
Px (Fx )dP(x)
Z
eY x
x
B
x
where
Z\{x,y,z) €
C
6
E^
Now we show
that
P
x
satisfies the
F}
Ey
[xa(x)P (B\x) P'
where \ A denotes the characteristic function
P
x Z, and define probability
by setting
P(F) =
and
onY
(-\x)
x E^,
(C\x)
dP
(x)
of A.
statement of the lemma. For each
A
Ex
€
G Ey, we have
P{A x B
marg XxY P(A x B)
x Z)
XA {x)P{B\x)P'{Z\x)dP{x)
XA {x)P{B\x)dP{x)
P(Ax
Since the probability measures
rectangles
A
B).
marg XxY P and
P
agree on the 7r-system of
x B, which generates the entire d-field on
Theorem they
are equal. This
is
similarly true for
all
X x Y, by Dynkin's tt-A
marg XxZ P and
See Parthasaraty (1967) for the results of probability theory in this proof.
P'.
jonathan weinstein and muhamet yildiz
36
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.
in:
Truman Bewley
Fifth World Congress, Cambridge UK: Cambridge
2k
CP
2 6 2003
Date Due
Lib-26-67
MIT LIBRARIES
3 9080 02524 45