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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Finite-Order Implications of
Any
Equilibrium
Jonathan Weinstein
Muhamet
Yildiz
Working Paper 04-06
February 6, 2004
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://ssm.com/abstract=500202
IVSASSACHUSETTS INSTITUTE
OF TECHNOLOGY
FEB
1
9 2004
LIBRARIES
FINITE-ORDER IMPLICATIONS OF ANY EQUILIBRIUM
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
Abstract. Present economic
theories
make
a
common-knowledge assrmip-
tion that implies that the first or the second-order beUefs determine all liigher-
order beUefs.
We
analyze the role of such closing assumptions at finite orders
by instead allowing higher orders to vary
of underlying uncertainty
is
arbitrarily.
sufficiently rich,
Assuming that the space
we show
that the resulting set
of possible outcomes, under an arbitrary fixed equihbrium,
outcomes that
best reply. For
siu-vive iterated elimination of strategies that are
many games,
this
solvable, every equihbriimi will
imphes that, unless the game
equihbrium
is
never a
is
be highly sensitive to higher-order
thus economic theories based on such equihbria
over, every
must include
may be
all
strict
dominancebeliefs,
and
misleading. More-
discontinuous at each type for which two or more
actions survive our elimination process.
Key
words:
liigher-order uncertainty, rationalizability, incomplete infor-
mation, equilibrium.
JEL Numbers:
C72, C73.
Date: First Version: September, 2003; This Version: January, 2004.
We
thank Jeffrey Ely, Stephen Morris, and the seminar participants
and Yale
for helpful
comments.
1
at Berkeley, Stanford,
2
"Game theory
tures to be
...
is
common
deficient to the extent
it
assumes other
knowledge, such as one player's probabihty
assessment about another's preferences or information.
game theory
the progress of
tions in the base of
common
I
foresee
depending on successive reduc-
as
Icnowledge required to conduct use-
ful
analyses of practical problems.
of
common knowledge assumption
reality"
fea-
Only by repeated weakening
will
the theory approximate
Wilson (1987)
Introduction
1.
Most economic
theories are based
closed after specifying the
first
on equilibrium analysis of models that are
and second-order behefs,
i.e.,
the beliefs about
underlying uncertainty and the beliefs about other players' beliefs about underlying uncertainty. These models assume that the specified belief structure
common
all
knowledge,
conditional on the
i.e.,
of the players' higher-order beliefs are
sumptions may easily
fail
modeled, these theories
liefs
first
and the second-order
common
is
beliefs,
knowledge. Since these as-
in the in the actual incomplete-information situation
may be
on equilibrium behavior
is
impact might indeed be large
misleading
large.
in
when the impact
of higher-order be-
There are examples that suggest that
some
this
situations (see Rubinstein (1989), Fein-
berg and Skrzypacz (2002), and also Milgrom and Weber (1985)). To overcome
this
fundamental deficiency, one
specifying
more orders
assumption.
more orders
that this
is
of beliefs,
As Wilson
We
model
at higher orders,
hope that by specifying more and
reality."
We demonstrate
how many
orders of beliefs
would "approximate
show that regardless
are specified, the closing assumption
beyond that of
to close the
and hence weakening the common knowledge
(1987), one might
of beliefs, the theory
not the case.
may want
is
of
required to gain any predictive power
iterated elimination of strategies that are never strict best reply.
FINITE-ORDER IMPLICATIONS
3
Consider a situation where players have incomplete information about some
payoff-relevant parameter.
Each player has a probability
distribution about
the parameter, which represents his first-order beliefs, a probability distribution about other players' first-order beliefs, wloich represents his second-order
and so
beliefs,
this
on.
Imagine a researcher who has computed an equilibrium of
game, where a type of a player
would
like to
make a
U
set of all actions that are
out
if
an
infinite
hierarchy of his
prediction about the action of a player
equilibrium. Fix a type
beliefs agree
is
of player
i
as his actual type,
is
whose
(t,)
for
the
first-order
the set of actions that the researcher cannot rule
he only knows the first-order beliefs and assumes that the player plays
according to the equilibrium. Similarly, write A^
(t^)
the researcher cannot rule out by looking at the
first
A°°
i
and
according to this
and write A]
played by some alternative type of
with U. This set
i
beliefs,'
{ti)
for the
many)
the set of actions that
k orders of
hmit of these (decreasing) sets as k approaches
set of all actions that
bitrarily
for
finite
beliefs.
Write
infinity, i.e.,
cannot be ruled out by the researcher by looking at
the
(ar-
orders of beliefs. This definition can be put another way.
Consider two researchers
who
agree on the equilibrium played.
of type
The
is
certain that player
is
willing to agree with this assessment for the
i
is
not have any further assumption.
tj.
The
One
researcher
other (slightly suspicious) researcher
set A'i
first
k orders of beliefs but does
{ti) is
precisely the set of actions
that will not be ruled out by the second researcher.
In a model that
by the
is
first
closed.
is
closed at order
A-,
all
higher-order beUefs are determined
k orders of beliefs and the assumption that
made when
the model
We wish to emphasize the sensitivity of the model's predictions to the
closing assumption. In a given equilibrium, the
for
is
each possible set of beliefs at orders
'For technical reasons,
we assume
1
model predicts a unique action
through
k,
namely the equilibrium
that beliefs at each finite order have coimtable support.
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
4
action for the complete type implied
by
and the
this set of beliefs
sumption. But in the general model, every other action in Af
a type whose
first
k orders of beliefs
Therefore,
the closing assiimption.)
will
be exactly as
we cannot
closing as-
(t,) is
played by
this type (but will fail
any action in A^
rule out
[ti]
without resorting to the closing assumption.
Our main
result gives a lower
of underlying uncertainty
range,
every action
i.e.,
we show that
eliminating
lar,
^°°
(ij)
includes
strict
(ij)
We
(t,).
all
is
On
the other hand,
A^
To
in
When
explain
why A]
process. Let
ii
(tj)
We
ties,
therefore Af°
(i,)
extend this charac-
where the action spaces are one-dimensional compact
classical
main argument
includes
all
and
in the
proof of the lower bound, we
now
actions that survive the
first
vary over the set of types that agree with
concerning the other players' type
implies that there are types
equilibrium action profile.
i^
own
action
concave in
economic models.
concerning the underlying parameter) but
(i.e.,
In particu-
there are no
same outcome, and
utility functions are strictly
many
illustrate the
tj.
a subset of actions
is
[ti)
a subset of rationalizable actions.
and the the
—as
k iterations of
k iterations of eliminating strictly dominated actions, and
terization to nice games,
continuous
first
actions that survive iterated elimination of actions that
best reply.
first
full
played by some type. For countable-action games,
precisely equal to the set of rationalizable outcomes.
intervals
assume that the space
enough so that our fixed equilibrium has
these elimination procedures lead to the
is
Af
actions which are never a strict best reply under
that survive the
hence A°°
is
rich
for
includes aU actions which survive the
A'^ {U)
all
cannot be a
is
bound
with any
may have any
profile.)
beliefs
Given any action
to his fixed belief about the parameter
Our
round of elimination
ti
at first order
(i.e.,
beliefs at higher orders
full
range assumption
whatsoever about other players'
a^ of
and some
i
that
belief
is
a strict best reply
about the other players'
FINITE-ORDER IMPLICATIONS
actions, there
is
a type U
who has these behefs
play the strict best reply,
as part of
a^,
in equihbrium,
This argument
in equilibrium.
an inductive proof of the main
5
and therefore must
will
be formalized
result.
For general games, Nash equilibrium has weak epistemic foundations (Au-
mann and Brandenburger
(1995)) in comparison to iterative admissibihty (Bran-
denburger and Keisler (2002)) and rationahzability (Bernheim (1985), Pearce
(1985)). Yet, in application, researchers frequently use equilibrimxi analysis
further focus
on a particular equilibrium, invoking refinement arguments and
such, so that they can
power
—beyond that
make
for
Our
predictions.
result
of om' elimination process
from the assumption that
is,
and
is
(implicitly)
shows that the predictive
—obtained
made when
in this
the model
is
way comes
closed.
That
any such prediction, there are types that are ruled out by the closing
assumption and that behave inconsistently with the prediction in the focused
equilibrimn. Therefore, our result suggests that the closing assumption deserves
a close scrutiny, and needs to be justified at least as
sumptions of the model.
It
much
as the explicit as-
would be highly desirable to investigate whether the
types that behave inconsistently with the prediction of the closed model can be
excluded by a weaker set of assumptions.
From an evolutionary
nals, if
point of view,
when
there are privately observed sig-
a myopic adjustment process converges,
the incomplete-information
game with these
its limit is
a Nash equilibrium of
signals as private information.
result suggests
then that the limit behavior of such a process
elusive signals
about the distribution of other players'
nals about these signals,
sophisticated
and so
may depend on
Our
the
signals, higher-order sig-
on. Incidentally, our result has counterparts in
and Bayesian learning models: the learning of sophisticated agents
leads to equilibrium
if
and only
if
the
game
is
dominance solvable (Milgrom and
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
6
Roberts (1991)), and in a specific model, any sequence of rationalizable action
can be a sample path in Bayesian learning (Nyarko (1996)).
profiles
Our
result also points to a close link
between higher-order reasoning and
the equilibrium impact of higher-order uncertainty.
When
there axe no ties,
assrmiing /cth-order mutual knowledge of payoffs and that a fixed equilibrium
(with
full
range)
of rationality
played
is
is
equivalent to assuming /cth-order mutual knowledge
and common knowledge of
payoffs." This implies that
equihbrium impact of high-order uncertainty
failvires
of rationality
is
is
large,
when the
the impact of high-order
also large. In that case, predictions
may be
unreliable
without a very accurate knowledge of players' reasoning capacity.
It is well
known
that
some Nash
equilibria
may be
discontinuous in product
topology and with respect to higher-order uncertainty, as in the electronic-mail
game of Rubinstein
discontinuity
is.
(1989).
There
is
an
interest in rmderstanding
how severe this
Monderer and Samet (1989,1997) and Kajii and Morris (1998)
have analyzed the weakest topologies that make the equilibrium continuous
over
all
games
(see also
These topologies are
Milgrom and Weber (1985)
cjuite strong,
we formally
not clear whether the equilibria used
is
establish) that,
if
the space of underlying uncertainty
sufficiently rich, every equilibrium is discontinuous (with respect to
topology and higher-order
or
it
appUcations will be highly sensitive to higher-order uncertainty. Our result
implies (and
is
a continuity result.)
but since they focus on the worst case games,
such as the electronic-mail game,
in
for
beliefs) for
every
more actions survive our elimination
The
relationsliip
straightforward; A'^
game
at every type for
product
which two
process.
between assumptions about rationality and payofF uncertainty
may
differ
from both rationaUzabihty and
iterative admissibility.
is
not
FINITE-ORDER IMPLICATIONS
As a precedent
to our main result, Brandenburger and Dekel (1987) show that
every rationaUzable outcome
rium
on
is
the outcome of a subjective correlated equihb-
(see Section 3 for a discussion.) Battigalli
this result to
7
and
Siniscalchi (2003)
djmamic games and investigated the imphcations of the
first-order beliefs
and common strong
extended
restrictions
belief in sequential rationality
rationahzable outcomes, which coincide with
all
equilibrium outcomes.
It
on the
seems
that, using their methodology, one can obtain sharp predictions in sequential
games using
lower
bound
relatively mild assumptions.
Note that
games, our
in sequential
usually weak, and assumptions about sequential rationality yield
is
strong predictions (Battigalh and Siniscalchi (2002) and Feinberg (2002)).
Our next
tion 3
is
section contains the basic definitions and prehminary results. Sec-
we develop our main notions and prove
the heart of the paper. There
our main theorem. Our main theorem
terization in Section 4,
is
and to mixed strategies and to the spaces of uncertainty
that are not necessarily rich in Section
and discuss
tinuity results
extended to the nice games as a charac-
5.
In Section
6,
we
present our discon-
their methodological implications for global
and robustness of equihbria. Section 7 contains a very negative
result
games
about
Com'not oligopoly as an application. Section 8 concludes. Some of the proofs
are relegated to the appendix.
Basic Definitions and Preliminary Results
2.
Notation
y-i
=
Given any
1.
list
Yi,
.
.
.
,
y„ of
(yi,...,yi-i,yi+i,...,2/„) e YL,,
and (yj,y_0
Likewise, for any family of functions fj
by
/_, (y_,)
=
(/j {yj))j^,-
:
Y=
sets, write
=
Y] -^ Zj,
{yi,
we
fJi ^i> ^-z
.
.
.
=
Yljjti ^j^
,yi^l,y^,y^+l,
define /_i
:
Y^i
.
—*
Given any metric space {Y,d), we write A{Y)
.
.
,yn)-
Z^i
for
the space of probability distributions on Y, suppressing the fixed cr-algebra on
f
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
8
Y
which
consider a
game with
of underlying uncertainty
and
sets
The support
(j-algebra in product spaces.
We
open
at least contains all
of
we use the product
singletons;
denoted by suppTr.
it is
finite set of players
A''
=
{1, 2,
.
Q
The source
n}.
,
.
a payoff-relevant parameter 6 E
is
.
where (0,
d) is a
compact, complete and separable metric space, with d a metric on set 0. Each
player
i
has action space Ai and utility function w,
Dekel (1993), a variant of an
earlier construction
with an additional assumption that the players'
comitable (or
finite)
support.
^
A =
E, where
>
where
0,
A (-^fc-i)
that have coimtable (or
by Mertens and Zamir (1985),
each
beliefs at
finite)
is
on
A'o,
X Xfc_i
the set of probability distributions on Xk-i
We endow
support.
each
A
Xk with
player
second order
beliefs
(about
all
the weak
z's first
order
by a probabihty
beliefs (about the underlying uncertainty 6) are represented
t]
order have
finite
= A(X,_0
Xq = Q and Xk
topology and the cr-algebra generated by this topology.
distribution
and
Types are defined using the auxiliary sequence
"^
of sets defined inductively hy
each k
for
^
We endow the game with the universal type space of Brandenburger
Yli Ai.
{Xk}
x
:
players' first order beliefs
and the underlying uncertainty) are represented by a probability distribution
onXi,
etc.
Therefore, a type
a player's /cth-order
of a player
t^
(
YikLi
^ (Xk-i)
are coherent,
i.e.,
in
)
i
and
t,i €
This assumption
1.)
Our type space
redundant type
is
which
the players
different orders agree.
any player
is
is
is
a
beliefs contain information
need the usual coherence requirements.
of
i
We
T
made
it is
We
member
their
write
own
f^^j
A (Xk-i)-
about his lower-order
T =
common knowledge
know
of
beliefs
will use the variables t,,tj
YiieN
-^*
^^^
f
Since
beliefs,
we
^^^ subset
that the players' beliefs
and
G
their marginals
from
Ti as generic types of
as generic type profiles. For every
U e
Ti,
there exists
to avoid technical issues related to measurability (see
Remark
dense in universal type space, and any countable type space with no
embedded
in our space.
FINITE-ORDER IMPLICATIONS
a probability distribution
(2.1)
and
1
tf
t}
=
on the
=
margQ/vt^, where S^k-l
set {tf~^}
margg^r^jY
is
_
)]'^\{>}«^ti
strategy of a player
ti
and any
x T_i such that
(^^')
the probabihty measure that puts probabihty
is
and marg denotes the marginal
Kt^
type
on
5j.-i xmarg0^[^(_Y,_2)]'^\«'^«.'
given any distribution
A
Kt^
9
x T_j, we can define
on
t^
G
T, via (2.1), as long as
always coimtable.
i is
any measirrable function
profile s_i of strategies,
we
st
T^ —^ At.
:
write 7r{-\ti,s_i) G
Given any
A (0
x A^i)
the joint distribution of the underlying uncertainty and the other players'
for
actions induced
mixed strategy
we
write
BRi
by
ti
and
profile cr_i.
(tt)
s_i;
vr
is
(-ItijCr^,)
For each
i
E
similarly defined for correlated
N and for each belief
for the set of actions Oj
=
{sl,S2,
.)
is
vr
e
A (0
x A^i),
G Ai that maximize the expected
value of Ui {9,ai,a-i) under the probabihty distribution
s*
Conversely,
distribution.
a Bayesian Nash equilibrium
iff
A
tt.
at each
strategy profile
ti,
s*{t,)eBR,{-K{-\t,,s*_,)).
An
equilibrium s*
is
said to have full range
iff
(FR)
s*{T)^A.
The
following assumption implies that every equilibrimxi s* has full range.
Assumption
ai,
1 (Richness of 0). Given any
there exists a probability distribution u on
i
E
N
,
any
/x
G
A [A^i),
and any
with countable support and such
that
BR,
Lemma
Proof.
1.
The
Under Assumption
1,
[u
x
i^l)
= {aj
.
every equilibrium s* has full range.
proofs that are omitted in the text are in the appendix.
D
JONATHAN WEINSTEIN AND MUHAMET
10
Elimination Processes.
only
witMn
and
We will use interim notions and allow correlations not
between their strategies and the under-
players' strategies but also
lying uncertainty
0.
Such correlated
Siniscalchi (2003)
YILDIZ
rationalizability
is
introduced by Battigalli
and Dekel, Fudenberg, and Morris
(2003).
Clearly, al-
lowing such correlation only makes our sets larger. Since our main result
lower
bound
in
terms of these
sets, this
a
is
only strengthens our result. Moreover,
our characterization provides yet another justification for this correlated rationalizability.
Write Mi
Towards defining
for the set of all
rationahzability, define sets S^
iteratively as follows. Set
set of all
for
each
5°
measurable functions /
t_i.
=
[U]
:
Ai. For
6 x T_i
Let also S'i"^ be the set of
Note that E^7^
i^
the set of
all
for the set of all all actions a^ of
in E^"-^.
The
>
& N, U E
-^ A^^ such that /
all
Ti,
0, let 5*^7^
(6*,
T^ to A^.
=
/.-;
0, 1,
C M_j
t_,)
G
.
.
.,
be the
S*^"^ [t_j]
probability distributions on 5^7^-
possible beliefs of player
U
i
[ti\, i
each k
allowable actions that are not eliminated in the
S\{t^=
x
measurable functions from
first
k
—
\
i
on other
players'
rounds. Write
BR,{'K{\U,a^,))
that are best reply against
set of all rationalizable actions for player
i
some
of his beliefs
(with type U)
is
CX3
sr
[u]
=
n ^'
[^^]
•
fc=0
Next we define the
set of strategies that survive iterative elimination of strate-
gies that are never strict best reply,
Wf [U] =
Wl"
denoted by W°°
[U]
,
similarly.
We
A, and
[U]
=
[a,\BR, (7r(-|i„a_,))
=
{a,} for
some a_, e
A
(h/-^7')}
>
set
FINITE-ORDER IMPLICATIONS
where W^^^
C M_i
f{6,t^i) 6 W^''^
[t_j]
the set of
is
each
for
all
functions
t-^. Finally,
we
/
11
x TL^ -^ A-i such that
:
set
oo
fc=0
Notice that we eliminate a strategy
on the remaining
belief
not a strict best-response to any
is
if it
smaller set than the result of iterative admissibility
of weakly dominated strategies)."*
yield strong predictions.
In
some games,
For example, in
leads to backwards induction outcomes.
games
3.
tions
iterative elimination
(i.e.,
iterative admissibility
finite perfect
may
information games
and
weak predictive power.
Equilibrium predictions with finite-order beliefs
are interested in
made
sumption
Let us
it
Nevertheless, in generic normal-form
(as defined in Definition 1 below), all these concepts are equivalent
usually have
We
Clearly, this yields a
strategies of the other players.
how
is
against the failure of assump-
common knowledge
at high orders, such as the failure of the
at high orders.
fix
robust equihbrium
We now
an equilibrium
s*
formalize our notion of robustness.
and a type
of a player
t^
librium, he will play s* (U).
Now
fcth-order beliefs of player
and knows that equilibrium
i
as-
imagine a researcher
searcher can conclude from this information
is
that
i
i.
who
According to equionly knows the
5* is played. All
will play
first
the re-
one of the actions
in
In particular,
gies will
gies.
if
we use non-reduced normal-form
of an extensive-form game,
be outcome equivalent, in which case our procedure
To avoid such
over-elimination,
strate-
will eliminate all of these strate-
we can use reduced-form, by representing
equivalent strategies by only one strategy.
many
all
outcome-
JONATHAN WEINSTEIN AND MUHAMET
12
Assuming, plausibly, that a researcher can verify only
know
player's beliefs, all a researcher can ever
is
YILDIZ
finitely
that player
many
orders of a
one of
will play
i
the actions in
Ar[s\U] = f]Al[s\U].
k=0
We
now ready
are
to prove our
games where each player
Proposition
1.
range, any k
E N,
main
has a countable or
i
games,
result for countable-action
finite action
space A^.
For any countable- action game, any equilibrium
£ N, and any
i
i.e.,
with full
s*
ti,
wnu]^Ai\s\t,]csnu]:
in particular.
wr[t^]^AT[s\t,\csm
Proof.
The
Appendix
Af
A\
inclusion
For
=
fc
C
games and
for general
[s*,ti].
[s*,tj]
0,
all
.
.
,
.
t^""^) ^'^d
respectively. Let
W':-'
[I]
L=
^
=
h
(t^j,
{/|3/i
U Wt-'
:
(/,
established by Proposition 8 in the
equilibria.
is
We will now show that W^
.
.
.)
are the lower
=
(/,
where
h)
and higher-order
€ TL,}. The induction hypothesis
[{I h!)]
C
G Wj^
[ti].
\ti\
C
given by the full-range assumption.
write each t_i as t_t
i,
t^|\
h)
is
[ti]
the statement
For any given k and any player
(tij, t?_j,
Sf
At-' [s\ [I
h)]
(V [l h)
e
is
/
=
beliefs,
that
T_,).
h'
Fix any type
s* (ti)
Qi
e
=
ai
ti
and any
and the
A'i [s*,ti].
first
a^
We
will construct
k orders of beliefs are same under
Now, by
best reply for type
ti if ti
ers' strategies, i.e.,
BRi
definition, for
a^^))
=
and
some a^i G A{W'll'),
{a,}. Let
ti,
ii
such that
showing that
a, is
the unique
(T_j
on the other play-
P {-[ti, a_^)
be the probability
assigns probability distribution
(tt (-l^i,
ti
a type
FINITE-ORDER IMPLICATIONS
distribution
x L x A^t induced by
on
a_0 G suppP
esis, for
each
some
Hence, there exists a mapping n
h.
(^,
/,
and
Kt^
suppP
:
By
G W^'^
a_,), a_,
(-It,,
a^t.
13
{-{ti,
the induction hypoth-
[I]
C
At',^ [s\ (IJi)] for
a^i) -^
Q
x T^i,
/i:(^,/,a_,)^ (^,/,Ma_„^,/)),
(3.1)
such that
sU(l,h{a^i,e,l)^=a_i.
(3.2)
We
U by
define
i^U
=
^(•|^t,o--j) O/i
x T_j by the mapping
the probabihty distribution induced on
probabihty distribution
P (-It,, o^i).
Notice that, since
and the action spaces are countable, the
which case
of
/.i,
the
first
marge^^/vj-^
k orders of beliefs (about
= msigQ^^P {-{U^a^i)
where the second inequality
P
(-[ti, cr_t).
is
by
o
(3.1)
P (-1^,, s\) =
of
7 to
TT
That
of
ti
is,
Ki^
last equality
suppK;j-^
{\ii,
s%)
.
is
o 7-1
due to the
t'l
and the
ji
has countable support
comitable, in
(•|tj,(T_j) is
By
well-defined.
construction
U and
are identical mider
/))
U:
= margQ^^P {\U,a_^) =
margg^L/^ti,
by
definition of
and the
ki^
last equality is
{6,
:
and
= P (.|t„ a_,)
o
fact that
/,
s*_^
h)
\-^
on
^r^ o 7-1
/i is
[6,
/,
s*_^ (/, /?))
x L x A_i
,
we can
is
= P {.\U, a..,)
,
the inverse of the restriction
Therefore,
= margexL^
{'lU,
s%)
the equilibrium beliefs of
about
Kj^ is
Moreover, using the mapping 7
check that the distribution induced by
where the
(6',
n'^
suppP
set
measurable. Hence
trivially
/i is
\
= marge^^P (-1^^, c7_,) =
U about
it
{-lU,
a-i)
x A^t are identical to the
x A_i when U assigns probability distribution
(T_j
beliefs
on the other
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
14
players' strategies. Since o^
ai in
the only best reply to these
is
beliefs,
ii
must play
equilibrium:
s* {U)
(3.3)
e BR, {n
{]{,,
s^))
= BR,
{n {-lU, a_,))
= (aj
.
D
Remark
beliefs
1.
Notice that the countability assumptions about the finite-order
and the action spaces are used only to make sure that kj
defined probability distribution, or
measurable, our proof
games
in
which
/,i
is
is
valid. In
measurable;
The conclusion that VV^
that
we know a
information.
[tt]
Suppose
is
measurable.
the next section,
/i
C A^
player's beliefs
/i
may
also that he has
In fact, whenever
first
that
aj will
be played in equilibrium
that
is
closed at order k and
if
fcth
order and do not have any further
an action
a,
that survives k rounds of
Put
it
—
we cannot
differently, if
for
some
rule out
we have a model
an action survives k rounds of iterated elimination
of strategies that cannot be a best reply for a type within the model, then
cannot rule out action
a^ as
the closing assumption.
first
is
can be spelled out as follows. Suppose
k orders of behefs are as specified. Then,
s*.
jj,
present another class of
iterated elimination of strategies that cannot be a strict best reply
type whose
a well-
not be measurable in general.
[s* ,ti\
up to the
we
is
an equilibrium action
for that
we
type without invoking
Hence, any prediction that does not follow from the
k steps of this ehmination process comes from the closing assumptions,
rather than the assumptions that lead to a specific equilibriimi
s*.
This suggests
that, contrary to the current practice in economics, a researcher needs to justify
his closing assimiption at least as
rationale for equilibrium selection.
much
as the other assumptions, such as the
FINITE-ORDER IMPLICATIONS
The
part
Af
[s*,ti]
for all equilibria (see
somewhat
involved,
Notation
[ti]
proved in the Appendix for general games and
is
Proposition
it is
it is
common knowledge
C
[s*,tf^ [9)]
edge of
6'
=
Si
=
Sj
for
9
=
i
who has
has k
the proof
(9),
that,
if
{9].
each
strictly
—
is
A;th-order
is
That
is,
if
a player
a player j has k
knowledge of ^
Ist-order knowledge of 9
j plays an action in S^~^.
=
9.
=
^
9 =
is
i
has fcth-order knowls*) will survive
—
{9}.)
kth
restric-
[s*,tf^'^ [9)]
C
Ist-order knowledge of
But any type of player
certain that every other player j
Hence, in equilibrium, he
Since his equilibrium action
such a belief (with support contained in
is
Notice that
as follows.
Towards an induction, assume that Aj"^
j.
who
i
dominated strategies under the
j will play an action in Sj~^ (under
9,
the general case
9.
then his equilibrium action (according to
^,
tion that
(9)]
=
that 9
means
[tf^^ [9]]
round of iterated elimination of
[tf''^
for
straightforward for the complete-information case.
For the types of the form tf^
A'^
Although the proof
8).
For any 9 E Q, write tP^ (O) for the type of a player
2.
certain that
C S^
15
S'iJ^),
it
is
is
certain that
a best response to
must survive the
A;th
round
of elimination.
Brandenburger and Dekel (1987) show that, given any rationalizable outcome
of a game,
by adding payoff
irrelevant types,
we can construct a type space
with an equilibrium that yields the original rationalizable outcome.'^ Since we
need to choose a different equilibrium
is
for
each rationahzable outcome, there
a large multiplicity of equilibria, and thus equilibrium as a solution concept
does not have more predictive power than rationalizability has.
One may want
to ignore this multiplicity by focusing on a particular equilibrium or relying
on one
""See
of the
many
refinements developed in the last few decades
Bergemann and Morris (2003)
for
an important application of
tliis
adea.
—
in order
JONATHAN WEINSTEIN AND MUHAMET
16
to cope with the usual multiphcity problem.
abandon
artificial restrictions
on type space,
Our
YILDIZ
result
shows that, once we
this unpredictability reappears as
high variability of any fixed equilibrium with respect to very high-order beliefs,
albeit with
There
is
somewhat smaller scope due
an intuitive relation between these two
general can be
embedded
beliefs at finite orders
we can
to the stronger elimination process.
results.
Since type spaces in
in the universal type space, given
and a
any fixed
set of
fixed equilibrium behavior in a fixed type space,
envision a coherent type whose lower-order beliefs are as the former
but whose higher-order beliefs assigns high probability to the
will give a best reply to
the equilibrium in the latter.
latter.
That type
Hence, multiplicity of
equilibria in various type spaces tends to yield high variability of equilibria
with respect to the higher-order
however, does not yield a proof.
beliefs in universal
This
types in Brandenburger and Dekel are
is
all
type space. This intuition,
because the distinctions
among the
payoff irrelevant, and thus their type
spaces are not contained in the universal type space.
More
importantly, the
equilibrium behavior in various type spaces need not be different witliin a fixed
equilibrium. In fact, Brandenburger
proof; hence,
if
and Dekel use a
single type space in their
their type space were contained in the universal type space, the
equilibrium behavior in this type space would be uniquely determined by the
equilibrium in the universal type space.
Om: next example shows that
be
strict.
Hence,
(i)
some
either of the inclusions in Proposition 1
rationalizable strategies
may
may
not be in A°°, showing
the distinction between the results of Brandenburger and Dekel and ours, and
(ii)
A°°
may
include
some weakly dominated
strategies, distinguishing our result
from the characterization of Brandenburger and Keisler (2000).
FINITE-ORDER IMPLICATIONS
Example
Take
1.
=
A''
{1,2},
6=
{^o,^i},
and
let
17
the action spaces and the
payoff functions for each 6 be given by
(Note that 9
aO
a'
a
0,0
0,0
a
0,0
1,1
not payoff relevant.) Define s* by
is
=
s* iU)
otherwise.
a}
Clearly, for each
U, while
>
A:
4^ [s*-tf^
Proposition
we have
1,
{9o)]
=
W^*^ [U]
=
{a^,a'}, and
Af [s*;f ^
yields a characterization
1
{a^} and S^
=
[ti\
(^i)]
=
{cP,a^} for each
{a'}.
whenever the payoffs are generic in
the following (standard) sense.
Definition
non-zero
We
1.
say that the payoffs are generic at 9
a G R^\ and
u, {9, a[, a_j) or
(ii)
distinct a,, a[, a^i,
Y.a,
^ (^0
u^ {9, ai, a_,)
Wlien the payoffs are generic
then any action that
some
belief (at
is
at 9
1.
=
a'_j
Y.a,
it is
there do not exist
such that
"
(i)
u^ {9,a^,a^i)
('^0 ^^ (^' ^'' ^-r)
common knowledge
=
—
0.
that 9
each round), and hence the two elimination processes
1
=
9,
generic at
some
9,
r„* fCK
A^[s\ti
will
be
yields the following characterization.
For any finite-action game and any equilibrium
if the payoffs are
i,
not strictly dominated will be a strict best reply against
equivalent. In that case, Proposition
Corollary
and
and
iff
then for each
\k
5f
[tr
i
and
{0)]
.
k,
s*
with full range,
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
18
That
finite
games, a researcher's predictions based on
in generic finite-action
is,
orders of players' beUefs and equihbrium will be equivalent to the predic-
tions that follow from rationalizabihty. This characterization will
be generalized
to the following widely-used class of games.
4.
We
will
now
Nice games
consider a class of "nice"
games (Mouhn
(1984)), which are
widely used in economic theory, such as imperfect competition, spatial competition, provision of public
Af
[s*,ti]
=
Definition
is
[U] for
2.
A game is said to be nice iff for each
=
(oj,
a_j)
and
strictly
i,
concave in
use the strict concavity assumption to
make
function for any fixed strategy profile of the others
his
We
own
action. (Single-peakedness
is
is
A^
=
full
[0, 1]
will
show that
range.
and
u^ {9, ai,a_i)
a^.
sure that a player's utility
is
always single-peaked in
not preserved in presence of uncertainty.)
use the continuity assumption to
response
We
etc.
each k whenever equihbrium s* has
S^
continuous in a
We
goods, theory of the firm,
make
siire
that a player's strategy best
continuous with respect to the other players' strategies.
complete information types, our results in this section
weaker condition that Uj(^,-,a_j)
is
will
single-peaked with a
For the
be true under the
maximand
that
is
continuous in a_,. Now, since our players have always unique best reply, our
ehmination processes
(4.1)
will
be equivalent, yielding the functional equation
W=
S.
.
FINITE-ORDER IMPLICATIONS
Moreover, our next
we only need
lemma
19
ensm'es that, despite our uncountable action spaces,
many
to consider coimtably
actions for types with countable sup-
ports, allowing us to circumvent the measurability issue discussed in
Remark
1.
Lemma
S-t
2.
For any nice game, for any
i,
U, k, and any a, G S^
there exists
[ti\,
E 5^7^ such that
=
BR,{TT{-\t,,s_,))
Together with
Proposition
Let also
suppni
Q
C
x
Lemma
(4.1),
2 gives us our
2.
For any nice game,
T
be a countable subset of
X
T-i.
eN,
Then, for any k
main
be
let s*
Q
{a,}.
x
result for this section.
any equilibrium with
T
such that for each
E N, and
i
full range.
ti
G
ti
€ %,
Ti,
in particular,
sr
Proof For any
W_~^ such
that
a^
G 5f
o^ is
[u]
=
= Ar
[u]
Wj"
[u]
,
by
[s\u]
Lemma
a strict best reply against
2,
tt {-{ti,
there exists s_i G S*^"^
s_j).
Since «£ has comit-
able support, P(-|ti,s_i), the probability distribution induced by
on
G
=
k^^
and
s-i
X L X A^i, has a countable support:
suppP
(-Iti,
5_,)
=
{{6,
Hence our proof of Proposition
1
necessarily in Tj) such that 5* [u]
1,
s_^ [6,
1,
applies.
=
Qi
and
h))
I
[6,
That
t-"
=
1,
is,
h) G suppACj,}
there exists
tf* for
each
m<
.
U G
k.
Ti
(not
D
jonathan weinstein and muhamet yildiz
20
Extensions
5.
For ease of exposition, we have so
with fuU range. In this section, we
Mixed
gies,
we
A
we
Strategies. Since
will focus
mixed strategy
(tt
extend
profile a*
each
[\ti,a*_^)) for
the set of types
by supper*
(tj)
all
games are
equilibria in nice
who
i
is
any measurable function
and
ti.
Writing T^'
play pure strategies,
= {sf
{t^)}.
We
the set of
ii
{T"' )
iff s'^*
whose
all
we
=
define a
{ti\
3.
full
supper*
iff
mapping
-^ A(y4j).
Ti
:
Isuppcr*
We
sf
:
C
=
1} for
Tf
—^ Ai
(ij)|
*
(tt)
say that a* has
full
set
1
under a* by some type
k orders of beliefs are identical to those of
equihbrium has
pm^e strate-
in
then use this "pure part of" a* to extend our
actions that are played with probability
first
Proposition
= A and
ai
Bayesian Nash equilibrium
previous definitions and results to mixed strategies.
range
mixed strategy
oiir results for
on the comitable-action games. Using interim formulation,
define a mixed strategy as
BRi
will
on pure strategy equilibria
and beyond the full-range assumption.
equilibria
5.1.
far focused
range under Assumption
1, i.e.,
when
For any countable- action game, any
equilibrium a* with full range, any k
<
co,
i
Clearly, every
t^.
sufficiently rich.
is
(possibly
G N, and any
mixed strategy)
ti,
Wt[t.]CAl[a\t,]CSf[t,].
Proof. In the proof of Proposition
restrict the
uerr'.
range of
/i
1,
insert
s'^'
and the domain of 7 to
everywhere
s*
appears, and
x Tf*. Notice that, by
(3.3),
FINITE-ORDER IMPLICATIONS
That
first
is, if
a* has
full
range
is
(e.g., if
k orders of a player's behefs, then
21
and we know only the
sufficiently rich)
for
any
a^
W^ [U],
G
we cannot
rule out
A;
>
1,
assumption can replaced by the weaker assumption that A^
C
U;^ supper* (tj).
that
5.2.
is
ttj
played with probability
Without
large changes.
full
range. Our
according to a*
1
For
.
the full-range
range assumption allowed us to consider
full
A researcher may be certain that it is common knowledge that the
set of parameters are restricted to a small subset, or equivalently, the equilibrium
considered
change.
may
We
much
not vary
will
now
as the beliefs about the underlying uncertainty
present extension of our
Local Rationalizability. For any
i
e
N,k e^,tieT^,
S'^~^ [B]
C M_j
is
=
U
the set of
such that f {9,t^i) G 5^"^ [B]t_i\
procedure as iterated
•
•
result to such cases.
x 5„
C
A, define sets
strict
BR,{n{-\U,a.,)),
measurable functions /
all
for
each
i.,.
dominance, except that the
k increases. Hence we define the set of
initial set is restricted
and W°"
[B]
t,]
{a, e A,\BR, [e\a_,)
= fj^o Um=fc ^^r
our process, which
is
become
larger as
by
oo
m=k
may be much
version of W°°, similarly, by setting
the same
{^[jsr[B-Mk=0
Notice that the set S'^^lB^ti]
is
locally rationalizable strategies
oo
sr[B-x] =
x T_j -^ A^i
:
Notice that this
to a subset. Unlike iterated strict dominance, these sets can
W^ [B-U] =
S'f [B;ij],
by setting
S'^[B-U]
where
x
_Bi
main
W^
[B; U]
= {aj
larger
=
for
than B.
We
define local
Bi,
some a.i G
[^; U]- Notice that
we
no longer an elimination process.
A
(w^"^ [B])}
consider
all
,
actions in
JONATHAN WEINSTEIN AND MUHAMET
22
Proposition
4.
For any equilibrium
s*, if the
YILDIZ
game has
countable action spaces,
then
Wt [s*
if the
game
(T)
C Al
[s\
=
t,]
statement implies that,
last
C
[s\t,]
S^
[s*
(T)
;
[s*
5f
for nice
(V^,
t,]
then with notation of Proposition
is nice,
Sf [B; U]
The
C A^
t,]
;
{T) ij
2,
t,)
games, even the
U)
;
BC
for any
(Vz, k,
;
A:,
s* (T),
.
slight
changes in
very higher-order behefs will have substantial impact on equilibrium behavior,
unless the
which a
game
is
locally dominance-solvable.
slight failure of
common knowledge
to substantially different outcomes
6.
It is
well
known
—as
assiunption in very high orders leads
in Section 7.
Continuity of Equilibrium
may be
that equilibrium
product topology. In this section we
will
6.1.
A.
this
violated in every equilibrium on a very large
set.
sequence (a'")^gf^
is
there exists k such that d
we
will
We
Equilibrium in pure strategies.
A
write
write
A'
D {B) =
[s\t]
Definition
much weaker than
show that even
respect to the product topology.
is
We
discontinuous with respect to the
introduce our notion of continuity with
respect to higher-order beliefs, which appears
property
There are important games in
3.
=
An
uct topology) at
sup{d{a,b)
Yl^
\a,b
^f [s\U], 5^
equilibrium
t iff
<
for
t''
some a €
m>
^
iff
for
>
0,
BC
A,
each
Given any subset
e
for
each
G B}
for
the diameter of B. As usual, we
fl,
S^
e
[s\t]
=
s* is said to
Vfc]
^
[s*
k.
[t,],
etc.
be continuous (with respect
each sequence {i[m\)^
P [m] -^
weaker continuity
consider an arbitrary metric d on
said to converge to
(a"^, a)
continuity with
of type profiles
{i[m]) -^ s*
{t)]
.
to prod-
FINITE-ORDER IMPLICATIONS
An
equilibrium 5*
t iff
each
for
e
>
said to be continuous with respect to higher-order beliefs at
is
there exists k such that for each
0,
[r =
The
23
latter continuity
r
ym <k]^d (s*
concept
is
(t)
,
i,
s* {t))
<
e.
uniform continuity with respect to the prod-
uct topology (on the type space) of discrete topologies on each order of beUefs.
Of
course, continuity with respect to discrete topology
other topologies.
Lemma
The next lemma
For any equilibrium
3.
(1) If s* is
continuous at
beliefs at
(2) s* is
t,
presents
s*
then
some
and for any
s* is
t,
is
much weaker than
basic facts.
the following are true.
continuous with respect to higher-order
t.
continuous with respect
to
higher-order beliefs at
t iff
D
i^A'^ [s* ,t\)
-^
as k ^' CO.
Ifs*
(3)
{s*
is
continuous with respect to higher-order
with
5°°
[t]
Lemma
range
full
=
is
3.3
and Proposition
2 imply that
{s* (t)}, yielding the following discontinuity result.
<
if
an equilibrium
t,
then
(One can obtain
[t]
=
{s* (t)} with
I.)
Proposition
5.
For any nice game, every equilibrium
continuous with respect to the higher-order
each type profile
files.
=
then
continuous with respect to higher-order beliefs at
a similar result for count able- action games by replacing 5°°
|M/°°[f]|
A^ [s* ,t]
t,
m.
For nice games,
s*
beliefs at
t
beliefs
s* with full
range
is dis-
(and the product topology) at
for which there are more than one rationalizable action pro-
In particular,
if
a nice
game
possesses an equilibrium, s* that
is
continuous
with respect to higher-order beliefs or with respect to the product topology, then
the
game
is
dominance
solvable.
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
24
6.2.
Mixed-strategy equilibria in finite-action games. Endow the space
mixed action
of
profiles,
A (A),
with an arbitrary metric
tions of continuity with respect to product topology
mixed-strategy equiUbria a* by replacing
Definition
A
4.
s*
Now, continuity
in
t iff
Extend the
and higher-order
with a* in Definition
mixed-strategy equilibrium a*
with respect to higher- order beliefs at
d.
is
defini-
beliefs to
3.
said to be weakly continuous
there exists k such that for each
i,
product topology implies continuity with respect to higher-
order beliefs. For finite-action games, the latter in turn implies weak continuity
with respect to higher-order
There we
6).
we show
beliefs, as
in the
Appendix
show that strong and weak continuity of a*
also
<
to Wgher-order beliefs imply that \A°° [a\t]\
1
at
t
(see
Lemma
with respect
and A°° [a\t] C supp(o-*
(t)),
respectively. Tliis yields the following result.
Proposition
For any finite-action game and any equilibrium a* with
6.
range, \W°°
[t]\
<
beliefs at
or
(ii)
at t
t,
and a*
{t)
is
I
whenever
a*
is
(i)
That
6).
is,
is
continuous with respect to higher-order
weakly continuous with respect to higher-order
beliefs
pure.
Proof Either of the conditions
Lemma
a*
full
(i)
Hence, by Proposition
for sufficiently rich
and
3,
(ii)
{W^
implies that
[t]\
<
|yl°° [a*, t]|
|^°° [a*,t]\
0, every equilibrium
is
<
I.
<
1
(see
D
discontinuous with respect
to higher-order beliefs (and the product topology) at each type profile for which
two or more action
not be a
profiles survive iterated elimination of strategies that can-
strict best reply.
These type
profiles include the generic instances of
FINITE-ORDER IMPLICATIONS
25
complete-information without dominance solvability. At each such type profile,
even the weakest continmty property
Example
fails if
the equilibrium actions are
Consider the coordinated attack
2.
game with
Attack
No Attack
1,1
-2,0
0,-2
0,0
Attack
No Attack
where there are two pure strategy equihbria: the
efficient
Attack) and the risk-dominant equilibrium (No Attack,
each action
is
a strict best reply, no action
process. Therefore,
Lemma
coordinated-attack
game
1
is
piu-e.
payoff matrix
equilibrium (Attack,
No
Since
Attack).
eliminated in our elimination
and Proposition 6 imply that when we embed the
in a rich
type space as a type
must be discontinuous with respect to higher-order
Rubinstein's (1989) electronic-mail
game
profile,
every equilibrium
beliefs at that
type
profile.
presents a type space in which any
equilibrimn that selects the efficient equilibrium in the coordinated-attack
game
must be discontinuous with respect to higher-order uncertainty. In that example
there
is
also a continuous equilibrium,
profile for
each type
equilibrium
is
an
profile.
which
selects the risk-dominant action
Our example shows that the
artifact of the small
type space
utilized,
latter continuous
and
in fact in a rich
type space, no equilibrium could have been robust against higher-order
and thus every equilibrimn theory would have been
sensitive to the
beliefs,
assumptions
about higher-order uncertainty, strengthening Rubinstein's position.
Following Carlsson and
Van Damme
(1993), global
games
literature investi-
gate the equilibria in nearby type profiles that are generated by a model that
is
closed at the
able,
first
order.
At these type
and the resulting equilibrium action
profiles,
the
game
is
dominant
profile converges to the
solv-
risk-dominant
equilibrium as these type profiles approach the coordinated-attack game. In this
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
26
way, they select the risk-dominant equihbrium.
result uncovers a difficulty
every equilibrium must be discontinuous at the limit-
methodology:
in this
Our
ing type profile, and an equihbrium selection argument based on continuity
is
problematic
—as there
is
another path that we could have taken the limit in
which we would have selected the other equilibrium. This
the equilibrium outcome
is
is
despite the fact that
robust against higher-order beliefs in these nearby
type profiles themselves (by Proposition 8 in the Appendix.)
On
a positive note, Kajii
librirmi
That
is
is, if
and Morris (1997) show that the risk-dominant
common
robust to incomplete information imder
the
common
prior puts sufficiently high weight
prior assumption.
on the
original
game,
then the incomplete information game has an equilibrium in which the
dominant equihbrium
is
played with high probability according to the
prior. Similar positive results are
and Ui
equi-
risk-
common
obtained by others, such as Ui (2001), Morris
(2003). This suggests that,
when
there
common
a
is
prior,
it
may put
low probability on the paths that converge to other equilibria.
7.
Application: Cournot Oligopoly
In a linear Cournot duopoly, the
sition 8 (in the
game
is
dominant-solvable, and hence Propo-
Appendix) implies that higher-order
beliefs
have negligible im-
pact on equilibrium. (This has also been shown by Weinstein and Yildiz (2003)
and
is
also implied
by a result of Nyarko (1996).)
Cournot duopoly with three or more
firms,
than or equal to the monopoly production
is
On
any production
rationalizable,
sition 2 implies that a researcher cannot rule out
firm no matter
more disturbing
how many
fact.
the other hand, in a linear
We
Focusing complete- information types,
when
is
less
and hence Propo-
any such output
orders of beliefs he specifies.
general oligopoly models we will show that
level that
will
level for a
now show
a
t*^^ {9), for fairly
there are sufficiently
many
FINITE-ORDER IMPLICATIONS
any such outcome
firms,
s*
will
be in
S^
[B;
it^^ {&)) Therefore, by Proposition
t^^"^ (^)]
27
for every
of
any outcome that
is
than monopoly outcome as a firm's equihbrium output.
General Cournot Model. Consider n
cost c
>
P {Q; 6)
we assume that 6
sume
P (O; ^) >
that
limQ^oo
P
firms with identical constant marginal
Simultaneously, each firm
0.
output at price
6,
B
even a slight doubt about the model
4,
in very high orders will lead a researcher to fail to rule out
less
neighborhood
(Q; ^)
=
is
0,
0.
where
produces
i
Q = ^^ q^
(•;
^) is strictly
G
decreasing
when
Q
sell its
For some fixed
We
{^}.
7^
Therefore, there exists a unique
and
at cost q^c
the total supply.
is
a closed interval with
P
qi
also as-
and
positive,
it is is
such that
p{Q-~e)=c.
(In order to
have a nice game, we can impose an upper bound
than Q, without affecting the
equilibria.)
We
assume
that,
on
[0,
for q, larger
Q],
P [,6)
is
continuously twice-differentiable and
P'
It is well
function,
u
known
{q,
Q;6)
that,
=
q
+ QP" <
0.
under the assumptions of the model,
{P
{q
+
Q) —
c), is strictly
the profit
(i)
own output
concave in
the unique best response q* [Q-i) to others' aggregate production Q_i
decreasing on
some A <
[0,(3]
0); (iii)
with slope bounded away from
equilibrium outcome at
t'-^'''
(O), s*
(i.e.,
is
q; (ii)
strictly
dq*/dQ^i < A
(iP^ (^))'
is
for
unique and
symmetric (Okuguchi and Suzumura (1971)).
Lemma 4. In the general Cournot model, for
n< 00 such that for anyn > n and any B = \s\
A with > 0, we have
any equilibrium
(tf^ {d))
-
t
Sr[B;t''''{e)]
=
[0,q'']
i^ieN),
e,
s\
s*
,
there exists
[tf^ [9))
+
e]"
C
.
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
28
where
q^
n be any
Proof. Let
any n >
A;
>
0,
By
n.
5"^'
monopoly output under
the
is
Define
integer greater than
B=
(iii),
=
Q^ =
-
{n
1)
—
A
1)
<
1,
(0)
=
-
(n
=
=
I^
+ l.
-
and
1)
^)
+
1/
where A
|A|,
<
with q°
g", if
is
=
q'^
q* {{n
=
g^ and Q*
Q*^
=
-
{n
-
1)
as in
By
(f.
g%
1)
—
1) s* {t'^^ {9))
is
Proof.
BC
q^
By
is
the
(i)
is, 5'=
s* [T] as in
any
>
=
than —1. Hence
[S;
t^-^'
(^)]
=
[O,
4, this
we have
4.
lemma
>
k.
D
yields the following.
Q=
\9
—
e ,9
+
e]
the best-response function q* {Q^i\9)
is
where Q-i
=
Then,
(VzeiV),
[0,q'']
P (•;^).
a nice game.
Hence,
yielding the desired equality.
decreases
g^^]" for each k
the others' aggregate output in equilibrium.
Lemma
Q''
[0,g^^]".
and that
0,
so that
'
monopoly output under
above,
for
and Q^ increases with k and takes value
Ar[s\tf^{9)] =
where
(ii),
Take
.
a continuous and strictly increasing function of 9 at (^Q-i,9^
(n
(ii).
.
In the general Cournot model, assume that
for arbitrarily small e
is
q'"')
Q* {Q"-^)
strictly less
finite k,
That
Together with Proposition
7.
{n
some
at
1) g^^ at
1
some
Q* {Q^-') and
Therefore, 5°° [5;t^^ [O]]
Proposition
q'-')
1)
the sfope of Q*
with k and becomes
Q*
-
Q''
g^
for
(•;
[g^ q^y\ where
q* ((n
Q'
Since (n
[<?°,9*^]"
=
[B; t^'^ {9)]
q'
P
Lemma
4
By
the hypothesis, there exists
and Proposition 4 imply
FINITE-ORDER IMPLICATIONS
In Proposition
that 5
a payoff-relevant parameter.
is
ficiently
the assumption that q* (Q-z; 0)
7,
many
dominance
Our proposition
be invalid whenever we
To
plete information model.
common knowledge
that
it is
that 9
=
9
=
is
|^
—
^|
level that is
the
a
first
<
c
is
is
c
and
his
confident that
it
only willing to concede
and agrees with the A-th-order
>
and
is
willing
arbitrarily large finite k.
any output
states that the skeptic nonetheless cannot rule out
not strictly dominated.
8.
It is
from the idealized com-
an arbitrarily generous skeptic; he
to concede the above for arbitrarily small
Our proposition
not implied by strict
The former
while the latter
6,
that
He
9.
is
see this, consider the confident researcher
common knowledge
mutual knowledge of
guaranties
suggests that, with suf-
slightly deviate
slightly skeptical friend in the Introduction.
is
responsive to
is
any equilibrium prediction that
firms,
will
29
common
Conclusion
model
after only specifying
or second order beliefs, using a (mostly implicit)
common knowledge
practice in economics to close the
assumption. In this paper,
we have
investigated the role of this assmiaption in
predictions according to an arbitrary fixed equilibrium.
and upper bounds
shown that
it is
for
Finding strong lower
the variations with respect to this assumption, we have
this casually
made common knowledge assumption
any prediction that we could not have made aheady by
strategies that can never be strict best reply. In
games
that drives
iteratively eliminating
like
Cournot oligopoly,
this implies that
no interesting conclusions could have been reached without
making a
common knowledge
precise
as sequential games, our lower
sharp predictions using
assiimption.
In
some other games, such
bounds are weak, and one may plausibly make
much weaker
assumptions. Therefore,
it is
essential for
assuring the reliability of theories to pay special care to closing assumption and
justify
it
at least as
much
as the other assumptions.
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
30
When
there
there are two or more actions that survive our ehmination process,
an inherent unpredictability wliich cannot be avoided without mal<;ing
is
an assumption on
many
infinitely
orders of beliefs, as
by some types whose
played with probability
1
arbitrarily high orders.
In that case, equilibrium
with respect to higher-order
there are no
there
ties,
and
beliefs
in
all
of these actions are
finite-order beliefs agree for
necessarily discontinuous
is
product topology. Moreover, when
a one-to-one relationship between this sensitivity to
is
higher-order beliefs and sensitivity to higher-order assumptions about players'
rationality. It
the
then becomes very
common knowledge
of rationahty as a
Appendix
Proof of
Lemma
support, and
let
/i
=
7
(7 o (•slj)~
Proposition
Proof. For k
some k -
T-iC
I
>
j
=
= vx^.
i^x/i.
0,
Hence,
LxH where L =
any
i,
s* [U)
and
(u^Z^
Let v be as in Assumption
Notice that
the proposition
0, i.e., for
good approximating assumption.
Take any 7 6 A(T_i) with countable
Oj.
A {A-i).
For any equilibrium
8.
=
G
o (slj)
as the type such that kj^
i/x
and any
i
analyzing these situations to justify
Proofs and further results
A.
Take any
1.
difficult in
is
= BR^
(tt
each
for for
A (X,_i))"~
i_,;,
and
A'^^
H=
take any
U with
(A.l)
^T_i
ff^
=
t™^ for all
m<
margexL^t^
^L
the proof,
X
(kj^
H
=
k.
= BR,
[s*,t_j]
(jj'^,^
members
(;/
t,
x 7)0/3"^
=
ai.
D
and k <
00,
{v x n)
and any
Assume
true by definition.
spaces of lower and higher-order beliefs with generic
Now
i,
=
Kf^o/3~-^
[\U, s*_^))
any player
s* ,
=
s*_^
tt (-1^;,
Define
1.
ti
tiiat it is
C
5^7^
and
true for
[^-^1-
A (X;_i))"
I
=
Now,
are the
h, respectively.
Clearly,
margexi'^t,-
because of the coherency requirement, which has no impact on the rest of
and
Kj
put probability
1
on the subset T_i.)
,
FINITE-ORDER IMPLICATIONS
For each
G
(0,/)
OxL with margexL^^t"
(i?,/)
=
31
H) >
k,;({(6I,0} x
be the the probabihty distribution on A-i induced by the behef
on
and
P
(•|6', Z,
si-)
s*_^
conditional
1)}
x
{9,1), i.e.,
P (a_,|0,
(A.2)
there exists a-i 6
for each
such
sU)
A (M_,)
=
^u {{{0.
1,
h)
\sU
h)
many
{9,t^j)
Note that according
.
{I,
such that u-i ({s_i|s_j
and a^i ({s_j|s_i
{9,1),
5'_~~
G
fixed S-i
1,
Since there are only countably
at each a_i.
each a-i, the probability that a_i
is
=
=
a_,}) /k^^ {{{9,
1,
h)
=
s_i (0, t_j)})
=
=P
(a_i|0,
is
P (a_i|6',
always
otherwise for
1
/,
Z,
slj.
By
Oi
s*_^
some
and
for
induction
a.,(^[s-,\s^,{9,l,h)&S'lt[{lM]])=^-
(A.3)
Moreover,
each
for
{9,
7T{{9,a-,)\U,a_^)
a_j) with
is
=
>
t| (9)
0,
fl^g^-g^P{a-,\9,l,s*_,)dKt,{0,l,h)
=
/
l{e=e}-P {a-^\9,
1,
s*_^) Kt. ({(61,
l^g^^Kt^{{{9,l,h)\s*_i{l,h)
=
where l/g-gi
t\ (9)
=
=
((0,a_O |ii,slj
TT
0} x H) dmargQxiKt,
P (a_j|0,
1,
,
the indicator function for {0
slj) does not
t] {9)
=
ij {9)
=
depend on
0,
we
h,
trivially
{9, 1)
= a^^})dmavgQ^Ld%{9,l)
=
^}; the
first
by integrating over a-i appropriately; the second equality
When
a_j})
to a-i, at each {9,1) in the support
played
>
with margexL^j- (^,0
(9,1)
{9,
H)
we then have
hypothesis,
0.
tj
0, let
and the third equality
have n
a-i)
{{9,
\ti,
is
is
a-^)
is
obtained
due to the
fact that
equality
due to (A.l) and (A.2).
=
n
{{0,
a-^) \U,
slj
=
Hence,
s* {U)
where the
Proof of
Lemma
G BFU [n
last inclusion is
Lemma
5.
2.
{-It,,
s*_,))
= BR,
{n {-lU, a-,))
by (A.3) and definition of
It follows
5^^
C S^
[U]
[ti\.
D
from the following lemma.
For any nice game and for any
i,
ti,
k, the
D
following are true.
.
JONATHAN WEINSTEIN AND MUHAMET YILDIZ
32
=
Sf
(2)
For each a^ G 5f
[ti]
some af,a^ S
[a^,a^] for
(1)
Ai, which depend
there exists s_,
[ti],
G S.~^ such
=
BR,{n{-\t^,s^i))
Proof.
We
will use
that part 1
each
for
induction on
true for
is
some
—
A;
For
k.
i.e.,
1,
This implies that Sj^
j.
topology).' Moreover, by the
singleton,
member
of BRi{iT {\ti,s-^))
i
Sj^
[S_~
',ti]
—
5f
payoff of type
ti
/3j
strict
{n
{-jti,
s_j))
concavity assumption,
(s^i) that
maps each
S_~
s_j
f
j
=
[a*^, a'^]
This readily proves part
for
in
Aj
Assume
=
[0, 1]
for
s-i))
(tt (-1^^,
it is
convex as
We
this
it is
claim that
Part 2 follows from the definition
1.
=
by setting Ui{ai\9,t-i,S-i)
and
BRi
df G Ai.
a*^,
each {9,t-i) G supp^t; and
strictly concave,
an upper-semi-
compact and convex,
is
some
is
G S_~^ to the unique
S'^~^;ti
is
for each cr_i
G
for
each s_i G S_~
,
Ui{9,ai, s-i{6,t-i)) at
A
iSj^
),
the expected
is
(A.4)
/
Now, take any
a closed interval
is
is, /3j
[ti\.
Clearly, Ui
[tj\
true by definition.
1 is
compact and connected, and hence
define function Ui{-\9,t^i,s-i)
Qi.
5-""^
continuous. Since
is
Towards proving our claim,
each
part
0,
;ti] is
unidimensional. That
Pi
=
that
{a\].
Maximum Theorem, BRi
and hence the function
implies that P^
/c
on U.
a closed, convex metric space (with product
is
continuous function of s-i. But by the
is
.
a^
>
Ui {ai\9, t_,, s-i) dKt^ (9, t-^) da-^ {s-^)
df.
Then,
for
each {9,t-i,s-i), by definition of
concavity of Ui{-\9,t-i,s-i), we have U^{a^\9,t-i,s-^)
<
J7,
aj^
and
[d'^\9,t-i,S-i).
It
strict
then
follows from (A.4) that d^ yields higher expected payoff than at for each (T_j
G
A
D
(
S'^jM, and thus
^Proof:
space of
all
a^
Firstly, Yl(e
5;
[ti].
^j~^
t
i)
measurabihty
convex.
is
^^
i^A
measurable functions /
these two spaces, namely 5^7^'
Similarly, a^
i^
:
Si
[ti]
for
each
aj
<
af
^ compact space by Tychonoff's theorem.
x T_i -^ R'^\^'^
is
But the
closed. Hence, the intersection of
compact. Convexity of 5^7^ follows from the facts that
preserved under point-wise multiplication and addition and that the range
is
FINITE-ORDER IMPLICATIONS
Proof of
2
Lemma
and the
any
e
Part 2 follows from the definitions, and Part 3 follows from Part
3.
D {A°° [s*,i]) < D [A'' [s*,t])
fact that
>
and any sequence
such that
i'"
[fc]
=
33
f^
for
6;^
>
that converges to
m
each
each
for
<
For each
0.
k and d [s* {i[k\)
To prove Part
k.
there exists
fc,
t [k]
D {A'' [s*,t]) /2 -
>
s* [t))
,
take
1,
e^
so that
0<D(^A''[s\t]^<2d{s*{i[k]),s*it))
(A.5)
But, by definition, for each
k
—
>
Hence,
oo.
s* is
if
m
and each k > m, i^
continuous at
t,
then as
the right hand side of (A.5) converges to
part 2 that s*
Lemma
6.
is
[k]
=
—
oo, s* (i
fc
That
0.
+
>•
i'",
D
is,
2ek.
and hence i™
[/c])
—
continuous with respect to higher-order beliefs at
For any finite-action game,
—
(^A'' [s*, t])
>
i™ as
»
and thus
s* (f),
+
—
[k]
showing by
0,
D
t.
the following propositions are ordered with
logical implication in the following decreasing order.
(1)
a*
is
continuous with respect
(2)
a*
is
continuous with respect to higher-order
(3)
a*
is
weakly continuous with respect to higher-order
(4)
A'^[a\t]
C
mixed
Lemma
3.1, (1)
write T,B
= {a
A (yl)
disjoint
the
B
and C, ds^c
minimum
beliefs at
[a* ,t]\
<
of ds^c
imphes
|supp (a)
(2).
C
To show that
B}, which
= min {(i(aB,Q;c)
among
such that whenever
whence supp(a*
(f))
(t))
—
{a}.
t.
all disjoint
B
\o^b
is
(2) implies (3), for
a compact
and C.
Clearly,
i"^
nsupp(a*
=
t'" for
(i)) 7^
m
each
—hence
<
(3).
The
laist
statement in the
k,
(3)
f'"
lemma
if
But
d[a*
i'"
0.
,a*
(4)
for
<
because
also follows
rfmin for
then
that there
(t))
each
A,
each
< dmm,
(2) implies
(t)
BC
for
Write
d{a,a')
imphes
=
>
each
Then,
set.
£ "^BjO^C G E^}
a E A'^ [c*!^] and each k, there exists i such that
supp(a*
beliefs at
1.
the supports of a and a' have non-empty intersection.
exists k
t.
strategies can be considered as pure strategies with values in
A{Ai), by
G
t.
supp{a*{t)).
Moreover, (2) implies that \A°°
Proof. Since
product topology at
to
m
c^min/2,
for
<
from the
each
k and
latter
JONATHAN WEINSTEIN AND MUHAMET
34
observation because a*
{t)
YILDIZ
cannot be arbitrarily close to two distinct pure action
D
profiles.
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