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DEWEY
HB31
.M415
Technology
Department of Economics
Working Paper Series
Massachusetts
Institute of
Generic Uniqueness and
Continuity of
Rationalizable Strategies
Muhamet
Yildiz
Working Paper 05-17
May 6, 2005
RoomE52-251
50 Memorial Drive
02142
Cambridge,
MA
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection
http://ssm.corn/abstract=722082
at
GENERIC UNIQUENESS AND CONTINUITY OF
RATIONALIZABLE STRATEGIES
MUHAMET
ABSTRACT. For
a finite set of actions
YILDIZ
and a
rich set of fundamentals, consider the
rationalizable actions on a universal type space,
topology.
there exists a unique rationalizable action profile.
(1) Generically,
Every model can be approximately embedded
A
rationalizable strategy
is
unique rationalizable action
Key
endowed with the usual product
continuous at a
in
(2)
a dominance-solvable model.
finite
type
if
and only
if
there
(3)
is
a
for that type.
higher-order uncertainty, rationalizability, universal type space,
words:
continuity
JEL Numbers: C72, C73.
Introduction
1.
This paper shows that,
if
one considers
possible payoff
all
and
belief structures,
then rationalizability generically leads to a unique solution. Moreover, when there
is
multiplicity, refining rationalizability implies ruling out
some nearby dominance-
solvable models as the true model. Formally, consider a finite-player, finite-action
game with some unknown
The
payoff parameters.
set
A
of action profiles
is
endowed
with the discrete topology. Assume that each action can be strictly dominant for
some parameter
value, e.g., that the
restricted a priori.
Endow
the
domain
game with the
is
not
of Mertens
and
of possible payoff structures
universal type space
T
Date: October 2004.
I
thank Jonathan Weinstein
joint work,
for long collaborations
and we had discussed some
discussions on the topic while
I
to us during a lunch discussion.
visited
I
on the
closely related ideas.
topic; this
I
work
is
thank Stephen Morris
Cowles Foundation; the main ideas of
thank Daron Acemoglu, Glenn
Rothschild for invaluable comments, and
partly built on our
Dov Samet and Aviad
Ellison,
this
for extensive
paper occurred
Bart Lipman, and Casey
Heifetz for earlier discussions.
MUHAMET
2
YILDIZ
T
Zamir (1985) and Brandenburger and Dekel (1993), where
usual product topology of weak convergence.
Main Result.
it is
continuous function
s*
U
on the open, dense set
U
That
for each
is, if
t
.
>
is,
A, such that
s* (t) is
U C T
then
each remain-
for
a unique rationalizable action profile, and the action profile
is
means that each type
Since a rationalizable strategy profile must
s*.
profile in
U
it
must be continuous on U. Continuity
has an open neighborhood on which
constant. This leads to an interesting picture: the universal type space
open
a fixed action profile
sets
is
and a
the unique rationalizable action
set of type profiles,
agree with s* on the open and dense set U,
of a collection of
and
In particular, every rationalizable strategy is continuous
given by a continuous function
of s*
prove the following.
there exist an open, dense set
we exclude a nowhere-dense
ing type profile, there
is
That
U—
:
6
t
endowed with the
Genetically, there exists a unique rationadizable action profile,
genetically continuous.
profile at
I
is
and
their boundaries, such that in each of the
the unique rationalizable action
discontinuities occur only
is
on the boundaries
of these sets,
profile.
s* is
comprised
open
Multiplicities
where the unique
sets,
and
rational-
izable action profile potentially changes. Ubiquity of multiple rationalizable actions
in usual
game
theoretical models suggests that our
put our models on these boundaries.
here
not negligible, as
is
What
answer,
is
it
includes
assumptions
This also shows that the nowhere-dense set
many
does this mathematical result
let
common knowledge
of the
tell
models
in
economics
literature.
us about economic modeling?
For an
us examine the universal type space more closely. In this space, a type
a coherent hierarchy of beliefs about the payoff parameters, where the first-order
beliefs are
about the parameters, the second-order
Here, U, the set of
all
type profiles with unique rationalizable action
cause the rationalizability correspondence
(2003))
beliefs are
and the action space
is finite.
I
is
about the
profile, is
first-order
open simply be-
upper semicontinuous (Dekel, Fudenberg, and Morris
show that
U
is
dense, using a result of Mertens and Zamir
(1985) and a construction by Weinstein and Yildiz (2004), whose
the seminal works of Rubinstein (1989) and Carlson and van
main
Damme
idea can be traced back to
(1993).
RATIONALIZABLE STRATEGIES
beliefs,
and so
The
on.
most type spaces
universal type space contains
closed" subspaces (henceforth, models). For example,
in
3
it
as "belief-
contains a family of models
which the players observe the parameters with noise, where the
and
level of noise
the prior beliefs vary across the models, as well as the complete information model
with no noise.
beliefs at
it
each
becomes
If
we
a prior and
let
the size of the noise go to zero, the players'
order converges to that of complete information. In that case,
finite
difficult for
the modeler to distinguish these models from each other in
when one can only observe
the interim stage,
partialy).
fix
The product topology captures
the posterior beliefs (possibly only
this difficulty of identification.
topology, a sequence of types converge to a fixed type
if
the beliefs at
In this
all
orders
converge. In particular, the above models converge to that of complete information
as the noise vanishes.
In the ex ante stage, the modeler can, of course, find the above models quite
different;" the prior
may have
substantial impact on strategic behavior even in the
presence of strong information. Unfortunately, however, in most applications, the
modeler faces the situation only in the interim stage.
The ex
ante stage
often
is
constructed by the modeler in order the capture the situation in a coherent model.
Indeed, the central question of this paper
that he has selected one model to analyze
finite
how the modeler should proceed given
among many
indistinguishable models.
To
from that angle, assume that the modeler can make observations
interpret the result
about
is
but arbitrarily
many
orders of beliefs, so that after the observation
he knows that the belief at each observed order
is
in
some
open
arbitrarily given
neighborhood. Now, genericity of uniqueness means the following:
(i)
the modeler can never rule out the case that each player has a unique rationalizable action
"The
identification
requires the belief at
and
problem
all
in the
ex ante stage can be captured by the "box topology", which
orders to converge uniformly. In this topology, the above models do not
converge as the noise vanishes. For, with noise, the limit of A;th-order expectations as k
the ex ante expected value of parameters (Samet (1998)).
—
>
oo
is
MUHAMET
4
(ii)
YILDIZ
whenever the players do have unique rationalizable actions, the modeler
could
make
sure that that
is
the case by making a sufficiently precise obser-
many
vation (by choosing sufficiently small open sets at sufficiently
Continuity of a strategy means that the modeler can
play according to the strategy
icity of continuity
possibility that
nalizability,
ity
is
if
his observation
the player will
sufficiently precise.
and uniqueness implies that the modeler can never
he could have learned what the players
will play
by making a more precise observation. In that
Then, generrule out the
according to ratio-
sense, the rationalizabil-
a strong solution concept.
Genericity of uniqueness provides a
ability
finite
is
know what
orders).
new perspective on refinements
(and equilibrium). Towards establishing
this, I further
show
of rationaliz-
any
that, given
type space and any rationalizable strategy in that type space, one can slightly
perturb the players' perceptions about the payoffs to obtain a nearby dominancesolvable
model
in
which the given strategy
type in the original model, there
whose
We
can therefore regard a
many indistinguishable
computing the
make any
situations
uniquely rationalizable.
(For each
be a type in the dominance-solvable model
beliefs are arbitrarily close to that of
orders.)
in
will
is
finite
the original type for arbitrarily
many
type space as a model that summarizes
by abstracting away from the
beliefs at very high orders.
By
details that are
used
specifying these details, one could
rationalizable strategy uniquely rationalizable. In the detailed model, one
must take the unique rationalizable strategy
as the only prediction,
refinement of rationalizability (or equilibrium) he believes
refines rationalizability
by ignoring some rationalizable
in.
no matter what
Therefore,
strategies,
when one
he simply ignores
the dominance-solvable models that are indistinguishable from the model at hand
but lead to the ignored strategies as unique solutions. In that sense, refinement
selection
The
among
is
a
payoff and information structures, rather than an epistemic issue.
last result leads to
extensions of two seemingly opposing results. Firstly,
tend, in a weaker form, the results of Carlsson
and Van
Damme
I
ex-
(1993) and Frankel.
RATIONALIZABLE STRATEGIES
Morris, and Pauzner (2003) for supermodular
games
5
to all finite-action games. For
supermodular games of complete information, they showed that any perturbation
within a canonical class leads to a dominance-solvable model
erate signal values at which the strategies jump.
—except
For arbitrary
for the
degen-
finite- action
games
with arbitrary payoff and information structures (with possibly infinite type spaces),
I
show that there
exists a perturbation that leads to a
The dominance-solvable model
model.
tions are introduced.
remain
will
when
so,
This suggests that multiplicity
higher-order uncertainty at
all levels.
nearby dominance-solvable
will
further small perturba-
become
rare as
we
allow
(As we successively introduce higher-order
uncertainty in the form of "small" noise, the domain of dominance-solvability will
grow, while the domain of multiplicity will shrink.)
Second, extending a discontinuity result of Weinstein and Yildiz (2004) for equilibrium,
that
I
obtain a characterization: a rationalizable strategy
lies in
a finite type space
for that type.
all
of
them
find the
common
with a
common
if
we
if
there
is
a unique rationalizable action
either all rationalizable strategies are continuous, or
above discussion misleading. The examples
The above
prior as well.
Lipman
counterintuitive results
(2003),
restrict ourselves to the finite
the nearby dominance-solvable models
of a larger finite type space with a
for
I
show that ah
models with
In the next section
2x2
I
games. In Section
5.
the latter
common
prior.
In particular,
prior,
and the above characterization of
if
we
restrict the
domain
prior.
provide examples of nearby dominance-solvable models for
3, I
introduce the model and preliminary results.
results are presented in Section 4.
Section
may be due
the finite type spaces can be taken as part
common
common
models
of the above results remain
continuity with dominance-solvability remains intact even
of the strategies to types with a
refer to
type space contains the models without
prior, while the universal
models. Using a result by
intact
and only
continuous at a type
are discontinuous.
Some may
a
At such a type,
if
is
Section 6 concludes.
The proof
of a central
lemma
is
The main
presented in
muhamet
6
2.
In this section, using
2x2
when incomplete information
multiple equilibria
Example
Examples
games,
will illustrate
I
introduced.
is
I
2x2
is
When
c
=
0,
9
is
common
two Nash equilibria
exist
strategies.
on
=
t
and the support of
9,
6,6
ft
0,6-1
9
+
ctj^
9
1,0
0,0
is
where
9 contains
knowledge.
in
6-
Oil
unknown but each
is
(r)i,rj 2 )
an interval
If it is also
With incomplete
positive, multiplicity disappears:
whenever x >
z
and so
where a
<
<
the case that 9 £
for
information, this
Damme
parameter
on.
This
is
Xi
is /3 t
e
>
for the fragile case of e
0,
and
for
an open
set of
=
0,
1
(0, 1).
<
b.
there
no longer possible.
^
is
small but
1/2, there exists a
whenever x < 1/2. and
x
uniqueness prevails in an
parameters
for the distributions
a reflection of a more general fact that dominance-solvability
When
the degenerate signal values with multiplicity, such as xi
—as
is
show that when e
each signal value
holds for an open set in the universal type space.
avoided
;
1/2.
While multiplicity holds
set of
G {1 2}
independently distributed
[a, b)
unique rationalizable action. The rationalizable action
open
i
pure strategies and one Nash equilibrium in mixed
Under mild conditions, Carlsson and van
a,
player
Without incomplete information, the players are able to "coordinate"
different equilibria.
it is
the games with
will first consider
02
Assume that
a real number.
observes a noisy signal x
from
multiplicity disappears
game
«2
where 6
how
— analyzed by Carlsson and van Damme (1993).
Consider the
1.
yildiz
in the next
example.
the type space
=
is finite,
1/2, are also easily
RATIONALIZABLE STRATEGIES
Example
where
9
2.
9\
=
=
e/2, 9 2
3e/2,.
then
— 9 m -i
Xi
,
.
The
distributed uniformly on 0.
is
.
=
9 for
some
on
9.
= 9 m+
Xi
9
G
[xi
—
(2k
+
1) e, Xi
+
(2k
+
>
is
it
0,
model
In this
—
*
verges to a model where 9 G [0,#] becomes
e
>
0,
some
The
limit
game
is
1.
Ex
if
9
it
is
(i.e.,
(9
—
1/2) / [(9
+ e/2)
ante,
=
9m
,
The
Moreover, for any
= x^
As
e
—
common knowledge
0,
the
sig-
an integer), the game
game
con-
before the players
characterized by multiple equilibria.
el is
#m-i},
,
common knowledge
except for a nowhere-dense set of parameter values for which 9 m
m
•
the players' /cth-order beliefs
0,
converge to that of fcth-order mutual knowledge of 9
take their action.
•
mutually known at the /cth-order that
Hence, as e
1) e].
>
•
with probability 1/2. 3
\
that the signal values are in e neighborhood of true value.
nal value Xi and any integer k
fixed 6
{9 ,9i,
players observe 9 with noise:
with probability 1/2 and
signals are independent conditional
9m-i
=
G
In the previous example, consider the case that 9
= — e/2,
#0
7
But when
=
1/2 for
is
dominance-solvable
in
pure strategies are
with the unique rationalizable strategy
Si (%i)
=
i
(
(One can
easily check this starting
Since the generic
2x2
a
2
if
x
x
>
1/2
<
1/2.
<
ft
if a*
from the two ends.)
games with unique equilibrium
dominance-solvable already, the above examples cover
all
2x2
games, except
for
the games with no equilibrium in pure strategies, such as the Matching-Penny game.
In such a game,
player
i
if
the dominance considerations had led to a unique strategy for a
and there were no payoff uncertainty, then
action and play a best response, against which
strategy.
Every action
is
use the covention that 6-\
random
opponent would foresee the
=
z's
would have wanted to play another
rationalizable in these games.
plete information in the above
I
i
his
The introduction
of incom-
form does not render these games dominance-solvable.
6q
variables that takes values 1
—
e
and &m
= 6m—\ + £•
In previous formulation,
and —1 with equal probabilities of 1/2.
rj
i
is
the
MUHAMET
8
YILDIZ
Nevertheless, the next example shows that these games, too, can be perturbed to
obtain a dominance-solvable model
EXAMPLE
6
is
common knowledge and
=
Take
{# 0)
0i,
•
,
—7
7 £
=
to (xj.^j)
ef [2(1
(0,
—
(0 m ,0
As
e)]).
before,
—
=
as £
set {{e, 7)
|0
For e
0.
<
<
7
02
CVl
9,0
6-1,6
Pi
0,0
0,6-1
ej
then there
in (0. 1),
=
6
m _i) and
6,
game
is
a?2
(1
[2
is
=
-e/2, 9 l
Conditional on 6
within ^-neighborhood of
is
Now, consider the pay-
prior).
it is
= 6m
probability
,
no pure strategy equilibrium.
e/2, 6 2
uniformly distributed on 0.
is
on the signals (xi,^).
1
—without a common
0m-i}, where
and assume that 6
1,
different belief structure.
matrix
off
ff
3 (Matching Pennies
—using a
=
3e/2,.
.
,
9
M -i =
&
<
Players have different belief
each player
i
to (xi,Xj)
=
7
.
common knowledge that
assigns probability
where
(9 rn -.i,9 m ),
the players' signals are
and the game converges to the the complete-information
—
But
for the
open
e)]} of parameters, the incomplete-information
game
0,
every strategy
is
rationalizable.
dominance-solvable, and the unique rationalizable strategy profile
is
as in the
=
6
following table:
X
t
4 (22)
(Clearly,
/3 1
when x =
l
9
,
00
01
02
03
04
Pi
«1
a
1
Pi
Pi
«2
fto
P2
Po
0'2
player
and «2 are dominant actions
player
i
j at Xj
i
do,
the player
iteratively in this way.)
i
Q'
2
for players 1
(6,
Xj)
=
and
•••
06
07
08
Oil
Pi
Pi
P2
P2
CV9
assigns high probability
assigns high probability to
=
05
2,
(#1. #o)-
1
'
•
—
'
7 to 9
,
when
=
#1,
Given the dominant action
for
One computes
s*
respectively.
has a unique best response;
it is
on.
When,
:r,
RATIONALIZABLE STRATEGIES
example the players do not have a common
In this
game
elimination process in this
depends only on the
ability
M orders of
first
Mth
stops at the
9
This
prior.
is
round, and hence the rationaliz-
beliefs.
Using Lipman's (2003) method,
we can then construct an incomplete-information game with a common
with types whose
game
M
first
and
prior
The new
orders of beliefs are as in the original game.
be dominance-solvable from these types' point of view, as
will
The
not crucial.
in the following
example.
EXAMPLE
assume
that
(1)
that, in addition to
correlated with 9
is
Player
7/i
4 (Matching Pennies
n each player
:r
and takes values
i
prior).
=
1,
(2)
;</i
=
3,
in {1, 2, ...
common
the prior probability of
=
(1
ji(6,
prior
(6>,
- if
xj
/M
etc.
3,
number y with y >
players have a
(01,01, 0o)
=
(3)
;yi
,
2K}
k, e.g., yo (1)
and /^ (0i,0 o
=
X\,X2, 1)
some
integer
,
0i)
K>
number y with y >
2,
y2
(2)
follows.
=
2,
Let
= r/M.
Define
/I
y-2
etc.
/i,
in the previous
i
variable k
M.
k; e.g.,
(k) of the
Now, the
(8,xi,x 2 ) be
example,
iteratively
e.g.,
by
ol^x (0, £1,2:2)
= Lk
p,(9,x 1 ,x 2 ,k)
=
x 2 ) according to player
,
for
random
Player 2 observes the value
about (9,Xi,x 2 ,k) as
/j
In the previous example,
partially observes a
observes the value y\ (k) of the smallest odd
1
smallest even
f-h
—with a common
~1
a/j,
ik
(9,x 1 ,x 2 )~^2]i{9,x 1 ,X2j)
Kk
for
ii-
each
(9,
is 1 if
k
is
addition to
As
e
—
>•
0,
x u x 2 ) and k e
i/ z
odd and
,
if
k
is
even.
,
2A"} where
Once
L>
again,
z
the belief hierarchy of each type with
of 9
ji
=
x^.
(1
it is
each player observes a signal x that
common knowledge
(2.1)
2
{2, 3, ...
(xj,
is
-
7) /t,
common knowledge
\xi, yi
(k))
=
/_/,
that, in
within ^-neighborhood of
9.
y % (k)) converges to that of the
Moreover, one can check that
{{9,x u x 2 )
a = l/L 2I< -\ and
{(9,x u x 2 ) Xi )
\
MUHAMET
10
<
for each yi (k)
2A'.
4
That
YILDIZ
new model
the posterior beliefs in the
is,
are identical
to that of previous example, except for the case that player 1 observes that y\ (k)
2K + 1.
%i — 6 mi
It
from
follows
each (x ,y (k)) with
(2.1) that, for
i
m
where
there exists a unique rationalizable action
s {x l ,y {k))
l
l
where
< 2K —
yi (k)
l
=
=
s*(xi),
the unique rationalizable strategy of
s* is
game
particular, the
is
example. 5
in the previous
i
In
dominance-solvable from the point of view of the types with
(xi,y (1)), which approximate the complete-information model.
t
way from
Notice that, in this example, the types whose belief hierarchies are far
may have
those of original model
consider the types with
(k)
y,
multiple rationalizable actions; for an example
> 2K —
m
•
i
An
x
•
game with
£ N, where
parameters
topology.
To
to
n1
is
The game
For yi (k)
Ji{-,yi (k)
Lipman
5
r^ is
^i
-
=
.
.
.
A=
n}, finite set
,
a n ), and utility functions Ui
:
G
A\ x
A—
x
»
IR,
The
9.
finite set
A
is
j/i
player 2
=
player
1,
€
(L
{3,
-
2
.
,
.
.
l)
IK },
aV^
-
player
i
A
space.
knows that k
1
knows that k e {1,2}, and
y, (k)
Jl(-,y l (k))
=
(k)
endowed with the
=
1,
type of a player
and
/}
+/x(-.2)
/i(-,l)
knows that k £
{yi
is
^
(•),
which
= L 2 a/i 2
(fc)
proportional to
is
i
proportional
"
2,
discrete
—
1,
^
j/i
(•).
()
is
(&)},
(See
(2003) for a complete proof.)
{QmiVi {k)) with
y, (fc)
on {6
,
.
.
endowed with the universal type
For any
+
Use induction on
with
is
2,
2 (•).
1)
.
,
continuous in
see this, notice that, for
.
a2
(oi,
some m.
for
a compact, complete and separable metric space of payoff-relevant
and
6,
proportional to
and
=
= 8m
N = {1, 2,
finite set of players
of action profiles a
t
Model
3.
Consider a
and x
<
2/\
m
to check this.
For
yi (k)
< 2K. Assuming
—
Player
?7i.
= m ,Xj = 9m —i}. By
i
m
=
0,
by
the statement
knows that
?/j (fc)
<
(2.1), s* (8
is
true for
2J\ —777
+
1,
m
)
is
m—
dominant action
1,
is
s* (9„,.).
each
consider any {8 m .yi (k))
and assigns very high probability
assumption, he must assign high probability on j playing
against which the only best response
for
s*
(^m-i),
RATIONALIZABLE STRATEGIES
an
is
infinite hierarchy of beliefs
=
u
where
i
G
t\
about
9,
A (0) is
t\ G A (0
A (A")
on. Here,
A (0) n
x
weak* topology.
I
i
on 0, representing the
assume that
common knowledge
it is
with each other). The set of
denotes the set of
all
all
type profiles
=
t
topology, so that a sequence of types
and only
if t^
converges to
m —
iff t
t
>
each
t\ for
im
—
»
ti
.
than
t
.
?'.
6
,
.
t n ),
and T_
T
Each
%
=
l
each
For each type
That
is, it is
strategy of a player
belief
A (0
n G
x A-i),
the expected value of u
Remark
1.
In
my
x
—
and K m
one-to-one,
i
is
any function
BR
(9,
Z
a
(
and only
if
K, ti
>
ti
tt
s,
:T ^>A
x
if t
.'
l
a-i)
formulation,
=
(/l (zi)
[ii]
In general,
flj^, Xj,
•
.
-
,
write x
=
Similarly, for functions,
fi-i (xi-i)
x-xFR
I
\—*
K ti
im —*
ti.
tz
For each
common knowledge
[i„]
,
(xi,
I
.
.
.
,x n )
write / (x)
fi+i (x l+ i) ,...,/„ (x n )),
and F_, [x_
t ]
=
and
=
=
^
the se^ 0I
s
a,,
G
—
.
>
,
.
ti,
t njn )
x T_j) be
(0,£_j).
an isomorphism.
G
A
A^
and
for
each
that maximize
t
k.
that the payoffs are given
X
(x,,x_ t ) £
(/i {x-A
.
,
about
tz
is
i
(£i jTn
A (0
Kti G
let
,
by a fixed continuous function of parameters. This assumption
Notation:
-0
under the probability distribution
it is
Tn
x
•
•
denoted by t im
ti,
(n) denotes the set of actions
,
•
endowed with the product
is
hm converges to a type
i.
T\ x
"
Ylj&
A sequence of type profiles t (m) =
k.
for
(t\,
Mertens and Zamir (1985) have shown that the mapping
'I
,t\),
and so
beliefs,
T=
Tf,
the unique probability distribution that represents the beliefs of
Fi
,
.
that the beliefs are coherent
such types are denoted by
profiles of types t- % for players other
.Y_,
.
each player knows his beliefs and his beliefs at different orders are consistent
(i.e.,
A
.
X, endowed with the
probability distributions on
all
beliefs of
a probability distribution for {9,t\,t\,
is
)
about 9 and the other players' first-order
the space of
is
{tltl...)
a probability distribution
representing the beliefs of
if
11
,
.
.
.
,
is
without loss of
= X\ x
•
•
x
Xn
and
/„ (x n )) and /_, (i_,)
for set-valued functions,
I
write
=
F [x] =
]1 J#! -^ [ij].
do not restrict the strategies to be measurable.
Measurability restriction could lead to a
non-existence problem, which can be avoided in the present interim framework (Simon, 200x).
MUHAMET
12
generality because
YILDIZ
we can take a parameter
action profiles to the payoff profiles. For example,
where 0j
=
allows
possible payoff functions, and here 6
all
[0, 1]'
for
each
i,
and
let
u
=
(9, a)
x
maps
to be simply the function that
8i (a) for
is
9=
we can take
each
Q\ x
a. 6).
(i,
•
•
•
x 0„
This model
simply an index for the profile of
the payoff functions. This model clearly satisfies the following richness assumption,
which
made by
also
is
Assumption
Carlsson and van
Damme
(Richness Assumption). For each
1
(1993).
and each a i} there
i
exists Q
a%
£
such that
a
>
Ui(d %ai,a-i)
and 6
a
*
That
^
is,
9
ai
whenever a
2
^
ai
Ui (6
,
a'i,
(Va-
a-i)
Oi,Va_i)
a[.
the space of possible payoff structures
tion can be strictly
^
dominant
for
rich
is
some parameter
value.
enough so that each
ac-
In developing a unified
theory of games, one would want to avoid a priori restrictions on the domain of
payoff structures.
When
there are no such restrictions and the actions represent the
strategies in a one-shot, simultaneous-move
satisfied.
be
When
indifferent
he does not
actions represent the strategies in a
between any two strategies that
by the
are ruled out
strategies themselves,
make any mistake
Hence, Assumption
1
differ
1
is
automatically
dynamic game, a player
will
only on information sets that
assuming that the player believes that
(or does not "tremble") in playing these strategies.
may appear
to rule out
that the player thinks that each player
set
game, Assumption
all
these games.
may make
But
possible
it is
a mistake at each information
with positive probability, as game theorists typically assume in their analyses of
such games.
The
latter case
is
modeled by another game. In that
necessarily be indifferent between those strategies.
satisfied for a
reduced-form representation,
of such mistakes a priori
and allows
all
if
case, he will not
Indeed, Assumption
1
will
be
one does not rule out the possibility
payoff vectors at terminal nodes.
RATIONALIZABLE STRATEGIES
Rationalizability. For each
by letting
iteratively,
7r
A (0
G
That
is,
x T_i x
a,;
G 5*
cii
j4_i)
and
i
set
£,;,
if
[£,]
Sf
and only
=
write S'^7
=
1
[t-i]
-1
n.7#l Sf
rationahzable actions for player
all
K u and n (a_; G S^-
=
[t]
—
S*
(with type U)
i
S^
[t 7]
for
7r)
for
1
[t-i])
k
>
some
=
1-
that puts positive probability only to
f,
and S k
fo]
sets
G B/?^ (rnarg@ Xj4
the actions that survive the elimination in round k
6. I
and define
i:
if a;
such that marge xt^tt
a best response to a belief of
is
=A
[£,;]
13
(As described in Footnote
1.
x
[t{\
•
•
•
x S*
[t n ].)
The
set of
is
oo
sr
m=ns
"
•
fc]
fc=0
A strategy profile s T —
iff
s (i)
G
iS
00
[i]
strategies
is
Remark
2.
in
many
each
for
denoted by
When
£ (resp., s t
R
there
a strategy
^4 (resp.
>
:
,
is
and
G 5°°
(U)
R=
Ri x
•
Sj
T —
:
2
[U] for
•
^4,) is
each
Rn
x
»
/,
2
).
said to be rationahzable
The set
of rationahzable
.
incomplete information, rationalizability can be defined
different ways, leading to different sets of rationahzable strategies.
I
use a
version of interim correlated rationalizability (Battigalli (2003), Battigalli and Siniscalchi (2003)
and Dekel, Fudenberg, and Morris
rationalizability
ity.
is
among the known
the weakest
(2003)).
The
interim correlated
interim notions of rationalizabil-
Dekel, Fudenberg, and Morris (2003) show that, for arbitrary type space and
independent of whether correlations are allowed,
a type with belief hierarchy
t 2J
then a
z
is
if
an action a
is
2
rationahzable for
interim correlated rationahzable for
tz
.
Using a weak notion of rationalizability strengthens both positive generic uniqueness and the negative discontinuity results; these results will remain valid under any
stronger notion of rationalizability.
izability slightly different
To
2
j4_j.
By
definition,
my
on functions /
formulation,
be rationahzable, which only strengthens
equivalent
when n
ti
has a
I
formulate rational-
from Dekel, Fudenberg, and Morris (2003). They define
rationalizability through the beliefs
x T_ x
simplify the exposition,
finite
support.
my
if
:
x T_^
—
A^i, rather than
anything, allows
results,
more actions to
and the two formulations are
MUHAMET
14
YILDIZ
Mathematical Definitions and Preliminary Results.
Definition
1
(Genericity).
The
A
smallest closed set that contains T".
each
A
t
€ T, there
contain any open
be nowhere- dense
set.
A
set X"
statement
C
T, denoted by V,
dense (in T)
is
sequence of type profiles
exists a
set T" is said to
V
closure of a set
is
empty,
said to be genetically true
is
=
T,
(m) G X" such that
t
the interior of T'
iff
T'
iff
the
is
for
i.e.,
—
£ (???)
»
£.
T' does not
i.e.,
if it is
true on an
open, dense set of type profiles.
An open and
is
dense set T"
C T
large in the sense that
is
nowhere- dense. In that case, T\T"
is
simply the boundary of
Clearly, topological notions of genericity
notions of genericity. Since this paper
may
is
Definition
for
each U G
T-,
Let
T
',
denoted by 9T".
from measure theoretical
seem to be appropriate.
ratio-
(To see
how
A subset
V
C T
is
said to be belief-closed
iff
CGxf,. A belief-closed T C T is said to be finite iff V
supp(«^)
many members and
be the union of
all finite,
are referred to as finite types.
T
differ
T\V
(1980).)
2 (Finite Types, Models).
contains finitely
T[.
Oxtoby
widely
T",
about the topological properties of
nalizable strategies, the topological notions
these notions are related, see
complement,
its
t]
has
finite
belief-closed subspaces T"
will use the
I
support for each
terms model and
C
t,
T.
=
(tj, tf,
Members
belief- closed
.)
£
of
T
.
subset of
interchangeably.
Lemma
1
(Mertens and Zamir (1985)).
Definition 3 (Dominance-Solvability).
solvable
if
and only
if
15°°
Definition 4 (Common
(with full support)
if
[t]\
=
Prior).
and only
if
1
for
A
f
A
each
model
is
dense,
model
t £E
i.e.,
V
f=
C T
is
T.
said to be dominance-
T"
VCT
is
said to adraii a
common
there exists a probability distribution p G
A (0
prior
x
T")
RATIONALIZABLE STRATEGIES
=
such that supp(p)
6' x
V for some 0'
C
and Kti
=
p
:>
I
x
(-|0
x T'_^
{t,}
for
each
UeT!.
The
T CPA
denoted by
mon
full
on
is
2
Tm C T
(-
m)
:
:
and
X"
— Tm
T'CT
(£,
A
5 (Continuity).
correspondence
tt
,
£
vn
then
F T—
:
>
2
A
in the product topology of
is
upper semicontinuous
i'
G
iff
T
—
as
t
>
strategy s t
U
=>
Since
.
model with supp(/«ti )
finite
Then, for each m. there exists a
T[.
m)
m—
>
model
common
is
said to be continuous at
Si (t iiTn )
A
%
is
->
x A. Since
t
A
is
0' x T_
finite
x
model
mapping
t7
iff
5, (ij)
endowed with the
said to be upper- semicontinuous
each
=
oo.
discrete topology,
constant on a neighborhood of U.
s z is
is
be a
prior with full support and a one-to-one
-*
ti,m
continuous at
e
t%
such that r
>
each sequence of types
is
finite
support (see also Feinberg (2000) ). 8
full
common
(3.1)
Si
than
for each
that admits a
Definition
for
by Lipman (2003) shows that, given any
(Lipman (2003)). Let
some 0' C
for
is
and admits a com-
belief-closed
is
prior
because the common-prior assumption does not put any restriction
finite-order beliefs other
Lemma
t
result
e T(\T'
{t z
common
support", one can obtain a nearby finite model that admits a
This
prior.
Tf PA =
formally,
;
The next
prior}.
"with
type profiles that comes from a model with a
set of all
endowed with the
has a neighborhood
rj
with
if its
A
(bounded)
graph
is
closed
discrete topology,
F
[t'\
C F
if
[i]
for
F
each
n.
Lemma
3 (Dekel, Fudenberg,
(2004)). 5*°° is
and Morris
non-empty and upper-
semicontinuous.
s
Lipman
{{{9,
t) \t t
=
(2003) uses a partitional model.
£,}
\ii
€
T, } as
If
the partition of player
one takes
i,
fi
=
x
T
as the state space
then the condition in the
and
lemma immediately
implies his weak-consistency condition, which characterizes the finite-order implications of the
common-prior assumption.
MUHAMET
16
YILDIZ
Dekel, Fudenberg, and Morris (2004) proves upper-semicontinuity of interim correlated rationalizability in their framework. For the sake of completeness.
a proof in the appendix.
this
lemma
Lemma
Together with the observations in the following lemma,
provide a main step in the proof of the main result.
will
Given any non-empty, upper-semicontinuous F,
4.
provide
I
let
Uf =
=
{t\ \F[t]\
1}.
Then,
(1)
Up
(2)
there exists a continuous function f*
open;
is
each
t
— A
Uf
>
such that
F
[t]
=
{/* (£)} for
is
continuous
£ Uf, and
any function f
(3) for
:
:
T
—* A,
if
f
F
£
(t)
[t]
for each
t,
then f
on Uf-
Proof. Define /*
F, each
:
Uf —* A by F
=
[t]
£ Up has a neighborhood
t
Since F[t'}
^
Therefore,
t^
{/* (£)},
rj
0, this implies that F[t']
open.
is
By
F
with
definition, /*
=
£ Up. By upper-semicontinuity of
t
[/.']
CF
[t]
=
{/* (t)} for each
{/* (t)} for each f £
(£')
=
/*
(t)
for
each
t'
£
so that
rj,
rj,
tj
t'
£
rj.
C PF
and hence /*
-
is
continuous. Finally, any / as in part 3 coincides with /* on the open neighborhood
r\
and hence
is
continuous at
t.
4.
In this section,
able strategies.
I
Whenever there
strategy
is
I
analyze the continuity and uniqueness properties of rationaliz-
show
is
that, generically, there exists a unique rationalizable action.
a unique rationalizable action for a type, every rationalizable
continuous at that type.
also true: a rationalizable strategy
is
Results
is
For
finite types,
continuous at a
a unique rationalizable action for that type.
I
I
show that the converse
finite
further
type
if
and only
show that
for
if
is
there
any model.
RATIONALIZABLE STRATEGIES
there
is
sults,
I
a perturbation that leads to a dominance-solvable model. Using these re-
then present characterizations for the topologies generated by rationalizable
strategies.
LEMMA
The next
m
Under Assumption
5.
—
oo and S°° [t(m)~\
*
can be found
in a
G T, and any a G
with type profiles i (m) G
Tm
=
{a} for each m. Moreover,
Tm
common
will present the
which a
t
is
[t],
there exists
such that t(m)
—
>
i
can be chosen to be
uniquely rationalizable. Moreover the
Since the proof of this result
prior.
proof in Section
5,
new type
is
somewhat
involved,
I
important implications of the
after exploring the
for this paper.
Equivalence of continuity and uniqueness.
1.
ous at a
finite
type U €
=
This characterization remains intact
[t»]
1-
|
is
00
prior with full support.
Proposition
\S?°
,
S*
dominance-solvable model or in a (possibly dominance-insolvable)
model with a common
4.1.
i
given any type and any rationalizable action a t for that type, one can
is,
find a nearby type for
lemma
any
for
1,
for this analysis.
Tm
dominance-solvable or with a
That
be the main tool
result will
a sequence of finite models
as
17
restricted to
Proof.
The
"if"
prove the "only
Si (ti) 7^
Srfc.m]
Si (ti)-
Lemma
a2
.
=
Under Assumption
T CPA
x
if
and only
by imposing the
a rationalizable strategy s G
t
if i t
'
if"
part, take
By Lemma
common
{cii}-
Since s
last
if
l
=
a
z
for
is
continu-
Lemma
each
i.e.,
prior assumption.
3
any rationalizable strategy
(t it , Tl )
t
the domain of strategy prohles
and part 3
Si
t
hm
,
of
Lemma 4. To
and any a 6 S°°
there exists a sequence of types
5,
R
has a unique rationalizable action,
part immediately follows from
To prove the
5).
,
T
1,
z
t irn
s { (t i>m )
—
»
iz
with
s
[t,]
t
with
(t h , n
)
G
does not converge to
statement of the proposition, one picks
t
PA (by
vn G T^
MUHAMET
18
Proposition
1
establishes that, at a finite type, either
are continuous, or
into
two groups.
and
all
YILDIZ
all
of
them
are discontinuous.
The
For the types in one group, the
all
rationalizable strategies
set of finite types
game
is
can be put
"dominance-solvable",
rationalizable strategies are continuous at these types.
For the types in
the other group, there are multiple rationalizable actions, and each rationalizable
strategy
is
discontinuous at each type in this group. Since there are typically multiple
into the second
rationalizable actions, the finite types in applications typically
fall
group. Assumption
some
strategy
may be
1 is
not superfluous; without Assumption
is
rationalizable
continuous at a type with multiple rationalizable actions.
Under weaker assumptions, Weinstein and
equilibrium
1,
Yildiz (2004) have
shown that every
discontinuous at a type for which multiple actions survive iterated
elimination of strategies that are never a strict best reply.
Proposition
the equilibrium and strictness requirements in their conclusion.
1
drops
This extension
is
important because equilibrium need not exist in general, and in some important
games, such as perfect-information games, there are multiple rationalizable actions,
but no action survives the elimination process above. The strictness requirement
is
not binding in generic complete-information games.
4.2.
Genericity of Uniqueness. Let
U = {teT\\S co
be the set of type
1,
Lemma
profiles
5 implies that
semicontinuous,
U
is
[t]\
=
1}
with unique rationalizable actions. Together with
U
is
dense in universal type space.
also open.
Since S°°
is
This yields the main result of the paper:
excludes a nowhere-dense set of types, there
is
Lemma
upperif
one
a unique rationalizable action for
each remaining type, which must be continuous in player's belief hierarchy.
PROPOSITION
2.
Genetically, there exists a unique rationalizable action,
generically continuous.
That
is,
there exist an open, dense set
U
and
it is
and a continuous
RATIONALIZABLE STRATEGIES
function s*
:
U—
rationalizable strategy
Proof. Since 5°°
U
show that
—
a sequence t(m)
i(m) G
U
U ^ T =
for
>
set
U
Proposition
and
its
and every
observe that, by
£
with S°° \i(m)]
=
U D
T.
Hence,
U
T, showing that
of the proposition
is
2,
{s* (/)} for each
s*
:
G U. In particular, every
t
continuous on the open and dense set U.
first
each m.
continuous function
By
is
=
[t]
upper-semicontinuous, by part
is
[t]
dense,
is
S 00
A, such that
>
19
U -* A
with 5°°
Lemma
by part 3 of
we can
But 7
By
=
[t]
any
5, for
some a G
{a} for
dense.
is
Lemma
Lemma
of
1
= T
1
by
part 2 of
\i\.
Lemma
Lemma
{s* (t)} for each
On
multiple rationalizable actions.
continuous.
By
Proposition
On
definition,
1.
Therefore,
The
last
open and dense
every rationalizable strategy
1,
1 is
common-knowledge type
profile.
When
T
into finitely
|0|
=
the original
1,
when T
game
their boundaries
many open
{t\S°°
is
T\U = Uae4 9U a
i.e.,
,
set
which
the unique rationalizable action
S°°
[t]
for
any such
profile s
each
t
t
€
0U a At
.
any
t
,
G
is
= {a}}
where
(aeA),
Ua
is
U a The
the closure of
.
open
sets
U, while their boundaries cover the boundary
a nowhere-dense
profile.
Since S°°
dU a n dU a
with multiple rationalizable action
must be discontinuous,
One
sets
[t]
dU a = U a \U a
form a partition of an open, dense
of U,
not
is
0.
Ua =
and
is
not superfluous.
Proposition 2 uncovers an interesting structure of the universal type space T.
can divide
part
the boundary, each type has
For example, a complete-information game can be modeled with
U=
By
U, each type has a unique rationalizable action,
discontinuous at each finite type on the boundary Assumption
dominance-solvable,
exists
4.
is
rationalizable strategy
consists of a single
17.
partition the universal type space to an
boundary T\U.
open. To
is
there exists a
4,
6
t
U
€ T, there
t
5"°°
4,
set.
is
On
each open set
Ua
a
,
upper-semicontinuous, a G
'
,
both a and
profiles,
as there are sequences
a'
are rationalizable.
At
every rationalizable strategy
t (a,
m)
—
>
t
and
t
(a',m)
—
>
t
MUHAMET
20
with
s (t (a,
Here,
=
m))
a and s
(t (a',
m))
=
a',
YILDIZ
where
t (a,
rationalizable strategies are rendered discontinuous at
all
the generically unique rationalizable theory changes
In
m) G U a and
summary, Proposition 2 establishes
that,
if
t (a',
m) £
by the
t
a '.
fact that
prescribed behavior at
its
f/
one excludes a nowhere-dense
9
t.
set of
types, then there will be a unique unified theory of rational behavior for the remain-
ing types, and
it
will
be continuous with respect to players'
or multiplicities arise only
on the nowhere-dense boundary of the open and dense
U, where the unique unified theory above changes
ers.
beliefs. Discontinuities
set
prescribed behavior for play-
its
Hence, from a theoretical point of view, for generic situations, rationalizability
leads to quite robust predictions:
beliefs sufficiently well.
for this prediction;
This
is
We
we can know the
do not need to know
common knowledge
players' actions
of rationality suffices.
The usual
a theoretical robustness, however.
specify the players' beliefs with such a high precision that
practical problems with
if
One may have
may be
it
to
impractical to
prediction with any reasonable level of precision. For example, a finitely-
repeated prisoners' dilemma
solvable
their
about the strategies
their beliefs
dominance-solvability and other robustness results do apply here.
make any
we know
if
game with many
we introduce small
trembles, but
predictions will dramatically change
when
become dominance-
repetitions will
it
is
well
known
that the equilibrium
the probability of an "irrational" type
exceeds a very low threshold, such as 0.001, as shown by Kreps, Milgrom. Roberts,
and Wilson (1982). Moreover,
in application,
nalizable actions, suggesting that our
we
typically have a large set of ratio-
common knowledge assumptions
lead us to the
boundary of U, and the present economic theories are about these nowhere-dense
set of types.
It is
also a general possibility that
t
that there cannot be such a finite type;
t
£
dU a \\J a '^ dU a
,
£
it
dU a \ U a '~ a dU a
implies that
t
£
for
some
~
(
1
a es°=[tl^
there are multiple rationalizable action profiles (as
action profile that remains rationalizable on an open neighborhood of
strategies
may be
continuous at
t.
a.
But
a f° r
t
t,
Lemma
eac h
i
5 implies
£ T. At any
£ T\U), but a
is
the only
and some rationalizable
RATION ALIZ ABLE STRATEGIES
Also, the result
is
21
true for a (strong) topological notion of genericity with respect
to a (canonical) topology.
As discussed
earlier,
it
need not be true
of genericity. This caveat applies the following
remarks as
Remark
spaces, there
3 (Redundant Types). In
some type
for other notions
well.
may be
distinct types
with identical belief hierarchies. In such type spaces with "redundant types", there
may be
equilibrium strategies that are not rationalizable for the corresponding belief
hierarchy in the universal type space
6.
One needs
if
one
insists
on independence of strategies from
a larger type space to capture the strategically relevant information
encoded in the redundant types (Ely and Peski (2004)).
when
there are "redundant types",
if
On
the other hand, even
the belief hierarchy of a type
rationalizable actions of that type are contained in Sf°
[t-,]
is
rationalizable for
all
types that
U, then all the
(Dekel, Fudenberg,
Morris (2003)). Proposition 2 establishes that, generically, \S°°
a unique action
is
come from
[ti\\
=
1,
and
and hence
arbitrary spaces but
have the same generic belief hierarchy. Then, the universal type space
suffices to
capture the strategic behavior of types with generic belief hierarchies.
Remark
4 (Epistemic Types). In a strategic situation, a player's beliefs can be
put into two groups: the beliefs regarding the payoffs, called Harsanyi type, and
the beliefs regarding the payers' actions, called epistemic type.
In the traditional
methodology, pioneered by Harsanyi, one specifies the former
beliefs as parts of the
problem and
from the former using
infers the latter beliefs, as parts of the solution,
In traditional type spaces, there are often a multitude of
rationality postulates.
epistemic types consistent with a given Harsanyi type and rationality. In epistemic
literature, the distinction
establishes that there
type
if
we assume that
mon knowledge
is
between these two types has been blurred. Proposition 2
indeed a unique epistemic type for a given generic Harsanyi
players are rational throughout the model. Hence, under
of rationality, generically, there
is
com-
no distinction between Harsanyi
types and epistemic types, and a player's Harsanyi type uniquely determines both
the decision problem and
its
solution.
MUHAMET
22
Remark
5 (Unified Theories).
an outcome
scribes
for every
A
YILDIZ
strategy profile in this paper simultaneously de-
model embedded
in the universal type space.
then be regarded as a unified theory. Proposition 2 implies that,
mon knowledge
and each
cases,
of rationality, then
we can have only one
of his unified theories will
if
It
can
we assume com-
unified theory for generic
be continuous (prescribing the same be-
havior for indistinguishable models) at generic type profiles. Kohlberg and Mertens
(1986)
and Govindan and Wilson (2004) seek equilibrium refinements that depend
only on the reduced- form representation and are independent to certain "irrelevant
transformations," including the introduction of mixed strategies as pure strategies,
a transformation that
is
ruled out here by the richness assumption.
I
take a comple-
mentary approach to the same conceptual problem they have addressed. Towards a
unified theory of games, they focus
while
4.3.
show that generically there
I
on developing a uniform equilibrium refinement,
is
only one such theory.
Nearby dominance-solvable models.
with a large set of rationalizable strategy
player's interim beliefs
Since
U is
dense, for any usual
a model such that
profiles, there is
and payoffs are similar to that
if
a
of a player in the original
game, then he has a unique rationalizable action. The game
from
game
is
dominance-solvable
In that sense, one can find "dominance-solvable"
this player's point of view.
games nearby any economic model, although
belief structures in these games.
I
now show
will
may be
it
difficult
to describe the
that one can indeed find a nearby
dominance-solvable model in the usual sense.
PROPOSITION
3.
there exist a dominance-solvable
that r
(t,
m)
—
>
t
Proof. First, take
t
(m) G
T
with
t
1,
for
any model X" C T, and any integer m,
model
Tm
and a mapping r
Under Assumption
as
m—
any
t
m)
:
T"
—* T m
such
oo.
»
£ T. By
(m) —*
(,
t.
dominance-solvable model
Lemma
By Lemma
T m,k
with
1,
there exists a sequence of type profiles
5, for all
integers
member i(m,
m
and
k) such that
k,
there exists a
t(m,k)
—
»
t(m)
as
RATIONALIZABLE STRATEGIES
k -»
T Lm = Tm m
Define
oo.
and t
'
=
{t,rn)
t(m,m).
23
T{t,m) ->
Clearly,
t.
Now,
7
T" by
define
77"=
Since each T'-
m
is
t,Tn
U
dominance-solvable, so
T,i
T"\ For each
is
Proposition 3 states that, given any model,
t
eT',t
(t,
m) G T m
we can perturb the model by
.
D
intro-
ducing a small noise in players' perceptions of the payoffs in such a way that the
new model
will
U
dominance-solvable. Moreover, since
is
is
open, the perturbed model
remain dominance-solvable when we introduce new small perturbations. The
next result states that,
when the
original type space
model can be taken to be part
support.
finite
10
of a
is finite,
the dominance-solvable
model that admits a common
prior with
Moreover, we can do this for each rationalizable strategy profile
model, so that
st< is
st' in
full
the
the unique rationalizable strategy profile in the perturbed
model.
PROPOSITION
Under Assumption
4.
nalizable strategy profile st>
any integer m, there
t (, s T/,m)
:
T
T S7 m
"'
(1)
T' -^
'
-» T^'-' n and f
is
A
models
exist finite
(•,
s T >,
for
1,
any
with
T ST
m)
:
'
,m
and
'^ in
model T' C T, any
£ 5°°
st' (t)
T -» f
T ST
dominance-solvable, and
finite
T Sj m
"'
s T'<
[t]
for each
t
ratio-
G T' and
,
and one-to-one mappings
m such that
admits a
common
prior with
full
support,
(2)
S°°[T(t, ST ,,m)}
(3)
t
(t,
st<
,
rn)
Proof.
By Lemma
model
T Us T'- m
As
in the
away types
to the
*•
5.
t
for
with r
(£,
(t,
each
t
>
t
G T" and m, there
ST',m) G
matching-penny game,
in the
method
—
= S°°[T(t,s T ,,m)} = {s T>(t)},aixd
and f
as m. — oo for each
sy/, m) —
T t,ST
'
,m
*•
exists
a
finite,
as in the proposition.
t
G T'
dominance-solvable
As
in the
this result does not rule out the possibility that
common-prior model have multiple rationalizable actions. (This
of proof.)
is
proof of
some
far
rather due
MUHAMET
24
Proposition
define the finite
3,
model
rpSrpl
Since r
(£,
whenever
Sjv
m
,
m) —>
>
each
t for
some
in for
types, so that t (-,st',
m)
m, we can assume that
t
Since
finite
model
e
by
is finite,
Tm k
'
st',
r* T,,m p ick
_
y
ym,«.
Strategic Equivalence
4.4.
I
any
distinct tjf, r
is finite,
=
k^
^
f
(•
,
sT
,
,
sy m)
,
5, for
r
(£', .s^/
^
an(j T
777.)
=
r'
m)
uniformly for
all
m without loss of generality)
each integer
^
>
.
we can change the index
(Since
all
7^
k,
there exist a
and a one-to-one mapping
prior
goo
(t,
m can be chosen
m > m.
9 in Section
common
^"
rpt,S T l,m
I
one-to-one for
is
Lemma
that admits a
fs T ,,,n
,m _
X", for
m)
by
''
one-to-one for
_^ fm,k such that goc
rps T ,,m
f
T Sj"' m
7
T ST m
Since T"
fa.
is
(-
€
£
YILDIZ
fc )
(-, J7l)
_^ £ ag
O T
(.•
,
fc
Sjv,
and Strategic Topologies. Now,
I
_^
t' (-,k)
:
^ for each
m)
will use the pre-
vious results to characterize the strategies under which the rationalizability corre-
spondence and the rationalizable strategies are continuous. 11
show that these
will
I
topologies are closely related to the product topology.
Fix a player
i.
Define
Ua
(4.1)
Under Assumption
>
by
1,
= te|3r
Lemma
3,
[*»]
=
each
Ua
nowhere-dense, consisting of the boundaries
topology above
will
Oi&Ai
{*}},
is
'
open, and by Proposition
dU a%
be generated by the sets
U
ai
of
open
sets
U
ai
.
Each
and some partition
way, the latter partition will be formed by partitions of the boundaries
Let
7~ s
is
Tf
be the topology on
T,
the smallest topology on
of
Mi
2,
is
strategic
M
dU at
.
t
In a
.
generated by the rationalizability correspondence;
T
r
with respect to which Sf°
is
continuous.
%s
is
A correspondence F T —> 2 A is lower semicontinuous iff for each a £ F[t] and each sequence
t (rn) —
there exists m such that a £ F (m)] for each m > m. A correspondence is continuous
:
>
if it is
t,
[t
both upper and lower semicontinuous.
RATIONALIZABLE STRATEGIES
the smallest topology that contains
Tf =
i
25
the sets of the form
all
=B
{U\Sf°[U]
i
BiCAi.
},
Clearly, the set
P? = {Tf
T
a partition of
is
the union of some sets in
U =
ai
Since
M
of
%
G "Pf
T^
Types U and
.
Finally, let
Tf =
I
for
the
empty
each «
Pf
consists of the
2
,
5.
7~ ,s
Ti\T( G
fl
There
f
p|
j\ an(j
j\\U ai
Z
for each a,,
iff
G T^'
and a partition
'
5?°
some
for
7^
[U]
s
,
=
Sf°
T Bl
t
G
[t' ].
z
Vf
which defines
1
Tf
1,
Ua
sets
.
7
and continuity of the correspondence S°° on T;.
exists a family of closed sets
Moreover, under Assumption
rja,
',
open
iff ti,t[
TS
Tf then T \T/ G
T[ G
If
that can be written as
be the relative topology on
'
the smallest topology that contains both
is
set.
are strategically equivalent
T-
all sets
Vf and
strategic equivalence for finite types
Proposition
the set of
is
t
i
are said to be strategically equivalent
t[
U and £
Equivalently,
Ts
The topology
.
t
CA }\{0}
\Bi
t
and
'
A
at e
l;
such that 7~ 5
complement
its
for each a
t
.
the smallest topology that contains both
is
Ua
where
Ca
Ca \
is
*
as defined in (4.1).
Proof. Define
Cm ={U\oi eSt°[U]}
a,G A
'
Since S°°
is
upper-semicontinuous,
Ca
and T^\C a,
for
that contains both C"2
Ub,c.4,
r BiU ^>
G
7~ 5
C
7j.
Since
Ca
'
s
,
so
2
.
is
By
Ti\C ai
.
by
Lemma
{
.
be the smallest topology
definition, for
-
1,
%
Let
t
proposition because, under Assumption
az
each a
in 7;
is
closed.
is
B C ^, T B = (Ua lG B, CU ') \
s
Therefore, 7j = 7~
This also implies
other hand, for each
that 7~ s
.
'
'
l
7"
Hence,
.
each
C
ai
C Tf On
.
C a! ) e
'
(Ua,$?B,
o,,
^
=
the
showing
the second statement in the.
5,
Ca
>
fit,
= U a nT,
'
for
each
D
.
Proposition 5 links the topology 7~ s
,
generated by the rationalizability corre-
spondence, to the product topology, by showing that
Ts
{
is
generated by finitely
MUHAMET
26
many
YILDIZ
product topology.
sets that are closed in the
semicontinuous. Under Assumption
This
for the finite types, the link
1,
7~ s
intersect each other only
closed sets that generate
an open and dense
their interiors constitute
boundary of each
Now,
that l~
T ss
let
ss
is
T
the smallest topology on
strategies are continuous. It
_1
sz
When
Lemma
due to
i
ai)
=
{ti\ S i
(
=
ai)
U]
=
is
{U}. In that case, {tj
open
sets
Si
6^,0,6
there are two distinct rationalizable actions a u
Ua U
in
U a\
s"
topology 7^ 55 for each
A, and
a G
r
1
n
(ai)
M
G
tt
t
(§i (ti))
s"
.
all
1
(a')
That
a\
)=s
s i (t i
two types are equivalent
[ti],
{t,}
one can find two
and s" 1
Mi
is
If
is,
T ss
{ti}
with
t2
Vf
G
s
,
z
One can
Vf s
is
Vf s
easily
U and
Ua
>
is
Under Assumption
open for each
a^
1,
and
t'
t
is
are
will
be the partition of
show that
55
7J
is
formed of the open
sets
But by Proposition
.
establishes the following link between 7^
of Ti where
{t t }
the players treat these types equiv-
and hence
M
=
generated by the open
is
t
2,
the
Ua
'
under Assumption
a nowhere-dense set, consisting of the boundaries of open sets
6.
(U))
they are not equivalent, then they
with this equivalence relation.
PROPOSITION
(s,
%,£/?,.
i)
in this sense,
smallest topology that contains
1,
the sets
iff
(t'
i
any rationalizable theory.
and the singletons
rationalizable
G %. Hence, each singleton
be treated differently by some rationalizable strategy. Let
Ti associated
Ri, so
the discrete topology on their boundaries. This topology
said to be strongly, strategically equivalent
alent ly under
G
sz
A,.
G Sf°
= U ai U
all
closely related to the following strong notion of strategic equivalence:
When
The
stronger.
5.
,
l
G Ri with s~
upper-
formed by partitioning the
is
the smallest topology that contains
is
is
their boundaries, so that
with respect to which
x
rationalizable strategies s l ,s l
''
7~ 5
set.
on
is
be the topology generated by rationalizable strategies
x
is
This
set.
because Sf°
is
U
ai
.
This
55 and the product topology.
there exists a partition
A/, is
{U a '\ai G A,}u{A/ }
2
a nowhere-dense set (with respect to
RATIONALIZABLE STRATEGIES
27
the product topology) and such that
(4.2)
vf
(4.3)
7f
That
Ua\
aj
we can
is,
G
and
Ai,
s
= {U^cneAijuiiujlueMi},
s
=
{U^^uma^'CM,}.
dU ai
their boundaries
are strongly strategically equivalent
for
some
Oi,
and a type
generated by open sets
G
tr
Ua
'
dU a
'
and only
if
is
in the following way.
if
is
Two
=
s, (t'i).
This condition
is
T ss
itself.
equivalent to
5'°°
at
is
t
As the opposite bencht{
and
t\
are said
there exists a rationalizable strategy
iff
U
their boundaries.
mark, consider the weakest form of strategic equivalence: types
(U)
distinct types
strategically equivalent only to
a stringent condition.
to be weakly strategically equivalent
sets
they are both in an open set
and the discrete topology on
Strong strategic equivalence
Si
many open
partition the universal type space into finitely
[iJnSf
[fj
^
0.
with
s;
When two types
are not weakly strategically equivalent, they are treated differently under every rationalizable theory.
is
The next
result
shows that
Ua
also closely related to the closed sets
>
this notion of strategic equivalence
that only intersect each other on their
boundaries with respect to the product topology:
PROPOSITION
Under Assumption
7.
strategically equivalent if
Proof. If ij
57°
[i'i\-
then by
and
i\
ifi
t
,
i[
E
and
finite types i t
U
ai
for
some
i[
G
T, are
3,
Lemma
a l G S°°
5, t z ,t-
G
U
ai
Conversely,
.
and a G S 00
1
[t,]
l
,
\t'^\
,
weakly
at.
are weakly strategically equivalent, then there exists a 2
Then, by
Lemma
and only
two
1,
if if
£i;
£-
G
Ua
showing that U and
^
G
Sf
for
t[
\ti
some a u
are weakly
strategically equivalent.
Dekel, Fudenberg, and Morris (2004) analyze the topologies under which s-rationalizability
exhibits the basic properties of e-optimization in usual Euclidean spaces.
such topologies as strategic topologies.
They
call
muhamet
28
Proof of Lemma
5.
Now,
prove
will
I
Lemma
A
5.
yildiz
5
substantial part of the proof utilizes the following
stronger notion of rationalizability, used also by Weinstein and Yildiz (2004).
Wf [U] = A and,
= {aj for some
Strict Rationalizability. Let
and only
if
marg0 X T_
t
B^ (marg eXj4
=
vr
_ i 7r)
and n (a_ G S ^"
1
Kt t
=
1
[t-i])
;
1-
each k
for
{
G
ix
A (9
>
Wf [U]
a G
0, let
z
if
x T_, x A-i) such that
Finally, let
CO
fc=0
be the
nated
set of all strictly rationalizable actions for
if it is
C
S^
For some tu
.
given any belief-closed T", consider any family
V
£
di
is
[ti]
z
G
5 has three
z
[t,]
C
1^°°
main
Lemma
2,
common
is
prior.
following
that
One can show
dense
(i,
sets.
£ N, such that
x
:
—
T'_ t
*
A_i
shows that, when we focus on
6)
Lemma
prior,
Lemma 5
7) will state that for
is
any
true
finite
a nearby finite type for which the action
yields
Lemma
Lipman
(2003),
namely
and the second step one more time, one can show that the common- prior
They show
for each
i
Finally,
t,.
Finally, using the result of
requirement can also be met (as stated in
The
Ai,ti E.T?,
each
12
empty.
Combining these two steps immediately
strictly rationalizable.
5 without a
C
may be
and do not require a common
type and any rationalizable action, there
is
elimi-
is
which are presented as the following
Lemma
second step (namely,
.
fc]
steps,
step (namely,
first
T The
z
[ti\
on functions /
for
strictly rationalizable strategies
tt
[ti\
t
V
Then,
[t-i].
The
three lemmas.
V
tt
The proof of Lemma
each
Notice that an action
Wf
a strict best reply to a belief of
with / {8,t-i) E V_i
for
.
not a strict best-response to any belief on the remaining strategies of
the other players. Clearly, W±
each
t%
a, 9),
lemma
if
az G
that,
then
if
W°°
is
similar to the
W* [U]
G=
is
,
Lemma
main
result of Weinstein
one can change the
Oj x
•
x
Gn
where
empty only on category
9).
G,;
=
and Yildiz (2004).
beliefs at order k
[0, 1]
1 set, i.e.,
for
each
i,
+
and
1
and higher
u; (9, a)
union of countably
=
9j (a)
many nowhere-
RATIONALIZABLE STRATEGIES
so that aj
S k+l
can indeed make a the only member of
{
lemma,
make
use their construction but
I
rounds of iterated dominance
Sk
show that
I
comes from a
also
ii
(i.e.,
is
new type
sure that the
come from
new type
for the
[t t ]
probability only on types £_j that
case,
The lemma
played by the new type in equilibrium.
is
29
finite
U.
t{
To prove
this
assigns positive
models that are solved after k
on these models). In that
singleton- valued
model that
finite
states that one
+1
solved after k
is
rounds
of iterated dominance.
LEMMA
W* [U],
Under Assumption
6.
U such
there exists
for each i,k, for each i%
1,
that
=
(i) t\
^ +1
and U G T?
each
E.T
t
solvable
m—
some
for
For any a G Wf"
1,
.
model
Tm
with type
t i>m
= 0,
Proof. For k
let i
be the type
that each j assigns probability
/
By Assumption
<
S]
1,
1
,
and
for each a,
G
k, (ii)
,
x-xTf
= Tf
G T"1 such that S°°
,
=
[i{\
Now fix any k >
and h
=
{l\3h
(t
:
h
k
_ l1
(I,
t
^
and any
1
.
,
.
such that \S k+l
[i z
.
=
[t]\
1 for
exists a finite,
dominance-
m =
t i<m
]
The
:
W"M
according to which
=
{a,},
and
is
{a,}
and
—
>
ii
as
aj
},
where 9
a]
is
it is
common knowledge
as defined in
vacuously true that
it is
i^
Assumption
=
each
for
i\
belief-closed.
Write each t_ as t_
z
z
=
(/,
h)
and higher-order
are the lower
.)
h) G T_,-}.
and each a_, G
i.
profile
to {6
Clearly, the type space {i}
k.
L=
<
t
oo.
>
1.
{a.-}
I
and integer m, there
[t ?;j
z
=
[ii]
model T~u
finite
for each
t\
T
G
inductive hypothesis
[£_*],
there exists finite
W*
[L, [a_ 4 ]]
is
where
=
(i^
t
,
2
£
_^
beliefs, respectively.
that for each finite £_j
[a_,]
=
U,
[a-i]]
=
{a_<}
L,
/
/i [/,
a_j) g
r!.~
=
.
.
,
f^"
Let
(/,
h)
i[a_il
such
for
each
that
(IH)
and
t
G
T'-'t a -l
=
J*-*
1
"-'
1
j *-i[a-i]_ Take any
,
x
a
•
z
•
•
G
= 5^
x r*-*^*
W*
[£»].
1
is
I
[L,
,
k
a finite model with \S
will construct a
type
ii
[t}\
=
1
as in the
lemma.
1
MUHAMET
30
By
BRi (marg 0Xy4 _.7r) = {a
definition,
marge x r_
=
7r
t
^u and
mapping
define
jjl
(a-i
tx
S^
E
t
where type
i_j [a_J
=
(l,h
a_
[I.
[t-i])
«t-.
is
2]
=
—
measurable, and
beliefs
= marg 0xL
marg exL Ki.
= marg
Moreover, by (IH), each
(6, t-i)
1
a unique action a_j E Sz^
such that marge xT^i 71
7
(0,
:
1
[6
is
,
J,
&
/, ft [/,
[I,
a-i])
a_,]
,
—
"
•-*
=
o^
(marg exLxA _.7r)
1
,
x T_ by the mapping
\i
2
p
_
C
7r)
4
leaves
and
tx
(marg QxLXy4 _
xl^
2
U by
as in (IH). Define
(about (6,1)) are identical under
x r_ x A-i) such that
x T_,, by
J
well-defined. Since
«£. is
A (0
Using the inductive hypothesis,
1-
>
Since supp(marg 0xLx
it.
E
tt
(M.MU-ilJ,
the probability distribution induced on
bility distribution
—
1
/i:(0,i,a-i)>-*
(5.1)
some
} for
supp(marg 0xLx4i 7r)
:
YILDIZ
£z
o
7r)
supp(/c t
)
-
(0, /)
and the proba-
x A_,
is finite,
intact, the first
fc
/i
is
orders of
:
pT l
i
= marg GxL
(marg0xLxA _.7r)
marg e xL K ir
E supp («£.), which
t1"
(^,Z,^[Z, a_i]
of the form
(
,
1
a -i €
-5-7
a_i)
But 7
.
aj), so that proj 0x£xAi o
7
o
[*-*])
^
is
=
(// (0, /,
1»
^
a_
r
6,1,
I
Thus, there exists a unique
[t-i [a-i]]
>% and
is
fr
))
=
7
[I,
a-i] j, has
A (0
E
=
is 7?
h
x T_; x A-i)
_1
where
k^ o 7
(#,
/,
h
[/,
a_*])
=
the identity mapping, where proj
the projection mapping. Therefore,
marg 0xLx A _.ir
=
K-t o
.
_1
7
o proj
= marg 0xLxyl _
=
1
xLx4 _
i
(margexiXj4 _.7r)
o
^
o
7r.
i
This, of course, yields
marg 0x A _^
But a
%
is
the only best reply to this
= marg 0x4 _.7r.
belief.
Therefore, S\ +
[ii]
=
{a;}.
7- 1
o proj
1
xLx/1 _
RATIONALIZABLE STRATEGIES
Now,
T
will define
I
tl
as in the
31
lemma. Define
T U = ^.jy
Tt-ila-i]
(J
I
\(e,i_i[a_i])€ supp(« t-.)
=
2?
^-
U
(0,t_i[a_i])€ supp(Kf
i
For any
Clearly, r*' finite.
since J
1
*-*! -*)
hand, supp(K Hence, T**
C 6
)
ti
construction, for each
To prove the
l
that
t
00
G
since a,
hm
last
M'^
=
VFf
m =
i\ for
each
for
some
finite
model
rpm
_
k, tl
j>t i|m
as
Tm
(t,
m
k,
Tu m
<
(i)
=
k
[tj]\
C 6 x
1,
[t
vn
showing that
tj
with
-
|5°°[t]|
On
7*..
eac h (6,t-i
for
+1
=
[u]\
W°°
=
]
m
|S
Sf°
-»
[U,
i%.
m+1
[t]|
m =
]
By
1.
For each m,
.
=
first
for
1
such
ti iTn
Clearly, for each
{ch}-
the
and by
1,
=
[t,]|
[tj]
the other
G supp(ft£.).
[o_j])
and hence |S fc+1
and
t_< [o_J,
part of the lemma, there exists
first
and S"1+1
states that
part,
each
m G Tf'
f,,
T ---.
4
G
t
S"
any rationalizable strategy within a
within a nearby
model
with a G
such that
m —>
m
Pick
00
[t]
Tm
1,
any
V to r
(t,
[r (t. a, ?n)] for
model
is
model.
finite
C T and any
model T"
and a one-to-one and onto mapping r
and t G
a G 1F°°
for
finite
finite
a,
m) =
(ti (tj
,
a±,
each (t,a,m), and
m)
(ii)
m)
(•,
,...
,r n
integer
that
(t n
,
m,
maps
an m)) G
,
T(t,a,m)
^
G
U G
as
t
oo for each (t,a).
Proof.
and
a)
<
by the
,
Under Assumption
there exist a finite
each
/
[/,-]
m >
strictly rationalizable
7.
\S
some
for
dominance-solvable model in the lemma.
ie
^-j
The next lemma
Lemma
?;
1
2
each
for
i\
{r~ },
[a - l]
-
statement in the lemma, take any a G
C
[£;]
G Tp\
tj
10
S?+1 [Q ={«,}, |S*
Finally, since
belief-closed.
is
0V0-
rj-'
x T[y
j^-'l -^
G
t_, [a_ t ]
G
tj
supp^J C
x T%, as
"'
)
^ G 2?\{*i},
belief-closed,
is
[a
The new type space
1
a!
G S?
[tj].
Tm
will consist of
types r t
(tj, a,,
m),
for
i
TV,
T/,
Let 5 Z denote the probability distribution that puts probability
on [x] and 9' be the
finite set of all
positive probability.
I
will define
parameter values that some type
tj
1
G Tj assigns
r(-,m) by simultaneously defining the
beliefs of
MUHAMET
32
eac:.h
Sf°
77
(ti,ai,m) about 9 and the others' types
there exists a belief
[U],
that a z 6 £/?, (marg e x4 _
Define
.
77 (t i:
az
,
£
(9, t-i,
2
(i_j, a_j,
a_j)
\-* (9,
m)
a
r_i
',
(#
(i_j, a_j,
:
(#,
fl
,
£
7r
=
The
"a
each
(i9,
77 (£j,
1— »
r_
(6',
z
2
(i_ n a_j,
?
7T
=
— <fyi,a_
=
—
7T*"
the belief of
1/m),
ai
,
in
77 (ti, di,
77 (ti,
m)
which case a
holds, in which case a t
is
a
I
T*
strict best reply,
each
for
77
(/,;,
a l}
-
'
=
(
t
au
m) about
space,
and f_ i>m
m) correspond
:
to a mix-
1/m
there
a point mass at
is
X T_, X
??i)
a_
,
=
z
),
A_
?
)
which
a_i at each
and
J
(9,
generated by the
r_, (i_j,
a_
t
,
m)).
lience
marg 0Xj4 _
1
is
tt*^.
777,/
x A_i
also a mixture.
is
same uncertainty
as
U does when
With
tz
probability
holds the belief
=
a
'
the unique best reply. Then, by the Sure-thing Principle, a
t
is
is
^.m
m), define the belief
,
m))
\
faces the
i.e.,
a best reply.
With
probability 1/m, the equality 9
BRi (marg Gx/1 _7T) =
{a
z
}.
Hence,
6 \V°°
a*
[77
9
(ti,ai,m)]
777).
will use induction to
(ti,
4
occurs with the probability
tti))
(#,a-i)>
+
i)
2
A (0
(i_j, a_i,
t
a
°
new type
(£;, a;,
r_j (£_j, a_ I;
777
(1
and marge x^tt
^
"J
% and with probability
s_ with s_ (r_
margexA
is,
€
since a^
support and such
finite
1,
"
(^
beliefs of 77
Clearly, proj QXj4 _. (7 (f_ vn (0,£-i: a_,)))
That
=
[£_*])
fixed type profile in the
1
Kr.iU^m) O 7" G
m))
+
(
m)). For the new type
profile
with
j4_j)
5^
6
(o_i
'
x
T!_ {
« i T _ t-_
ilS _ ilWl ))
i
— 1/m,
1
r_i (t_j, a_j,
pure strategy
"
4
), tt
r^i (t_j, a_j, m)).
K
where 7
" at
some
according to
(9, i_j, a_j)
eA(9'x
a*
-
i
-<5(
is
ture: with probability
of
u
Now,
(t_i,a_t-,m).
r_,-
m) by
^(t^m) =
where r_
ir
7r
,
n ti
YILDIZ
ai,m) converges to
t\ (ti,Oi,m)
=
show that
£*,
as
m—
>
77
(77;,
a^m) —
marg KT .(t. iOiim
marg **
'°
'"
^,
each
i.e.,
fcth
00. Firstly, the first-order belief
)
=
— d>,
777
->
»
= marg e «*,
+
(
\
=t\.
1
-
777
)
/
order belief
is
marg G 7T'"
RATIONALIZABLE STRATEGIES
>
Fix some k
r
r (t^a^m)
k
l
the set of
l
each (t,,^) 6 7] x
k
-»
t
=
t\ (ti,ai,m)
L be
Let
0.
f
for
-*(^
-»
lim
m—>oo
=
iT
*-i (t
.
all beliefs i'l"
iaiim)ir
marg 0xL 7r
t
"a
=
marg e xl^'"'
»
=
f ij
at order
—
k
and assume that
\,
Them,
A,-.
+
T x(f_. ia _. |m ))
o
'
1
;
f 1
-J
marg e xL 7r'-°-
1
"a o
f I,
£
lim
m—>oo
-
projg^
o
'
7r
*?
[To obtain the penultimate equality, observe that proj 0xL (f_, im
=
proj QxL (e,r_,(,t_ J ,a_ J ,777))
one can choose
Finally,
Tm
case
a t and
m
r^
(0,
9
a
'
^
a
9
'.
rendering t
(£;,
x
some
there exists
?n such that r,
m. Since there are only
In the previous lemma,
new model
will also
if
8.
(t z
Proof.
is
by
for each
Take
Tm
(•,
m)
m), and
and r
(•,
Lemma 3 and part
[i\.
Since T'
W°°[r(t,m)].
:
r, (t,, a[,
since r,
(tj,
a2
m)
many
1
).]
one-to-one, in which
is
each
for
m) —*
,
m)
a,, r?i) 7^ T; (^, a^,
and r
for
each
is
distinct
On
and m.
t2
ij
types, one can choose
the original model T'
V
(ii)
m)
simply defined on type
S°°
^
new model. This
for
1,
any
finite,
Tm
-»
r
(t,
as in
profiles.
m)
such that
2
(^, a,,
(ctj,
—>
Lemma
t
as
7.
Since T"
m —>
>
aQ and each
m uniform.
fh
is
uniform
for all
C T
W°°
Tm C T and a one-to-one
[r (t, 777)]
=
S°°
[t (t,
m)]
=
00.
is
dominance-solvable, r
dominance-solvable and r
t.
strict
stated in the next lemma.
is
2 of 4, there exists fh such that for each
is finite,
£,,
dominance-solvable, then the
model
Since T"
is
the
—
m)
dominance-solvable model T"
(i)
=
£_,, a_,))
show now. For any two
will
I
exists a finite, dominance-solvable
(t,
k
(9,t _~
(6>,
be dominance-solvable, and the rationalizability and the
and onto mapping t
[t]
,
finitely
Under Assumption
and any m, there
S°°
,
a u m)
t[,
rationalizability will coincide in the
Lemma
-
t_,,a_ 2 m))
(,
enough so that r(-.m)
large
other hand, for any distinct U and
m>
1
does not have redundant types, as
a[.
° *:£„,
777
Moreover, by
>
777..
S°°
Lemma
(£,
—
t,
m)]
=
m)
[r (t.
»
00
7,
S'
m)
(•,
[/]
=
D
MUHAMET
34
YILDIZ
Together with the result of Lipman (2003) and upper-semicontinuity of 5°°.
model can be approximately embedded
implies that a dominance-solvable
model with a common
LEMMA
and any m, there
Proof.
T
= S 00
[t]
By Lemma
8,
[t (t,
77?,)]
any
for
1,
(t,
m), and
r
(ii)
(t,
=
[r(l,m)]
m) —
such that
»
m—
as
t
»
— Tm
T"
:
common
that admits a
:
5°°^], and such that f(t,m) -^
— Tm
T'
i
with TV 00
»
—
m
as
Tm C
[f (t
?i(ti,m) plays a strict best reply to his unique belief, one can perturb ?$(£*,
by assigning positive but small probability
wlhch
fj
where
(ii,
?77.)
is
the
S°°
[t],
and
.
(f
.
=
(ti , Tn)|fc) )
r (f
(iii)
(£, 77?)
,
one mapping f
/
—*
00.
(•, /)
fm k
>
:
But since
T'"'
Hence,
solvable.
m>
I,
when
By
I
>
I,
setting
S°°
—
A:)
f
>
(£,
'
->
'
m)
1
Tm
fc
-
fc
''
J "1 ^, such that
as
fe
[f (i,
such that f (f (f
dominance-solvable, by
is
(t, 77?)
,
k)
[f (f (f (£,
T m = Tm m m
'
'
,
/)]
m)
,
=
k)
S°°
,
/)]
[f (f (t,
=
and T(-,m)
S*°
=
each f (f
for
m)
Lemma
f
prior
k)
,
,
Lemma
m)
,
fc)]
(r,
C T
k
-
m)
fc),
,
each
/,
and a one-to(i,
m)
,
fe)
3 and part 2 of
4,
when
-
fc ' ;
I
>
J for
some
dominance-
is
f(-,m) o f(-,m)
o
(-,777)
2, for
-> f (f
I)
rm
and
[t],
l
= S~[r(f (t,m),fc)] =
common
(*,
777.)],
€ Tj™ puts positive
tj
—> 00. But by
that admits a
m)
x T'" on
m)) €
m)} or 5°°
1
>
W°°[f{f(t,m),k)]
(ii)
Tm k C T
this implies that S°° [f (f (f
r.
Tm —
:
rf
x
there exists a finite model
as
[f (t,
parameters on which some type
finite set of
and one-to-one mappings f(-,k)
supp(« T
W°°
puts zero probability without affecting
(£_.$,
Hence, there exist sequences of dominance-solvable models T"
probability.
(1)
at each (0, r_j
=
m)]
,
Since each type
oo.
»
(i)
cxd.
>
each m, there exist a dominance-solvable model
for
and a one-to-one and onto mapping f(-,?n)
5 DO
Tm
model
and a one-to-one mapping t (*,m)
for each
C T
dominance-solvable model T'
finite,
exist a finite, dominance-solvable
prior with full support
S°°
in a larger
prior without affecting the rationalizable strategies.
Under Assumption
9.
this
for
one completes the proof.
Proof 0/
Lemma
5.
there exists t(m) €
Take any
T
t
€ T, and any a G S°°
such that a € iV
r
°°
[t(m)]
and
[i]
t
.
By Lemma
(m)
—
»
f
as
???
7, for
—
»
00.
each
777.
But by
RATIONALIZABLE STRATEGIES
Lemma
since a G
6,
solvable
model
t(m,k)
—
£(m,
Lemma
prior
i (to,
'
fc
Tm =
and
—
>
2"™'™
profile
If
fit
the
bill.
(•, /)
-v oo, and S°°
A:)
as
/
[r (< (to,
We
then obtain a model with a
and
T™ = Tm m m
>
such that S°°
fc),
[t
dominance-
=
(in, k)]
{a} and
=
we only need dominance-solvability, then £(m)
oo.
and a one-to-one mapping r
Jfc)
(to,
/
Now
each m,k,l, there exist a
9, for
m and k, there exists a finite,
[£(m)], for each
with a type
t(rn) as k
>
to)
r ,n
W°°
.',:,
suppose we need a
finite
,
'
T m k — T m k -\
'
:
Tm k
model
/)]
common
-
>
=
S°°
prior,
[i]
=
>
1
common
that admits a
such that r
{a} for every
by setting
t
(m)
=
(i
(m,
common
fc)
—
/)
,
and
t (to, A:)
r{t
By
prior.
(to, to)
,
/.
m)
D
>
.
6.
Conclusion
Usual game theoretical models typically have a multitude of rationalizable actions.
The
predictions of these models then crucially
sumptions about the players'
beliefs
for all rationalizable strategies.
The
—except
depend on the model's
as-
for the
few predictions that are true
may
be, however, a property of the
multiplicity
present models, rather than a property of rational behavior. Indeed, theoretically,
rationalizability generically leads to quite robust predictions: there exists a
rationalizable outcome,
The
finite
and
it is
unique
continuous with respect to the players' beliefs.
models accomplish what one would expect from a model. Each of them
summarizes dominance-solvable situations by abstracting away from the
would have mattered mostly
for
computing the
beliefs at very
details that
high orders.
By
specifying these details appropriately, any rationalizable strategy could have been
made
uniquely rationalizable.
some
ruling out
But then,
refining rationalizability
of these nearby models as the true model.
refining rationalizability, a researcher ought to explain
tantamount to
In that case,
why he can
when
rule out those
nearby payoff and information structures that are nearly indistinguishable from
his
model
at the interim stage, rather
refinement.
than providing epistemic arguments
for the
MUHAMET
36
Appendix
DEFINITION
the graph of F. For each
B/
= argmaxB^
(tt)
F X — 2 Y Gr (F) = {{x, y) \y
A (e x Gr (s'^ )) — 2' 4 by
>
:
:
—
fc
is
5j
0,
>
tion, fix a k
A
Thus,
(by compactness of
(e
is
also
x
x T- l x ^4_ 2
is
)
ous function of
Fv
But, since
x
since
7T
G
A
(
:
S~_~
x T- t x .4_,
com-
also
is
)
continuous and bounded
a continuous function of
[u x {ai,a-i,6)\ is
A
Now, by
definition of
weak convergence, margexT-i
Since T{
tt.
h-»
(tt, cjj)
Gr (B%)
Gr [S^~
x
is
t
(
closed. Since
t
is
I
)
)
S^
0. Moreover, since
marg0 xT._.7r J
[ij] is
Gr
such that
non-empty
(5f )
is
,
A (0
tt
(by
x T-
a,
By
marge xr.,^ =
for
each k
<
closed for each k
t
— T
*
)
2
^t,,
oo,
t2
,
so that
Gr
Gr
is
a continu-
ip
closed.
is
{Gr (Bf )).
Moreover,
one can easily construct a
Sf
=
is
[£;]
is finite,
(5°°)
=
(S!f)
{Gr (Bf))
ip
oo and Aj
<
71"
Consider the continuous
.
definition,
for each
is finite),
Gr (Bf)
compact,
is
x
x T-i) (Mertens and Zamir (1985)),
A (0
J.
A
(s'l' )) x
continuous,
is
closed (and A^i
is
Gr S_~
Finally, since
f
compact and ^
is
1
(<^>
:
<fi
1
(© x Gr
isomorphic to
there also exists a continuous function
ip
Gr
x
is
A
(S'l' )) x
compact.
mapping
compact,
is
Moreover, u
compact.
Gr [St^j C
will
J
x A), so that
1
Gr
x
I
weak convergence). Therefore, by Berge's Maximum Theorem, Gr (Bf) C
definition of
A
.
/
upper-semicontinuous and non-empty.
is
the inductive hypothesis,
Since
Gr [S_~
x
I
By
closed.
is
closed and non-empty.
pact.
denotes
upper-semicontinuous and non-empty by definition. Towards an induc-
show that Gr (£*)
is
V
a'
and assume that SJ~
0,
F [x]}
= argmaxBB^ (margQ X/1 _,7r)
a'
For k
6
,
'
(a^,a_j,0)]
\ui
3
1
B*
define
k,
Proof of Lemma
A.
For any correspondence
6.
YILDIZ
5?°
non-empty.
[U]
f\-<oo
-'
(
=
nfc<oo ^i
r C^f)
*
M^
s c-l°sed.
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.
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