1985
1-29.
(1985):
@
a
of
number
of
correspond-
possible
anyone
to
belong
types,
player
the
.
such
of
.
environment.
as
the
rules
and
game
physical
social
his
about
Ui
of
well
i's
main
as
function
own
parameters
the
as
payoff
of
parameters
cial
player
cru-
in
some
it
that
i
of
summarizes
himself
attributes
and
player
social,
psychological
"
ci
vector
the
can
we
.
..
as
regard
certain
representing
171]:
physical,
1967,p.
Harsanyi,
words
his
In
vector.
attribute
the
calls
he
a
[see
which
vector
one
certain
by
player,
to
concerning
beliefs
and
parameters
all
summarize
was
idea
Harsanyi's
manageable.
more
much
incomplete
in
information
with
making
to
games
useful
be
proved
very
-
which
of
the
introduced
of
concept
[1967
1968]
beliefs.
mutual
Harsanyi
type
to
sequences
infinite
with
work
of
having
the
overcome
an
In
difficulty
to
attempt
the
.
g~es,
.
etc.
of
on
on
parameters
the
on
beliefs
their
own
his
on
beliefs
beliefs
the
players'
other
beliefs
his
of
own
other
the
parameters
the
beliefs
his
on
games,
beliefs
players'
Qn
the
on
beliefs
his
the
of
on
parameters
the
on
other
the
the
players
of
beliefs
games,
His
of
beliefs
his
the
parameters
the
on
probabilities)
subjective
beliefs
games,
(i.e.
"an
each
for
beliefs"
of
infinite
player:
hierarchy
handle
to
one
tions,
naturally
led
is
func-
the
and
the
uncertain
payoff
spaces
strategy
defIning
parameters
the
about
all
are
which
in
In
i.e.
infonnation,
players
games
incomplete
with
a
analyzing
game
1.
Introduction
and Shmuel
Copyright
1
no.
of
to
i
given
any
beliefs
his
Mertens
14,
Theory
Game
Journal
International
from
Reprinted
allow
J ean-Franfois
355
Vienna.
Verlag,
Physica-
...
CHAPTER
17
Formulation of Bayesian Analysis for Games
with Incomplete Information
Zamir
356
ing
Game
to
the
and
Economic
alternative
Theory
values
of
his
attribute
vector
type
assumed
to
players'
know
his
actual
Can
of
idea
be
possible
to
be
in
general
In
other
.
.
player
Each
ignorant
about
is
the
other
formalized
mathematically?
values
defined
of
the
of
set
parameters
the
of
"states
of
the
the
words:
game
can
world"
in
Starting
one
which
from
identify
a
every
point
a
set
8
mathematically
contains
all
Y
well
but
actual
take.
types."
this
all
own
could
Cj
characteristics,
beliefs
yes,
If
would
exactly
the
what
and
any
we
call
defined
we
"the
do
Section
lead
2
space
of
called
T,
in
players?
all
beliefs
of
to
this
some
point
paper.
The
by
generated
in
Y?
space
it
and
This
is
derIDed
Y
there
is
roughly
includes,
8"
states
space
the
the
world
space
(up
satisfy
to
arising
of
some
all
from
possible
8.
types
appropriate
Furthermore,
of
there
a
player
in
homeomorphism)
is
such
the
a
a
well
game,
such
following
two
Y
T
and
of
of
beliefs
universal
possible
all
beliefs
hierarchy
construction
speaking,
that
mutual
infinite
relations:
[T]n;
(i)
the
(ii)
X
set
of
all
probability
distributions
(8
on
[T]n-l).
X
=
=
T
Y
S
y
The
s
first
E
equality
S
and
type
of
-
(n
says
an
is
players.
a
state
types,
of
player
a
1)
that
n-tuple
just
a
This
is
of
one
joint
the
for
Y
player.
probability
exactly
consists
E
world
each
The
distribution
the
of
second
on
formalization
of
S
the
a
state
of
relation
types
and
notion
nature
says
of
of
that
the
'type'
a
other
as
by
used
.
Harsanyi.
in
an
actual
situation
many
of
the
points
in
will
be
considered
im-
Y
Typically
by
possible
of
are
players.
is
uncertain
of
t
turns
other
instance
the
words
the
others).
what
case
This
if
is
all
leads
then
players
to
relevant
know
the
one
notion
only
is
of
some
parameter
what
in
we
8
but
beliefs
call
Y.
out,
finite,
In
for
about
subspaces
I
the
(This
of
Y.
subset
all
as
both
it
and
can
be
most
easily
of
seen,
its
that
beliefs
even
if
subspaces
we
will
start
be
with
sets
a
of
set
high
S
which
is
cardinality.
On
the
Y
other
hand,
many
most
of
possible
by
the
states
proving
work
of
that
any
games
on
the
with
world.
beliefs
In
incomplete
Section
subspace
3
of
can
Y
subspace
which
is
arbitrarily
close
to
it
in
information
we
be
the
assume
provide
some
for
"approximated"
by
Hausdorff
finitely
justification
distance
a
finite
this
beliefs
between
closed
sets.
In
and
Section
4
by
later
consistent
such
that
his
fact
his
consider
the
beliefs
private
own
of
is
each
the
intuitive
the
consistency,
to
equal
the
of
can
test
probability
We
the
hypothesis
distribution
of
any
he
error
is
in
his
in.
In
conclusion.
such
a
test
each
on
player
has
represents
states
of
(subjective)
prove
show
that
the
world
of
the
the
it
to
probability
it
based
on
world
is
belongs
the
in
consistency
player,
which
corresponding
that
that
each
state
to
a
the
distribution
then
states
Harsanyi
world
the
and
sense
that
set
the
all
formally
in
consistent
by
probability
concept
Y
situation
of
on
consistency.
the
the
discussed
state
conditional
this
knowledge,
compute
global
a
distribution
derIDe
common
also
speaking,
probability
meaning
is
yes
of
player
only,
if
a
We
situation
information
compute
there
information.
actual
consistent,
concept
Generally
if
captures
an
the
AumannjMaschler.
situation
given
of
we
and
consistent
0
of
cornitting
Fommlation
357
of Bayesian Analysis
theorem
is
This
own
is
his
what
on
infonned
is
player
Harsanyi's
type.
[Harsanyi,
Finally, in Section 5, we define a game in strategic form determined by the beliefs
space (or subspace). This will be typically a game "with vector payoffs", but the Nash
equilibria are well defined. For a consistent beliefs subset, the Nash equilibria will be
the same as those of a certain extensive fonn game in which the state of the world
is chosen according to the (uniquely detennined)probability
distribution, and each
aI.
et
B6ge
done
direction
same
the
in
works
that
out
pointed
It
should
be
were
by
1967, part II, p. 321] which is in the background of most models of games with in'complete infonnation.
2.
The
Universal
Beliefs
Space
If
who, being interested mainly in the equilibrium points of games with incomplete infonnation, incorporated the strategy choices of the players as part of the space of
parameters on which the infInite hierarchy of beliefs is built.
un-
are
which
of
the
n
members
,
.
.
1=
te
of
.
players
set
a
infonnation
involving
{1,
},
The main objective of this section is to prove Theorem 2.9 which establishes the
existence of a space of infInite hierarchy of beliefs. We consider a situation ofincomple-
spaces
strategy
and
the
of
fu11listing
a
as
S
of
point
a
set
parameter-space.
as
the
S
to
space.
compact
a
in
how,
see
us
let
restrictive,
too
not
is
assumption
this
that
To
Remark:
see
typical
a
S
Assumption:
is
.
the
refer
shall
We
functions).
payoff
S
of
some
of
think
may
(we
certain about the parameters of the game they are playing which may be any element
and rather general model if incomplete information, the space S will in fact be compact: Observe that 8 has most generally to include all the parameters of the game
including the parameters of the utility functions of the player. So let So be th~ set of
of the game. Clearly 80 may ~e assumed
(by
of 8 E 80),
action
his
be
A
The set of outcomes
can
The
compact.
are
is
=
So
and
X
X
C
set
the
with
be
then
identified
compact
if
Az
Az
n
.
.
independent
i
let
player
so as to become
each
For
compact.
set (enlarged
be
to
if
it
enlarging
necessary)
Z
possible values of all the parameters
(continuous)
a
i
is
applying
bounded
be
to
assume
to
want
instance
(for
may
we
-+
ui:
C
which
R,
by
~eal
of
function
utility
Van-Neumann
Morgenstern
player
map
i=l
possible
all
of
set
the
and
that
course
of
is
case
special
A
is
v.:hich
X
metrisable.
addition
in
be
probability
of
space
the
denote
will
II
compact
X,
compact
For
space
any
(X)
So
whicb
Swill
then
finite
are
andA
in
i
=
So
8
then
is
games
comp'act.
[[0,
1]C]n
ui:
-+
take
C
may
we
Hence
Petersburg
paradox).
1]
theory to all countable lotteries, in order to avoid the St.
[0,
the Van Neumann-Morgenstern
of
set
the
in
closed
clearly
is
[1
t
topology.
weak*
the
with
on
endowed
measures
X,
'"
by
theorem. ]
weak*-compact
hence
C
of
dual
Alaoglu's
the
of
ball
unit
the
theorem
(X),
all measures of norm ~ 1 since the function 1 is continuous-, and the set is by Riesz
Game
Definition
and
2.1:
Economic
Theory
coherent
A
beliefs
hierarchy
of
(K
K
level
1,
.
2,
.
)
.
is
a
sequence
358
=
(Co,C1"",CK)where:
1)
is
Co
a
compact
subset
of
S
and
for
.
1,
k
.
.
,
is
K,
a
compact
subset
of
=
Ck
ti
X
(as
[ll
topological
spaces).
(We
by
denote
and
the
(Ck-l)]n
Ck-l
Pk-l
projections
ofCk
and
on
the
copy
i-th
of
respectively.)
n
(Ck-l)
,K.
(2)
(Ck)=Ck-l;k=
1,...
Ck-l
Pk-l
E
let
V
then
(3)
Vi:
=
(ck)'
Ck
ck
ck-l
Pk-l
ti
ti
HI)
the
marginal
distribution
of
on
(ck)
is
(ck-l);
Ck-2
ti
H2)
the
marginal
distribution
on
of
copy
the
ti
mass
of
i-th
(ck)
is
n
the
unit
(Ck-2)
ti
at
=
(ck-l)
(ck»'
(Pk-l
1.
{Pk-l[(ti)-1
t
(Ck);k=
Vi,
1,...
,K,
VtEti
(t)]}
(4)
=
as
interpret
a
set
of
up
beliefs
to
level
k
and
thus
a
point
consists
in
Ck
of
beliefs
up
to
level
1)
-
(k
hierarchy
of
Ck
(Le.
a
point
in
and
for
each
player
i
We
Ck-l)
probability
distribution
on
hierarchies
beliefs
of
up
to
level
(k
-1)
(i.e.
t~
En
Condition
says
HI
that
player
k-level
i's
beliefs
coincide
with
his
t~
(Ck-l»'
1)
level
-
dition
beliefs
says
H2
In
the
that
next
C
whatever
concerns
player
i
definition
we
knows
we
would
like
to
up
hierarchies
his
own
formalize
previous
the
Any
obtain:
to
E
s
bution
on
C
S
and
c
point
which
of
coherent
is
2).
Con-
space
the
of
determines
states
of
uniquely
a
the
set
of
ti
type
the
-
beliefs.
of
ti
parameters
(k
level
order
properties
C
world
in
E
(k
player.
each
in
the
type
The
sense
that
each
is
player
a
probability
distri-
knows
his
type.
own
ti
other
words
if
(C)
En
a
is
type
certain
of
ti
player
is
This
in
motivates
all
points
the
in
the
following.
is
(n
an
3)
tuple
+
abstract
S-based
An
then
i,
ti.
type
of
beliefs
i
2.2:
(ti»
space
(Supp
of
Definition
support
player
(BL-space)
In
f,
S,
where
(ti)?",
C
is
a
set,
compact
(C,
S
is
some
compact
is
space,f
a
continuous
1)
Ii:
ti,
mappings
,
continuous
are
i
Sand
.
C
.
n,
.
1,
mappingf:
C
(with
=
satisfying:
topology)
-*
ti
(c).
(C)
Sup
E
and
"*
=
(c»
c
E
C
c
(*)
(ti
ti
weak*
to
the
respect
(C)
n
-*
confusion
The
only
may
space
C
the
state
is
a
of
space
we
shall
which
in
s
nature
denote
each
S
of
result
the
point
also
but
no
E
When
c
of
all
BL-space
E
C
beliefs,
simply
contains
by
a
beliefs
full
on
C.
description
beliefs
not
etc.
on
S.
ti
In
fact
if
we
interpret
as
player
i's
(subjective)
probability
distribution
on
C,
then
a
EX"
hk+1
hk
t1,
=
(s,
ti,
.
.
.
,
t~,
.
.
.
,
tk,
.
.
.
.
.
,
(c)
,
,
t;,
tk+1'
X"+l'
h"+1:
,
.
.
.
,
t;+1)
EXk+1
where
c
Vi,
and
C
want
E
then
any
~
k
=
ho
follows:
as
derIDed
are
Assume
inductively
h
mappings
(c).
(c)
f
k
k
X
on
h
of
the
is
.
projection
h
words,
(c)
(c)
h"
Pk
=
(hI
k
~
1
(c)
(c))
~
X"
h"
that
such
C
:
a
.
.
.
0,
mapping
2,.
,
~
done
be
will
for
This
X
C
continuous
each
h:
derIDing
by
mapping
(S).
~
(ti)~=1)
a
now
defrne
we
To
natural
certain
BL-space
abstract
S-based
S,
(C,
any
f,
c
c
-
.'
J
=
J
dJ.l..
on
functionf
continuous
(c)
I{J)
C,
df1
f(E)
(fo
Jl
~:
(0
lor
to
which
such
n
n
maps
n
n
that
E
E
.
mapping
any
(C)
(C)
(C)
f1
~
~
(0
the
induced
canonically
n
denote
n
the
by
mapping
by
namely
I{J,
(C)
~
two
between
we
compact
continuous
a
C
If
and
C
spaces
mapping
is
C
I{J:
C,
~
(Xk-1)'
=
n
'
k
Tk
t1
d
tZ,
=
=
t~
x
for
where
.
.
each
i
each
.
,
.
t
.
.
.
.
k
and
.
.
.
.
E
),
,
(s,
,
,
,
.
X
in
a
denote
shall
typical
We
X
write
we
as
point
shall
space
(8).
(fl)
to
we
want
whenever
and
S
X
Note
the
generated
is
that
generating
specify
by
=
well
a
derIDed
is
which
Defme
is
so
when
compact
S
X
S.
space
[Tz]n,
X
X
also
~[Tz]n;
Take
C
derIDe
to
we
derIDed
is
~
=
k
Xk=Xk-1X[Tkr=SX
then
..
.
.
.
.
.
Xk
'"
C
The
other
in
Tk
(Xk-1)
=
n
Xo
t~,
d,
(s,
(c)
hk:
k'
k'
X
T
spaces
the
we
S
Given
define
by
discrete
a
the
probabili~
i
which
dition
in
positive
assigns
that
basically
player
says
(
one
associated
beliefs
is
c
with
fmd
we
inductively
of
C
each
that
Proceeding
i.
player
E
(tj)'-L"
hence
and
on
a
defines
also
of
beliefs
distribution
probability
But
i.
player
ti
combined withfit
=
tZ)
=
let
Fommlation of Bayesian Analysis
359
defrnes a probability distribution on S, which is the first level
J-r-l
on the first level beliefs of the other players. This may be called the second level
infinite hierarchy of beliefs for each player. The condition (*) is a consistency concase) only to points of C in which he has the same beliefs. In other words he is certain
of his own beliefs.
Let us write now fonnally the above mentioned observation:
=S
Z=1
k=1,2,...
00
Z=1
"'cEC,
where ~' is
the
mapping
~':
II
(C)
;>
nee)
II
canonically
each
i,
mapping
S
.....
e
(ti)f=l)
8,
(C,
f,
properties)
will
induced
NR-
C.
in
non-redundancy
whichfis
for
measur-
C)
given
the
points
the
of
redundant
an
to
homeomorphisms
also
41
S
f
i
41'
.....
i1
)
e
t1
41'
,..
l~
'"
(C)
by I{)'.
8
preserve
formally.
to
isomorphism.
satisfies
distinct
two
formally,
an
is
f,
satisfy
subsets
8.
is
proceed
we
it
(ti)f=l)
to
F.
(of
on
which
now
and
consider
continuous
hence
each
said
is
BE
a-field
smallest
the
be
structure
beliefs
something
for
a
to
that
aErospace
like
is
topological
8,
1)
F
let
more
one
(C,
separates
'rj
(B)
(ti
(ti)7=1)
nonredundancy
to
one
F
(ti);=
f,
'rj
f,
by
such
S
from
would
I{)'
their
also
BL-space
a-field
measurable
i,
8,
the
to
only
n
to
where
we
is
~
8,
(C,
is
(c»
and
(C,
differ
onto
calledBL-morphisms
addition
be
BL-spaces
H
C
abstract
discussion
>
8
I{)')
(I{),
(BL-morphism)
(in
the
if
BL-space
able
of
:
h
A
2.4:
Definition
notion
and
pair
will
with
8-based
an
If
By our previous
of
a
which
8
comutes:
mapping
is
morphism
mappings
mapping
(NK.condition)
condition
concerns
(0
1
(ii)f=l)
These
2.5:
Proposition
this
continuous
l.
them
beliefs
dealing
S,
define
the
then
condition
whatever
diagram
a
I{)
A
In
is
(C,
2.6:
structure.
belicfs
define
following
the
Definition
BL-spaces
and
C
onto
aBL-space
to
To
,11,
l,
C
of
in
.
.
.
i
cal
~
=
c'
c'
identi-
are
and
c
c
for
h
clear
unless
h
that
intuitively
is
it
fore
:1=
:1=
pretty
(c')
(c)
There-
8.
on
and
8
information
all
contains
the
beliefs
concerning
possible
h
image
(c)
to
we
construction,
H
write
shall
we
8
the
want
underlying
emphasize
When
By
(8).
~
=
0S).
It
X
h
H
X
:
h
the
that
follows
Let
continuous.
is
C
defined
so
(C)
(8)
~
k'
h
induced
canonically
by
Tk+1'
tk+1
hk
hk
hk:
=
=
(Xk)
(Xk)
the
is
and
C
II
II
mapping
II
(C)
0
ti:
~
~
360
Game and Economic Theory
and
BL-space
we thus have:
between
the
Y
to
of
Y).
a
called
BL-subspace
-condition,
is
canonically
Y
Y
T').
S
and
f:
~
~
ti:
Yj/
Yon
the
also
projection
of
by
{Yk}k=O
limit
the
respect
projective
the
is
projec-
natural
to
(with
{Yk}k=O
Yk)
Y1,
"if
k
(Yo,
.
.
.
coherent
a
is
S.t.
spaces
beliefs
,
}
=
T
.
n
X
[T]
(8
2)
)
-
.
n
1
..
T
n
8
Yand
there
positive
and
For
2.9:
Theorem
are
integer
compact
spaces
any
theorem
NR
the
Pk
denote
We
Y
compact
3)
up
projections
Yk-l.
~
are
There
the main
be
satisfies
the
(with
BL-space
Yk
and
hierarchy
to state
will
of
which
BL-space,
S-based
an
is
tionPk-l:
We are now ready
(which
compact
Y
4)
such
abstract
S-based
Any
5)
S.
8
induced
v;:
the
by
mapping
~
BL-morphism
is
This
C.
BL-space
to
C
BL-space
from
V;
BL-morphism
of
speak
the
notation
our
shorten
shall
we
and
2.8
Lemma
of
In
Remark:
and
view
terminology
the
satisfies
C
since
itself
c'
it
NR-condition.
determines
uniquely
=
with
associated
infinite
each
hence
and
(e')
Ii
hierarchy
the
c
<p'
Cuniquely
e'
E
(e),
cP
that
In
with
combined
is
the
of
idea
the
for
determines
the
proof
words,
diagram
'P)
=
Ii
<.p'
(c).
(h
(c)
0
-1
hence
2.5)
Proposition
one
is
NR-condition
satisfies
invertible.
and
one
to
Ii
the
(by
'P)
=
(c»
«p'
we
C
have
C
c
that
Since
EX
Ii
"if
implies
computes
(h
E
(8).
(c)
0
'P
the
that
The
of
fact
8
2.6
Definition
diagram
replaced
is
underlying
(fl).
by
'P:
('P
~X
which
the
which
C
c
the
(8»
X
h
in
h
to
maps
E
mapping
(8)
(c)
C~
0
we
X
X
C
Ii:
and
C
:
h
notation
our
denote
Proof:
Using
by
(8)
(8)
~
~
'P.
then
NR-condition,
the
the
and
determined
is
satisfies
latter
uniquely
<.p'
if
by
1,
Lem11'1£Z
(ti)~l)
from
aBL-morphism
to
(ti)~l)
is
If
2.8:
8,
8,
(C,
<p')
(C,
(cp,
f,
fact true provided
subset
a
BL-homeomorphic
cp~
be
in
This
the
detennined
to
is
diagram.
above
via
seems
two
not
and
since
<.p
by
('P,
C
to
from
essential
one
is
there
then
C
actually
BL-morpbism
a
mapping
is
if
<p')
BL-isomorphic.
be
to
said
are
BL-spaces
(P)F=l)
(ti)~l)'
from
s,1,
to
aBL-morpbism
is
two
('P-1,
The
8,
('P')-1)
(C,
(C,
f,
('P')
and
exist
and
Inverse
the
BL
if
-isomorphism
a
called
mappings
cp
-1
-1
1,
from
BL-morphism
A
is
(ii)~l)
to
f,(ti)~l)
2.7:
Definition
8,
8,
cp')
(C,
(C,
(cp,
Formulation of Bayesian Analysis
361
Some thought on the diagramm of Defmition 2.6 leads us to the observation that
C satisfies the NR-condition:
Therefore:
of this section.
that:
l)Y=SX['nn
to BL-morphisms.
Yk'
Pk
The
I"
proof
=
Y
0.
in
(Y)
V
Yk)
hierarchy;
Y1,
.
(Yo,
k,
i)
,
\j
2.11:
Corollary
corollaries.
following
has
proposition
This
Yk'
Pk
(Yk+l)
=
k
0,1,2,...
2.10:
Proposition
Yk'
Pk
(Yk+l)
=
=
we
This
k
that
0,1,
(2),
namely
all
k,
(ii)
Yk
Yk)
'if
have
of
Now
definition
,
by
properties
Yk-l'
Pk-l
Yk
Y
a
well
natural
the
to
respect
--+-
{Yk}k=O
Yk
'if
of
k.
compact
is
inductively
that
Yk-l
Yk
=
Yo
follows
is
then
compact,
if
is
Yk
It
(a)
conditions
the
(b)
Note
X
noted
have
we
As
already
(X)
(Yk-l)}'
(Yk-2)
i-th
is
on
the
copy
ti
Yk-2
(Yk-l)
(Yk)
on
is
and
(b)
ti
ti
ti
Yk-l
Yk
=
I
(Yk-l)]n
'ifi
distribution
X
E
(2.1)
(a)
=
=
Yo
S
1,
{Yk}k=O
of
the
sequence
Define
Y
Construction
of
Y
T
and
we
shall
Then
satisfy
Y
Y
of
its
as
define
Ti
{Yk}k=O
Proof:
constructing
shall
We
by
(3)
S
generated
Universal
the
space
type
Y
called
be
BL-space
will
(and
n)
,by
Y
Y.
and
which
BL-subspace
Any
of
Y
T
Y
a
This
respectively.
and
map
f
can
Yand
7)
mapped
satisfy
Any
Ck'
Pk
=
Y
k
S.t.
0,
,
(C)
of
K)
C1,
.
(Co,
coherent
beliefs
For
,
any
6)
362
Game and Economic Theory
onto
set.
of
of Proposition
*-
particular
coherent
2.10
will
beliefs
immediate
the
=
follow
from
(Definition
2.1)
the
following.
now:
prove
hierarchy
..
beliefs
.
coherent
a
is
.
a
.
automatically
.
.
.
satisfy
(Yo,
k,
that
we
except
compact
derIDed
is
:
projections
Y
with
limit
projective
the
be
Let
it
compact,
S
Since
compact.
also
is
for
condition
that
follows
conditions.
closed
are
of
definition
the
in
and
that
also
compact.
is
also
n
then
if
compact,
is
at
mass
unit
the
n
of
(Yk)
of
distribution
the
marginal
{Yk
of
marginal
the
[n
.
.
.
k
for
and
2,
follows:
as
spaces
the
properties.
required
these
that
prove
i's
on
as
limit
coordinates.
player
projection
the
and
projective
and
in
sequence
the
theorem
the
prove
(and
n).
by
be
Twill
and
Universal
the
called
S
generated
satisfies
be
can
mapped
BL-morphically
6)
3)
and
between
BL-homeomorphism
a
induces
onto
be
continuously
5)
and
or
2)
and
1)
which
4)
=
K.
.
.
.
C
a
BL-subscpace
is
there
C
.
.
hierarchy
Y
I
(
Fonnulation
Lemma
2.12:
q
E
Let
(A).
II
whose
A
and
necessary
A
B
be
compact
and
marginal
distribution
sets,
sufficient
of
D
a
compact
condition
on
A
q,
is
q
that
the
Don
Proof:
A.
Since
D
X
CD
A
q
(D
1.
Define
L
the
necessity
is
fdq.
if)
=
A)
B,
=
is
L
a
where
prove
To
X
p
Band
E
(D)
II
the
DAis
=
obvious.
A
of
1,
A)
of
tion
of
existence
(D
363
Analysis
subset
for
is
Bayesian
projec-
sufficiency
the
assume
(D
linear
functional
defined
on
(the
C
J
q
q
A)'
continuous
real
functions
on
D
we
consider
a
function
on
D
as
of
If
space
A
A)'
linear
(fl,
f
function
D,
on
by
the
natural
F
defInition
(a,
b)
b)
\t
ED,
and
write
=
(fl)
a
a
linear
functional
positive
derIDed
functional
on
a
L
with
linear
of
subspace
/I
II
=
C
This
clearly
a
extension
(D).
q
is
then
is
Hahn-Banach
Lq
J
By
dq,
F
=
1.
(F)
L
q
Riesz
representation
fdp
if)
=
theorem
\tIE
C
J
there
p
This
(D).
is
is
the
L
a
of
nonn
probability
1
extension
(D).
C
p
E
(D),
II
q.
of
on
[II
Yo.
to
(y,
E
can
be
extended
to
a
point
E
pointy
,
any
.
that
.
show
.
Pk-1
Yk'
Yk-l
Pk
=
(Yk+l)
=
we
have
words
In
that
other
prove
us
(Yk)
let
and
Po
=
=
Y1
=
=
Yo
that
Assume
(Y1)
thus
X
S
Sand
since
a
(S)]n,
k
on
measure
required
the
2.10:
Proposition
Proof
of
D
functional
for
L
linear
inductively
by
s.t.
positive
proposition
Finally
a
It
q
to
holds
extended
k.
be
prove
can
L
We
theorem
Yk+l'
t;+l)
Yk
(a)
probability
of
definition
()
tk+l
in
and
conditions
the
atisfying
s
distributions
E
(b)
l'
1
tt+
t;+
of
.
.
.
n-tuple
an
of
the
existence
establish
to
have
So
we
,
tt+1'
that
the
marginal
disitribution
on
X
is
tk
(Yk-l)]i
X
[II
namely
iCy)'
Yk-l
Yk'
ti
is
the
(y).
element
which
II
0
assigns
mass
1
to
We
have
thus
supported
[II
by
its
subset
Yk-l
t~+l
Yk
(Yk-l)]n
(Yk-1)h
i.e.
X
.
.
.
X
X
on
[II
/
distribution
probability
a
to
extended
can
be
these
of
each
that
to
show
iCy)
marginals
(Yk-l)
where
2.12
it
remains
to
prove
that
()
tk+1
=
(Yk)
1.
Using
Lemma
(ti
{ti
(y)}
(y)
[ti
(y))
Sup
Yk-1
Yk
(Yk-l)]r
on
of
X
projection
[II
t
(y)
i
~
=
X
Supp
]
X
{ti
ti
(y))
ti
let
E
So
(y)
Supp
ti
(y)}
(y)
Supp
E
X
Since
~
Le'Yk-1
(y)).
that
(Pk-1
CYk-l)=ti
toti.(Pk-1
follows
1
(y)assignsprobability
(y))it
ti
ti
CYk-l'
Yk-l'
by
hypothesis
extension
=
there
(Yk)
an
is
induction
,
Since
ti
Yk-l'
Pk-1
replace
we
point
in
we
'
CYk-l'
Yk'
'i£
this
'i~
if
that
claim
We
E
'i;)
.
.
.
,
(y)
by
ti
a
point
which
is
also
in
(y))
proving
that
is
in
the
projection
of
obtain
(Yk-l)]i
Yk-l
Yk
and
X
on
[II
Yk'
(Yk-l'
thus
completing
the
proof.
364
Game
and
Economic
Theory
ti
To
see
that
.
.
.
,
E
(y),...,
t;,
note
cerning
that
iare
satisfied
.
since
.
.
,
E
The
t;)
j"*
con-
conditions
concern-
Yk'
CYk-1'
tk'
y
(y)
ing
conditions
Yk'
tk'
ti
ail
t:)
CYk-1'
are
satisfied
since
these
are
the
conditions
required
E
for
(recalling
Yk
ti
ti
(y))).
that
This
completes
the
proof
of
Proposition
2.10.
=
CYk-1)
(Pk-1
y
Remark:
Note
that
E
when
is
su~h
that
all
distributions
are
Yk
of
support,
finite
tf
y
the
extension
of
to
a
point
in
is
Y
k+
support,
can
be
pointed
y
,
any
.
.
)
.
an
extension,
also
with
and
for
each
EN,
i
consider
the
sequence
of
prob-
Yl
=
Y
(Yo,
For
and
explicitely.
at
E
finite
straightforward
1
(I
(y
abilities
ti
};'O,
.
(Y2),
d,
.
.
on
Y1,
.
Y2,
.
respectively.
.
By
the
definition
of
ti
ofti
this
sequence
satisfies
that
k,
"if
the
marginal
is
on
CYk).
{Yk}k':O'
CYk+1)
Yk-1
Since
(Y)
also
k,
it
follows
that
any
for
continuous
functionfK
real
on
V
=
Yk
Pk
ti
f
which
Y
depends
only
on
K
coordinates,
the
sequence
integrals
of
(k
(J
d
))k=
K
1
(ti
is
well
defined
and
constant
for
k
~
1.
K
Therefore
the
sequence
+
CYk))k=1
defines
positive
a
and
of
norm
1
on
the
space
of
all
such
functions
fK
hence
on
functional
lin~ar
on
By
the
closure
Riesz
of
space
this
representation
which
theorem
is
there
the
is
space
of
uniquely
a
all
continuous
functions
probability
determined
Y.
measure
in
(Y)
n
Definition
which
represents
this
linear
functional.
2.13:
on
distribution
the
probability
(y)
define
described
above
the
(Y).
n
ti
~
=
r
Let
(Y)
(ii)
(i
Y
in
determined
by
way
y
v:i
Y
EN,
and
E
eachy
For
by
ti
(i)
Remark:
Note
that
the
are
mappings
continuous.
Ti
Clearly
all
spaces
type
that
and
Y
3)
that
is
the
same
space
respectively
by
in
T
S
which
the
(and
fact
n),
and
satisfy
we
universal
by
satisfied
the
the
by
denote
T.
beliefs
rest
of
space
properties
this
section
and
is
the
universal
in
claimed
devoted
prove
to
Theorem
2.9.
So
far
we
construction.
.
r]
i=l
=
Y
S
X
(homeomorphically).
[X
1:
Property
n
have
are
generated
these
of
and
Y
space
copies
T
The
are
Proof:
First
let
us
establish
a
one
to
mapping
one
between
the
two
sets.
Each
y
E
determines
uniquely
some
Y
y
s
ES
(namely
(y)),
s
=
ti
deter'minesuniquely
(y)
also
Po
by
definition
of
Ti
E
V
i.
This
establishes
a
mappingf;
Y
-+
S
X
Ti].
[X
i=1
the
other
hand
by
its
definition
Y
On
can
be
represented
as:
T,
=
Property
T
Y
2:
BL-Space.
abstract
Y
have:
we
S,
'projection
~
,tk-1)
tk-1)
,tk-l)'
(t1'
(t1'
-
V
.
.
k
.
.
'"
IT(8 X [T]n-1) (homeomorphically).
thus
and
.
.
Therefore
.
.
-
,
i
i
7i
i
-i
.
'
I
1
)]'
to
t
of
1
[IT
X
on
distribution
probability
assigns
marginal
the
that
of
(Y
(2.1)
l
.
k.,{
-
itl)'
Y
follows
and
repeatedly
using
it
.
.
properties
sirice
But
.
b)
by
a)
E
,
(if,
y
on;~
(Yk-1)
oft:
(Yl)]i
IT
contains
the
distribution
of
[h
E
marginal
support
Y1'
.J
the
V
Yo,
on
k
that
..
of
implies
E
respectively
(Y)
and
Supp
(y)
(y))
(ti
ti
ti
Proof:
of
the
be
.
t~,
distributions
marginal
sequences
.
and
Let
.
.
(d,
.
)
(if,
'"i1,
)
,
=
Y
'\/
then
2.14:
Lemma
(y).
Supp
E
ifyE
iVy
(y)),
(j)
(ti
ti
ti
'
proof.
the
of
rest
the
for
be
will
needed
the
of
important
an
The
which
f
property
establishes
lemma
m.appings
following
i=1
Y
T')
a
Hausdorff
is
and
is
X
S
compact
space.
(X
n
.
i=1
a
is
since
homeomorphism
it
therefore
and
continuous,
and
X
one
to
one
is
(X
TZ)
n
,
.
i=1
the
and
ti
(y).
ti
.
i
1=0
of
continuity
=
(Y)
(t1'
k
of
the
2.14,
Lemma
of
consequence
immediate
an
As
ti
Po
Y
Po:
Y
the
X
So
is
the
X
S
continuous.
S
clearly
mapping
projection
tz:
~
~
n
.
Y
Y
i
'\/
is
Also
is
compact
hence
IT
compact).
(since
and
continuous
are
(Y)
ti:
Ti
~
Yk'
be
that
on
functions
continuous
can
the
This
approximated
implies
mappings
by
Y
on
Now
function
continuous
theorem,
Stone-Weierstrass
that
note
any
by
i=1
f.
Y
of
inverse
the
be
to
is
verified
easily
which
X
S
g:
[X
Tz]
~
.
n
.
Y1,
Y.
(Yo,
a
have
we
some
to
.
sequence
SO
E
corresponding
.
.
mapping
y
)
in
Thus
from
derived
point
any
a
S
E
any
determines
X
uniquely
[X
Tz.
TZ]
tZ
'
n
.
,
(Yk));=1'
{Yk};=O
(tk)k'=1
which
on
the
satisfied
are
sequence
sequence
by
(ti
(Yk-1)'
the
on
k
(Yk)Ell
only
'\/
f
conditions
are
and
conditions
the
i
certain
Butfor
b)
a)
(Yk-1)
(Yk)
and
offonnula
and
conditions
IT
(2.1)
b)
a)
E
ti
Y={yo,(t1
S-based
an
is
4:
Property
Formulation
of Bayesian Analysis
365
(Yk));=1,...,(tn(Yk));=1IYoES,\/k'ii
are satisfied},
i=1
366
Game
Proof:
and
We
shall
Economic
prove
Theory
that
ri
Vi,
is
homeomorphic
to
n
(8
X
(X
rf».
.
.
tl
n.
rl
E
is
an
element
in
(Y),
n
hence
in
n
(8
X
(by
(X
Property
1).
r'
»
'ii
Supp
.
.
=>
Therefore
there
is
=
Lemma
.
2.14,
E
,
(s,
ti.
(ti)
71,
'in)
,=1
ri
ti
ri
(X
ri).
each
X
maps
which
i
!+i
We
want
to
show
now
that
is
this
homeomorphism:
ri)
(X
X
being
-r
Ii
i*i
S
compact
thatfi
being
Hausdorff,
it
is
sufficient
to
prove
is
one
to
offi:
mapping
this
we
shall
exhibit
the
Given
inverse
J1E
(8
II
and
one
Ni
For
mapping
natural
8
ri»
by
But
on
(X
marginal
X
its
(8
n
to
to
E
of
(
Each
i+i
and
(X
X
onto.
r/»
we
i'Fi
want
to
ti
show
the
(y)
and
on
S
E
S.t.
the
marginal
ri
(y)
of
on
Y
ri)
(X
X
ti
ofy
existence
By
is
Property
1)
it
J1.
is
enough
to
is
define
Y1,
.
Yo,
.
on
.
)
.
.
.
distributions
,
respectively
which
d,
(d,
marginal
(2.1)
marginal
k
the
marginal
distribution
of
(S
on
J1
X
X
J1k
>
will
satisfy
the
k-1
be
let
each
For
1,
at
aving
n
of
element
an
defme
and
distributions.
.
correct
mass
h
of
which
b)
and
k
a)
V
conditions
unit
sequence
a
i+i
of
a
[n
X
(Yl)]j)
1=0 i+i
(that
is
the
factor
space
of
whichdoesnotinvolvecoordinate,i).
Let
Yk
the
marginal
distribution
of
on
J1
by:
ElI
X
=
(Yk-l)
tk
t~
8
and
define
inductively
a
J1k
ii,
k
2.
follows
It
that
has
the
required
properties.
This
completes
2.
2.15:
A
closed
subset
of
which
satisfies
Y
Definition
C
Property
t!
C
is
I,
ViE
supported
by
(y)
C,
E
Y
called
beliefs
closed
(BL-closed)
or
a
of
subspace
beliefs
(BL-subspace)
of
Y.
be
proof
(2.2)
V
will
the
the
(t~);'=l
=
Po
from
readily
>
(tl,...,tk-1)
construction
(J1)
=
d
c
C
y
an
(c)
define
a
mappingy:
where
(tZ);;~1)
(tk);;=1'
sequences
respectiv-
S
point
any
that
uniquely
'if
(tZ)t=l)
(tk)t==l'
determines
k
and
'if
and
which
i
satisfy
.
.
.
a)
b)
(s,
,
>
Remark
1:
k
all
for
2.1)
conditions
(of
b)
a),
satisfying
ely,
(Yk);=O
(tk);'=1
sequence"
dist~butions
on
of
a
is
the
of
each
and
s
a
each
shall
,
for
We
.
defined
Y
abstract
8-based
=
-*
an
2.4).
.
by
C
be
(Definition
.
C
condition
BL-subspace.
Let
non-redundancy
E
Proof:
the
(8,
under
E
Y
BL-closed
the
are
BL-spaces
abstract
8-based
The
5:
(by
of
subspaces
Property
BL-morphism),
Formulation
point
(c)
by
therefore
in
of
Bayesian
(s,
.,
.
defmingYK
367
Analysis
'tf
=
E
c
ewe
are
(tZ}f=l)
(t~)f=l"
Yk'
~
defining
a
~
C
mappingYK:
and
hence
an
induced
mappingYK:
(C)
II
II
(Y
K).
YK
construct
We
mappings
these
inductively
onK:
ti
.
.
i,
define
.
Cletyo
=f(c)
ES
and
for
~
0
,
1,2,
K
(c)
E
c
'tf
C
:
'tf
where
II
is
the
mapping
=
=
ti
(YK-1)'
;\-1
YK-1
~
(C)
II
II
canonically
induced
by
(Yk-1)
Yk-1:
Yk-1'
Using
(*)
condition
(c)
in
fact
of
(2.2),
Definition
satisfies
the
required
it
follows
condition
and
E
c
that
C
the
above
defined
"J
hence
corresponds
to
a
point
in
Y.
Y
ti
Furthermore,
sincefand
are
continuous,
it
follows
inductively
'tf
K
are
thatYK
~
y:
and
(C)
C
then
it
clear
C
derIDed
fr,om
the
is
Y
cobstruction
continuous.
that
If
satisfies
we
denote
(2.2)
i.e.
it
is
a
~
BL-subspace
of
proof
is
the
Y
=
Y
hence
C
mappings
continuous
Y.
At
this
readily
follows
we
2.16:
the
Any
have
to
notice
the
proposition
whose
defmitions:
BL-closed
of
subset
is
an
S-based
abstract
BL-space
(with
Y
Proposition
point
from
following
.
n
respect
to
the
Tl)
of
projections
s
=
Y
(X
X
on
its
factor
spaces).
-
i=l
the
terminology
of
Lemma
2.8
and
the
remark
that
it,
follows
mapping
the
~
Using
C
we
constructed
is
the
BL-morphism
from
C
to
by
induced
identity
clearly
(since
S
satisfies
the
NR-condition).
Using
the
on
~e
Y
C
y:
same
notation
the
above
Y
y
(C)
is
space
of
invertible,
infmite
and
hierarchies
hence
BL-isomorphism
by
generated
between
C)
and
H
Therefore
if
C
h
=
satisfies
the
~
(the
clearly
C.
constructed
NR-condition
we
TIlls
use
concludes
the
Property
6:
subs
Proposition
pace
proof
For
C
to
property
of
any
of
2.5
thaty:
Cis
C
aBL-isomorphism.
5.
coherent
beliefs
hierarchy
(C)
S.t.
deduce
(Co,
0,.
k
Y
.
.
Cl,...,
there
CK)
isaBL-
,K.
=
Pk
~
Ck'
ti
Proof:
By
(4)
condition
E
of
Defmition
2.1,
'tf
(CK):
ti
Supp
X
5
),~
Projection
[II
X
on
(ti
ofCK
(CK-1)]r
CK-1
i
tK
It
follows
(for
by
instance
Lemma
2.12)
that
there
is
an
extension
to
of
a
probability
ti
ii
distribution
on
X
CK
X
[U
.,
.
X
Take
[II
all
possible
(CK-1)]n'
(CK-1)]1
£
CK-1
...
ti
such
extensions
for
each
E
(C
i,
to
define
.
C
Prove
that
'V
tic
K)'
Co,
.
.
.
,
C
C
is
K
+
K
+
1
1,
a
coherent
beliefs
hierarchy
of
level
K
and
+
proceed
1
K'
in
the
construction
of
to
construct
a
limiting
which
C
£
Y
as
Y
inductively
requiredBL-subspace.
be
the
IT on
are
results
be closed
=
Co
Sand
hi
Po
under
of
distribution
the
C,
E
iVy
0
I
some
S
of
(s)
a
construct
to
like
would
We
ace
H..
which
i
assigns
of
function
information
player
private
the
be
~
other
a standard
Borel
a
probabilities
In
is
our
can therefore
B
of
;
model?
a};
set
our
~
which
p
all
I
in
(B)
I
space,
(s»
(h.
V
I
h.
Hi
S
incorporated
11T
the
property.
hierarchy
required
let
this
property:
the
element
E
be
{1T
indeed
do
To
(y).
0
with
the
S
s
hi:
Let
follows:
own
know
each
if
For
his
instance
nature.
the
on
which
state
depend
player
may
of
information
has
each
S
some
about
private
player
information
incomplete
to
addition
in
which
in
that
is
of
A
information
incomplete
situation
common
very
2.17:
Remark
the proof of Theorem
standard
n
by
Borel
a
the
Po
at
Y,
C
to
a situation
generated
theoretic:
or
a beliefs
have
will
(Co, C1) is trivially
such
a-field
countable
)lsES,t.EIT(h-:1
hi
~
state
each
Can
or
BL-subspace
mass
unit
a
BL-subspace
function.
required
(with
measure
finite
={(S,tl,...,t
is
ti
This concludes
space
purely
the
Borel
This
6.
C1
(y)
utility
obtain
is
S
If
2.18:
property
by
S we
standard
again
Remark
on
a
is
S
Y.
induce
the
BL-morphism
two
the
Since
of
BL-subspace
a
to
identity
homeomorphic
Y
Y
Y
BL-
Y.
that
and
is
satisfy
same
the
satisfy
argument,
namely
imply
6)
3)
By
Yk'
Pk
=
Y1,
to
can we construct aBL-subspace
in which each player knows his private information
and it is a common knowledge that such is the situation? This in fact can be done as
words
I
)
i=l,...,n}
to aBL-subspace
space
.
.
(Yo,
of
the
.
limit
projective
BL-homeomorphic
Cis
thus
V
k,
«7)
)
a
s.t.
aBL-subspace
is
there
beliefs
coherent
is
defined
of
hierarchy,
C
'9,
Y
Y1,.
If
the
which
S-basedBL-space
an
we
.
(Yo,
since
then
satisfy
6),
.)
.
Y.
T
The
of
C
BL-subspace
induced
Tis
onto
of
in
mapping
the
natural
way.
=
Y
Y
Y
the
and
hence
and
identity
the
be
to
BL-isomorphic
is
must
IJ;
-+
IJ;
0
I{):
I{)-l
Y
on
induces
which
of
the
therefore
identity
the
also
BL-morphisms
composed
S,
Y
a
a
is
that
of
the
fro~
follows
it
to
BL-morphism
there
proof
BL-subspace
from
5)
IJ;
Y
the
On
S.
on
the
induces
which
S-based
an
is
hand,
other
identity
BL-subspace,
4)
by
~
Y
Y
is
NR-conditions,
aRt-morphism
and
a
there
the
C
C
compact
satisfy
I{):
~
Y
Y
Tsatisfies
and
that
Assume
we
the
and
constructed
since
5).
By
4)
5)
.
=
Y.
Tis
induced
Tonto
from
The
accordingly.
C
mapping
Pk
Yk
=
=
Po
2»
(0
(0
hence
k
V
and
S
and
inductively
(using
1)
1),
By
y.
can
it
the
of
BL-subspace
some
onto
be
of
proof
C
BL-morphica1ly
mapped
5),
by
Y
T
Y
is
If
therefore
abstractEr-space
S-based
an
and
then
satisfy
and
1)
2),
.
Yand
T.
7:
minimality
The
properties
Property
of
368
Game and Economic Theory
result.
2.9.
Formulation of Bayesian Analysis
from
the
derive
we
one
the
as
same
the
is
a-field
this
Borel
and
Sand
in
set
ER)
Q
369
weak* topology.
To carry also the results onBL-subspaces
to the measure
those
where
case
the
proofs
above
the
rewrite
to
first
has
theoretic
one
setup,
for
and on abstractBL-spaces
the
of
extension
this
theory
complication,
technical
except
that
quite
convinced
for
concepts would be defined as analytic spaces instead of compact spaces. We are
causes no serious difficulty.
3. Approximation
of a BL-Subspace
by a Finite BL-Subspace
theorem.
of
open
any
Y,
cover
finite
BL-subspace
that:
states
3.1
the
in
Y,
of
allBL-subspaces
of
set
the
in
dense
are
Y.
on
subsets
closed
on
Hausdorff
topology
Y
The
of
BL-subspaces
[mite
In
other
Theorem
words,
.
*0}.
C*~U{OEOlonc
(ii)
*
S;
0
C*
Ion
0}
E
C
(i)
U
{O
Y
C*
t.
s.
a
there
is
finite
of
Y
0
and
C
BL-subspace
closed
3.1:
Theorem
For
of
any
In this section we prove the following approximation
Lemma
3.2:
Let
X be a compact
of
the
space.
diagonal
in
For
X
X
X
s.
known result (see e.g. Kelley, General
any
finite
t.
x
.'t
E
open
X~
3
cover
0
E
0 of X there
is a
which satisfies:
y)
=> Y
Ox'
V
E
(x,
E
x
V
neighborhood
we use the following
0
To prove this theorem
Topology 6.33, p. 199).
3.3:
Clearly
in Lemma 3.2 can be taken to be a basic neighborhood
V
Remark
of the
diagonal (for instance one which is generated by a finite open cover of X).
For
any
finite
open
cover
a
of
3.4:
compact
space
X
there
W
close,
is
a
finite
open
0
Lemma
cover
W
S.t.
x,
't
y,
z
EX,
if(x,y
)
are
W-close
and
(y,
z)
are
then
(x,
z)
are
a-close.
W satisfies
the
required
by some
X X X satisfying
fmite
open
the
property.
We shall denote by Ref (0) all such fmite open covers
given by Lemma
3.4.
"
Notation:
cover
X.
of the diagonal
is generated
of
be a neighorhood
3.2 and which
W
This
3.3 let
of.Lemma
W
conclusion
V
Proof: By Remark
a
K
of
be
a compact
continuous
space.
Given
functions
f1,
,
a finite
,
open
cover
of
Let
set
(K),
II
then
\;j
on
K
s.
t.
v
E
II
(f(.),
E
3
fn
°v
-
v
(f.)
I
]
I
Jl
that
(f
.)
]
~
j
1,
't
=
1,
.
.
.
,
n
implies
there
0
3.5:
finite
.
is
.
Lemma
EO.
Jl
v
0
such
370
0
Game
Using
Economic
Lemma
Theory
3.2
for
(K)
II
X
let
be
a
neighborhood
of
the
diagonal
in
=
V
Proof:
and
(K)
II
(K)
satisfying
the
conclusions
of
the
lemma.
In
view
of
3.3,
Remark
can
V
X
form:
the
of
Le.
1,...
I~
I
=
I
V
Cfj)
Cfj)-J1
V
1;j=
{(v,J1)
,n},
the
of
baSic
a
be
to
taken
be
neighborhood
open
diagonal,
II
[,
,
are
continuous
functions
on
K.
This
finite
set
of
satisfy
functions
.
.
.
,
where
n
ft
required
properties.
cover
open
finite
a
is
there
cover
finite
-+
measurable
is
a
K
K
:
0
,3
S
0
s.
t.
E
(!()
II
that
U
U,
E
some
(x»
.
S
x
which
K,
Ufor
X
then
E
U
E
\I
for
mapping
(x,
J1
51
U
K
with
the
property
E
of
if
0
a
Given
3.6:
Lemma
then
open
II
of
(!(),
the
((J1,S(J1»EOXO.
be
.,f.
.
Proof:
the
continuous
functions
by
determined
Lemma
3.5
for
the
Letfl"
n
cover
Let
be
a
finite
open
cover
of
K
S.t.:
,
n.
.
.
.
1,
1,
~
U
some
claim
that
this
finite
open
cover
is
the
required
one.
fact,
In
E
let
E
0
be
open
the
set
and
containing
satisfying
the
(!(),
II
let
J1
U
We
=
I
I
fj
fj
U
for
U
X
-
E
implies
U
E
(x)
y)
(x,
j
(y)
U
O.
finite
conclusion
of
Lemma
3.5
and
I{),
J1
n:
.
.
.
1,
\I
=
then
measurable
a
such
take
mapping
j
,
(I{)
I{)
I{)
0
-
(x)
-
(x»!)
(Max
(fj
~
J1
f.
Ifj
I
J1
I
=
I
I
J1
J1
Cfj)
(J1)(jj)
(fj)
-
xEK
tp
(X,
E
UX
(x»
U
for
some
-
(x)
UEUandhence
(tp
EK,
x
~
1,
(x»
V
[,.
But
I
Vi.
I
If..
I
tp
I
the
definition
by
which
.
.
n
imply
,
j
.
~
for
1,
(f..)
1
(J1)
-
that
(f..)
follows
I
tp
of
that
I
E
(J1,
0
(J1»
X
o.
n
fj
=
I
I
J1
It
Lemma
3.7:
Let
=
.
i,
where
X
is
a
V
compact
space.
any
For
open
finite
Xi
Xi
p
,
covers
t.
s.
ectively
res
X
.
.
Xl,
cover
open
an
is
X
which
is
finer
than
i=l
I
=
O.
V
V.
X
of
n
n
n
V
1,
0
V
.
finite
.
are
.
there
.
X
,
open
of
cover
of
j~l
,
,
open
a
be
=
=
Uk}
On}
0
U
rectangular
.
.
.
let
{Ut,
and
.
{OJ
.
,
Let
.
Proof:
finer
than
As
usual
by
denote
the
projection
X
and,
is
-+
which
Xi
O.
cover
(U.)}.
E
n
V
covers
xi
I
Vi={V
IxjEX.}.
'I
xi'I
'
I
I
I
I
=
fmite
the
take
Then
x.
X
let
\I
EX.
(U.)
p.
{po
Pi
Notation:
We
shall
denote
by
R
refining
0,
pea,
Xl,
provided by Lemma 3.7.
.
.
.
,
the
Xn)
set
of
all
such
product
covers
Ct
Ct
is
not
that
note
C
:
detennined
C
-+
tp
,
tp
0
subsets
closed
on
the
topology
Hausdorff
of
)
o
1/1
(in
tp
'*'0
pp
C
0
=
(1/1
Ct
0
=
tp
to
(X)
ing
'*'0
1/1
converges
C
'*'0
Ct
=
ma
Ct
3.8:
Proposition
c
the
Y).
tpo'
Y
1/1
such
X
:
a
that:
have
we
-+
mapping
determined
solely
by
Ct
Y
is
.
.
denote
will
we
.
Since
.
C
of
by
which
(X,
subspace
,
(fmite)BL-
,pn)
pI
=
to
some
homeomorphic
itis
3.5,
Proposition
tpo'
X
where
.
.
.
By
(C).
,
(X,
,pn)
pl
is
so
-space
BL
also
and
abstract
-based
S
some
is
.
tion
.
.
(X,
,
,
pn)
pl
defmi-
our
then
n
to
from
-+
the
by
denote
we
If
mapping
i
(X),
by
pii
Xi
ti
";j
pi
t
t
t
li
In
lQ
>
=
=
=>
0
.
Remark
t~
t'!)
.,
thatpii
ex)
,.
(t?
onXby
(x)).
tie
(tpol
pii
pii
ti
distribution
following
the
derIDe
probability
each
1
i
For
for
and
~
EXi,
ti
avoid
to
denoted
all
will
which
tpo
be
-+
defines
by
also
mappings
uniquely
ri
ri
PO
S;
=
tpo
therefore
on
that
also
Y.
only
depends
Remark
X
Clearly
(Y)
ti,
"ti
1
1
=
3
t~.
and
where
i
EP~
j:
";j
f
(i
=
.
.
.
.
tpo
.
.
,
,
tn)
(to,
(to,
tn)
Y
Y
:
tpo
the
Define
by
-+
mapping
i=O
X=
Considering
Proof:
n
..
inPJ
=
{tJ}
let
and
let
i
otherwise.
point
and
any
non
is
intersection
this
empty
Xi
";j
l
1
I
'if
=
=
be
.
if
fIXed
.
.
P
in
point
t~
let
1,
i
any
n.,
.
n
~
,.n,
.
.
0,
(C)
,
1,
i,
j
j,
\;j
pi
Ct.
cover
the
finer
than
open
OJ'
+
S;
be
will
of
Such
.
.
.
partitions
O.
P
30
p=
l)tuple
(n
E
,
E
~
(po,
pn)
p
pi
=
'if
of
measurable
a
.
.
.
partition
be
,p~)
let
i
.
and
.
.
s.t;
,
(p~
,
To,
Tn)
Ti
pi
P
,On)
A
R
Y.
(00,
of
.
.
let
a:
-subspace
BL
.
a
be
Let
E
closed
C
E
,
";j
(OCt
of
instead
~
0/3.
oCt
A
.
~
ct
write
will
we
.
instead
.
elements
Accordingly
.
.
of
'.
{3
a:,
,
{3,
by
oCt,
O{3,
0
the
confusion
no
denote
will
result
When
refines
~
order:
we
iff
may
O{3.
oCt
O{3
Ct
Y
=
A
all
the
with
of
covers
finite
of
net
partial
open
increasing
the
Consider
{O,Ct},
=
i.
of
set
is
player
the
(Y)
type
Ti
ti
i=O
=
=
=
Y
n
.
for
i
and
S
where
X
write
we
Theorem
.,
.
1.3
1,
By
TI,
To
n.
section,
3.1.
Theorem
Fonnulation of Bayesian Analysis
371
Having done these preparations we proceed now to prove the main result of this
XXi.
additional notation.
t
'*'0
(C).
'"
~
'Y
X.
to
-close
show
to
have
tp
C,
E
,
oa
for
notation
the
(x).
use
to
(x)
convenient
ti
pi
~
in
stated
(*).
the
satisfies
property
fact
in
this
that
prove
us
Let
V).
1
Ak+
~
is
a
the
E
given
to
corresponds
which
required
the
0
~V
Ak
0
V
induction
A
for
E
(*)
(by
satisfying
the
denote
if
E
we
by
Finally,
{3
i=l
n
=
W
V
WI,
0,
XVi,
let
.
and
.
.
of
inement
VoX
,
(V
)
-
~
~
~
n
0
ref-
V
common
-be
Let
Lemma
V
any
4.2).
t
to
(close
is
T
(see
t
and
(t
tJ;
,
E
V
)
i
i
i
i
-
Yk'
Wi)
Y
for
some
X
V
WE
W
W
of
(y»
W(shift
measurable
E
E
(Le.,
tJ;
any
(y,
y
tJ;
Wi
i)
ViE
for
that
such
cover
or'Yk
finite
a
be
let
~
i
Ref
open
V
and
V
let
1,
i,
(V
i=
1
I
0
V
than
finer
is
V.
X
X
oa.
n
T£+l
Yk
0
be
V
finite
of
cover
finite
a
let
and
of
cover
that
such
open
open
Ak+l
Vi
+
it
let
for
~
i
V
a
1:
k
for
us
and
k
true
is
be
let
1
and
Let
prove
(*)
a
E
=
~
=
the
a
k
For
our
from
obvious
is
statement
Assume
a.
definitions,
that
taking
on
induction
shall
We
k:
prove
(*)
by
ifJ'Y'
'Y
~
Ak,
A:
x
~~,
is
V
V
V
3
Va
x.
to
E
E
C;
E
E
Oa-close
(x)
tp
tp
0
Since
cover
some
refined
is
cover
any
of
number
finite
a
involving
only
by
oa
a
Ak
I
=
A
k
the
of
a
coordinates).
first
terms
in
defined
is
a
V
E
E
{a
oa,
a
~
k
define:
V
and
a-coordinate)
the
being
(s
y
tk
tk
tk
(Yk-1)
=
(t~,
V
k-th
to
t;),
of
coordinate
the
as
refer
shall
We
i.
.
.
En
.
,
tk"
=
Y
t1,..
in
V
s
where
.
a
write
and
.
point
generic
k,
ES
(s,
asy
.)
,
k
Y
For
next
the
of
limit
the
as
of
defmition
the
recall
we
argument
and
Y
(projective)
~
ifJ{3
A
A
Va
x.
to
is
x
V
thatV
such
3
E
E
Oa-close
(x)
C,
E
E
tp
tp
a
ifJ
the
to
to
It
on
identity
uniformly
converges
that
prove
sufficient
is
mapping
J
that
of
choice
a
yield
tp}.
{t!}
=
ifJa
P
I
=
and
a
finer
than
.
a
is
There
.
.
partition
,
{tp
(Jfl,
pn)
J
the
and
.
.
.
S
by
,
pO,
pn
a
.
ifJ
So
all
of
set
be
let
oints
the
{t~}.
p ecial choice of the P
is
that
we
x
E
be
0'
~V
max
hypothesis),
tp
ex)
ifJ
tp
will
it
E
=
mappings
let
and
Yk+l'
x
~
(*)
~
For
such
'Y
Let
measurable
finite
partitions
the
of
choice
the
but
a
special
also
uniquely
by
by
372
Game and Economic Theory
a
Le.,
C, Le.
coordinates, it is sufficient to prove that:
Y.
et
L
be
Y
answer
In
,
-
Y?
tributionP
in
dis-
"prior"
each
given
player's
are
subjective
players,
the
of
Following
different
players?
what
are
procedure
any,
of
to
Nothing
as
y
Y.
bution
(y))
tl
stat~
each
Y,
E
y
Y
At
all
E
completely
y
"states
be
of
may
thought
Y
far:
so
the
Summing
up
4.
Consistency
i-close
V
clude
is
that
(x),
(f.{J
ti
ti
(1"O
t
(x))
I
i
a
=
con-
and
V.
since
ti
V
f.{J0
is
'rf
(x)
V
(1"O
t
(x))
a
J
-
t.
J
J
'
I
z
=
=
V
0
pi
pi
-
a
W).
1,
'rf
'rfti
ri,
ti
0
=
V
-close
have
x
C.
c.p
c.p
Y
Y
i-close
on
V
(defIDed
is
E
'rf
c.p:
c.p
rl,
tZ
fl.
-
0
.
.
.
If'
0'
I
=
V
V
Since
that
c.po
(x)
c.p
(3;;;':
0
c.p
;;;.:
By
to
.
~V
-
tl
I
is
E/
ti
and
0
are
ti
which
is
V
to
ti
Since
possible
We
pz
their
Definition
y
=
Y
,
probability
distribution
P
E
II
(Y)
is
said
to
be
the
to
(distri-
universalBL-space
consistent
point
S
3.1.
we
(V.)
J
Therefore
to
t~.
Y
and
0
ti
(x)
(on
C;
of
set
z
to
players.
each
-close
Thus
beliefs
the
(x))
E
-+
is
-close
which
in
V
'rf
there
to
x.
generated
by
the
fact
probabilities,
i-close
Ref
E
x
C)
0
is
Theorem
on
that
compact
of
the
all
a
and
C
defmition
According
for
for
is
to
Bayesian
discuss
we
probability
distribution
.
of
x
to
x,
this
between
sense
from
proof
hand
the
in
then
to
situations
probability
beliefs
1/1
if
of
section
world?
the
this
C
conditional
if
the
-close
which
there
on
for
is
(x)
.
V
next
the
other
to
denote
started
c.p-l
distributions
Are
it
mutual
t!
(see
some
(subjective)
from
We
(c.p
to
state
world"
(x))
the
the
(C),
c.p-l
ask:
and
completing
Extend
to
own
of
that
to
is
iny
equal
relations,
his
then
c.p
from
we
actual
V(close
the
constructed
the
also
p;,
shall
on
E
t~
probability
beliefs
of
is
is
knows
which
.-close
in
points
derived
What
to
C).
E
know
we
that
the
and
those
Harsanyi
what
developed
P
x
particular
V
questipns
namely
levels
of
C
We
is
n.
of
theoretical
these
information,
we
characterize
defmition
space
two
certainly
i
i-close
far
and
one
x
(f.{J
have
.
A
we
.
4.1:
structure
(f.{Jo
the
games
so
are
determined?
In
the
E
i,
1,
p=!pidP
'rf
we
to
(tl)
of
.
i
i,
game
(x))
(x)
as
of
.-close.
player
x
f.{Jo
'rf
BL-subspace
private
on
by
said
i;;;':
E
for
definition
it
is
was
-close
ti
Can
of
beliefs
defines
that
closed
section,
on
the
'xE
C
t(0=
piadoshow
a
this
case?
Formulation
Analysis
373
0
(x)
.
the!efore
remains
(x))
tl
(x).
Thus
by
derIDing
hence
'rf
ti
(C),
c.p-l
respectively
which
ti
since
S.
This
by
relevance.
if:
(4.1)
Game and Economic Theory
374
the
on
distribution
prior
P~
Y
a
as
be
regarded
may
P
words,
In
other
type.
having
.
y
The following proposition proves that this definition in fact captures the intuitive
meaning of consistency we have in mind, namely: If P is consistent then for each
player i, his subjective probability pi equals the conditional P-probability given his
the
generated
I1
in
measurable
of
a-field
sub
the
then:
consistent,
4.2.:
I1
is
E
P
Proposition
If
(Y)
then
tion
ti,
T
be
let
I1
sets
projec-
by
(Y),
(Y)
(ti)
as posteriors. Formally, with the appropriate measurability structure on Y and on
\j
subset
A-borel
the
for
proposition
the
prove
shall
we
clearly
more
idea
the
see
To
Proof:
first
simple
y
I
=
T
P
Y.
i,
(A
of
y,
(ti»
(A)
\j
\j
pi
(4.2)
c
subset
of
partition
a
i
each
i,
player
namely
(y».
Y
IP;
=P~}
=
=
{YE
(y)
(ti
Ti
(ti)-1
of
various
subsets
to
in
types
of
.
Y
Y
for
any
define
(Y)
I1
The
projections
y
Ti
~
ti:
Y:
finite
Proof
for
case in which Y is finite and then provide a proof for the general case which asks for
more careful measurability considerations.
With this notation, the statement of Lemma 2.14 can be rewritten as
(y).
Supp
Y
(P~)
Y,
i,
V
V
E
C
Ti
(4.3)
Y is finite,
(4.2)
becomes
(4.4)
C
Y.
y
I
A
(y»
E
Y,
y
(A
=p
(A)
\j
Ti
\j
pi
When
(P).)
Supp
E
lEY
(Y).
(y»P
n
(A
~
y
=
"£
P
(y»
n
Ti
Ti
pL
So
(A
(4.1)
write
as
lEY
~"£
P;
=
VACY.
(Y)
P
(A)P
(A)
>
we
Now
if
satisfied
be
will
(This
(y»
P
O.
y
(Ti
Actually we want to prove this whenever it has any meaning, namely whenever
each
4.3:
The
types.
of
fromB.
constant
that
from
readily
,
ti
pi
-
Y
-
=
A
set
the
to
This
measurable
B).
applied
(A)
n
pi
pi
-
The
to
right-hand
side
equal
(A
n
B
B
-
Y
of
tion
F
that
given
i.e.
A)
(ti),
(ti)
A
pI
set
measurable
is
A
Y,
in
indicator
(A)
(the
and
a
is
from
transitiop
remains
probability
Y,
for
that,
any
Ti
=
F
Thus,
any
letting
we
(Y)
(ti)
Ti
ti
Y
also
Remark
to
applied
Baire
(resp.
thatpi
.
L
on
functions
between
bracketing
Y,
functions
by
with
f
f
f
and
functions
ernicontinuous
s
therefore,
Y,
class
by
from
-+
Y
extends
continuous
equation
(4.1)
upper-
ti:
pi,
Y
Notice
the
that
first
regularity
and
by
the
continuity
,
pi
Y
I
(A)=P(A
(y)),
Ti
pi
is
general
y
J
pi
(A)dP(Y)=J
P
(A)P
(A
n
(y))
(Ti
ri
=pi
Y
y
=
Also,
again
so
by
(A
(4.3),
(A),
(y))
ri
pi
{
(A
n
yE
(y))
ri
ri
pi
Y
=
(A
n
(y))
ri
pL
0
Y
what has to be proved.
case:
all bounded functions, and finally from those to all bounded universally measurable
universally measurable) function yields a similar function.
IA
dP(y).
B
Y
everyBL-subspace? The following example answers this question negatively.
Clearly the first question
BL-subspace Yhas thus fou~ points corresponding to the four possible couples of
-'
But
by
Supp
(4.3),
has
players
situation
to be asked is: Does a consistent
two
which
of
two
a
the
concludes
of
proof
This
4.2.
Proposition
disjoint
.
on
so
of
the
is
distribution
full
this
that
support
fact
the
C
(pL)
Y
I
show
we
if
B
that
n
(A
so
will
this
that
P
from
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the
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denote
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entry
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of
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each
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of
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(5)
proceed
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t 21
approximation
our
view
subset
see
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no
r
12
and
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in
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the
state
each
Assume
finite
y=
11
1,2.,...
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[
U
ci
C
=
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C
i
For
Y.
376
Game and Economic Theory
types:
1
2,
i.e.
)
also:
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e ci
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,
and
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2
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have
since
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fmite,
a
Bayesian
377
Analysis
limiting
set
Will
be
reached
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ei
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of
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This
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satisfies:
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course,
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containing
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to
However
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have:
then:
finite
with
P
some
fact
in
is
C
=
i..
rand
then
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set
this
Denoting
and
i
all
for
hence
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y
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by
Ci
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Proposition
4.4:
consistent
for
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E
(P),
If
y
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support,
may
y
y
y
y
y
y
containing
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y).
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the
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observe
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(pi)
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Proof:
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C
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y
(P)
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y
(y)
proving
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equality:
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is
inclusion
second
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(1).
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induction
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proving
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Note
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in
the
situation
described
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Proposition,
C
is
a
common
knowledge,
y
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i.e.
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can
be
knowing
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The
set
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and
an
outside
only
observer
from
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following
some
player
each
computed
proposition
consistent
shows
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distribution,
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for
determined
in
and
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the
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but
that
there
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uniquely
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probability
distribution
which
on
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knowledge.
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and
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and
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Proposition
P;
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.
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of
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4.5:
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support
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)
z
(z)
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P
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>
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=>
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probability
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of
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to
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real
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probability
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finds
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his
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player
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e
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on
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no
finds
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reject
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hence:
on
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C),
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each
then
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and
set
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reach
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is
4.7,
corollary
C
player
By
"
He:
"The
consistent.
is
world
the
state
actual
of
y
the
hypothesis:
following
testing
problem
of
now
can
world
the
state
the
The
presented
be
consistency
question
the
as
of
of
of
it
as
the'players
known
be
to
assumed
be
in
usually
is
it
Bayesian
the
assumed
may
by
is
it
because
not
distribution,
also
but
mathematically
so,
only
prior
because
speaking,
of
view
In
think
to
sense
makes
it
Corollary
consistent
a
4.7,
a
as
distribution
of
=
C
consistent
the
and
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on
distribution
i,
and
detennined
V
is
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any
then
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is
If
common
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a
probability
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ci
Y
a
is
P.
then
the
state
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bution
the
world,
it
containing
C-subspace
is
of
If
A
4.7:
in
consistent
state
Corollary
some
the
is
is
distri-
consistent
ifit
support
of
y
a
called
be
will
in-
world
the
state
consistent
be
is
it
(otherwise
C
to
said
E
of
y
=
be
will
with
a
called
P
a
or
BL-subspace
consistent
C-subspace.
shortly
C,
Supp
Any
A
4.6:
Definition
which
on
BL-subspace
consistent
a
exists
there
C
P
distribution
distribution
consistent
p'.
=
=
(C~)
(C~)
if
Note
for
P
that
alsoP'
then
consistentP
some
0
other
for
0
any
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for
is
true
C~,k+l'
Co
statement
same
for
also
true
is
it
for
is
statement
the
Since
true
trivially
378
Game and Economic Theory
consistent).
A combination of Proposition 4.4 and 4.5 yields:
knowledge.
approach.
O.So we have:
distribution is computable by each player.
Especially if we make the rather weak assumption that each player assigns positive
of Bayesian
Analysis
379
a
is
a
common
or
not
C-subspace
in
is
world
the
state
real
the
Whether
Theorem
4.8:
of
Y
'V
(P;
have:
we
then
Supp
E
i),
Formulation
containing
it and the consistent
di~tribution
on
any,
is
there
if
beliefs
others
on
each
of
beliefs
the
from
only
derived
knowledge
is
common
a
player
con~istent
that
distribution
emphasized
knowledge.
be
should
It
Remark:
the C-subspace
are also common
prior
If it is, then
it (the priors)
the
knowledge.
not
from
a
type
"new"
of
beliefs
on
the
mechanism
selecting
the
state
of
the
reasonable
any
P~
"inconsistent"
in
belief
highly
describes
The
a
case
Et
Supp
y
world.
.
and
of this
consider
"impossible"
If players
(Le.
wrong
are so much
has-probability
conclusions
mistaken
0)
are
quite
the
in their
real
state
expected
the
so as to
world,
following
then
examples
show.
11
12
21
22
[
.
=
.
Y
thus
two
has
of
which
each
two
are
There
4.9:
Exumple
beliefs
of
as the
players
(objectively)
word.
types,
meaning
The
subjective
probabilities
of
each
player
on
Player
the
2
-
Player
1, type
of
the
others
are
given
by:
2
Player
type 1
-
types
type 2
3
5
0
2
5
1
1 (1,0)
12
[11
2,
type
2
(2/3,
1/3)
21
22
]
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then
12,
(only)
with
21,
the
players
the
accepting
So
He
1/3,
1/6).
(1/2,
distribution
consistent
probability
by
=
C-set
fmd
will
Both
(P;)
the
22}
{11,
players
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{22}.
y
=
=
is
world
the
of
actual
the
If
state.
y
{II},
(pI)
Player
be
commiting
(type
II)
error.
since
be
should
it
as
1,2,
i
for
Supp
in accepting
it).
1
probability
he is right
Note
however
that
inspite
of its being
inconsistent,
21,
{II,
C
consistent
same
=
=
to
the
both
led
12
the
state
players
-subset
22}.The
y
that
believes
each
with
player
(although
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committed
both
players
...
P;
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in
fact
12,
state
the
for
Note
that
yEt
y
will
next
example
shows
that
even
this
is not
guaranteed
in
an
inconsistent
state.
]
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accident.
of
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Note
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34,43,44,
find
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1/2)
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with
22,
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12}
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C1
44
45
43
42
35
34
33
32
25
24
23
22
y
C2
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13'
14
15
12
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,~
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,
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correctly
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in
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with
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player
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therefore
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conclusion
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Consider
4.10:
example
Example
Economic
and
Game
Theory
380
namely:
r:j r:j
J
(r
0
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3
real
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,
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state
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of the type dependent action sets over all his types).
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23
33
43
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estate
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add
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conclude
16
product
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22,32,33,34,42,43,
42
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)
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any
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there
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Example
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Y
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be
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Fonnulation
Analysis
381
types
each player.
for
3
5
0
0
2
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1
pl
0
1
2
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2
y
.';3--;
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14
l
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44
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on it. (The verification of this is rather simple via
in a gamesituation with incomPleteinformation.Weproceed now to define a game
Up
tonow
we
constructed
and
discussed
the
universalBL-spa
anditsBL-s
382
Game
We
define
.
and
first
The
a
players
Economic
Theory
vector-payoff
set
game
is
{1,
I
2,
in
.
.
.
,
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.
strategy
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is
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set
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i
The
y
~
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rJ:
which
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.
A
payoff
player
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resulting
from
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n-tuple
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.
vector
the
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is
,
.
a
,
.
payoff:
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to
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The
u.
.).
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.
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r
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sense,
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should
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Although
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in
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the
concept
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way,
ti
-ai)
'(ji
usual
T(measurable
i
(a)
,~)
N.E.
if
V'i,
~
V'
V'
is
u
.
u
.
E
a
(a
Ti,
ESi,
(al,.
namely:
(
,
(
i
i
I
=
=U,...,u
n
i+1
,...,a..
i-1
-i
1
,a,a
-i
a
)
)
I
Remark
5.1:
game
in
to
Y
which
the
are
When
the
number
finite
payoff
BL-subspace,
player
for
of
the
i
player
i
is
a
above
vector
defmed
with
game
number
is
of
an
n-person
coordinates
equal
Ti
(namely
It
is
easily
seen
that
as
far
as
N.E.
I
game
this
[see
a
types
of
concerned
is
I).
were
(
a
(
h
is
Harsanyi,
equivalent
to
1967,1968,
what
is
Section
by
called
p.
15,
496].
Harsanyi
"Selten
This
is
an
game
ordinary
G**"
Ti
Tn
J2
Tl
.
.
.
X
person
game
which
in
each
"player"
a,
f'E
selects
I
I
I
I
X
I
I
X
and
selects
his
i,
1)
-
(n
partners,
one
from
each
i
according
to
his
i
=1=
then
T
strategy
subjective
probability
Remark
is
distribution.
5.2:
the
When
same
as
for
payoffs
Y
the
is
one
finite
we
player
i
we
can
defined
to
be
an
define
above
a.
but
ordinary
game
instead
of
in
vector
ti
2;'
u
=
E
r
,.
i
I
form
we
.
Ti,
'i;j
where
strategic
payoffs
is
a
which
defme
the
strictly
positive
r
i
(
i
-ri
('
t
(EI
Clearly,
of
independently
the
we
constants
choose,
this
game
has
the
i
r
constant.
t
N.E.
In
points
as
particular
our
if
vector
we
payoff
take
game
V'i,
to
(and
hence
as
the
1
satisfy
.L
r
=
r
.
i
corresponding
we
Selten
get
a
game
game).
equivalent
i
(
(
,
(IE
T'
to
p.
that
by
341].
As
Also
"Ii'
suggested
all
far
these
as
N.E.
games
Aumann
and
points
Maschler
are
have
the
for
concerned
same
the
their
N.E.
points
inconsistent
case
game
is
as
that
[AumannjMaschler,
independent
suggested
ofthe
by
Selten.
parameters
same
"
1968.
14
lli.
I,
II,
Game
Theory,
September,
Disarmament.
Gradual
on
Final
report
of
A
Incomplete
With
Games
Information,
Repeated
Survey
,
1967,
N.J.,
of
Aspects
Maschler:
mathe-
of
introduction
The
of
matter
a
P
C.
on
distribution
just
is
should
the
Pas
in
believe
in
players
that
claim
not
we
any
do
way
the
form
extensive
C
defined
via
by
situation
the
game,
analyzing
By
"a
in
form".
standard
form
game
game
extensive
this
calls
=
that
C.
(P)
Supp
and
N
.E.
of
the
of
from
It
defmition
the
definition
the
follows
Proof:
games,
readily
Y
-
.
.
a
receives
and
"if
.
chooses
then
i
i,
payoffu'
player
(s,
,
sn).
EAl
Sl
,
.
.
.
Y
-
A
informed
to
each
then
P
according
C
chooses
move
chtmce
of
is
player
E
y
pi.
extensive
of
Princeton,
Mathematica,
by
Theoretical
M.
and
Harsanyi
fact
the
in
Journal
Baysian
prepared
Game
In:
Results.
5.4:
Remark:
game
Parts
Players.
International
of
Solutions
On
ACDA/ST-1l6,
probability
Harsanyi,
Remark
following
Bayesian
by
Games.
Eisele:
Ill.
R.i.,
unlike
the
Played
Incomplete
193-215.
Th.
and
Recent
prior
as
Information
With
Games
1979,
W.,
Contract
a
points
and
c.:
(4),
Chapter
Aumann,
NE.
7),1967
5,
(3,
1.
8
Boge,
same
Science
Management
Harsanyi,
the
form
ho.s
C
defined
vector
the
distribution
strategic
Then
on
C.
by
game
payoff
consistent
Y.
a
Theorem
the
Let
subset
consistent
be
Let
5.3
be
P
C
EHarsanyi):
of
Fonnulation
of Bayesian Analysis
383
For a consistent subset C one has the following theorem, due to Harsanyi, that
allows us, in looking for N .E. to replace the strategic form game by a certain extensive
form game:
form:
matical convenience. It serves to find the original N .E. points naturally defllled by C
via subjective probabilities.
Furthermore, by Corollary 4.7, since the "priors" are common knowledge, the
above described game in extensive form is also a common knowledge which gives
even more justification for using it in analyzing the situation of incomplete information.
References
.