Typical Types
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Economists usually represent incomplete information by assuming a fixed set of
types for each player. Each type has beliefs about states of the world and other
players’ types. Each mapping from a player’s types into beliefs over states of the
world and other players’ types are informally assumed to be common knowledge.
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http://www.econ.yale.edu/~sm326/typical.pdf1
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3 3
# % % B Q% % Q
1 Q 4 Q
4 Q Q 4 #
5 Q 2 6% "
This approach is quite common in the literature. See, for example, Cremer and McLean
(1988).
#
This topology is employed in this context by Mertens and Zamir (1985) and Lipman (2001).
$
As in, for example, Geanakoplos and Polemarchakis (1982) and Rubinstein (1989).
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2. The Uniform Topology on Higher Order Expectations
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Write for the set of coherent types.
{E ' × X
'
2.2. Higher Order Expectations Types
Here we describe one very simple notion of closeness of types that is easy to
visualize. For each type , let A
? E be the th order expectation of # #
A 1 4 A 4 A
4 # %
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A ? E
h ? E ? '
'
]
ErcBn
#
3
Q M %
D ?! ? ?
B?−
, Aξ T
1 &% : & # 0 n () n ( .
n
B 1 We write
|
|
1 5
# 5 λ#
'E
B Q # R
?B
a , ∞
i
E
' B E Q i j ?Q
? Q '6(.
;
/ ? E+1 2 : % / 5
# # ' . A
. In fact, this is the unique strategy profile surviving iterated deletion of
strictly dominated strategies, since by induction on , we have that if pure strategy
survives rounds of iterated deletion, then for all ?
E
E' E
<
! "
= |% | | | ' 0
. 9
%
0
% '. %
E
'
E
HM %
Q E
B EQ
2
?
? ? ( ? )
?
? E %
Bn cr
E / / %
%
E
'
A
E E −A E
E %
Λ
i j # $%
: $ 0
# 5
& ;>
?Q E E
E 9 2 |% / AE
E
'
sMS
1 E 0 Q E n
? ? E D ? -? δ? cr
His expected payoff to behaving as if he were type -
-
'
H M%
Q -
B Q E − −
?
B? cr
?
? () .
2
? ? - D ? - ? n+1 - >
A
1 - 1 E A
? -| '
D
? Q. ? - ? ?+1 -t% δ? 4 D %
-D #
HM %
D 2
1 -
0 1 E 0 1 - Aξ Q -|
u
A -ξ -t , t, λ '
n n - ? -δ?
=
(
? B? ,s
?
A
? ? (| ) ? (|) ()
Q - A Q -|
) B
−
? -
∞
n
n n -| ? -| # - '
'
'
A
- - −A E
A
λB ξ Q Et Q -
E , t, λ
?
∗
- A
λ? ξ ? -| ? -|
@
∗∗
- i ∈
M
λ ξ -| A Q -|
? E
?
j
n
n n ( Aξ ? -|
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( " '
) *
A
T
=
#
t
#
Q =
|
such that A
? ()
∈ : ∃ ∈ [0 1]
? -|
% Q # N 0
@
'())-. Q # 5 %
'B '6CC6..
1 % # 'B '6CC6..
8
% 0 0
% 1 1 '
9 '(. A |
;>
= 2 2 = # | → |
# B
1 %
# ? A ' B. : %
∗
A ?
|
|
k
-|
? () − ≤ ? -| A ? | + k ≥ 3 '66.%
: δ > B% k εk Q Q 2? '66.
Q n -| − A 2?+1 -t
→ n → ∞
# ? -|
+
'6. A
|
∈ 2 % A ? () 1 ? - ∗ k
and consider a type | with
A ? 2?+1 B
Q + k 1 % 1 B % Q % B , if ? is odd
→
A k
→
By construction, |
∞, but each ∈ | as # 1 % ' B '6CC6.. @ % 9 # , we exhibit a common knowledge discrete type space containing
a type | with - ' . Let Q ' ' {Q 2 }4 ' {f } A  Q EEQ '
EEQ '
| Q
Q
Q Q
Q Q Q % |Q ' | '
f% % k
% # % # ∞ =
1
Q
A
X (k,k ,s) =
'f
% = % Q = + 1 = 1
Q − % | % Q k k
-
1% 0% s=1
s=0
1 E - % Q E Q Q
% %
? Q E - Q
for all ≥ B. So
A -Q QQ Q @ Q % + ,
$%
D % 0
1
9% !
" !" @% $
'()*+,*-. 2 B E
'()-?. % % & # % # 0 0 2 0
% !
" !
" 9 )
% 3 #% 0
% # 0
9% / 0
/ '> '()-).% 8
=
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B @ '())-% 6CCC.. # 0
% , $% @% %
/ '% % 8
BA
'()--.. # !" % ':
'())). 3
B '6CC(.. :
'())). % , -
" .
%/
B E
'()-). % !0
" # T
ST %
:
∆(
)
t = (δQ, δ , ....)% f (t) # ∆ (S T ) 0
9 % 3
= '())7.
F # ; ' .
# #
# B E
'()-?.
9 % f 1 $0
% # 0
G 5
(C
+
"
# 2 G $% $% H # /0
%
# # # +
-
# '
? . '(. % % $ 3 4 '6.
1 + .%% + .
# ? 5 # ;
/ - .
Morris and Shin (2001) show that the optimal action is this game for type t is to
set his action equal to
-
nQ
n!QA ? -| % | 0
% '. %
AE
'
E H M%
Q (
) (
− s −λ
- :
E '
a
?Q
-
E
λ λ
? !Q
11
-
?=1
% ?!QA n - )
() %
? -| − A ? E )
,
so the strategy distance between a pair of types is
- [01)
?Q
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'()-?.. + . # '
% .% 2 % % 5
% ;
% 5
0
+
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& B
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I(J 3
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="
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