la theorie des jeux repetes

!
"
# $ ""
# ""
"
!
%
!
$
"
! "
"
" !
"$
/ 0 1 -"
#
%
! !
&'
'"
-"
!
$
.
"$
"
.
"
"
% $
'"
.
"$
"
'
# "'
( )*+ )**,
!
!
#$"
!
"$
!
'"
"
%
"
."
$&'
"#
!
#
- /
"
"
!
"
# %2
!
% 3"
4
"
! ""
#
/
1%
56
"
-
"
"
"
'
-
!
"
!
#
!
# !
'
""
"
!
1#
"
"
"
!
$
!
-"
!
( )*+, # "
" %
!
." !
"
"
"
7
!
"
"
%
"
( )8 ,% 9
"
!
"
!
"$
! " %
"
"
"
. $
.
"
!
"
!
"
!
!
!
!
! %
";
#
" #
"
.
"
"
"
"
.
"
' !" . "
%
!
!
%
"
"$
@"
"
A
!
!
!
!
!
C
"
.
"
!
2
"
! ""
! %
" !
"
! %
"
'"
"
%
#;" ;
;
"
%
%B
!"
)*+
"
# "'
! "
""
:
"
" 4
"
"
)**%
#
!
"
. ";
"
"
"
"
"
% 3"
";
!
'
-
;
!
" ! ""
" !
!
"
"
!
-
!"
"
? %
!
"
&'
"#
"
:
"
( ))>,
"" .
# %
#
!
!
"$
#
"
=
$
"!
!
-
"
'
"
"
"
"
$
"
B
<
!
""
!
$
"
;
"
""
"
# "'
"
! ! "#
$!
% &
"
#
"
/
"
D
"
- % 3" $
"
'
-
. "
"
$
!#
"
!
!
""
."
!
"
"
! "
'"
1
!
4
!
!
!
$ !
"
!
!'!!/
"
!
'
!
B
%
4 !$
%
!
$
%
#
"
!
"
"
'"
:
"
%
%
!
%
(
%
6
!
"
5!
"
!
"
C
%
"
!
!
!
#
!
!
" !
.
"
!
!
#
"
!
" # ""
#
!
!
!
.
$
"
E
9 "$
""
!
'"
$
# "#
:
% 3"
%
&
!
!
#
5
"
" # " "
"
"
#
1
%
#
# !
%
%6
!
" A
# "#
%
!
"
#
$
C
. "$
/E
%
1%
' %
4
$
'
". %
'"
FG
4 $
!
3"
#
!
!
: %%
#$
'"
"
# "
!% B
!
-
"
!
'"
1%
#
"
!"""!
$
= Χ{
−
≠
"$
}
!
$
!
#"
%
'"
' '"
"
/ = Χ{ ∈
"
"
.
1%
}
!
!
. "
!
/ 5 1
!
$ !
/ 5 1
%
!
"
&
&
$#
!"""!
$
!
≠ !
$ #"
!
% !"""!
%
-
%
!
∈
>
'
'
#
'"
!
'("
#∈
!
'
∈ %
6
#
/%1
"!
)(
C
'"
" # ""
"
#$"
' '"
!
"
!
$
% B
% !"""!
!
%
'"
!
!
!
! "
'"
!
!
'"
!"""! !"""! # ∈ %
%6
!
!
:% $
!
$ !
!
'
! !
"
!
!
#$" !
!
. %
H%
'"
/
"
%%%
!
6
& $#
#
- '
"
9
5!
%
!
$
∈
"
"
!
∈ %
$
#
%9"
*!
.
"
"
" # " "
!
!
*%
>
F/ %%%
" "
'" &
%%% J1∈&
"
"→∞ ∈&%
Σ
"
!
*
'
≤ I% 3"
$"
$∈&
>
'
I " #
"
!
λ∈(K , λ 0/ 5λ1 $∈&%
!
G "H∈&
!
!
!
!
!
"$
#
!
"
"-
"
"
!
"- %
#
%B
L
! !
"
!
4! #
! #
!
#
"
# "'
)!* #+
#
"
+
6
!
!
$ # "'
%
"$
# "'
6
" # "
"
!
"
""
%
#
%
!,
(4
O
!
∈
"
∈
O
"
O
/ 1%
O
#
B$
"
#
"
"
( )):,1%
"
&
#≥
O
−
&
O
−
! ! #
/ 1
%
"
";
#
!
! !
!
"
Γ
6
#
. !
#
# ! #
!
!
# "'
4
"
L
"
%%%
" !
"$
% 3" !
!
!
N6 9
!
!
"
.! #
6 9
!
! ! #" B
!
6 9
!
#$"!
"
ΓF/6 9 1
"
'"
!
Γ
"
%
%
#
N "$
6
'"
" #
M
6
.>
" %
! ! #
" E# "
"$
-
!"""! !"""! # ∈
$ !
'"
"
Γ
'
""
-
L
%
!
"!
" #$" " !
" ! !
! "" !
"$ '
'
!
O
# "
# "
! "! "
!
#
! P%
/7
5
!
O
%
"
; # "'
# "
'
5
!
!
6
-
( )L ,
"
'
"
O
−
.
!
%
;
!
$
# "'
9
/
J
-
!
.>
$
1 "$
!
-
# "'
!
$
Q
#
6
!
"!
6
%
"
!
# "$
"
#$"
52
$
"
# "'
$
! !
'
."
%
!
$
"
!
8
! ! #
!
$
"
"
%
"
-
" 7 "J
$ # "'
6
%
! .! #
!
%
!
"
!
"$
'"
"
" !
.!
+
! ""
%
.
" $ !
# " !
" $
$ # "' #
!
%%%1%
#
"
"$ !
"
!
"
/ /K1 / 1 %%% / − 11 "$
"
5!
!
%
/ 11 ∈
!
!'!!
"
/1
/ 1 %%%
"
"
"
6
Γ
%
!
%
!
'"
" # ""
.
"
$
#
!
!
!
"
/!
6
!
"
6
!
"
"
/ 1 = / / 1 %%%
-
!
"$
"
!
'
%
!
-
""
( )L:,
"$ #
!
!
<" !J '
!
"
-
# "'
( )*+,1%
# "'
"
$
1
!
$
$
#
!
/B
"
( )L:,
6
'
!
!
'
% B'
$
. "$
#
# "'
"
#
%
-
+
#!
# "
3"
8
O
−
&
" !
!
O
O
−
.
"
"
%
.
% $
$
# "'
6 %6
"$ # " '
6
/ # "'
!
. "$
'"
!
∈
5 %6
B
!,
!
" $ !
#
" !
-
-"
σ
!
*
σ
Σ
Σ
#
#
"
!
"!
! "
-"
+
.
#$"
"
$
,
"
6
!
!
,# F
!
"
" ! #
%
$ !
/
1
)
!
"
$ !
σ
#
G
## ,! !"""H
!
"$
!
"
"
!
%
FK "
,#
8
$
%
≠ -∈ %
"
!
."
!
%
! σ #( ,Fσ #( -,
$
"
"$
#
# "
/+" 0
σ∈Σ
%2
"
"
!
# Σ
∈ %
."
%
# " #
#!""""#
'"
"
!"
' !"
σ ,#! σ #!"""! σ
"!
!
!
"
"$
#
/Σ Χ = Σ 1%
"
*
-"
!
!
! #
R
"
.
# "
"#
!
#
!
'"
"
"
#!""""#
%9Σ
."
'"
!!
!
σ #!"""! σ
#!
σ σ !"""!σ !"""!σ # "
"""" 6
&!
"$
.
"$
σ #!"""!σ #!"""!σ ## "
#
#
!
."
Σ
#
σ ,#! σ
"!
#
!
"
RΣ
!
%
!
-"
!
%
σ
."
'"
σ
.!
%
!
"
.! #
"
-"
" K
"
"$
!
"
"
%
1
+
4B
-"
#
= Χ/
R
"
5
σ
# "
.
-
!
""
%6
#
"
"!
3
"
"
!
!
# , 1! (2
+
#
" '
σ #% 6
!
"$
"$
!
-"
%
"$ !
!
-"
σ #( ,≠σ #( -,
"
"
B
"
-"
"$
'"
!
# ""
5' !J%
" !
%
.
;&'
!
.
; !
"
$
# "
5
( )*+ )**,
% $
/ 01
$
!
# $ ""
! . $
!
! -∈ %
' !"
"
%
-"
' !"
!
%
!
! #
&"
! !
'"
$
/ 0 1 -"
$ !
%
#
""
# "'
"
"
!" %
-
"!
1 '#
-(
)"
B
"
!
'! " &
!
" '"
.! #
%
#
.
)
2
$ # "'
K
"
!"
6
"
' !"
%
B
# "'
!
$ !
"
-"
" !
!
" %
' !"
!!
!
#
"
' !"
# "'
5
!
"
!
"
"
!
!
#
!
%
"
!
"
#$
!
!
!
"
#
! "
!
"$
$
$
%
*
$
"$
?!
"$
%
!"
/
"
5
1%
!
"$
"
" !
$
5
.
!
$
5
.
%
B
#
!
-
"
""
" $ !
12
" !
3
(9
3
3
!"""H R G #
#
"
3
3
!""!
!
"
."
# "$ # " '
%
6
Γ ! ! #" G 3!
!
∀ H
"; # " '
/ #0 ∞= K R
$ !
" # ""
6
!
-
%
'"
"
"
!,
-
4
"
$
σ / 1 = /σ / 1 %%% σ / 1 %%% σ / 11
σ
"
%
"
""
"
:
'"
σ = /σ /K1 σ / 1 %%% σ / 1 %%%1 ∈Σ
9
"
-"
4(σ , ∈
-
"
%
"
!
R
."
4(σ, = {.(σ,/ 1} = K
∞
∞
.(σ ,/K1 = /K1 = / /K1 %%%
/K1 %%%
%
"
!
. "
!
FK%
."
"
!
-"
'
"
. "
/K11
$ !
. "
≥ %
/K1 ∈
"
" !
σ/ 1
!
'
.(σ ,/ 1
"
#
"
.(σ ,/ 1
" $ !
"$
/!$
FK1% & "
= ( .(σ ,/K1 .(σ ,/ 1 %%% .(σ ,/ − 1)
"
"# 4
.(σ ,/ 1 = σ / 1( .(σ ,/K1 .(σ ,/ 1 %%% .(σ ,/ − 1)
!
.
"$
" $ !
!
"
-"
σ/ 1 "
"
5 /σ 1 =
:
3"
!
>
K
!
.! #
%
'
3
4(σ ,
#
- "
δ
/.(σ ,/ 11
∞
=K
'"
!
.
'" %
)
"
#
"
! !
.4
!
!δ"
!
; !
"
6
!
! !
!
"
%
"
%
5 / 4(σ , 1" 5
"
"
Γ
Γ∞
!
.
!
"
"
!
!Σ!5!δ# RΣ
"$
!
! ! # "
!
'"
5 5 !"""!5 !"""!5 #
%
!,
34B
Γ ∞ F/6 Σ < δ1 "
"
σ; ∈ Σ
∈6
"
σ
"
# "'
6
4
5 σ# ≥ 5 σ6 ! σ$ #
B
"$ # " '
S
"
"
# "'
"
6
-
6
$ !
!
$
!
# "'
'" %
5
! -
6
#
!
!
!
'"
# "
"
B
6
#$
'
-
""
"
5
!
!
5
- ";
'"
'
σ
"
-
# "'
$
!$
"
";
'"
%
K
# "'
# "'
/
σ = /σ Q σ − 1 "
Σ
# "$
"$
"
5
"
$"
. $
%
$ # "'
'" %
!
"
'"
!
""
!
!
""
# "'
! #
$ # "' %
"#
%
( )8L,%
5
%
"
#
5
9"
# "
"
# "' % $
5
" $ !
!
"
/% '" ,"
$ # "'
"
! #$" $
#
! + # ! !* #+
$ "
! !
" ! "
"
1
σ
"
5
!
#
!,
44"
"
σ
"
# "'
!Σ!5!δ#
Γ∞
"
4
1σ
# "'
6
";
Γ∞ 4
'"
5 σ# ≥ 5 σ6 ! σ$ #
1σ
# "'
6
4
5 /σ 1 ≥ 5 /σ ; Q σ − 1
∈6
"
-"
!
Γ∞
" 47!
!
4 %2
!
" !
"
4
";
!
"
-
'" %
!
!
'"
?!
"
"
!
!
"
";
"
4,!
!
#
-"
!
4
4 ! %%%
$
σ6 ∈ Σ %
F : %%%
; # "'
!
4
σ6 ∈ Σ
∈6
"
!
"
"
"
;
# "'
#
!
! "! "
!
!
%
!
!
"
! ""
! %
;&'
"
# "'
!
#
#
--
$
%
' ,#
!"
"
S
%
"!
1'#
$&'
"
" !
%
#
!
'
'"
"
"
-"
.
"!
"
"
%
!,
5 4
"
"
-
/4,! 4 !"""!4 1
%
51
!
!
4, . "
"
'"
#
!
!
C51
"
-"
!
4
FK%
4,
:1
/8 1
$
"
4 -# "
"
"
!
%
"
!
4 / FK %% 1%
>51
"
;
4
'
4,
:
"
$
!
"
4%
"
"
#
%
.
"
-"
%
!
!
"
!
"
";
!
"
4
"
%9."
" -"
!
"
" !
!
"
"
!
/
%9
1"
. "$ !
%
-"
!
.
"
!
!
3"
! #
"
!
-"
"
" !
""
!
!
!
C
";
!
C
.
"
!
""
"
8
"
!
"
;
""
!
"
; # "' #
#
!
4
&'
F /σ
-"
#
σ
#
1% $
#
"
/σ
#
'"
!
σ
≠1
#
$
" $ !
'
."
%
" # ""
"
! ! " !"
$
" !
! # "$
"$ # " '
!
."
" ! %
""
$
"
%
"
%B
.
!!
$
.! #
!
%
#
#
:
$
"
!$
"
!"
!"
! "
% 3"
#
"
"
!
"
!
" # "" " !
"
B
-
""
"
99σ /σ ,#! σ #! σ #!"""! σ #!"""0 !
"
4 /. τ# 0 τ∞= K
4
!
"
/4,! 4 !"""!4 1
0
σ
≥
#
$ #( $
#,F . τ# / - "
,F. τ$ #
!
σ
≥
#
$ #( $
!
1%
"#
$ #≠ σ
∃
"
σ #(
!
$ #( $
!
$ # σ
,
. τ$
$ #≠ . /τ − 1
*
"
$ #≠
"
."
≠
,
!
"
" ."
1%
1 "
#
."
/τ$ 1
5 "
/"
!
.
. ,#! . #!"""! . τ$ #1
# ! #
!
!$
.
. τ$ 1 " - "
"
!
& !
"
!
";
-
."
" ! -
'"
"
. τ#%
" $ !
4 ."
5 %
!" .
%
" # ""
."
."
!
99
!
.
;
%
%
"
-
%%% " / 0 1 -
!
"
./ %1
."
!
!
!
."
!
." τ
4
σ /4,! 4 !"""!4 1%
/4,! 4 !"""!4 1
0
"
!
-"
!
.
.
."
%
99 σ 4,! 4 !"""!4 #
/ %1
"
4 !"""!4 # "
/4
$
1%
!
$ #( $
!
σ #(
"
!
∃ ! * ∈ ! ≠* " #
#,F . ,# / - "
$ !
."
$ #,F. τ# / - "
$
τ≤
FK %%%
!
,F. τ$ #
σ #(
9#
$ #F. τ$ #
"$ '
. * /τ − 1 "
!
4
FK σ ,# F., ,#%
#
"
FK %%%
"
% 3"
"
# "'
6
'
!
. "
99 σ 4,! 4 !"""!4 #
99σ 4,! 4 !"""!4 #
%6
#
σ 4 !"""!4 # σ 4 !
∈6
" %
!
"
#
4,1%
/ %1
"
"
!
#;"
!
"
"
%
C
"
"
σ 4 !"""!4 #!
#
""
!
?!
!"
!
"
!
"
'"
"
"
!
-3
#
'
∞
=K
!
"
4 / FK %% 1
!
!
(
"
- "
5 /4 Q τ 1 ≥
!
"$
" ! P $
"
. τ#
!
"
"
4
" .
"
5 /4 1 =
∞
=K
4
(. / 1)
δ
'
"
τ
!
$ !
"
4
[
/. /τ 11 ≤ δ 5 /4 Q τ + 1 − 5 /4 1
"
∞
=K
/:1
(. /τ +
δ
+ 1
"
]
/:1
)
$
$
"
%9! ! P
"
"
"
!
$
'" %
>
"
'
!$
" E
5
# %
"
4
!
"
"
/. /τ 11 + δ 5 /4 1
! 5 /4 Q τ + 1 =
$
!
#
'"
"
/. /τ 11 −
%9 "
$
L
.$ %
;
!
"
/ Q .− 1 "
{ }
!
& -
.
!
"
$
!
'
$ # "'
/. 1 =
!
!
"
%
# "
""
" $ !
%
%
%
)
"
.
τ
Q .− /τ 1 + δ 5 /4 1 %
.
L
!
.
{ }
%
$"
- "
"
$
(. /τ + 1)
δ
>
'"
$
*
" ! P $
.
$
$
!."
"
. 5 /4 Q τ 1 =
!
!
'
!
" # "
! " M#
5 4 &τ , # 5 4 # #
>
"$
"
"
!
"
!
!
" 4%
'
(
σ 4,! 4 !"""!4 #
+
∈ !
+
/. /τ 11 ≤ δ 5 /4 Q τ + 1 − 5 /4 1
!
*
#
"
!."
0
)
".
C#
!* #+ '" ,"
' , "+#
!
!
'"
$
!
.
$
'
!
$
"
"
!
%
" . ";
.
#
'"
#
"
'"
"
! -
4/ !
19σ
!
"
-
- ."
" "
!
#
!
;
#
!
$
$
#
$
#
"
.
!
%
"
#
"
.
; !
#
"
!
'"
$
"
. "$
- C/
!
1 8%
'
" $ # "'
%
""
. "
!
"
/C1
σ
! !
9
#$
!
#$ !
!! N
#
"
#
" !
!
∞
</ ∞1 5 </ 1 FK%
$
"
'" % 3"
""
"$
"
!
!
5 σT !σ − # $ 5 σ #
!
#
$
%
σT
→∞
/C1
" σ%
! ⇐1 9 "$
-
"
"
! . σ− %
""
'" % 3" $
B
" !
"$
σT
#
/9
+
$
"
# "$
!
4"
]
'" %
!
8
'
'!
"
#
(5
,, "
$
τ ,! !""" (
,! !""!
[
/. /τ 11 −
.
"
"
ε UK
/C1
"
E!
!
"
!
# 4
-
L
M%
. "$
"
!$
!
.
"
"
!
'
"
-
"# "
!
"
!
# $. "
#
.σ
#
"
"
5
!
5
σ #
σ
."
"
!
A
. εV:%
."
# "$
# "
-
A
#
!"""! !"""!
A
#
. σT
"
εV:%
$
'"
"
!
$
# ""
:%
$
"
#
"
"
'"
!
σ
"
-
%
!
" !
%
!
!
"
!
.
#
'"
$
$
'
!
.
"
"#
%
#
!
# "
%
; = 4(σ , V σ ∈ Σ
49
"
Γ
!
!
'" ! #
"
!
'
"
!
"
'" %
";
'"
}
"
"$
'"
% ; ⊂ 9∞ % 9 4∈ ;
Γ∞
"
"
$ # "'
!Σ!5!δ#%
Γ∞
4
%9
#
Σ ⊂Σ!
$ # "'
"$
σ%
# "'
{
-
! !"
"
6
"
. "$
!
"
!
$
"
!
/C1
!
"
"
11
!
%
9
!
"
"
-
!
"
!Σ!5!δ#
+
-
.> "
"
4
$ # "'
<)
Σ
# "'
! !5#" 43 / 3! 3! """"! 3!"""0
$
"
'"
"
+
"
%
9 %
6
3
!
43 "
"
. 3"
"
"
"!
.
!
$ !
&
Σ
"
"$
"
"$ # " '
"
" !
'"
-
'" %
.
6
"$
'"
%
'"
;
!
FK %%% %
!
!
$ # "'
"$
!
σ 4,!4 !"""!4 # " # 4 43
$ # "'
# "'
"
!
"
!
! !
$ # "'
,
4 # "
%
-4
'! " &
-"
%&"
"
$&'
"
#
"
"
" /=8>1
!
!
" /=8 1%
!,
64
!
"
" /4 !"""!4 1
/σ 4 !"""!4 #!""!σ 4 !"""!4 #0 R"
"
5 !
" σ
!
"
" !
*
"
.
"
%
!,
74
# "'
!
"
6
!,
84
!
"σ
"
5 5 ?σ @" & "
σ 4 !"""!4 #
"
F %%% %
"
5 ?σ @F
"
" /4 !"""!4 1
"
5 !
/5 ?σ@ σ∈ Σ 0
"
F %%
# "'
"
- #
$
/σ !""!σ 1 " # 4
"
"
!
"
- !
.
. 5%
8
% 3"
!
!
'" % 6
"
"
&'
"$
"
'"
"
# "'
"
4
!
"
!
. "$ ' !
"
!
!
"
.
!
" %
'
Γ∞
- A
4&
!Σ!5!δ#!
" '"
" "
σ ∈Σ
11 3 4
∈
%
4(σ, ∈ ;
σ
/.(σ ,/ 11 ≤ δ [5 /4(σ ,Q + 1 − 5
! "
.5 !
%9
!$
"
# "$
5
"
"
#
"
% &"
."
5% B
4?σ @ 4 % 6
'" !
!
"
4
# "
"
# "'
!
"
/σ 0 ∞= ∈ Σ
F %%%
" # " 5 σ #!
Σ
'"
%
C%
"
"
σ% B
"
5
!
- !
/>1
# "'
"
/>1
.
'
σ
]
" %
" # ""
!
"
(
/.(σ ,/ 11 −
! !
2
"
"
4 ∈; %
5
!
!
"
4
5 4# 5%
σ
"
4 !"""!4 #
"
"
%
9σ $
" $ # "'
"
"
J∈G %%% H
τ " # 4
.* τ## $
!
5
.* τ## $
*
!
"
0
.* τ## B δ?5 4*!τ% # $ 5 @
4
.* τ## B δ?5 4* !τ% # $ 5 @
"
/4,!4 !"""!4 1%
/4 !"""!4 1
*
#;
"
"
B$
- " "
σ* ∈ Σ % σ
> ! ! !
'
# "'
!
"
∈6%
C
$
$
"
"
#
!
#
"$
!
!
" !
"
#
" %
"
L
!
"
"
! ! %
'
σ !""!σ #
44
σ !
4 !""!4 #
+
"
σ ∈ Σ
4
"
4 4 σ#
4 4 σ #∈;8
"
$
- " "
>
4
[
/. /τ 11 −
]
/. /τ 11 ≤ δ 5 /4 Q τ + 1 − 5
τ%
∈6 F %%
B$
"
-"
# σ 4 !""!4 #
! !"
% B
"
!
5 4 # 5 /4 !""!4 1
'
# "'
!
"
"
" %
!
"
" !
";
'"
!
.
"
"
'
"
54
/9
>
!
! ⇐1 9
# ""
"!
!
'" %
σ !""!σ # 2
. -
/. K /τ 11 −
4
% 3"
-
∈
! . ;
"C
4,∈ ;
τ(
[
/. K /τ 11 ≤ δ 5 /4 K Q τ + 1 − 5
! "
"
/4 !""!4 1 " !
]
/L1
>%
"
" # . ! # 4∈ ; !
"
"
)
σ !""!σ #"
"
4
"
[
/. /τ 11 −
]
/. /τ 11 ≤ δ 5 /4 Q τ + 1 − 5
∈6 F %%
"
"
"$
! !"
"
/L1
"
ι%
+
# σ/4,! 4 !""!4 1
$
# "'
Q !$
"#
"
# 4,
.
∈ ; %
!
"
#
"
#
!
$ # "'
"
"#
! 1
!
#
." ! !
'! "# ' 1"#
# ,1
% '
!
!."#
%
"
#
"
# %
!
#
#
"$
'"
"
-
"
# ,,
1"&1"#
!
"
%
'
6 4
σ !""!σ # 2
. -
" C
4 !""!4 # 9 4 4 σ #
σ 4,! 4 !""!4 #
8
;
"
;&'
.
!
%
"
!
!
"
"
!
!
. ";
'"
"!
";
%
"
"
!
"
#
# " !
;
"
- %
;&'
!
"
#
!
%B
% 9
"-
"
%
"
"
!
"
$ !!
!
!
:K
"$
#
"
"
""
# "#
#
"
"
!
! !
#
'"
!
! ! # "
.
%
S
N
4,∈ ;
.
" ! # ; " !!
!
";
"
+
!
#
!
%
9
B
""
"
"
"
"
%
"
!
9 #
"
"
( )*+,%
#
!
$:' $;
6
!
!
%
"
#
.
$ !!
"
"
!
!
"$ !!
$
"#
%
.
!
* "
!
)
#
"
! !
#
"
$ !
!
"
"
!
%
:
( )8 ,
#
%
-
8 "#"
. F/. !""""!. 1 "
#
!
'
!
!
7
π . & .− # R "
"
$ ! "
!
"
!
@. %
""
!
!
."
(
" !
!
" ! !
!
'! " &
)
!
%
6
"
" #
"
". # $
"
&'
$
"
! % $
!
!
"
""
"
"
#
"
%
!
"
!
!
"
"
.
#
"
4 π . # ?8 Σ
"
"
!$
.
! P
# $ ""
"
=
.#$
"
"
#
"
"
"
.−
!
"
!
#
%
4 8 "#
!
4 8 ,#B
"
'4
!
→∞
!
%
8 2# K
?,!C@
"#
δ
π C&,!"""!,# 0
−δ
2
π 2&,!"""!,#:,
"
" C
$
& !
- C
"
#$ !
$
'
'
!
""
. C
% B
!
"
!
"
"
"$
! ! !"
% $ # "'
6
56
%6
! #
"
% $
!
'"
!
#
'"
!
"
3% 6
!
%
#
#
#
". $
!
!
! !
.
#
# "
"
# "'
"
"
#
# !
#
.T
6
" "$ # " '
# "'
. "$ # " '
!
!
!
$
5
#
. !"""!. # "
:K
%6
!
. "
#
%
"
#
#
#
!.
'
#
2
.T
!
. $
$
" !
"
!
"
-
%
::
"
.C "
"
"
"
!#
"
"
#
%
"!
""
%9
. !"""!. # #
π 2#" B
"
:K
"
π 2# π 2!"""!2#
2%
!
""
-
$ # "'
-
7
( )8 ,
"
.
!
!
.
!
!
%
#
! "% ""
7
%
!
!
"
"
$ !
"
"
"
#;
"
!
"
!
%
'
"
"
#
%
#
" "
!" !
:>
%
# "
"
. "$ # " '
!
::
#
7
!
% B- # $
! ""
!
%
:C
.
"
$
!
!
"$ # " '
#
"
!
%
.
#
#
! ""
:
'"
"
!"
"
"!
"
!
!
#
"#
$ !!
"
!
! . "$
%
9 "$
.
-
! ""
!
4"
4 / .T ! .T %%%% .T %%%H
!
π /. K 1 B π /.T 1
!
:
!
::
:C
:>
"
!
!
#
.K
! ""
:L
#
"
.T
/!$
.
π /. K 1 Uπ . # 1%
!" !
"
4, / . K . K ! !"""! . K !"""0
! ""
" #
"
4 !"""!4 !"""!4 #
!" !
! 4 4
"
#
F %%% %
"
σ 4,!
"
"
"
$ !
"
σ 4,! 4 #"
"
#
!+ %
6
# "
# %9"
$
$ .
( )*K, # ";
7
!!
7
#
"
"
7
& "
( )*>,% ""
!
. $
"
% EB
"
-"
"
"
"
% 3" $
""
% 3" $
"
! - "
M& "
/( )):, >L1%
:L
!
#
! ""
$
!
.
%
.!
#
.
#
! # "$ # " '
6
"
#
W
!" !W
!
!"
E "
"
""
-"
'
""
! "" #
'
" !
"
7
" !
!
.
# "$
#
:C
"
.! #
56
%
M
!
"
#
"
'"
!
!
"
π /. K 1 F
6
"
G2H
"$
!
$
'
"
! ""
!
!
!" !%
4 / .T ! .T !""""! .T !"""0
(4
-.
4, / . K . K ! !"""! . K !"""0
=
"8
π /. K 1 B π /.T 1
!"""! !
+
σ 4,! 4 # +
.
.
(
δ
π /. K 1 $ π /. K 1 ≤
4 9"
−δ
? π /. K 1 $ π /.T 1 @
"
"
"
!
$
""
$
"
/ 1
"
/ 1
δ≥
π /. K 1 − π /. K 1
π /. K 1 − π /.T 1
!
! ""
#
#
"
#
" !
%
! ""
%
!
! P/
!
!
"
/:1
!"""!
"
"
!
$ !
"
,
4 . "$
"
%
# ""
"
!" !%
4 8
π /. K 1 B π /.T 1
!
4
"
$ !
6
"
!
"
!
!
$
1%
!
'
/ 01
$
"
"
"
%
'"
.K
! ""
"$ # " '
"
"
"
"$
"
.!
$
"
"
.
'"
% 3"
#
$
"
!
!"""!
-
!
'" %
6
# "
.−K " 6
5
$
π / 2Q .−K 1 "
4, / . K . K ! !"""! . K !"""0
+
!"""! !
σ 4,! 4 #
δ:
2
.
+
:>
"
.
.
δ∈?δ! @!
# π /. K 1 B π /.T 1 !
4
δ
GH
σ 4,! 4 !"""!4 #
π /. K 1 − π /. K 1
π /. K 1 − π /.T 1
π /. K 1 − π /. K 1
π /. K 1 − π /.T 1
# "'
.
δ∈?δ! @! " !
"#
%
#
B
1.!
"
$
"$ # " '
"
C%
# "
"$
"
/
!
'"
$ !
!
(
S
1+#
!'!!=
" "
"
%
-
"
#
"+# '"
"!
!#
"
%
&, 1
-
# " !
"
"
% 3"
'? W
#
#
# +>
( )*+,
#;
""
.
!
"
7
&'
"
:
π1 (qc)
-
! "
!
%
π2 (qc)
, 1
# "
"
'"
'"
<
!
/ 1
!" !
#
% 3"
!
7
;
!
"'
# "' %
4
!
-
!
"
!
" W!
# !
! !
-
/" !
:L
1
"
"
- /" '? 1
# !
"
"
#
"
"
! ""
"
% 3"
#
!
$ !
"
B
""
"
" !
" '?
-
"
"
!
#
"
.K
#
" '"
!
"
"
%
-
1
#
"
. ! #
"
/
/. 1
"
!
'"
/.K1"
'" #
""
! ""
#
!
!
%
&'
#
"$
'"
#
"
!
#
#
#
"$ !
"
"
!
%
# $ ""
"
'
#
.
-(
/. !.,1
π .# "
# "
π .# "
""
#$
" #
4 / .T ! .T K ! .T K !""0
.
4 !"""!4 #
9
. !.,# 9 +
.
-
6 .
!
%6
# "
'
! .T = /. %%% . 1 "
"
"
"
"
.T K = /. K %%% . K 1 "
'?
!
'
G .T ! .T K ! .T K !"""H
4
#
"
<
."
2
!
2
9
2
(
., . "
π . # $ π . # δ ? π . # $ π . #@
'#
π . # $ π . # ≤ δ ? π . # $ π . #@
)#
., B .
π . # $ π . # δ ? π .,# $ π . #@
'-#
π .,# $ π .,# δ ? π .,# $ π . #@
)-#
!
"
%
&'
$ #
/C
C$1
#
"
! !
( )*+,%
""
!
$
/>
!
#
:+
$
'"
>$1 $
# .
" ! P
$
!
"
!
9"
"
!
"
;
-
. #
$
"
"
!
/C1
/>1
"
"
!
+
!
;
/>1
/ "
"
"
! "" #
"
!
'
'" % B
"
!
!"
'
#
"
/C$1
/>$1%
.%
.
<
σ 4,! 4 #
"
#; !
'
!
% B$
"
"
!
$ !
- "
.T K
" $ !
σ
"
$
%
"
"
"
!
#
"
"
#
"
".S
!
# "
.
.,
" #
C%
"
"
- #;
1
%
"
"
"
"
!
" '?
"
! !
!
% 6
#
.
+
"
"
$
.,
"
C
" '"
'"
- .
"
/. ! .,1
4
" !
%
.
"
!
!
4,
"
" !
.
!
"
# " #
;
1
1
/>$1
&'
%B
#
"
"
" σ
"
"
4 / .T ! .T K ! .T K !"""! .T K !"""0
4
"
!
#
"
4, / .T K ! .T K !"""! .T K !"""0!
.
"
$"
!
341
2
9"
!
.
/6. 1% 9; "
"
#
#
"
-
"
$
π .,# 1
−δ
%B
"
! ""
δ
/
"
'"-
!
!
δ:
π .,# 1%
−δ
"- /δπ . # %
!
"
"
"
π /. 1 +
!
δ
−δ
π /. K 1 = K
." !
'? %
:8
#
!
"
"
.
!
"
# "
!
" '"
$ !
"
"
:+
"
"#:
""
"
!
"$
"
"
"
"
# "
"
"
. =
$ !
"!
"
56
"
.
!
! "
"
" !
! ! % & - ! "! "
."
6
#
/
!
%
" . /X1F 5 X
!% & "$ # " '
$ 1
"
! #
" . π =
'
"
/
"
Y
%
+ 1Y
. F9V: /! #
1%
" #
!" !
" .
:8
# %
#
%
" # ""
" #
"
#
$
Z
"
.
" !
:8
! ""
!#
" !
+
! ""
#
'"
'" # "
"
:+
"
'?
!" !%
! P
:
#
!
"
!
"
.
""
!" !
$
"
- ( ,, " !
#
'" "
" !
'"
"
!
"
M# $
!
"
'? %
! !
#
" '?
#
"
%
"
'! " &
"
! E" !
!
"
1 '" " .
6
"
!
!
%
-
!
"
"
#
"
/ 1
"
"
" # "" "
"
"
!
"
:%
4
"
!
:*
!
"
" '"
"
-
"
# "
-
. "
%
"
. &'
( )**,%
δ
π /. 1 − π /. 1 ≤
!
$ !
"
"
−δ
(π /.
".
/L1
# "" #
" .
'"
4
9
-
δ+ =
π /. 1 − π /. 1
π /. 1 − π /. 1
"
4
δ+ =
/ + 1Y
/ + 1Y + >
'
δ≥
/+1
/81
4 A
<
D
+
/ + 1Y
/ + 1Y + >
.
%
#
'
" !
"
!
"
(
−/
'
.K =
S %
$
)
"
!" !
" %
#
δ "
"
− / − . K 1. K =
"
4
δ
/ − . K 1. K −
−δ
Y
+ 1Y
/
/*1
4
+ −δ/
/
"
:*
:
1
+
− 1Y
− 1/ +
+ 1Y − δ /
.K
! ""
!
.
$ !
:
!
#
!
"
#
"$ #
− 1. K
+
!
!
" ! ""
"
>
6
"
/ + 1Y
/ + 1Y + >
.K #
<
(
!"
" %
#
!
. -
! !
9 δ<
=
=δ +
"δ+ !
:*
)
1−π
π /. K 1 − π /. K 1 =
$ !
δ
−δ
/)1
.! "
"
(π / .
!
K
1−π
)%
:)
#
"
!
S
"
$ !
"
.K " -
#
. #!
'
! ""
%
'
5 A
.
+
<
!.
-
.
2
-.
9
,, " !
" !
@
"
%
"
" #
-
+>
"
"
!
" '?
! ""
"
=
+
?#" "
!
! "
"
!
'
!
" δ
.
. δ+%6
!
"
/"
'"
# " "
"
1%
6 A
<
D
"
!
'"
!
-
+
=
!
<
+
. -
(
.
δ≥
/
δ≥
/
/
− 1Y
=δ
+ 1Y
&
! !
"
#
'
. +
!
"
"
"
'
.
≥E
.
/
/
− 1Y
+ 1Y
"
"
$&'
"
/ + 1Y
/ + 1Y + >
"
.
"
! "! "
"
:E
!
!" !
.δ %
'
.
%
X "#
"
+ 1Y
=δ "
+
/
"
+ 1Y
+
"
"
'"
! %
CK
"
"
"
"
!
!
δ+
'
! % &
!
"
"+#" (
1'" "
,"
# )
!#
6
'
##
"!
1"&1"# A #
#" "
1
' #B
"!
+>
-
3
4
5
6
7
(2
-2
(22
7!
"
δ+
K%L:)
K%L8
K%+K)
K%+>:
K%+8
K%+)L
K%8 +
K%8L
K%***
K%)+:
7!
"
δ
K%:*
K%CCC
K%C)
K%>L
K%L
K%L+:
K%+K>
K%++)
K%*8L
K%)+
$
! !
" !
" '"
" %B
".
'?
."
#
"
"
# "!
"
"
-
'"
!
$ !
"
"
. "$
'"
%
'
7
A
9
!
(
δ≥
δ≥
.
>
/
+ 1Y
/ + 1Y
>/ − 1Y
'
"
"
:)
Z
:)
.
≥E
.+ " !
"
"
:E
&
# "
!
.
#
"
'
#
"
%
"
! !
#
" !
#
" #
"
C
"
+ 1Y
/
"
#
"
≥ E%
>
" '?
"
. ! ""
/ + 1Y
>/ − 1Y
δ <
.+
( )**,
#
δ >
" !
"
9
" '?
#
!
'? %
"
-
."
!
'" "
.
! ."
"
"
"
'
( )*8,
K%:* . K%
"
$ !
"
;&'
% 3"
#
"
! !
"
$
" #
"
."
!
."
! !
" " '"
%
!
! "
"
" '?
M1 # "" #
"
"
"
!
$ !
.
9
',
#
#
2 9
.
"
!
/E" !
"
;
#
!
"
%
!
!
!
!
.
"
%B
+
.
"
#
9
"
"
"
!
'"
"
-
""
"
%B
"!
"
"
"
"
"
! "
!
!
"#
% 3"
"
$
!
""
"
!
#
"
""
! !
. "$ # " '
"
' '"
"
"
$ '
!
!
%
;
"
"
'
!
# $. "$ # " '
"
#
."
"
!
! %
! ! %
"#
% "
!
!
$"
"
"!
!
. "$
!
. "$ # " ' %
""
9
' '"
"$ !!
CK
""
! ""
""-"
! !"
"
#$"
( )**, !
$
!
"
"$
"
!"
.
#
" Q !$
!
% B
"
"
! ""
C:
! ""
#$
! ""
"
.
!
!
%2
#
"
!
%
$
.
!!
-
""
""
"
" ! ""
5
# $ ""
'" % 8
D
'
"
"
" %B
!
"
'"
"
! !
.
&
'? %
!1
#" "
#$"
9" #
/# # 1
"
6
$
[
/. 1 − π /. 1 = δ [π /.
"
7 A,"
'"
-
#
"
%
' '
"
K
"
"
!
#
"
"
B
"
!
!
+>
. "$ # " '
"
" !
"
'?
-
"
" '?
"
-
"
"
.
9
9
"
!
# )
"
"!
"
" !
4
]
1 − π /. 1]
π /. K 1 − π /. K 1 = δ π /. K 1 − π /. 1
π
6
K
A (B
!4
π /. K 1 − π /. K 1 = π /. 1 − π /. 1
9!
π /. 1 =
#
[
−/
− 1. ]
>
:
−/
− 1.
)
:
>
& -
"
(
!
− / − . 1.
!
−/
+ 1.
#
"
(
=
"
) =(
:
π /. 1 = K
−
#≥
π /. K 1 − π /. K 1 = π /. 1 − π /. 1
"$ #
(
#[
'
−/
. . + .K =
+ 1. K
:
+
)
−/
− 1. K
>
4
:
. . + .K
CC
.
)
:
− / − . K 1. K
"
−
#[
−
4
#
""
" ." #
""
.K
" !
−
"
.
[
−/
+ 1.
)
:
= δ /. − . K 1
>
9
"
"
# " #
'?
!
.
.
]
$
-"
/. − . K 1 −
-
/& 1
4
)
4
. =
+ *δ
.K =
− *δ
"
(
'?
%
:
π /. 1 − π /. 1 = δ π /. K 1 − π /. 1
(
"
/
−
+ 1Y
+
/
−
+ 1Y
+
#
"
# $ ""
." #
−
%
!
/ + 1Y
%
>/ − 1Y
δ<
B
.K =
-
"
#K
− *δ
−
+ 1Y
+
/
6
.
=
'"
-
#
"
#
"
/ + 1Y / + 1Y
>
>/ − 1Y
+
"
:
+ 1Y
% /"
+
1
!
6Y 5 +6 0 [ K
1
:
/
δ<
/
" !
R
6[+1
! "
+ 1Y
+
R
/
/ + 1Y
/!$
>/ − 1Y
6[ +%
" !
. δ F
/
+ 1Y
%6
+
#
#
"
#F
/ + 1Y
>/ − 1Y
+ 1Y
+
"
δ
−
.
"
#
" %
!
$ !
C>
"
"
'?
"
!
"
" '?
.δ
4
π/
−
δ
1+
−δ
& - ! "! "
δ=
4
>
+ 1Y
/ + 1Y / + 1Y
≤
>/ − 1Y
+
"
/ + 1Y
" !
>/ − 1Y
- !
!
δ
−δ
π /. K 1 =
" !
−δ
!δ
−δ
"
'"
!
π /. 1
4
π /. 1 − π /. 1
π /. 1
& - ! "! "
δ F
/
/
π /. 1 = K %
-
4
4
δ =
−
"
π /. K 1
"
π /. 1 =
#F
""
π /. K 1 = K
#
9
" %
"
/#K # 1 $ !
π /. 1 +
6≥+%
" '?
'"
$ #
δ
1=K
:
'
/
&;1
δ≥
π/
'
4
− 1Y
+ 1Y
CL
$ !
"
.
CC
9
Abreu, D.(1986) «Extremal Equilibria of Oligopolistic Supergames,» Journal of Economic
Theory 39, 191-225.
Abreu, D.(1988) «On the Theory of Infinitely Repeated Games with Discounting,»
Econometrica 56, 383-396.
Axelrod, R. (1984) The Evolution of Cooperation, New York, Basic Books. (Traduction
française Donnant Donnant, Paris, Odile Jacob, 1992.).
Dasgupta, P., & E. Maskin (1986) « The Existence of Equilibrium in Discontinuous
Economic Games I : Theory », Review of Economic Studies 53, 1-26.
Debreu, G. (1952) « A Social Equilibrium Existence Theorem », Proceedings of the
National Academy of Sciences 38, 886-893.
Friedman, J.W. (1971) «A Non-Cooperative Equilibrium for Supergames,» Review of
Economic Studies 28, 1-12.
Friedman, J.W. (1992) «The Interaction between Game Theory and Theoretical Industrial
Economics,» Scottish Journal of Political Economy 39, 353-373.
Friedman, J.W. (1993) «Repeated Games, Supergames and Oligopoly,» Working Parper,
University of North Carolina.
Fudenberg, D. & J. Tirole (1991) Game Theory, Cambridge, MIT Press.
Glicksberg, I. (1952) « A Further Generalization of the Kakutani Fixed Point Theorem
with Application to Nash Equilibrium Points », Proceedings of the American
Mathematical Society 3, 170-174.
Greif, A., P. Milgrom, & B.R. Weingast (1994) « Coordination, Commitment, and
Enforcement : the Case of the Merchant Guild », Journal of Political Economy 102, 745-776.
Kakutani, S. (1941) « A Generalization of Brouwer’s Fixed Point Theorem », Duke
Mathematical Journal 8, 457-458.
Lambson, V. (1987) «Optimal Penal Codes in Price-Setting Supergames with Capacity
Constraints,» Review of Economic Studies 54, 385-398.
Leonard, R. (1994) « Reading Cournot, Reading Nash : the Creation and Stabilisation of the
Nash Equilibrium », Economic Journal 104, 492-511.
Myerson, R.B.(1991) Game Theory : Analysis of Conflict, Cambridge, Harvard University
Press.
Nash, J.F. (1951) « Noncooperative Games », Annals of Mathematics 54, 289-295.
Radner, R. (1980) «Collusive Behavior in Non-cooperative Epsilon-Equilibria of Oligopolies
with Long but Finite Lives», Journal of Economic Theory 22, 136-154.
Segerstrom, P. (1988) « Demons and Repentance », Journal of Economic Theory 45, 32-52.
Selten, R. (1975) «Reexamination of the Perfectness Concept for Equilibrium Points in
Extensive Games,» International Journal of Game Theory 4, 25-55.
C+