Yale Law School
Yale Law School Legal Scholarship Repository
John M. Olin Center for Studies in Law,
Economics, and Public Policy Working Papers
Yale Law School Other Scholarship
3-20-2003
Simultaneous and Sequential Anticommons
Francesco Parisi
none
Norbert Schulz
none
Ben Depoorter
none
Recommended Citation
Parisi, Francesco; Schulz, Norbert; and Depoorter, Ben, "Simultaneous and Sequential Anticommons" (2003). John M. Olin Center for
Studies in Law, Economics, and Public Policy Working Papers. Paper 279.
http://digitalcommons.law.yale.edu/lepp_papers/279
This Article is brought to you for free and open access by the Yale Law School Other Scholarship at Yale Law School Legal Scholarship Repository. It
has been accepted for inclusion in John M. Olin Center for Studies in Law, Economics, and Public Policy Working Papers by an authorized
administrator of Yale Law School Legal Scholarship Repository. For more information, please contact
[email protected].
!
" # #
# $ $ %#!
& ' ()
$ *()
+ $! $
(,-./
012!011!013! 45! 62)
. '
7 8 9% !
! # $ : :
(;! 133<* ; - ! 133<* =! 5222* ! 5225*
!5221*
&! !522>)
! !
$ !#$ &
! *
#
8
#
# $
(
&! 522>) # #
# ! # ? !
# # ! !
# $ ! #
# ! !
8 #
$
$ & !
" ' $
@ # #
# !
#
! $ @
$ # $
8 #
$
& $
A! $ ! (
)$ ! $ $
#
# "
# $
!
$ ! !
# (
)$ # !
- !
( &!
!522>)B
! ' ! ! $ &
$ + &
$
$ !
$
#
!
-$
$
#@
! ! $ # ()
$ B# ! ! @ ?
! $
@
$
!
!
# @
! +
$
#
# #
$ $ &
(
) $ ( )
!# & >!
$
! #
8
$
+ &
B !
! !
!
! # $ $
##
!
! @
@# ? + !
' ' ! # @
! & ##
! $
#
$ # ! #
# $ $
! $ $
##
## # +
@#
8 $!
@ + & #(!
@ ! ! )!
!
# (! $#!
#
* ! #
! )
@
$
51 #
! $
! # (!
# ! #
!) 55
!# $ $
# 8 !
' # #-
# ' #
& #! # @#
& ' '8 =(5222)
&!
(522>)! 8 @ C C @
B #
$ ! #$
! #
$
! # 1$ # #
5? # B# ! ! $
#$ # ' !
# 1$ # # 5? #
! #
! # ' '!
' # ;! $
# ' !
! $ #
! # !(!)
! $
?
$!
' '! ! #
# ' #
$!
& ' #8 $
! ' #
' '
#
' '
$
!(!) #
#$ $ ? #
(1)
∂9
[ [ >
∂[
.?
$
$
# ! $ #
!
!(!)D!(!)8 !
# $ & !(!) ! &
/
(5)
∂9
∂9
[ [ = [ [ =
∂[
∂[
! #
# A #
$
! $ DD
&
! $ &!Г!
/
&
∂9
∂9
∂9
∂9
[ [ + [ [ = [ [ + [ [ =
∂[
∂[
∂[
∂[
(>)
A #
! $
!Г! # (>) ;!
/DD
F
&
# !
(5)
(5)(>) #
+ (>) #
1# !()! #
1
#$ (
(1))! #
(>) ## ! #
;!
!
# (>) $#
()! (>) # 1
8 & #
GH ! F
$ $
&
# $ ? !
?
' '
B# ! $ ' #
$ $! ? $
$
#
!$
$
' '
& /. $$
$
!DC $$ $
!
$ ? ' '
" # # # ( )$
( ) $ #
# $
$
#
! # ( ) & ( )
' !
$
#$ # ( )
# ' # ( )!
$
! #
! ' '# !
!
' '
#7$
9
$
!!! !"
# #
. #
# #
!
$ .!(!)
8 !
!
(H)
∂9
S S <
∂S
8
!
$ 7 9 #@ & !
! $ 7$
#
9
! &
! @ #
# $ =(5222)
" ! !
$
( @ @ )
=(5222)
B $
# "
# $
! !
( # ) ( @ )%!
# @ @
# #
!
! # !
$
# (!
$
' )
# $
!$
! # # 8
!
$
!
# ?
# ! #$ !# #$
# $ !
' '! ! #$
# # ' #
!
+ & # @
(!;!133H)B ( # )!
5 # %()
1 %(
) # # # # # 1!(! )
$
# # # !
# $ &
: [ [ = 5 [ [ − & [ − & [
(4)
8 $
# #!
# .(!) 1
5 @ ! @
# /
(6)
9 [ [ = 5 [ [ − & [ − 8 [ [
9 [ [ = 8 [ [ − & [
I @ !
# # # ;! #
5%#! # # 1
8 !
# #
# # !
# # ! #8
! # !
@
! # #
$ &
#
/
(<)
∂9 ∂5
∂&
∂8
[ [ − [ −
[ [ =
=
∂[ ∂[
∂[
∂[
(3)
∂9 ∂8
∂&
[ [ − [ =
=
∂[ ∂[
∂[
B # #
' /
∂&
∂&
∂5
∂5
[ [ − [ =
[ [ − [ =
∂[
∂[
∂[
∂[
(12)
I $ # !
1 (12)
5 (12) !
(12) #
# #
&(12);! #$
&!
$
#
#
!
$ #
@ #
8 ! # $
$ @ $
#
! #
$ !
#$
$ ! &
8 @ #
# ( ! # #
)! & $
$
# (
!
&!5222)
#
& # #
# # ! ! #
$ $!
#'-
!
! #$ # #
' #
&' (
+ # # !
$ ! !
8 () $ *()
8 #
! !
! ! ##
! *!
@ #
& %
& !
$! B
&!
#
!# !
$ & #
" #B $ #!#@
##
!
B# !
5!
@ @ ! #
(
)
@
$ !
& $ ;(133<)+ #!
$
# ! $#
#
#
? () #
# !() #
( # # @ )
&
$ & #
! ! &
#
# # !
" $ & # ! C ##
C @
( ! @ ! $
#
# ) #!
!
!
( # )!
& ) (
@ ! # @
B!$
5 # #
! #
$! # !
$
!
1
# ' # #
#
# I !
# A # # !
!
! #
J #
'
# &
# + #
! !
# # ! #
#
# & ! #
# ! $
8 5
! # #
$!
& @
#
(133<)
$!;-
# @ # #
# !
# #$
# !
55 #!
$ $ ! # &
!
! #
#
%!7 & 9
! !#
5
#
! !
.
& "
$ #@ #@ #
#
B !$!
#
# + 8@8 &
##! $ #
!
! @
!
! # ! #
#
!
! #!#
! @
$ &
> !
! # 5!
# # $ @ # !
# + ! ! #
! $ @ !! # (
) @
&# ! ! # !
$
(!!
)8 ! #8 %
;(1333)# K ? # #
$
# #
!& # !$
!( !;(1333!116>@116G)! & # & !
# );
! !
$ !$
# @# $
!
#
# #
$
B# !
' # #
( !5225)
*%
!#
! & #
# $ $ # "
& ' () $
*()
# @
!
@
@
8
! #
! !
L
$ #@ #@
8
#
: & :# #
$ # $
#
(
!" # $#%
&
'' ( ( &
)& *+
,- . & / / %""" 0&1 2 &
3! $ !
2 4
" 0 + 2
! $33
5 , & 6 %"" 6 7 & 2 1 0 8
9 " $"!
0
" 0 0 2 &
!" !3 3"$3'
4& :
4 ;
!"
4 ;
# 0 0& 2
% %3!$ %3#
# 0 0& 1 0 ;8 ;< #"
% $#'
0 ,-& =
$
"#
!$ %%!
4 ; & +
# 2 5 )=> 0 ,&
8& ?7 5@ ,<A % & 3 ! !$ '
*+ : 5 & B
'3 #3
#3 2- 2
' !"
; 0 & 4 %""" C &&D *@ 1 0 E-- 2--
$
" $'"
; 0 .
()*+, +-+ C8&1 C8& 7=
; 6< )
# 7 & 61 2 6-& ?.-
2A *@ #"!" #" $ %#
; 6< )
#% ( & *@ ( < . & & 2 . B
& & ../) (+ ( E@ /< E@ /< 7= !$3"
6 %"" 0 2 01 5- & 7&
9" $ !
6 %""% ' , " " '" $'!#
2 ;
# 0 2& 21 2- 2 & ) -+ "-
,(!" ! ' $'#"
2 ;
B := 2 5 F ) ;- G - &
(0 " - E@ /<1 =
-D E %""" 0- E- H B , -+ I
""$ '
BJD+-
-D E 6 & , 5 %""! 6 1 H & 4D
*(
0 .
" K- ) & $0- - # !3'$!%
! *(*(- 1 2
2+& ;1 ;)0
M0 - @-& < < 2 : & ; &@ & - ))) 8
- = & &=
0 &-& + ;
) = 4&A
#%) & & - + 4
# &
#) @ <@ 0& 2
#) =& @- 4
0 & & + 4
#) =D& < - - 4
&- - = ;@ ;@ -+I -&-
+- @ & & = &
8-& 6< ;
#%) & & @
; && -- (("- -(( (
03 ( + + 4 -+"" -( 5-(2
0- ( - @ & - $@& 0 8
=&& @& = ? A &
) - 8 @ & +- - I I =
&-
0 - @ @ -- I & & +
& 8+< 8 &-+ D + @ D& +
") @ & - & - &- 0
!1
'3)
C & ;
) C < =- 8 &=
&- +-& @ - &= +& D =
8 = =
K- + -& +--$< - @ -
& + K-& & 8 = = &- @ = &-
- -
) & @& 8 -& + & + 2 + +@
& .- &$- @-& & D @&
+ @-& <@ + & + B =- 2
+ A = @= @ -& D + &
A + & 8 K- - A + +I
0 + &=&- @ & &
- K- + =- & @ & =
+ & D = 8
) - - L / 5 5 @-& = / & @-& = &==
@ = = 0 - 8- @-& =
K-= - -
8-= K- B
8-& & @ ) =@ +
8-& - 0 + = & +
@ &- =
6 -= - + - 8 -& + ,- & / %""") 0 -
& - M < M I @& + @ &=&- 0 @ I @ =
-- 8- 0 & @ @ -D < & +
+ $@ ) 8 - = - @ < 0
=- - M % @ % & -+ - E -+ -
+& + 8D % M % @ = =- % L N% E @ < @-&
L N% - L M % M & 8D N%
0 I- ) @ @- &
= & = -+ - Г L M % 4 & - &&
< % L M $ 0 - & 1 M $ *
@ 8D 0 & & &
1 M % $ L " 6 %
- K- &= 0 & K-+- =- L L N! Г L %N! 4 -&&
- & -&-D <
) K- @ A && + D& &-+
D + 0 - &-+ D -+ - *<@ K-
& = -&- - 0 -
K- + -& & &$- + -&
+= 0 & - + +=- @
-& & E= - K- +- +1 --
& K- & + $&= & =$&= +@ @ &
,- & / %""" @ -+ @ 8- +
-&-D $&= 8 ) -D %""") @
- +& K-$&= 0 - @ 8+
+ @ -+ - *+ &
B
#3 -& - &-& = +@ -+
==& & = 8 -
) 8 - 8 I-= & -
&
'' &+ - @ =
- & 4 =& 8 = &$- - =-&
@ @ &- =
2
%"" D K- <& & -
" - &
"+ !#$!
-- & =- = - =
) - & = + -& - @ @-& &1
= 3+@ - &,F & +
++ 67 + = 0
& D + & --
+ = 2 ) -- @ , L,
D + & & &- &= + = ++ =
0 & @ < 8 L
;8 P67Q ) , , D + -& & -
&-& + $& < 8 0
&=& & & + = -
) && - &=& 0& 2 2
#
&+& @ - @ & =-D - -&
= - @ < 8 ?-+ A @
@ & - = =&& -
0 K- & !' 2 2 ,- & 6 3 *- & & &
@ - - @ & -
& - &
0 & & & -+
D & $ &
%""% =- = & - &
& = = &
D
& ; & %""" = -& $ 8 =-
=& -&- 0 +- & &-
& @A - @ =& ;
; & %""" =& $ 8 &
& &$ 6
@ & -+ &&
0 & & - + =<& -
- $ - -= && & -& +
+- - & & = +I= &
K-
; & %""" & %""%