rB − C > 0
B
r
C
rAB
A
B
A
B
A
B
rAB
i
0.5ni
ni
r=
r 0.5
!
ni
i 0.5
0.25
0.125
r = 1/2
r = 1/8
r
r
r
r
G
G
G
G
G
r
r
r
β
α
gi
i
ĝi
i
ϵ
gi = α + βĝi + ϵ
z
z
β
gi
gi
r=β=
(gi , ĝi )
(gi )
r
r
r
ĝi
gi
Gi
rG =
(Gi )
(gi )
r
r
r
r
M
X
X
λ
N
N + λN
r
r
r
r
N + λN
r
M >> N
N + λN (M − N )/(M − 1)
M >> 1
r
r
r
r
r
r
r
r
r
t
(gi (t), ĝi (t))
(gi (t))
r(t) =
r
r
r
r(t)B − C
r
r(t)B − C
r(t)
r(t)
B
C
tA
r(tA )
tA
tA
tC
X
X
B
tA
C
r(tA )
X
X
r(tA )B −C
X
X
tT
tA
tT
X
X
X
tA
tA
X
X
r(tA )
r(tA )
r(tT )
tT
tA
tA
tT
X
r
r=
(gi (tT ), ĝi (tA ))
(gi (tT ), gi (tA ))
tA
tT
r
X
r
r
r
r=
(gi (tT ), Gi (tA ))
(gi (tT ), gi (tA ))
Gi
i
tA
tT
ĝi
Gi
gA
gT
gA
gT
r
r
rB
C
r
r B
r
rB
C
C
r
r
B−C
−C
B
fX
0
X
gXi (t)
X
X
t
t
gXi (t) = 1
gXi (t) = 0
fY
Y
gY i (t)
i
gY i (t) = 1
X
Y
Y
t
t
gY i (t) = 0
Y
X
Y
π
X
λX
X
λX
Y
λY
Y
B
C
B
C
λY
tA
X
Y
X
Y
∆gi =
wi
1
[
w
(wi , gi ) + (wi ∆gi )]
i
gi
gi
∆gi =
g′i
gi′
1"
w
i
#
(wi , gi′ ) + (∆gi )
gi (tT )
i
∆gi
∆gi
g
∆gi =
1
[
w
(wi , gi (tT ))] + (∆gi )
i
wi
wi = 2 (V − Ci gXi (tA ) + BĝXi (tA )) + ϵ
tA
wi
X
X
∆g Xi =
2
[−C
w
V
(gXi (tA ), gXi (tT )) + B
(ĝXi (tA ), gXi (tT ))] + (∆gXi )
ϵ
∆g Xi =
2
[(rB − C)
w
(gXi (tA ), gXi (tT ))] + (∆gXi )
r
(gXi (tT ), ĝXi (tA ))
(gXi (tT ), gXi (tA ))
r=
X
r
(∆gXi )
X
Y
Y
X
Y
Y
wi
(wi , gY i (tT )) =
0
∆g Y i = (∆gY i )
Y
X
Y
Y
X
∆g Xi > ∆g Y i ⇐⇒
X
λ X = λY
Y
2
[(rB − C)
w
(gXi (tA ), gXi (tT ))] > (∆gY i ) − (∆gXi )
Y
fX = fY
(∆gY i ) = (∆gXi )
∆g Xi > ∆g Y i ⇐⇒
2
[(rB − C)
w
(gXi (tA ), gXi (tT ))] > 0
(gXi (tA ), gXi (tT )) > 0
X
Y
∆g Xi > ∆g Y i ⇐⇒ rB − C > 0
rB − C < 0
rB − C > 0
r
πλX
r
r
πλX
r
r=
πλX
βĝ (t ),g (t )
(gXi (tT ), ĝi (tA ))
= Xi A Xi T
(gXi (tT ), gXi (tA ))
βgXi (tA ),gXi (tT )
βĝXi (tA ),gXi (tT ) = (ĝXi (tA )|gXi (tT ) = 1) − (ĝXi (tA )|gXi (tT ) = 0) = πλX
βgXi (tA ),gXi (tT ) = (gXi (tA )|gXi (tT ) = 1) − (gXi (tA )|gXi (tT ) = 0) = 1 − (1 − fX )πλX
r=
πλX
1 − (1 − fX )πλX
r
π
λX
r
N
fX
X
fY
Y
tA
X
B
(GXi (tA ))B
GXi (tA )
tA
C
π
N
X
λX
Y
λY
tA
X
Y
∆gi =
1
[
w
(wi , gi (tT ))] + (∆gi )
wi = 2 (V − Ci gXi (tA ) + BGXi (tA )) + ϵ
∆g Xi =
2
[−C
w
∆g Xi =
(gXi (tA ), gi (tT )) + B
2
[(rG B − C)
w
(GXi (tA ), gXi (tT ))] + (∆gXi )
(gXi (tA ), Xi (tT ))] + (∆gXi )
rG
rG =
(gXi (tT ), GXi (tA ))
(gXi (tT ), gXi (tA ))
r
Y
∆g Y i = (∆gY i )
X
Y
X
Y
2
[(rG B − C)
w
∆g Xi > ∆g Y i ⇐⇒
(gXi (tA ), gXi (tT ))] > (∆gY i ) − (∆gXi )
X
Y
(GXi (tT ), GXi (tA )) > 0
tA
X
Y
∆g Xi > ∆g Y i ⇐⇒ rG B − C > 0
rG B−C <
0
rG B − C > 0
r
tA
r
tT
rG
πλX
N
rG =
1 + (N − 1)r
N
N
π
λX
X
X
r
rG
rG =
1 + (N − 1)(πλX /(1 − πλX (1 − fX )))
N
π
λX
1
4