$2n-1 n

"

1 1 1 1 1
(0) + 21 + 22 + . . . = + + . . . = "
2 22 23 2 2



1 1
EU = u (0) + u 21 + . . . �!=
2 22






L1 {" L2 pi,1u (xi) > pi,2u (xi)
i i



A - B
B - A
ŹA - ŹB
ŹB - ŹA


n = 1
n = k
n = k n = k + 1
4" n

X = { } R X X � X



xRx e", =
Ź (xRx) >
xRy �! yRx
xRy �! Ź (yRx) >
xRy '" yRz �! xRz > e",
Ź (xRy) '" Ź (yRz) �! Ź (xRz) > e"
Ź (xRy) �! yRx e"
xRy '" yRz �! x = z >


xRz �! xRy (" yRz





x {" y - Ź (y {" x)
x {" x - Ź (x {" x)

xRz �! xRy (" yRz
x {" y '" y {" z x {" z


e" -
x =
4, y = 5, z = 2
Ź (x y) '" Ź (y z) - Ź (x z)
-



2 1

3 3

2 1

3 3

(x {" y) - Ź (y {" x)



101 100 120
X : Y : Z :
4 5 6
x {" y, x {" z z {" y




{" X = { } A ą" X A = "


A {"
C (A| {") = {x " A "y " A y {" x}
x A A x
C " A
A C (A| {") = "

C (A| {") = " x " A y " A
y {" x A
x, y A B x " C (A| {") y " C (B| {")
x " C (B| {") y " C (A| {")

x " C (A| {") "z " A z {" x y " A
y {" x x y
y x
x <" y
x A A,


A ą" X c (A) A


c (A) = "

x, y " A, B x " c (A) y " c (B)
x " c (B) y " c (A)

{"c x {"c y A x, y
x " c (A) y " c (A)
{"c c (A) {"c


A ą" X x, y " A {"c x {"c y
x " c (A) y " c (A) Ź (y {"c x) .

Ź (x {"c y) '" Ź (y {"c z) x {"c z
x {"c z =�! z " c ({x, y, z})
Ź (y {"c z) =�! z " c ({y, z}) y " c ({x, y, z})
Ź (x {"c y) =�! y " c ({x, y}) x " c ({x, y, z})
4" c ({x, y, z}) = "





X = { } {" " u (�) , u : X R
x {" y u (x) > u (y)
u (�) {"
" u : X R x {" y u (x) > u (y)

x {" y =�! u (x) > u (y)
=�! u (y) > u (x)

=�! y {" x

x �" y '" y �" z =�! u (x) o" u (y) '" u (y) o" u (z)
=�! u (x) o" u (z)
=�! x �" z
{"
{" u (�)
X = { }
{" {" {" {"
u " [0, 1]

n n + 1

X

R

x1 x2.






X P X

{" " u : X R p1 {" p2 u (x) p1 (x) >
x

u (x) p2 (x)
x

{"

p {" q ą " (0, 1)

ąp + (1 - ą) r {" ąq + (1 - ą) r
r
ą ą
p q
{"
r r
1 - ą 1 - ą

p {" q {" r " ą, � " (0, 1)
ąp + (1 - ą) r {" q {" �p + (1 - �) r
ą
r
�
ą
p p
{" q {"
r r
1 - ą 1 - �
b w X
b w b {" w ą < � " [0, 1] ��b + (1 - �) �w {"
ą�b + (1 - ą) �w

p, " ą" " [0, 1]
ą"
b
p <"
w
1 - ą"

ą" ą"
�b



{" " u : X - R p {" q p (x) u (x) >
x

q (x) u (x)
x
u (�) v (�) v (x) = a + b � u (x) b > 0


X = { }
p (x1)
x1
p (x2) x2
xn
p (xn)
u (xbest) = 1 u (xworst) = 0.
ą" " [0, 1] "xi " X
i
ą"
i
xbest
xi <"
xworst
1 - ą"
i
u (xi) = ą"
i

ą"
xb
1
p (x1)
1 - ą"
xw
1
ą"
xb
2
p (x2)
1 - ą"
xw
2
ą"
xb
p (xn) n
1 - ą"
xw
n


p (xi) ą"
i i
xb
xw

p (xi) (1 - ą")
i i
u (xb) = 1 u (xw) = 0 u (xi) = ą"
i

p (xi) u (xi)
i



p (xi) u (xi)
i


f2 (�) > 0 x, y " X x > y
�x {" �y u (�)
f2 2 (�) < 0. p X Ep =

xp (x) ą " [0, 1]
x
f [ąx1 + (1 - ą) x2] e" ąf (x1) + (1 - ą) f (x2)
�Ep p
u (�)
Ep = ąx + (1 - ą) y �Ep p
u (Ep) e" Eu (p)
u (�)

ą
x
p =
y
1 - ą
x " X

u (y)

u (Ep)

Eu (p)

u (x)

x z y
Ep = ąx + (1 - ą) y

z
u (z) = Eu (p)


RP = EV - CE

u (�)
�Ep {" p CE < Ep RP > 0
�Ep p CE d" Ep RP e" 0
�Ep <" p CE = Ep RP = 0
�Ep p CE e" Ep RP d" 0
�Ep z" p CE > Ep RP < 0

1 1
2 2

z"


1
10

1
{"


9
10





w0 wl wh
w0

EUL

u (x0)

wl
w0
wh

EUL > u (w0)




u1 u2

u1 = a + bu2, b > 0 u1 u2


Eu1 (p) = p (x) u1 (x)
x"X

= p (x) [a + bu2 (x)]
x"X

= a p (x) + b p (x) u2 (x)
x"X x"X
Ą = a + bEu2 (p)
Eu1 (p) > Eu1 (q) Eu2 (p) > Eu2 (q) .
u1 u2
" h, k > 0
u1 (x) = h + ku2 (x) "x

x1
x <"
x0
u1 u2
ui (x) = Ąui (x1) + (1 - Ą) ui (x0) , i = {1, 2}
ui (x) - ui (x0)
4" Ą =
ui (x1) - ui (x0)

u1 (x) - u1 (x0) u2 (x) - u2 (x0)
= Ą =
1 -
u1Ą - u1 (x0) u2 (x1) - u2 (x0)
(x1)
u2 (x1) - u2 (x0) u2 (x) - u2 (x0)
=
u1 (x1) - u1 (x0) u1 (x) - u1 (x0)

x

x
h k







u2 2 (x)
rA (x) = -
u2 (x)
rA (x) > 0 u (�) x
w0 wl wh
w0
u1
u2
EU1
EU2

wl
w0
wh

w0
1 2
RP1 < RP2 =�! CE1 < CE2
u2 u1 u2 w0

w0

x E� = 0 Ą x0 + x
� x �
x0
Ą
u (x0 - Ą) = Eu (x0 + x)
�
f (�) f2 (�) f2 2 (�)



RHS : u (x0 - Ą) H" u (x0) - Ąu2 (x0)

1
LHS : Eu (x0 + x) H" E u (x0) + xu2 (x0) + x2u2 2 (x0)
� � �
2

1
= Eu (x0) + E [xu2 (x0)] + E x2u2 2 (x0)
� �
2

1
= u (x0) + u2 (x0) E [x] + u2 2 (x0) E x2
� �
2

=0
=V�
x
1
= u (x0) + u2 2 (x0) V�
x
2

1
u (x0) - Ąu2 (x0) = u (x0) + u2 2 (x0) V�
x
2
1
-Ąu2 (x0) = u2 2 (x0) V�
x
2
1
Ą = - rA (x0) � V�
x
2

v (x) u (x) v (x) = f [u (x)] f (�)

v2 2 (x)
v (x) = -
v2 (x)

v2 2 (x) = f2 [u (x)] u2 2 (x) + u2 (x) f2 2 (x) u2 (x)
= f2 (x) u2 2 (x) + f2 2 (x) [u2 (x)]2
v2 (x) = f2 [u (x)] u2 (x)
v2 2 (x) f2 (x) u2 2 (x) + f2 2 (x) [u2 (x)]2
4" - = -
v2 (x) f2 [u (x)] u2 (x)
u2 2 (x) f2 2 (x)
= - - u2 (x)
u2 (x) f2 (x)
f2 2 (x)
= ru (x) - u2 (x)
f2 (x)

<0
4" rv (x) > ru (x) , "x
p z u (�) (p + z)
�
z

rA
x0 Eu (x0) = u (x0)

 u1 = a + bu2, b > 0
u2 = bu2
1 2
u2 2 = bu2 2
1 2
u2 2
1
r1 = -
u2
1
bu2 2
2
= -
bu2
2
= r2
-e-ąx



p q p
q



{"



Fp Fq p q p
q Fp (x) d" Fq (x) , "x

0.25 0.25


{"
0.5 0.3

0.25 0.2
P (X d" x) d" Q (X d" x) "x
P (X d" 0) = 0.25 Q (X d" 0) = 0.5
P (X d" 1) = 0.75 Q (X d" 1) = 0.8
p {" q EUp > EUq EUp - EUq
Ć Ć
EUp - EUq = u (x) fp (x) dx - u (x) fq (x) dx
Ć
= u (x) [fp (x) - fq (x)] dx
Ć
"
= u (x) [Fp (x) - Fq (x)]|" - [Fp (x) - Fq (x)] u2 (x) dx
-"

-"
d"0 e"0

Fp (x) d" Fq (x)
F OSD
p {" q EUp > EUq
u2 (�) > 0

� �
b b
udv = uv|b - vdu
a a a

q
p



ńł ńł
�ł �ł
�łHH 15 �łHH 0
�ł �ł
�łHT 1 �łHT 5
p : q :
�łT H 5 �łT H 10
�ł �ł
�ł �ł
ół ół
T T 10 T T 15
p q
p q
Fp (x) < Fq (x) "x " (0, 1)
p q EVp e" EVq

p q EUp e" EUq
u (x) = x
F OSD
EVp e" EVq =�! p {" q


SOSD
p {" q
Ć Ć
x x
Fp (t) dt d" Fq (t) dt
-" -"
p q

ńł
1

�ł
�ł5
3 2
10
5 3
p : 15 q :
1
�ł
20
ół30 9
1 3
9

SOSD
p {" q
1 2
Pp (X d" 5) = , Pq (X d" 5) =
3 3
Ć Ć
15 15
8
Fp (t) dt = , Fq (t) dt = 1
9
-" -"
Ć Ć
30 30
Fp (t) dt = 1, Fq (t) dt = 1
-" -"

 p q



p q

p q
q p
q = p +
� �
x x
Fp (t) dt d" Fq (t) dt
-" -"

ńł ńł
�ł �ł
�łHH 0 �łHH 1.5
�ł �ł
�łHT 5 �łHT 4.5
q : p :
�łT H 10 �łT H 9.5
�ł �ł
�ł �ł
ół ół
T T 15 T T 14.5
q p p {" q


=�!






w c

max u1 (c) + u2 (w - c)
c
"
u1 (c) + u2 (w - c) = u2 (c) - u2 (w - c) = 0
1 2
"c
u2 (c) = u2 (w - c)
1 2
u1 = u2
1
c = w
2
u2 2 (�) = 0
u2 2 (�) > 0



x
�
max u1 (c) + u2 (w - c + x)
�
c
"
u1 (c) + u2 (w - c + x) = u2 (c) - Eu2 (w - c + x) = 0
� �
1 2
"c
u2 (c) = Eu2 (w - c + x)
�
1 2

u2 (c) | > u2 (c) |
1 1

Eu2 (w - c + x) > u2 (w - c)
�
2 2
Eu2 (w - c + x) > u2 [E (w - c + x)]
� �
2 2

u2 (�) u2 2 2 (�) > 0



�
E [u2 (w + x)] = u2 (w - �) , E� = 0
� x
u2 (�) �
w - � < w u2 (�)
� (w)
u2 2 2 (w)
� (w) = -
u2 2 (w)


�
u2 (w - �) = Eu2 (w + x)
�

LHS : u2 (w - �) H" u2 (w) - �u2 2 (w)

1
RHS : Eu2 (w + x) H" E u2 (w) + xu2 2 (w) + x2u2 2 2 (w)
� � �
2

1
= u2 (w) + (E� u2 2 (w) + E�2 u2 2 2 (w)
x) x
2
=0
1
= u2 (w) + V� (w)
xu2 2 2
2

1
u2 (w) - �u2 2 (w) = u2 (w) + V� (w)
xu2 2 2
2
1 u2 2 2 (x)
� = - V�
x
2 u2 2 (x)
�
= V�
x
2

u2 2 (w)
rA = -
u2 (w)

d u2 (w) u2 2 2 (w) - [u2 2 (w)]2
rA = -
dw
[u2 (w)]2
2
u2 2 2 (w) u2 2 (w)
= - +
u2 (w) u2 (w)
2
u2 2 2 (w) u2 2 (w) u2 2 (w)
= - +
u2 (w) u2 2 (w) u2 (w)

u2 2 (w) u2 2 2 (w) u2 2 (w)
= - � -
u2 (w) u2 2 (w) u2 (w)
= rA (w) [rA (w) - � (w)]

2
rA (w) � (w) rA (w) � Ą
� = rA Ą = �
< 0 � > rA Ą < �
Ą
u (w - Ą) = E [u (w + x)]
�
dx
�
w = 0
dw

dĄ
u2 (w - Ą) 1 - = E [u2 (w + x) (1)]
�
dw
= u2 (w - �)

dĄ
= 0
dw
u2 (w - Ą) (1) = u2 (w - �)
4" Ą = �
dĄ
< 0
dw

dĄ
u2 (w - Ą) 1 - = u2 (w - �)
dw

>1
u2 (w - Ą) < u2 (w - �)
w - Ą > w - �
Ą < �

r2 (w) = r (w) [r (w) - � (w)]
� > r � > r

EL1 + EL2
L1 + L2



H -1 H +1
p : q :
T +1 T -1


p z" �-�
q z" �-�
�
CEp + CEq
p + q = �0
CEp + CEq = �-2�




w w d" 100
u (w) =
1
50 + w w > 100
2

w0 = 101

1
+14
2
q : Eq = 1.5
1
-11
2

u
100
w


u (w0) = 100.5

1 1
Eu (w0 + q) = u (w0 + 14) + u (w0 - 11)
2 2
= 98.75



1
+20
2
p : Ep = 0
1
-20
2
q
1 1
Eu (w0 + p) = u (121) + u (81)
2 2
= 95.75
q
1
Eu (w0 + p + q) = (117.5 + 105 + 95 + 70)
4
= 96.88




u (�) Ep < 0
ł Eł d" 0
Eu2 2 (w + ł) u2 2 (w)
- e" - "w
Eu2 (w + ł) u2 (w)

x w x
� � �

Eu (w + CE (x | w)) = Eu (w + x)
� � � � �


 u (�)
ł x CE (x | w + ł) d"
� �
CE (x | w)
�

x ł
�




1
 (x | w + ł) H" V� (w + ł)
� xR
2

"   (x | w + ł) -  (x | w)
� �
a"
  (x | w)
�
R (w + ł) - rA (w)
=
rA (w)
"rA
=
rA

ł = -� "r = -�r2 (w)
"r �r2 (w)
= - = � [� (w) - r (w)]
r r (w)
� > r
� = r
ł = � E� = 0 ł
� �


u2 2 2 2 (w)
t (w) = -
u2 2 2 (w)

"r 1
H" V�� (w) [t (w) - r (w)]
�
r 2




u (�)
r (w) > 0
� (w) > r (w)
t (w) > r (w) =�! u2 2 2 2 (w) < 0
x ł
�


y = f (x)
y + "y = f (x + "x)
f (x) + "y H" f (x) + "xf2 (x)
"y H" "xf2 (x)
4" "r H" -�r2 (w)

 u (�) w Si {w, u} x
�
Si

Eł d" 0




p
u (w) u (EV ) =
u (pL) w - (w - pL) = pL
w - CE
w - CE > pL
RN
RA

URA (EV )
EURN = EURA


w - L CERA w - pL = CERN w


CE = EV - RP
= W - pL - RP
pL + RP pL
p q q < p
q
p q

p q




 un (w) = ua (w) un (w - L) =
ua (w - L)




1
0
2
p :
1
-k
2
� E� = 0
� �

1 1
� 0
�
2 2
p2 : p2 2 :
1 1
-k � - k
�
2 2

p2 p2 2

w1 (x) = Eu (x + �) - u (x)
�

=u[E(x+�
�)]
� x
�
u2 2 (�) d" 0
x
2
w1 (x) = Eu2 (x + �) - u2 (x)
�
u2 2 2 (�) e" 0 x



1
0
2
p :
�1 1
�
2
�2
�

1
�2 1 0
�
2 2
p2 : p2 2 :
�1 1 �1 + �2 1
� � �
2 2
u2 2 2 2 (�) e" 0 p2 p2 2





0 p
p :
w + x 1 - p
x x
�w
u (w) e" u (p)
u (w) e" pu (0) + (1 - p) u (w + x)


0 = (1 - p) u2 (w + x) dx + [u (0) - u (w + x)] dp
dp (1 - p) u2 (w + x)
=
dx u (w + x) - u (0)
u2 (w)
lim = �!=
p,x0
u (w) - u (0)

1
=





w + y p
p :
w - x 1 - p
y p

u" = limy" u (w + y)

u (w) = pu (w + y) + (1 - p) u (w - x)
u (w) = pu" + (1 - p) u (w - x)
du (w)
= dp (u" - u (w - x)) + dx (- (1 - p) u2 (w - x)) = 0
dp dx
dp (1 - p) u2 (w - x)
=
dx u" - u (w - x)
u2 (w)
lim =
p,x0
u" - u (w)


a - be-rx a










dy
- v
dx

"V
"x
MRS = -
"V
"y

"v "v
v (x, y) "v = ,
"x "y

"v

y x


v (x, y) = ąx + y
= vx (x) + y
= vx (x) + vy (y)
x y x y "v (�)
v (x, y) = vx (x) + y vx (�) x

 x y

"v (�) x
v (x, y) = vx (x) + y
"v
"x 2
= vx (x)
"v
"y
x v (�)
"y
MRS = = f (x)
"x
"y = "x � f (x)
Ć Ć
dy = f (x) dx
Ć
y = f (x) dx - C
Ć
C = y + f (x) dx

=vx(x)
vx,y (x, y) =
vx (x) + vy (y)

MRS (x1, y2) MRS (x2, y2)
=
MRS (x1, y1) MRS (x2, y1)

MRS (x2, y1) MRS (x2, y2)
=
MRS (x1, y1) MRS (x2, y1)
x

QALY = qT
ln (QALY ) = ln q + ln T

MRS (q2, y1) MRS (q2, y2)
=
MRS (q1, y1) MRS (q1, y2)


(x, y) (x2 , y2 ) z (x, y, z) {" (x2 , y2 , z)

(x, y) z
x y z = z2 z2
(x, y, z2 ) {" (x2 , y2 , z2 ) (x, y) z
(x, y, z) {" (x2 , y2 , z) "z
L H Y
(L, H) Ą"p Y
(L, H) Ą"p Y
(L, Y ) Ą"p H
(Y, H) Ą"p L


(L, H) Ą"p Y T (L, H |) H"
[T (L, H | H") , H", Y ] <" (L, H, Y )
(L, H, Y ) , Y H = H". L


v (L, H, Y ) = v [T (L, H | H") , H", Y ] = v (T, Y )
H" Y

y
z
Ą"p y = "
y z z z
"
z z
"
z

z


y Ą"p Ą"p
z z y
v ( = f [vy ( , vz (
y, z) y) z)]
y z

f (�)

x1, x2, . . . , xn
y, ą" X
y z z

n = 3 x1, x2 Ą"p x3 x2, x3 Ą"p x1
x1, x3 Ą"p x2
�" X = {x1, x2, . . . , xn}
y, z
z
y )" = "
z
y *" = S

y z

z y

y z



x

x1, x2, . . . , xn "


v ( = vi (xi)
x)
i





u (�) v (�)
v (�) = f (u (�)) "f (�) st f2 (�) > 0

u (�) = f (v (�)) "f (�) st f2 (�) > 0


p
y, z2
2
�w
,z <"
y2 , z2
1 - p

p
y, z2 2
2 2
�w
,z <"
y2 , z2 2
1 - p
w
z
y z y
z = z2 z2
y Ą"u z u (y, z)
u (y, z) = a (z) + b (z) uy (y)
b (z) > 0 "z

x1, x2, . . . , xn


y z u (y, z)

u (y, z) = uy (y) + uz (z) + kuy (y) uz (z) �!=
z
u (y, z | z = z") = uz (z") + uy (y) [1 + kuz (z")]
b (z) = [1 + kuz (z")]

 k y z
B C
z1
z0 A D
y1
y0

u (y, z0) y
u (y, z0) = uy (y)
u (y, z) = u (y, z0) + u (y0, z) + ku (y, z0) u (y0, z)

u (A) = u (y0, z0) = 0
u (D) = u (y1, z0) = ą
u (B) = u (y0, z1) = �
u (C) = u (y1, z1) = ą + � + ką�

k
1 1
2 2
A B
I : ? II :
C D

1
EUI = [ą + � + ką�]
2
1
EUII = [ą + �]
2

ńł
�ł
�ł {" k > 0 �!=
<" k = 0 �!=
�ł
ół z" k > 0 �!=

k = 0


k = 0 k > 0

v (y, z) = k [u (y, z)] + 1
= k [uy + uz + kuyuz] + 1
= kuy + kuz + k2uyuz + 1
= (kuy + 1) (kuz + 1)
= vy � vz


y z
y, z

k = 0 u (y, z) = uy (y) + uz (z)
1
1
2
2
y2 , z
y2 , z2
<"
y, z
y, z2
y z

y z

y z
1 1
2 2
y2 , z2 y2 , z2 2
<"
y2 2 , z2 2 y2 2 , z2
y2 , z2 , y2 2 , z2 2
y2 , z2 A" y2 , z2 2
y2 , z2 A" y2 2 , z2
y z
k = 0

w
z
u (w
z, z) = Eu (ł, z)


y1, z
<"
w
z, z y2, z
yn, z
z w
z w
z
z
(ł, z) w

�
ę ł z
�
(w
, ę)
y z
y z

Eu (ł, z) = E [uy (ł) + uz (z) + kuy (ł) uz (z)]
� � �
Euy (ł) + Euz (z) + kE [uy (ł) uz (z)]
� �
Ó!
= uy (w
) + uz (ę) + kuy (w
) uz (ę)

x1, x2, . . . , xn u (x1, . . . , xn)

n n

u (x1, . . . , xn) = ui (xi) + k ui (xi) uj (xj) +
i=1 i=1 j>i
n n

k2 ui (xi) uj (xj) um (xm) + kn-1 ui (xi)
i=1 j>i m>j i=1
k
k = 0 vi (xi) = kui (xi) + 1


n

v (x1, . . . , xn) = vi (xi)
i=1

 n = 3
3 3

vi (xi) = kui (xi) + 1
i=1 i=1
= (ku1 + 1) (ku2 + 1) (ku3 + 1)

= (ku1 + 1) k2u2u3 + ku2 + ku3 + 1
= k3u1u2u3 + k2u1u2 + k2u1u3 + ku1 +
k2u2u3 + ku2 + ku3 + 1
�ł �ł �ł łł
3 3

�ł �ł
= k ui + k uiujłł + k2 ui�ł + 1
i=1 i j i=1
Ó!
�ł �ł
3 3

�ł
4" u (x1, x2, x3) = ui + k uiujłł + k2 ui
i=1 i j i=1
k = 0
n

u (x1, . . . , xn) = ui (xi)
i=1
X = { } x+ (xi, xj)
ij
x+ = {x3, x4, . . . , xn}
12
x1, . . . xn x+"
12
1 1
2 2
x2 , x2 , x+" x2 , x2 2 , x+"
1 2 12 1 2 12
<"
x2 2 , x2 2 , x+" x2 2 , x2 , x+"
1 2 12 1 2 12
k = 0
k = 0



x1, . . . , xn
xi Ą"u x i = 1, 2, . . . , n
-i
{xi, xi+1, . . . , xn} Ą"u i = 2, 3, . . . , n {x1, x2, . . . , xn-1} Ą"u
xn
{xi, xi+1} Ą"u i = 1, 2, . . . , n - 1
j xj Ą"u x-j {xi, xj}
i = j


e" 2
some xi
x
x-i U (x)
ńł
�ł
�łV (x) �!=
U (x) = -e-cV (x) �!=
�ł
ółecV (x) �!=


x1 Ą"u x-1 x-1 = y (x, y)
x

V ( = iVi (xi)  i = 1
x)
vi (xi") = 0 vi (x") = 1
i
V ( = 0 V ( ") = 1
x") x
x1 y
p
x1"
(x1, y) <"
Ć
x"
1
1 - p
y

p
A
n
f [1v1 (x1) + ivi (xi)] <"
Ć
i=2
B
1 - p


n

A = f 1v1 (x1") + ivi (xi)
i=2

n

B = f 1v1 (x") + ivi (xi)
1
i=2
x1 y y
Ć


xi Ą"u x-i
n n

U ( = ui + kijuiuj + kijmuiujum + k1n ui
x)
i=1 i j i j m i=1
k






k = 0




t
T

1
V (x) = xt
1 + r
t=0


xt e" yt "t y
x
xt > yt t {"
x y

xt = yt t t = {i, i + 1} y
x
i i + 1
x y


xt xt+!

xt+1
xt


xt xt+1 t


X = {x1, x2, . . .}

x1, x2, {x1, x2}
(x1, x2, x3 . . .) (x1, x2 , x2 . . .) (x2, x3, . . .)
2 3
(x2 , x2 , . . .)
2 3





"t
x1, x2, . . . , xt-1, yt , yt+1, . . . z" x1, x2, . . . , xt-1, xt , yt+1, . . .
y z" x


t
"

1
V (x1, x2, . . .) = V (xt)
1 + r
t=1
r > 0




t Ć
"
"

1
P V = xt H" xte-rtdt
1 + r
0
t=0






1 1
� = = � = =
1+d 1+r
B C
� > � d < r
�B B
>
�C C


 



ct t r
Xt Yt



t T t
T

1 1
c (Y ) = ct | ct d" yt
1 + r 1 + r
t=0 t=0


E (y) x " E (y) y x c (y) " E (y)
P Vc d" P Vy
X X2
2 2
"Y " E (X) st Y {" Y "Y " E (X2 )
X X2
2
X Y X Y
X2



Y, H = (y1, h1) , (y2, h2) , . . . , (yT , hT )
r


2
(yt, h") = (yt + Tt (ht, yt) , h")
t t

2
(yt, h") <" (yt + Tt (ht, yt) , h") <" (yt, ht)
t t
h" Tt h"
ht
t
T

1
v (y2 , h") = [yt + Tt (ht, yt)]
1 + r
t=0









U = u1 (y1, h1) + �u2 (y2, h2)

1
max U st Y = y1 + �y2, � =
1 + r

L = u1 (y1, h1) + �u2 (y2, h2) -  [y1 + �y2]

"L "u1
= - 
"y1 "y1
"u1
 =
"y1
"L "u2
= � - �
"y2 "y2
"u2 �
 = �
"y2 �

"u1 "u2 �
= �
"y1 "y2 �
� = 1

� = �

"y1 "u1
"h1
- =
"h1 "u1
"y1
"u2
"y2 "h2
- =
"h2 "u2
"y2
"y1 � "u2
"h2
- =
"h2 "u1
"y1
�
MU2
�
"y1 � "u2
"h2
- =
�
"h2 � "u2
"y2
= � � MRSy ,h2
2
�



=


b
�t = , b > 0
b + t

b
�t+1 b+t+1 b + t
= = < 1 "t
b
�t b+t b + t + 1

�t+1
lim = 1
t"
�t


t
1
� =
1 + r
�t+1 1
= "t
�t 1 + r
�t+1 1
lim =
t"
�t 1 + r



t xt yt xt - yt


-1095
r = 0.1, b = 10
t = 1
1
P V� (x - y) = 1000 - (1095) = 4.54
1.1
=�! x {" y
10
P V� (x - y) = 1000 - (1095) = 4.54
11
=�! x {" y
t = 0
1 1
P V� (x - y) = 0 + (1000) - (1095) = 4.13
1.1 1.21
=�! x {" y
10 10
P V� (x - y) = 0 + (1000) - (1095) = -3.41
11 12
=�! x z" y





"

V = a (t) � xt
t=0
a (t)

xs <" xt "s, t | s = t


xs {" xt "s < t

t


xs <" yt
xs+� <" yt+�
x < y



t
1
a (t) =
1 + r
{"�
xt {"� x2
t2
2
xt {"� x2 "t, t2 > �, �2
t2
xt �
�2 t t2
{"� xt {"0 x2
t2

xt+� {"� x2
t2 +�
x
�
=�!
=�!




�
b > 0 b st E� = b
b
�
b {" b



1 1
Ż �
b > E b
1 + r 1 + s
1 1
Ż Ż
b > b
1 + r 1 + s
s > r


�
c < 0 b st E� = c
b


1 1
c > E c
Ż �
1 + r 1 + s
1 1
c > c
Ż Ż
1 + r 1 + s
c < 0 r > s

x1 x2


u1 = x1 + �x2
x1 x2
u1 = E� + �E�
x1 x2


�

x1 + �x2 = (x1 + �") + � (x2 - ")



u2 = u (x1 + �x2)


x2


Eu2 = Eu (x1 + �x2) = u x1 + �x2
�




�


u3 = v (x1) + �v (x2)



� < 1
x1 x2
Eu = Ev (x1) + �Ev (x2) = v (x1) + v (x2)
� � Ć Ć
u2
�
u3 y1 y2 x1 x2

v (x1) H" v (y1) + (x1 - y1) v2 (y1)
v (x2) H" v (y2) + (x2 - y2) v2 (y2)
4" u3 H" v (y1) + (x1 - y1) v2 (y1) + � [v (y2) + (x2 - y2) v2 (y2)]
v2 (y1) v2 (�)

u3 v (y1) + (x1 - y1) v2 (y1) + � [v (y2) + (x2 - y2) v2 (y2)]
=
v2 (y1) v2 (y1)
v2 (y2) v (y1) + �v (y2)
= (x1 - y1) + � (x2 - y2) +
v2 (y1) v2 (y )
1

v2 (y2)
" (x1 - y1) + � (x2 - y2)
v2 (y1)


v2 (y2)
v2 (y1)


v2 (y2)
y2 > y1 - v2 (y2) < v2 (y1) - < 1
v2 (y1)
�


r = � + g�
r � g
�

u3


r = 0.05
� H" 0
g = 0.013
� = 1
�
� = 1 v (�) = ln (�)
r
r <" Ą (r)


t
T

1
P V = E xt
1 + r
�
t=1


1
r = 0.1
2
Ą (r) =
1
r = 0
2

t
1 1 1
E = (1)t + (1.1)t
1 + r 2 2
�
t
1 1
lim E =
t"
1 + r 2
�
t
rt
Ć
min (r)
�
t




t t
1 1
E =

1+� 1+E�
r r






rSRT P < rOC





t . . .
. . .
r r . . .


t
"

1 r
r =
1 + d d
t=0
d r

r
= S =
d



t
T

1
{bt - ct [pS + (1 - p)]}
1 + d
t=0
bt ct t p
1 - p S



T t = 0
C
C, H
H
C
(1+s)T
C, H
H
(1+d)T
C
C H
H
(1+d)T
C
(1+r)T
C
H
H
(1+r)T
(1+d)T
s = d = r =
C

(1+r)T
X <" Y s = d

 X <" Z d = 0

Y <" W s = r
r = s = d = 0

X <" Z W <" Y Y Y
Y <" W {" Z <" X =�! Y {" Z
Y <" Z


Z d < s



Y Q
"q2 > q1
(y2 , q1) <" (ąy2 , q2)

(y, q1) <" (ąy, q2)
0 < ą < 1

u (y, q) = auY (y) + buQ (q) + (1 - a - b) uY (y) uQ (q)
a = k1, b = k2, (1 - a - b) = k12
y q uY (y)

u2 2
Y
-y = k
u2
Y

p
(1 + ą) y
y <"
(1 - �) y
1 - p
y y
y y

ńł
�ł-yr r < 0
�ł
uY (y) = ln y r = 0
�ł
ółyr r > 0
ln y r
r = 1 =�!
r < 1 =�!
r > 1 =�!
r > 0 q " [q", q"]
p
(y, q") <" (py, q")
y
uQ (q") = 0 uQ (q") = 1

uY (y) = yr

u (y, q) = ayr + buQ (q) + (1 - a - b) yruQ (q)


u (y, q") = ayr
u (py, q") = a (py)r + b + (1 - a - b) (py)r
p
ayr = a (py)r + b + (1 - a - b) (py)r "y
b = 0 a = pr
u (y, q) = pryr + (1 - pr) yruQ (q)
r > 0

1
r
Hr (q) = [pr + (1 - pr) uQ (q)] , r = 0


Hr (q") = p
Hr (q") = 1
uQ
[Hr (q)]r - pr
uQ (q) =
1 - pr

[Hr (q)]r - pr
u (y, q) = pryr + (1 - pr) yr
1 - pr
= pryr + yr {[Hr (q)]r - pr}
= yr [Hr (q)]r
= [y � Hr (q)]r

r = 1 y
u (y, q) = yp + yuQ (q) (1 - p)
uQ (q") = 0 p = 0
u (y, q) = y � uQ (q)

r = 1




s
y, 1
y, q <"
y, 0
1 - s

 r = 1
QALY = y � uQ
= y � VQ
= y � s
u = V = s

(y, q) <" (py, 1)
u (q) = v (q) = p
r = 1


yruQ = syr + (1 - s) yr (0)
uQ = s

[yVQ]r = [syVQ]r + [(1 - s) y (0)]
1
r
VQ = s

uruQ = (py)r
uQ = pr

[yVQ]r = [py]r
VQ = p
uQ VQ



Ć
y
P V (y) = e-rtdt
0
r = 0



y
y = 0 q
y Ą"u q


u (y, q) = a (q) + V (q) y

u (y, q1)
u (y, q3)
u (y, q)
u (y, q2)
y



u (y, q1)
u (y, q)
u (y, q3)
u (y, q2)
0
y





u (y, q) = a (q) + v (q) uQ (q)
"2u
"y2
rA = -
"u
"y
v (q) u2 2 (q)
Q
= -
v (q) u2 (q)
Q
u2 2 (q)
Q
= -
u2 (q)
Q





A

Normal speech
U
A
No speech
y






arg max EU








t
T

1
QALY = qt
1 + r
t=0




weight = age � e-ą�age
ą = 0.04

1.5
1
8 55 age











weight
 p"
p"
(y, q")
(y, q) <"

0.01
q"
h
p"
(y, q")
(h, q") <"

0.01
(y, q) <" (h, q")
h
h




u (�, w) = �ua (w) + (1 - �) ud (w)
�

1
� =
0

p =
max EU = (1 - p) ua (w) + pud (w)

0 = (1 - p) u2 dw + pu2 dw - dpua + dpud
a d
dp [ua - ud] = dw [(1 - p) u2 + pu2 ]
a d
dw ua - ud
= = V SL
dp (1 - p) u2 + pu2
a
d
dw

dp

dw
= V SL
dp
ua (w) > ud (w)
u2 (w) > u2 (w)
a d


VSL
A
VSL
0 1
1 - p





wealth




p
dw
d
dp
> 0
dp
p
u2 < u2
d a

p

w
dw
d
dp
> 0
dw
ua (w) > ud (w) u2 (w) > u2 (w)
a d
u2 2 d" u2 2
a d




ud u2 ua u2
d a
u2 < u2
a, sick a, healthy










t t

1 1
max stu (ct) st ct d" W
1 + d 1 + r
st t




A
B
Age

r > d




ua - ud
V SL =
(1 - p) u2 + pu2
a d
ud = 0
ua
V SL =
(1 - p) u2
a
ud = 0
ua (0) = ud

VSL
ua
U
ua (w0)
va
(1 - p) ua (w0)
w = CEu CEv w0
Ż
W

u (�) w0 - w
p v (�)
w0 - CEv

u (�) w0
v (�) w0 - w

ua
U
va
ua (w0)
(1 - p) ua (w0)
w0
Ż
CEv w = CEu
W

v (�)
v (�)

 p, q p q
� � � �
w0 + p = u
p w0 + q = u
q
� �
RPp = w0 - u
p > RPq = w0 - u
q

u (�) v (�)
(u) (v)
RPp > RPp
(v) (v)
RPq > RPq
p q
RPp - RPq

(u) (v)
RP > RP
u (�)
u (�) v (�)
u2 2 (w) v2 2 (w)
- > - "w
u2 (w) v2 (w)

u2 2 (w) v2 2 (w)
- , "w " [w", w"] > - , "w " [w", w"]
u2 (w) v2 (w)




1

1000





1000

1
V SLi = V SL
1000
i=1
V SL = $5M


p1 p2
1
(p1 + p2) = p
2
N

p1 -N

1 - p1
0
p2 -N

1 - p2
0
pN

w2
wp
Ż
w1
w0
1
1 1 Ż
0 - p2 - p 1 - p1
1 - p

wi w0
i
w2 w1
wp
1
(w1 + w2) > wp
2





VSL
x1
x
Ż
x2
s0 s1 1
0 s
Ż
Survival


1
x > (x1 + x2)
2


u (h, T, w) u (h, T |w) = QALY =
Q

u (h, T, w) = Q � a (w) + b (w) , a (w) > 0

0 = Qa2 (w) dw + b2 (w) dw + a (w) dQ
dw a (w)
- =
dQ Qa2 (w) + b2 (w)
=�!


"u
= Qa2 (w) + b2 (w)
"w

Q = 0 b2 (w)


VSL

ua - ud
V SL =
(1 - p) u2 + pu2
a
d
Qa (w) + b (w) - b (w)
=
(1 - p) (Qa2 (w) + b2 (w)) + pb2 (w)
Qa (w)
=
(1 - p) (Qa2 (w)) + b2 (w)

Qa (w)
V S =
(1 - p) (Qa2 (w))

Q ę!- V SL ę!


8

100,000
4

100,000

= $137
= $50
1

100,000
= $155
= $50


4
$3.4M =
100,000
V SL =
1
$2.2M =
100,000
1

1000
0.001






( ) > ( )



Pareto improvement zone

X

Y

SQ
Benefits to A

Y X
A B Y



H
L

5
L
100,000
4
H
100,000



risk � V SL

Benefits to B

4
= $240
100,000
H

5
= $200
100,000
L

-30 +10





40

M
H
115 19.17 115 19.17
= =
6M M 6M M
50

M
L
115 28.75 115 28.75
= =
4M M 4M M
+2.08 -7.92





SQB
SQA
Benefits to A

SQA SQB





C W T P
NHB = "risk -  =
 "r


Benefits to B

W T P = W T A

$
y2
EV
y0
SQ
CV
y1
original
q0 q1
Q


y0 - y1

y2 - y0

EV e" CV

EV = CV


{"G= G ą" S S
R =
x {"G y xRy G {x, y}
G {x, y} " X G
X |X| e" 3 S e" 2 "


x {x, y, z} x
{x, y}

S



















x {" y
y {" z
z {" x






1 1 1
, , . . . ,
1000 1000
1000
1 1
, , 0, 0 . . . , 0
2 2

1
1000
(1, 1, 1, . . .)
(0, 0, 0, . . .)
999
1000

1
1000
(1, 0, 0, 0 . . .)
1
(0, 1, 0, 0, . . .)
1000
(0, 0, 0, . . . , 1)
1
1000
x = Ex = 1



D {" C
C {" D


(p1, p2, . . . , p1000) = (pi, pj, pij)
pi i pij

(x1, x2, . . . , x1000)


1
xi =
0

(pi, pj, pij) (pi + �, pj - �, pij)
|pi - pj| < |pi + � - (pj - �)|



UR UF

UR (p1, p2, . . . , pn) = EUF (x1, x2, . . . , xn)
� � �



n

UR = ipi
i=1
i < 0 "i UR

UR
UR

UR

UR UF UR
n n

UR (p1, . . . , pn) = kipi + kijpipj + . . . + k1n pi
i=1 i,j i=1

UR (0, 0, . . . , 0) = 0
ki = UF (xi = 1, x-1 = 0)
n = 2
UF (0, 0) = 0
UF (1, 0) = u1
UF (0, 1) = u2
UF (1, 1) = u12

UR (p1, p2) = EUF (x1, x2)
� �
=0

= (1 - p1) (1 - p2) UF (0, 0) +p1 (1 - p2) u1
+p2 (1 - p1) u2 + p1p2u12
= p1u1 - p1p2u1 + p2u2 - p1p2u2 + p1p2u12
= p1u1 + p2u2 + p1p2 [u12 - u1 - u2]

UR (p1, p2) = p1k1 + p2k2 + p12k12
UR UF



2
UR (�, �, 0, 0, . . . , 0) UR (2�, 0, 0, 0, . . . , 0)
2
UR {" UR
k1� + k2� + k12�2 > k12�
k1 + k2 + k12� > 2k1
k12� > k1 - k2
� 0 k1 - k2 0 k1 = k2 k1 = u1 =
UF (1, 0) k2 = u2 = UF (0, 1)
UR UF

n

u xi = u (x)
i=1
x

pi pj pij UR UF
UR

n

1
UR (p1, p2, . . . , pn) = (1 - kpi) - 1 , 0 < k < 1
k
i=1

UR (0, . . . , 0) = 0
UR (1, 0, . . . , 0) = UR (0, 1, 0, . . . , 0) = . . . = UR (0, 0, . . . , 1) = -1

1
u (x) = [(1 - k)x - 1]
k
u (x)
U
x
0
1
-
k



A B A B
C A D

B

(0, 1) <" (1, 0)
p {" q
ą ą
p q
{"
r r
1 - ą 1 - ą
p = (0, 1) q = (1, 0) p <" q
ą ą
(1, 0) (0, 1)
A"
(1, 0) (1, 0)
1 - ą 1 - ą




a s
Ąs
n

W (a) = ui (a)
i=1



W (a, s) = Ąsui (a, s)
i s



W (a, s) = Ąs ui (a, s)
s i





Eui
i



E ui
i


n

W (a) = g [ui (a)]
i=1
g2 (�) > 0 g2 2 (�) < 0



W (a, s) = g Ąsui (a, s)
i s



W (a, s) = Ąs g [ui (a, s)]
s i










"
g (�) = �
Ąs
1

2
1

2


1
EUA = (100 + 9) = 54.5
2
1
EUB = (110 + 4) = 57
2

B {" A



1
W (A) = (100 + 9) H" 7.4
2

1
W (B) = (110 + 4) H" 7.5
2

B {" A

" "
1
W (A) = 100 + 9 = 6.5
2
" "
1
W (B) = 110 + 4 H" 6.2
2

A {" B





"W
=?
"ui,s


W = Ąsui,s
i,s
"W
= Ąs
"ui,s





W = g Ąsui,s
i s


"W
= Ąsg2 Ąsui,s
"ui,s
s



W = Ąs g [ui,s]
s i
"W
= Ąsg2 (ui,s)
"ui,s




g2 (�) i


g2 (�)























x



w (x) =
f (x) = x
e (x) = x
p (x, e (x)) =


e = f (x) � e (x)
x
ua (w) ud (w)
S = { }


EU [s, w (x) - e] = f (x) � {p (x, e (x)) � ud (w (x) - e) + [1 - p (x, e (x))] � ua (w (x) - e)}
� �
x
e (x)
pe (x, e (x))
EU2 (s, w (x) - e)
� �
-pe (x, e (x)) =
U" (w - e)
U" = ua (w - e) - ud (w - e)
x
�










X = { }
S = { }
F = { }
F f (�) | S X



fA (sH) = $50
fA (sY ) = -$20

f {" f2

Ąs � u [f (s)] > Ąs � u [f2 (s)]
s s


Ąs
u (�) Ą" Ąs
f (�) Ą" Ąs Ą" u (�)






0.01
-50 -50
A : B :
+50 +50
0.99
B
B





< 5% > 95%





Assessed probability




{"

"x, y " Z st x {" y

x {" y x
y
f, g, f2 , g2 " F A ą" S
f (s) = f2 (s) "s " A
g (s) = g2 (s) "s " A
f (s) = g (s) "s " A
/
f2 (s) = g2 (s) "s " A
/

A
S
f = f2
g = g2
f = g
f2 = g2



Z a" X = { }

Actual frequency

f {" g f2 {" g2
f {" g A f = g A A f = f2
g = g2
A = {s1, s2} A = {s3}
s1
s1
x2
x
f : g :
s2 y
s2 y2
s3
s3
z
z
s1
s1
x2
x
f2 : s2 y g2 :
s2 y2
s3
s3
z2
z2
f {" g s1, s2 x, y, x2 , y2 f2 {" g2
A


A ą" S f, g " F f <" g | A
A A A
p = 0
A ą" S f (s) = x g (s) = y g " A
f {"A g x {" y

c {" d c2 {" d2
A B
c c
f : g :
Ż

B
d d
A B
c2 c2
f2 : g2 :
Ż

B
d2 d2

f {" g �!�! f2 {" g2
Ż
A B B



A ą" S
[f {" g (s) | A] =�! f g | A "s " A

[g (s) {" f | A] =�! g s | A "s " A
f g

f, g " F f {" g
x " Z S A
"a " A, f2 (s) = x "s " A, f2 (s) = f (s) "s " A =�! f2 {" g

"a " A, g2 (s) = x "s " A, g2 (s) = g (s) "s " A =�! f {" g2
f2 f A
A x
A f2 g f g

x f g f2 x A
f A A
f2 g


F = {f1 = f2 = }

- $1000 - cost
S =
$3000 - cost


Z = {-$2000, -$1000, $2000, $3000}

f1 (s1) = -2000
f1 (s2) = 2000
f2 (s1) = -1000
f2 (s2) = 3000





0.1
5M
1.0
A1 : z" B1 :
0.89 1M 1M
0.01
0
0.1 0.11
5M 1M
A2 : {" B2 :
0.9 0.89
0 0

t
0.1
5M
1 - t
1M
A (t) : B (t) :
0.89 - t 1M
0
t
0
0.01 + t
A (0) = A1 B (0) = B1 A (0.89) = A2 B (0.89) = B2
A (0) z" B (0) A (0.89) {" B (0.89)
t


p � 200 (1 - p) � 2000












X <" g (x|�) �!=
� <" f (�) �!=
g (x|�) f (�)
�
P (�|X) =
g (x|�) f (�) d�
�

Ć
h (x) = g (x|�) f (�) d�
�
x


p (�)
d (�) =
1 - p (�)
d (�)
p (�) =
1 + d (�)

d (�|X)
P (�|X) =
1 + d (�|X)
d (�|X) g (x|�) f (�)
=
1 + d (�|X) h (x)
g(x|�)f(�)
h(x)
d (�|X) =
g x|� f �
( ) ( )
h(x)
g (x|�) f (�)
d (�|X) = �

g x|� f �





p (�)
logit (�) = ln = ą + �X
1 - p (�)
= ln [( ) ( )]
= ln ( ) + ln ( )
ln ( ) X = 0 ą

T (x) g (x|�)
p (�|x1) = p (�|x2) "x1, x2 T (x1) = T (x2)
T (x)
g (x|�) <" U (0, �) � <" f (�)


x1, x2, . . . <" g (x|�)
T (x) = max (xi)
i
� e" 2
� e" 10

X <" B (p)

n

i
i
p (x1, x2, . . . , xn | p) = px (1 - p)1-x
i=1


xi
i
i
= p (1 - p)n- xi


n

T (x) = xi, n
i=1

T (x) {f (x | �) , � " &!}
f (x | �)
f (x | �) = u (x) � v [T (x)] "x " X, � " &!
u (�) x v (�) x
T (x)
�



1 (xi - x)2
T (x1, x2, . . . , xn) = xi,
n �
i


" T (x) g (x|�) " p (�)
p (�|x) x
p (�) g (x|�)



x1, x2, . . . xn <" Bern (p) p <" Beta (ą, �)
g (x|�) = px (1 - p)n-x

� (ą + �)
f (p | ą, �) = pą-1 (1 - p)�-1
� (ą) � (�)

ą
Ep =
ą + �
ą� Ep [1 - Ep]
Vp = =
ą + � + 1
(ą + �)2 (ą + � + 1)


f (p) g (x|p)
p (p | x1, x2, . . . , xn) =
K



x
" p (1 - p)n- x � pą-1 (1 - p)�-1
<" Beta (ą + , � + )


ą + � + x
E (p|x) =
ą + � + n

ą ą + � x n
= � +
ą + � ą + � + n n ą + � + n
= Ep + (1 - ) x

n n ",  0, E (p|x) x

E (p|x) [1 - E (p|x)]
V (p|x) =
ą + � + 1 + n
n ", V (p|x) 0
�


xi <" N �, �2 �2 � <" N �, �2

� nx
+
1
2
� �2
p (�|X) <" N ,
1 n 1 n
+ +
2 2
� �2 � �2

1 n
2
� �2
E (�|X) = � + x
1 n 1 n
+ +
2 2
� �2 � �2
= � + (1 - ) x
n ",  0, E (�|X) x V (�|X) 0

x p (�|x)
Ć
E [p (�|x)] = p (�|x) h (x) dx
Ćx Ć
= p (�|x) � g (x|�) f (�) d� dx
�
Ćx Ć
g (x|�) f (�)
�
= � g (x|�) f (�) d� dx
g (x|�) f (�) d�
�
Ćx �
= g (x|�) f (�) dx
x
Ć
= f (�) g (x|�) dx
x
=1
= f (�)




V� = Eh [V (�|x)] + Vh [E (�|x)]
�
�
�
E [V (�|x)]



A = { a1, a2, . . . , an}
� =
p (�) = �
u [ai (�)] = i �

a" (�) p (�)
Ep(�)u [a" (�)] e" Ep(�)u [ai (�)] "ai

Ć
Ep(�)u [ai (�)] = u [ai (�)] p (�) d�
�
a"
a" (�) �
�

ai
Ć
EOL (ai) = {u [a" (�)] - u [ai (�)]} p (�) d�
�
Ć� Ć
= u [a" (�)] p (�) d� - u [ai (�)] p (�) d�
�
� �

ai

a"

EV P I = EOL (a")
Ć Ć
= u [a" (�)] p (�) d� - u [a" (�)] p (�) d�
�
� �
�

EV P I = EU - EU
Ć Ć
= p (�) � max u [ai (�)] d� - max u [ai (�)] p (�) d�
i i
� �



�
h (x) = g (x|�) f (�) d� x
�
x



EV SI = Eh Ep(�|x)u a" (�) - Ep(�)u [a" (�)]
p(�|x)
Ć Ć
Ć

= h (x) � p (�|x) u a" (�) d� dx - p (�) u [a" (�)] d�
p(�|x)
x � �
x



EV SI = E EV - EV

EV P I | = EU - EU
EV P I | = EU - EU


EV P I | - E [EV P I | ] = EU - EU - E EU - EU

= E EU - EU
= EV SI

EVPI | prior

E [EVPI | post]
EVSI
EUprior EUposterior EUperfect


v
Ć
Ć
Ć
h (x) p (�|x) � u a" (�) - v dx = p (�) � u a" (�)
p(�|x) p(�)
x � �
v

e

Ć
Ć
Ć
h (x) p (�|x) � u a" (�) dx = p (�) � u a" (�) + e
p(�|x) p(�)
x � �



! !
A !



 � " {�1, �2}

1
�1
2
p (�) =
1
�2
2
a1 - a4
� p (�) a1 a2 a3 a4
�1 1
2
�2 1
2


a" a" a"
�1 �2
a1, a2, a3, a4 a3 a2 a1
a1, a2, a4 a1 a2 a1
a2, a3, a4 a3 a2 a3 a4
a1, a2, a3 a3 a2 a1
a3, a4 a3 a4 a3 a4










-15
p 1 - p

pu (-15) + (1 - p) u (35) e" u (0)
u (35) - u (0)
p d"
u (35) - u (-15)
un = w ua = f (w)
p
7
pn =
10
u (35) - u (0)
pa =
u (35) - u (-15)
pa pn.
ua (-15) = un (-15)
ua (35) = un (35)
ua (0) > un (0) pa < pn

U
un
ua


-15 0
35

p < pa
p > pn
pa < p < pn

p
pa
p p
-15 0
prior : perfect :
1 - p
1 - p
+35 35



pn
p p
0 0
prior : perfect :
1 - p 1 - p
0 35




ai � i


U
u (a2)
EVperfect
EVprior
u (a1)

Ż
�1 �2
�

a1 �1 a2 �2 a1
EV

u �



� <" U (-b, b)
a1 = 1 - �2
a2 = 0
u (w) = w

EUa = 0
2
Ć
b

1
EUa = 1 - �2 d�
1
2b
-b
Ć
b

1
= 2 1 - �2 d�
2b
0

b
1 �3

= � -

b 3
0
b2
= 1 -
3
�, a1
"
b d" 3
b


 b d" 1 a1 a2 EV P I = 0
b > 1 a1 |�| d" 1 a2
�ł łł
�ł śł
Ć Ć
-1 b
�łĆ 1 śł
1
�ł
EV |P I = � 1 - �2 d� + 0d� + 0d�śł
�ł śł
2b
�ł -1 1
-b �ł
a1 a2
Ć
1

1
= 1 - �2 d�
b
0
2
=
3b
EV P I = EV |P I - EV |
ńł
�ł
�ła1 b d" 1
"
a1| 1 d" b d" 3
�ł
óła | b e" "3
2

ńł
�ł0
b d" 1
�ł
�ł
"
2 b2
EV P I = - 1 - 1 d" b d" 3
3b 3
�ł
"
�ł
ół 2
b e" 3
3b

EV P I

"
0 1 3 b







1
3 0.99
+16 +4
A : z" B :
-2 +1
2
0.01
3
CEA > CEB



A
B









p L
pL L


p 1 - p p 1 - p
w - pL w - pL 0 pL
w - L w L - pL 0

L
pL

u = f ( ) + g ( )
f (x) = x, g (0) = 0, g2 (�) < 0
2
Euinsure e" Eudon t
p [w - pL + g (0)] + (1 - p) [w - pL + g (pL)] e" p [w - L + g (L - pL)] + (1 - p) [w + g (0)]
w - pL + (1 - p) g (pL) e" w - pL + pg ((1 - p) L)
(1 - p) g (pL) e" p � g ((1 - p) L)
g (�)

p (1 - p) LpL
(1 - p) L

L

(1 - p) � g (pL)

(1 - p) � g ((1 - p) L)

g ((1 - p) L)



(1 - p) pL pL
L - pL = (1 - p) L p


p 1 - p p 1 - p
w + (1 - p) L w - pL 0 pL
w w L - pL 0

2
Eubet e" Eudon t
p [w + (1 - p) L + g (0)] + (1 - p) [w - pL + g (pL)] e" p [w + g ((1 - p) L)] + (1 - p) [w + g (0)]
(1 - p) g (pL) e" p � g ((1 - p) L)












0.25

0.5

=�!
0.25

0.5

0.5



0.8 0.8
100
+200
=�!
0
0.2 0.2









V = Ą (pi) v (xi)
i
Ą (�)
Ą (�)
Ą (0) = 0, Ą (1) = 1

Ą (ąp) > ą � Ą (p) , ą " [0, 1]


Ą (pi) < 1,
i
45ć%

Ą (pq) Ą (pqr)
< , p, q, r " [0, 1]
Ą (p) Ą (pr)




1
Ą (p)

0 p 1

v (�)




v (�)

Gains
Losses






v (x) = Ą (pi) v (xi)
i



v (x) = v (x") + Ą (pi) [v (xi) - v (x")]
i
x" = mini {|xi|}



x p
L : , x z" y
y 1 - p
EV = px + (1 - p) y
EU = p � u (x) + (1 - p) � u (y)


V = w (pi) xi
i
x p
EY U = w (p) � x + [1 - w (p)] � y
y 1-w (p) w (1 - p)


px + (1 - p) y e" w (p) � x + [1 - w (p)] � y
w (p) > p

1
w (p)

0 p 1




V = w (pi) � u (xi)
i
w (p) > p u (�)


w (�)



u (�)
w (�)


1
w (p)

0 p 1




ńł
p1
�łx1
�ł
�ł
�łx2 p2
�ł
�ł
�ł
x3 p3 , x1 < x2 < . . . < xn
L :
�ł
�ł
�ł
�ł
�ł
�ł
ół
xn pn



Ą1 = w (p1)
Ą2 = w (p1 + p2) - w (p1)
i-1
i

Ąi = w pt - w pt
t=1 t=1



w (p)

1
Ą4
Ą3
Ą2
Ą1

0 p1 p1 + p2 p1 + p2 + p3






p " {P}




EU = min p (x) u (x)
p
i
v (�)

EU = v [p (x) u (x)]
p i


X = {x1, x2, x3}
P = {p1, p2, p3}

x1 < x2 < x3 pi = 1
i


1

0
1
p1


3
p
e
r
u
s
r
o
f
3
x
e
r
e
u
r
s
u
r
s
o
r
f
o
f
2
x
1
x
 EV
k = p1x1 + p2x2 + p3x3
= p1x1 + (1 - p1 - p3) x2 + p3x3

x2 - x1 k - x2
p3 = p1 +
x3 - x2 x3 - x2
EV

1
Constant EV lines
0
p1 1


EU = p1u (x1) + p2u (x2) + p3u (x3) = k

u (x2) - u (x1) k - u (x2)
4" p3 = p1 +
u (x3) - u (x2) u (x3) - u (x2)
EV

u (x3) - u (x2) < x3 - x2
u (x2) - u (x1) > x2 - x1
u (x2) - u (x1) x2 - x1
4" >
u (x3) - u (x2) x3 - x2


3
p
ICs
B

A p1

B A A {" B A
B


0.1
5M
1.0
a1 : {" a2 :
1M 0.89 1M
0.01
0
0.11 0.1
1M 5M
a3 : z" a4 :
0.89 0.9
0 0
x3 = 5M, x2 = 1M, x1 = 0
p1 p3
a1
a2
a3
a4


3
p
IC
IC
a4
a2


a1 a3
p1







ą ą
�x p
b1 : {" b2 :
1 - ą 1 - ą
p"" p""
ą ą
�x p
b3 : z" b4 :
1 - ą 1 - ą
p" p"


p"" {" p" p x
p""

b1 b2 b3/b4


ą = 0.11

10
5M
11
p =
1
1M
11
x = 1M
p" = 0
p"" = 1M

3
p
1
p = 1 q = 0.8 r =
4
q
p
y = 4000
x = 3000
c1 : {" c2 :
1 - p
1 - q
0
0
qr
pr
y = 4000
x = 3000
c3 : z" c4 :
1 - pr
1 - qr
0
0
p > q x < y
r " (0, 1)
x1 = 0, x2 = 3000, x3 = 4000
IC top
c2

IC bottom
c4


c1 c3
p1




10
5M
11
a2
1
0
11
a1
0.11
1M
1M
0.89
a2


3
p
10
5M
11
a2
1
0
11
a1
0.11
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V (p) = EU (p) = piu (xi)
i
V (p)
u (�) x
u (�)
u1 u2 u1 = f (u2) f (�)

V (p) = v (p1, p2, . . . , pn)
"
4" u (xi, p) = V (p)
"pi
p


p


p q

r (x, y) = -r (y, x)
q p

r (xi, xj) qipj e" r (xi, xj) pipj
i j i j

p

�ł łł

�ł
LHS : r (xi, xj) pj�ł qi = Ć (xi, p) qi
i j i
�ł łł

�ł
RHS : r (xi, xj) pj�ł pi = Ć (xi, p) pi
i j i
Ć (xi, p) x p
q {" p

Ć (xi, p) qi e" Ć (xi, p) pi
i i
EqĆ (x, p) e" EpĆ (x, p)
























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