https://www.nicovideo.jp/watch/sm38094624 https://youtu.be/eikXE_jw1rU

1. 3

2. 1.
2.

3. 1.
(a)
(b)
(c)
(d)

5.
N
P
m Rm
+
E : N → P × Rm
+
i ( ⪰i , wi) m = 2
wi = (w1
i , w2
i )

6.
N
i Xi = {xi ∈ Rnm
|xi = (xi1, ⋯, xin),
xij = (x1
ij, ⋯, xm
ij );
∑
j∈N
xh
ij = wh
i , h = 1,⋯, m}
i ui(x) = vi(g(x)i)
m = 2
N = {1, 2, 3}
X1 = {x11, x12, x13}
x11 = (x1
11, x2
11) x12 = (x1
12, x2
12) x13 = (x1
13, x2
13)
x1
11 + x1
12 + x1
13 = w1
1 x2
11 + x2
12 + x2
13 = w2
1

7.
N
i Xi = {xi ∈ Rnm
|xi = (xi1, ⋯, xin),
xij = (x1
ij, ⋯, xm
ij );
∑
j∈N
xh
ij = wh
i , h = 1,⋯, m}
i ui(x) = vi(g(x)i)
vi( ⋅ )
g(x) =
{
(∑j∈N
xj1, ⋯, ∑j∈N
xjn), ( ⋅ ) ∈ Rnm
+
w, otherwise

8.
f x1 x2 α ∈ (0, 1)
f(αx1 + (1 − α)x2) ≥ min{f(x1), f(x2)}
f
f(x1)
f(x2)
x1x2
0
αx1 + (1 − α)x2
f(αx1 + (1 − α)x2)
f
f(x1)
f(x2)
x1x2
0
αx1 + (1 − α)x2
f(αx1 + (1 − α)x2)

9.
f a
{x| f(x) ≥ a}
f
a
0
{x| f(x) ≥ a}
f
0
{x| f(x) ≥ a}
a

10.
N
i Xi = {xi ∈ Rnm
|xi = (xi1, ⋯, xin),
xij = (x1
ij, ⋯, xm
ij );
∑
j∈N
xh
ij = wh
i , h = 1,⋯, m}
i ui(x) = vi(g(x)i)
vi( ⋅ )
g(x) =
{
(∑j∈N
xj1, ⋯, ∑j∈N
xjn), ( ⋅ ) ∈ Rnm
+
w, otherwise

11.
wi = (w1
i , ⋯, wm
i ) ∈ Rm
+
S ⊆ N
yi = (y1
i , ⋯, ym
i ) ∈ Rm
+
∑
i∈S
yh
i =
∑
i∈S
wh
i , h = 1,⋯, m

12.
wi = (w1
i , ⋯, wm
i ) ∈ Rm
+
S ⊆ N
yi = (y1
i , ⋯, ym
i ) ∈ Rm
+
∑
i∈S
yh
i =
∑
i∈S
wh
i , h = 1,⋯, m
S = N

13.
g(x)
g(x) =
{
(∑j∈N
xj1, ⋯, ∑j∈N
xjn), ( ⋅ ) ∈ Rnm
w, otherwise
∑
i∈S
xh
i =
∑
i∈S
wh
i , h = 1,⋯, m S = N
∑
i∈N
xh
i =
∑
i∈N
wh
i , h = 1,⋯, m
∑
i∈N
∑
j∈N
xh
ji =
∑
j∈N
∑
i∈N
xh
ji =
∑
j∈N
wh
j
xh
ji j h wh
j i
∑
i∈N
xh
ji wh
j

14.
i ∈ N
Xi ⊆ Rmn
+
x*ii
= wi, ∀i ∈ N x*
x*

15.
i ∈ N
Xi ⊆ Rmn
+
x*ii
= wi, ∀i ∈ N x*
x*
x u(x)
uN(y) > uN(x) y x

16.
x ∈ X S ⊆ N
S x yS ∈ XS
uS(yS, xNS) > uS(x)
x ∈ X S ⊆ N
x
uN(yN) > uN(x) yN

17.
i ∈ N
Xi ⊆ Rmn
+
x*ii
= wi, ∀i ∈ N x*
x*

18.
i ∈ N
Xi ⊆ Rmn
+
x*ii
= wi, ∀i ∈ N x*
x*

19.
i ∈ N
Xi ⊆ Rmn
+
x*ii
= wi, ∀i ∈ N x*
x*
x*
S ⊊ N
ui(zS, x*NS
) > ui(x*) = vi(wi) ∀i ∈ S
zS ∈ XS vi
zij = 0, ∀i ∈ S, ∀j ∈ NS

20.
x*
S ⊊ N
ui(zS, x*NS
) > ui(x*) = vi(wi) ∀i ∈ S
zS ∈ XS vi
zij = 0, ∀i ∈ S, ∀j ∈ NS
uj(zS, x*NS
) = uj(x*) = vj(wj) ∀j ∈ NS

21.
x*
S ⊊ N
ui(zS, x*NS
) > ui(x*) = vi(wi) ∀i ∈ S
zS ∈ XS vi
zij = 0, ∀i ∈ S, ∀j ∈ NS
uj(zS, x*NS
) = uj(x*) = vj(wj) ∀j ∈ NS
vi zS yS ∈ XS
ui(yS, x*NS
) > ui(x*) = vi(wi) ∀i ∈ N
x*

22.
x*
S ⊊ N
ui(zS, x*NS
) > ui(x*) = vi(wi) ∀i ∈ S
zS ∈ XS vi
zij = 0, ∀i ∈ S, ∀j ∈ NS
uj(zS, x*NS
) = uj(x*) = vj(wj) ∀j ∈ NS
vi zS yS ∈ XS
ui(yS, x*NS
) > ui(x*) = vi(wi) ∀i ∈ N
x*
i ∈ S zS
j ∈ NS zS
uj(yS, x*NS
) > uj(zS, x*NS
)

23.
i ∈ N
Xi ⊆ Rmn
+
x*ii
= wi, ∀i ∈ N x*
x*
vi x*

25. See you next time