https://youtu.be/D3P3xnOJExY

1. ήʔϜཧ࿦#4*$ԋश
༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬

2. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
໰୊
ղ౴
ఆཧ

3. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
੒෼ήʔϜ
ϓϨΠϠʔͷू߹
ϓϨΠϠʔ ͷߦಈͷू߹
ϓϨΠϠʔ ͷརಘؔ਺ ͱͯ͠
੒෼ήʔϜΛ ճ ‫܁‬Γฦ͢ήʔϜΛߟ͑Δ
ճ໨ʹ͓͍ͯ ֤ϓϨΠϠʔ͸ ճ໨·Ͱʹબ͹ΕͨߦಈΛ͢΂ͯ஌্ͬͨͰߦಈΛબ୒͢Δ
‫܁‬Γฦ͠ήʔϜʹ͓͚Δઓུ
ͱͯ͠ ͱ͠ ͱ͢Δ͜͜Ͱ Λ ճ໨·Ͱͷཤྺͱ͍͏
ϓϨΠϠʔ ͷ ճ໨ͷߦಈ͸ ཤྺ Λ΋ͱʹܾఆ͞ΕΔ ͢ͳΘͪ Ͱ༩͑ΒΕΔ
͜ΕΑΓ‫܁‬Γฦ͠ήʔϜʹ͓͚ΔϓϨΠϠʔ ͷ७ਮઓུ͸ Ͱ༩͑ΒΕΔ
ϓϨΠϠʔ ͷ७ਮઓུͷू߹Λ ͱ͠ ϓϨΠϠʔશମͷઓུͷ૊ͷू߹Λ Ͱද͢
G = (N, {Si
}i∈N, {fi
}i∈N)
N
Si
i
fi
i S = S1
× ⋯ × Sn
fi
: S → ℜ
T T ∞
t t − 1
S = S1
× ⋯ × Sn
St−1 =
t−1
S × ⋯ × S S0 = {∅} ht−1 = (s1, ⋯, st−1) ∈ St−1 t − 1
i t ht−1 xi
t : St−1 → Si
i xi
= (xi
t)T
t=1
i XTi
XT
= XT1
× ⋯ × XTn

4. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
‫܁‬Γฦ͠ήʔϜʹ͓͚Δརಘ
ׂҾҼࢠ ͱͯ͠
ΛׂҾརಘͱ͍͏
‫܁‬Γฦ͠ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬
ઓུ ͕φογϡ‫͋Ͱߧۉ‬Δͱ͸ ͢΂ͯͷϓϨΠϠʔ ʹ͍ͭͯ
ׂҾརಘ
δ, 0 δ 1
fTi
δ (x) =
T
∑
t=1
δt−1
fi
(st(x))
x* = (x1
* , ⋯, xn
*) i
fTi
δ (xi*
, x−i*
) ≥ fTi
δ (xi
, x−i*
), ∀xi
∈ XTi

5. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
Ҏ‫ޙ‬ͷ෦෼ήʔϜΛ Ͱද͢
·ͨ ೚ҙͷϓϨΠϠʔʹ͍ͭͯ ઓུ ͕ ʹಋ͘ઓུΛ Ͱද͢
۩ମతʹ ΛҰͭߟ͑ͯΈΔ͜ͷཤྺͰୡ੒͞Εͨ෦෼ήʔϜ ʹ͓͍ͯ
ճ໨ͷߦಈ͸ ճ·Ͱͷཤྺ ʹରͯ͠
༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
ʹ͓͍ͯ ઓུ ͕෦෼ήʔϜ‫׬‬શ‫͋Ͱߧۉ‬Δͱ͸
೚ҙͷճ ͱ ճ·Ͱͷ೚ҙͷཤྺ ʹରͯ͠ ֤ϓϨΠϠʔ ͷઓུ ͕ಋ͘ઓུ ͷ૊
͕ ͷφογϡ‫ͳͱߧۉ‬Δ͜ͱͰ͋Δ
ht−1 GT
(δ)(ht−1)
xi
∈ XTi
GT
(δ)(ht−1) xi
|ht−1
ht−1 = (s1, s2, ⋯, st−1) GT
(δ)(ht−1)
t′

t′

− 1 ht′

−1 = (s′

1, s′

2, ⋯, s′

t′

−1) (xi
|ht−1)t′

(ht′

−1) = xi
(s1, s2, ⋯, st−1, s′

1, s′

2, ⋯, s′

t′

−1)
GT
(δ) = (N, {XTi
}i∈N, {fTi
δ }i∈N) x* = (x1*
, ⋯, xn*
)
t = 1,2,⋯, T t − 1 ht−1 i xi*
xi*
|ht−1
x*|ht−1 = (x1*
|ht−1, ⋯, xn*
|ht−1) GT
(δ)(ht−1)

6. ໰୊
ӈͷήʔϜΛճ‫܁‬Γฦ͢ήʔϜΛߟ͑Δ
ҎԼͷઓུ͕φογϡ‫͋Ͱߧۉ‬ΔͨΊͷׂҾҼࢠ ͷ৚݅Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ
ճ໨ͷߦಈ͸
ճ໨ͷߦಈ͸ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ৔߹
ϓϨΠϠʔ
ճ໨ͷߦಈ͸
ճ໨ͷߦಈ͸ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ৔߹
ͷઓུͷ૊͸෦෼ήʔϜ‫׬‬શ‫Ͱߧۉ‬͸ͳ͍͜ͱΛઆ໌ͤΑ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬Λ‫ٻ‬ΊΑ
δ
a
(a, d) c b
d
(a, d) f e
̍ʘ E F G
B
C
D

7. ղ౴
ӈͷήʔϜΛճ‫܁‬Γฦ͢ήʔϜΛߟ͑Δ
ҎԼͷઓུ͕φογϡ‫͋Ͱߧۉ‬ΔͨΊͷׂҾҼࢠ ͷ৚݅Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ
ճ໨ͷߦಈ͸
ճ໨ͷߦಈ͸ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ৔߹
ϓϨΠϠʔ
ճ໨ͷߦಈ͸
ճ໨ͷߦಈ͸ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ৔߹
ϓϨΠϠʔʹ͍ͭͯ ϓϨΠϠʔͷઓུʹରͯ͠࠷ద൓ԠʹͳΔ৚݅Λ‫ٻ‬ΊΔ
্‫ه‬ઓུͷ΋ͱͰಘΒΕΔརಘ͸
ճ໨ͷߦಈͰ ΛऔΔΑ͏ͳઓུΛͱΔͱ ϓϨΠϠʔ͕̎GΛऔΔͨΊ ΑΓେ͖͍རಘΛಘΔ͜ͱ͸ͳ͍
ճ໨ͷߦಈ͸ ΛऔΔΑ͏ͳઓུΛͱΔͱ ϓϨΠϠʔ͕̎FΛऔΔͨΊ ࠷େͰ ͷརಘΛಘΔ
࠷ద൓ԠʹͳΔͨΊʹ͸
δ
a
(a, d) c b
d
(a, d) f e
10 + 5δ
a 10 + 5δ
c 13 + δ
10 + 5δ ≥ 13 + δ ⇔ δ ≥
3
4
̍ʘ E F G
B
C
D

8. ղ౴
ӈͷήʔϜΛճ‫܁‬Γฦ͢ήʔϜΛߟ͑Δ
ҎԼͷઓུ͕φογϡ‫͋Ͱߧۉ‬ΔͨΊͷׂҾҼࢠ ͷ৚݅Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ
ճ໨ͷߦಈ͸
ճ໨ͷߦಈ͸ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ৔߹
ϓϨΠϠʔ
ճ໨ͷߦಈ͸
ճ໨ͷߦಈ͸ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ৔߹
ͷઓུͷ૊͸෦෼ήʔϜ‫׬‬શ‫Ͱߧۉ‬͸ͳ͍͜ͱΛઆ໌ͤΑ
Ҏ֎ͷཤྺͷ෦෼ήʔϜʹ͓͍ͯ ͷઓུͷ૊͕ͱΒΕΔ͕ ͜Ε͸φογϡ‫Ͱߧۉ‬͸ͳ͍ͨΊ
δ
a
(a, d) c b
d
(a, d) f e
(a, d) (b, e)
̍ʘ E F G
B
C
D

9. ղ౴
ӈͷήʔϜΛճ‫܁‬Γฦ͢ήʔϜΛߟ͑Δ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬Λ‫ٻ‬ΊΑ
೚ҙͷཤྺʹ͍ͭͯ
ઓུͷ૊͕ಋ͘ճ໨ͷߦಈͷ૊͕φογϡ‫ͯͬͳʹߧۉ‬Δඞཁ͕͋Δ
Ώ͑ʹ ճ໨ͷઓུͷ૊͕ ͱͳΔઓུͷΈ͕෦෼ήʔϜ‫׬‬શ‫ͳʹߧۉ‬Δ
͜ΕΛલఏͱͯ͠ རಘදΛॻ͚͹
͜ΕΑΓ ճ໨ͷઓུͷ૊΋ ͱͳΔ΋ͷ͕ ෦෼ήʔϜ‫׬‬શ‫ͳͱߧۉ‬ΔΏ͑ʹ
ϓϨΠϠʔ
ճ໨ͷߦಈ͸ ճ໨ͷߦಈ͸ ೚ҙͷཤྺʹର͠
ϓϨΠϠʔ
ճ໨ͷߦಈ͸ ճ໨ͷߦಈ͸ ೚ҙͷཤྺʹର͠
(c, f )
(c, f )
c c
f f
̍ʘ E F G
B
C
D
̍ʘ E F G
B Ў Ў Ў Ў Ў Ў
C Ў Ў Ў Ў Ў Ў
D Ў Ў Ў Ў Ў Ў

10. ఆཧ
ఆཧ
੒෼ήʔϜ͕ͨͩҰͭͷφογϡ‫ߧۉ‬ Λ΋ͭͱ͢Δ͜ͷͱ͖ ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ ͸ͨͩҰͭଘࡏ͠ ߦಈͷ૊ Λ༩͑Δ
ήʔϜཧ࿦#4*$ୈճิ଍ࢀর
s* T
x* s(x*) = (
T
s*, s*, ⋯, s*)
̍ʘ E F G
B
C
D

11. ήʔϜཧ࿦#4*$ԋश
༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬