1. The document describes a game theory model with multiple players and outcomes. 2. Equilibrium conditions for the players' strategies are derived by setting each player's payoff equal across different strategy choices. 3. The solutions to the equilibrium conditions are found to be mixed strategies with probabilities of 1/2, 1/3, and 3/4 for some players' strategies.

1. ήʔϜཧ࿦#4*$ԋश
ਓήʔϜͷࠞ߹ઓུφογϡ‫ߧۉ‬

2. ࠞ߹ઓུφογϡ‫ߧۉ‬ͷఆٛ
༗༻ͳఆཧ
ਓήʔϜͷφογϡ‫ߧۉ‬

3. ࠞ߹ઓུφογϡ‫ߧۉ‬
ϓϨΠϠʔ Ҏ֎ͷࠞ߹ઓུΛ ͱ͠‫ݻ‬ఆ͢Δ
Λຬͨ͢ͱ͖ ͸ ʹର͢Δ࠷ద൓Ԡઓུͱ͍͏
ઓུͷ૊ ͕ࠞ߹ઓུφογϡ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷϓϨΠϠʔʹରͯ͠
Λຬͨ͢ͱ͖Λ͍͏
ήʔϜཧ࿦#4*$ୈճࢀর
i p−i
∈ P−i
Fi
(p*i
, p−i
) ≥ Fi
(pi
, p−i
), ∀pi
∈ Pi
p*i
p−i
(p*1
, ⋯, p*n
) ∈ P
Fi
(p*i
, p*−i
) ≥ Fi
(pi
, p*−i
), ∀pi
∈ Pi

4. ࠞ߹ઓུʹؔ͢Δఆཧ
ఆཧ
ࠞ߹֦େͨ͠ήʔϜ ʹ͓͍ͯ
ͭҎ্ͷ७ઓུʹର͠
ਖ਼ͷ֬཰Λ༩͑Δࠞ߹ઓུ ͕
ଞͷϓϨΠϠʔͷઓུͷ૊ ʹରͯ͠࠷ద൓Ԡͱ͢Δ
͜ͷͱ͖ͦͷਖ਼ͷ֬཰Ͱ༻͍ΒΕΔ७ઓུ΋ଞͷϓϨΠϠʔͷઓུͷ૊ ʹର͢Δ࠷ద൓ԠͱͳΔ
ήʔϜཧ࿦#4*$ԋशୈճࢀর
(N, {Pi
}i∈N, {Fi
}i∈N) pi
p−i
p−i

5. ҎԼͷήʔϜʹ͓͚Δ‫׬‬શࠞ߹ઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
͜͜Ͱ‫׬‬શࠞ߹ઓུͱ͸
७ઓུΛදࠞ͢߹ઓུΛ‫߹͍ࠞͳ·ؚ‬ઓུͷ͜ͱ
ͭ·Γ
ϓϨΠϠʔ͸
ࣗ਎ͷ͢΂ͯͷ७ઓུʹରͯ͠
ਖ਼ͷ֬཰Λ༩͑ΔΑ͏ͳࠞ߹ઓུͷ͜ͱΛ͍͏
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

6. ҎԼͷήʔϜʹ͓͚Δ‫׬‬શࠞ߹ઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ࢀߟ
७ઓུʹΑΔφογϡ‫ߧۉ‬͸ͳ͍
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

7. ҎԼͷήʔϜʹ͓͚Δ‫׬‬શࠞ߹ઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
֤ϓϨΠϠʔͷࠞ߹ઓུΛ
ͱ͢Δ
ͨͩ͠
‫׬‬શࠞ߹ઓུ͚ͩߟ͑ΔͷͰ
ͭ·Γ
p1
= (p,1 − p) p2
= (q,1 − q) p3
= (r,1 − r)
p, q, r ≠ 0,1 p, q, r ∈ (0,1)
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
p
1 − p
q 1 − q
p
1 − p
q 1 − q
r 1 − r

8. ҎԼͷήʔϜʹ͓͚Δ‫׬‬શࠞ߹ઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
֤ϓϨΠϠʔͷࠞ߹ઓུΛ
ͱ͢Δ
p1
= (p,1 − p) p2
= (q,1 − q) p3
= (r,1 − r) p, q, r ∈ (0,1)
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
p
1 − p
q 1 − q
p
1 − p
q 1 − q
r 1 − r

9. ֤ϓϨΠϠʔͷࠞ߹ઓུΛ
ͱ͢Δ
͜ͷࠞ߹ઓུ͕φογϡ‫·ͭ
ߧۉ‬Γ‫࠷ʹ͍ޓ‬ద൓ԠͰ͋ΔͳΒ͹
ఆཧΑΓ
֤ϓϨΠϠʔͷਖ਼ͷ֬཰Ͱ࢖ΘΕΔ७ઓུ΋ಉ͡‫ظ‬଴རಘΛ༩͑ΔͷͰ
ҎԼͷ͕ࣜಉ࣌ʹ੒Γཱͭɿ
ɾɾɾ
ɾɾɾ
ɾɾɾ
p1
= (p,1 − p) p2
= (q,1 − q) p3
= (r,1 − r) p, q, r ∈ (0,1)
1 ⋅ (1 − q)r + 3 ⋅ (1 − q)(1 − r) = 2 ⋅ qr + 2 ⋅ q(1 − r) ⇔ 3 − 2r − 5q + 2qr = 0
2 ⋅ (1 − p)r + 2 ⋅ p(1 − r) = 2 ⋅ pr + 1 ⋅ (1 − p)r + 1 ⋅ (1 − p)(1 − r) ⇔ (2r − 1)(1 − 3p) = 0
1 ⋅ pq + 3 ⋅ (1 − p)(1 − q) = 1 ⋅ (1 − p)q + 3 ⋅ p(1 − q) ⇔ (2p − 1)(4q − 3) = 0
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
p
1 − p
q 1 − q
p
1 − p
q 1 − q
r 1 − r
Λऔͬͨͱ͖ͷརಘ
a1 Λऔͬͨͱ͖ͷརಘ
a2
Λऔͬͨͱ͖ͷརಘ
b1 Λऔͬͨͱ͖ͷརಘ
b2
Λऔͬͨͱ͖ͷརಘ
c1 Λऔͬͨͱ͖ͷརಘ
c2

10. ɾɾɾ
ɾɾɾ
ɾɾɾ
ΑΓ
·ͨ͸
ͷͱ͖
ΑΓ
ͷͱ͖
ΑΓ
Ώ͑ʹφογϡ‫ߧۉ‬͸
3 − 2r − 5q + 2qr = 0
(2r − 1)(1 − 3p) = 0
(2p − 1)(4q − 3) = 0
r =
1
2
p =
1
3
r =
1
2
3 − 2 ⋅
1
2
− 5q + 2q ⋅
1
2
= 0 ⇔ q =
1
2
q =
1
2
(2p − 1)(4 ⋅
1
2
− 3) = 0 ⇔ p =
1
2 ((
1
2
,
1
2)
,
(
1
2
,
1
2)
,
(
1
2
,
1
2))
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
p
1 − p
q 1 − q
p
1 − p
q 1 − q
r 1 − r

11. ɾɾɾ
ɾɾɾ
ɾɾɾ
ΑΓ
·ͨ͸
ͷͱ͖
ΑΓ
ͷͱ͖
ΑΓ
͜ΕΛຬͨ͢ ͸ଘࡏ͠ͳ͍
3 − 2r − 5q + 2qr = 0
(2r − 1)(1 − 3p) = 0
(2p − 1)(4q − 3) = 0
r =
1
2
p =
1
3
p =
1
3
(2 ⋅
1
3
− 1)(4q − 3) = 0 ⇔ q =
3
4
q =
3
4
3 − 2r − 5 ⋅
3
4
+ 2 ⋅
3
4
r = 0 ⇔ 2r + 3 = 0 r ∈ (0,1)
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
p
1 − p
q 1 − q
p
1 − p
q 1 − q
r 1 − r
((
1
2
,
1
2)
,
(
1
2
,
1
2)
,
(
1
2
,
1
2))

12. ɾɾɾ
ɾɾɾ
ɾɾɾ
ΑΓ
·ͨ͸
ͷͱ͖
ΑΓ
ͷͱ͖
ΑΓ
3 − 2r − 5q + 2qr = 0
(2r − 1)(1 − 3p) = 0
(2p − 1)(4q − 3) = 0
p =
1
2
q =
3
4
p =
1
2
(2r − 1)(1 − 3 ⋅
1
2
) = 0 ⇔ r =
1
2
r =
1
2
q =
1
2
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
p
1 − p
q 1 − q
p
1 − p
q 1 − q
r 1 − r
((
1
2
,
1
2)
,
(
1
2
,
1
2)
,
(
1
2
,
1
2))

13. ɾɾɾ
ɾɾɾ
ɾɾɾ
ΑΓ
·ͨ͸
ΑΓ
͜ΕΛຬͨ͢ ͸ଘࡏ͠ͳ͍
3 − 2r − 5q + 2qr = 0
(2r − 1)(1 − 3p) = 0
(2p − 1)(4q − 3) = 0
p =
1
2
q =
3
4
3 − 2r − 5 ⋅
3
4
+ 2 ⋅
3
4
r = 0 ⇔ 2r + 3 = 0 r ∈ (0,1)
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
p
1 − p
q 1 − q
p
1 − p
q 1 − q
r 1 − r
((
1
2
,
1
2)
,
(
1
2
,
1
2)
,
(
1
2
,
1
2))

14. Ҏ্ΑΓ
‫׬‬શࠞ߹ઓུφογϡ‫ߧۉ‬͸
((
1
2
,
1
2)
,
(
1
2
,
1
2 )
,
(
1
2
,
1
2))
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD֬཰
1ʘ2 C C
B
B
͕७ઓུD֬཰
1ʘ2 C C
B
B
1
2
1
2
1
2
1
2
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

15. ิ଍
ͱ͍͏ઓུͷ૊ͷ΋ͱͰ
ϓϨΠϠʔͷࠞ߹ઓུ ͱ
ਖ਼ͷ֬཰Ͱ࢖ΘΕΔ७ઓུ ͱ ͷ‫ظ‬଴རಘ͕Ұக͢Δ͜ͱΛ‫͓ͯ͘ݟ‬
((
1
2
,
1
2)
,
(
1
2
,
1
2)
,
(
1
2
,
1
2))
(
1
2
,
1
2)
a1 a2
F1
((
1
2
,
1
2)
,
(
1
2
,
1
2)
,
(
1
2
,
1
2 ))
= 2 ⋅
1
8
+ 1 ⋅
1
8
+ 2 ⋅
1
8
+ 3 ⋅
1
8
=
8
8
= 1
F1
(
(1,0),
(
1
2
,
1
2)
,
(
1
2
,
1
2 ))
= 1 ⋅
1
4
+ 3 ⋅
1
4
=
4
4
= 1
F1
(
(0,1),
(
1
2
,
1
2)
,
(
1
2
,
1
2 ))
= 2 ⋅
1
4
+ 2 ⋅
1
4
=
4
4
= 1
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD֬཰
1ʘ2 C C
B
B
͕७ઓུD֬཰
1ʘ2 C C
B
B
1
2
1
2
1
2
1
2
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

16. ิ଍
ͱ͍͏ઓུͷ૊ͷ΋ͱͰ
ϓϨΠϠʔͷࠞ߹ઓུ ͱ
ਖ਼ͷ֬཰Ͱ࢖ΘΕΔ७ઓུ ͱ ͷ‫ظ‬଴རಘ͕Ұக͢Δ͜ͱΛ‫͓ͯ͘ݟ‬
஫ҙ
ϓϨΠϠʔ͕
ϓϨΠϠʔ͕ ͷ΋ͱͰ
ϓϨΠϠʔ͸ ͔Β ͷํ͕‫ظ‬଴རಘ͕ߴ͘ͳΔͷͰ
͸φογϡ‫Ͱߧۉ‬͸ͳ͍
((
1
2
,
1
2)
,
(
1
2
,
1
2)
,
(
1
2
,
1
2 ))
(
1
2
,
1
2)
a1 a2
F1
((
1
2
,
1
2)
,
(
1
2
,
1
2)
,
(
1
2
,
1
2 ))
= 2 ⋅
1
8
+ 1 ⋅
1
8
+ 2 ⋅
1
8
+ 3 ⋅
1
8
=
8
8
= 1
F1
(
(1,0),
(
1
2
,
1
2)
,
(
1
2
,
1
2 ))
= 1 ⋅
1
4
+ 3 ⋅
1
4
=
4
4
= 1
F1
(
(0,1),
(
1
2
,
1
2)
,
(
1
2
,
1
2 ))
= 2 ⋅
1
4
+ 2 ⋅
1
4
=
4
4
= 1
(1,0)
(
1
2
,
1
2)
(
1
2
,
1
2)
(0,1)
(
(1,0),
(
1
2
,
1
2)
,
(
1
2
,
1
2))
1 ⋅
1
4
+ 3 ⋅
1
4
3 ⋅
1
2
ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD֬཰
1ʘ2 C C
B
B
͕७ઓུD֬཰
1ʘ2 C C
B
B
1
1
2
1
2
1
2
→ 0
1
2
→ 1
1
4
→ 0
0 0 0 0 0
1
4
→ 0
1
4
→
1
2
1
4
→
1
2
1
2
1
2
1
0

17. ήʔϜཧ࿦#4*$ԋश
ਓήʔϜͷࠞ߹ઓུφογϡ‫ߧۉ‬