This document describes a multi-agent system for modeling the behaviors and interactions of two agents, P1 and P2. It defines the possible states and actions for each agent, and a framework for modeling their joint and individual decision making processes over time. Specifically, it represents each agent's state as a point in 2D space, defines their possible actions as moving between states, and introduces a probabilistic model to describe how their next joint action is selected based on their current individual states.

2.
͜ͷࢿྉ͓Αͼಈը͸ɺҎԼͷಈըΛࢹௌࡁΈͰ͋Δ͜ͱΛલఏʹ࡞ΒΕ͍ͯ·͢ɻ
ˎήʔϜཧ࿦#"4*$෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
ˎήʔϜཧ࿦#4*$‫׬‬શϕΠδΞϯ‫ߧۉ‬
ˎήʔϜཧ࿦#4*$ԋश͓Αͼԋश
෦෼ήʔϜ‫׬‬શ‫׬ߧۉ‬શϕΠδΞϯ‫ߧۉ‬ͷఆٛͷத਎ͷৄࡉʹ͸
৮Εͯͳ͍ͷͰɺ͜ͷลΓΛ͝ཧղͷ্͝ࢹௌ͍ͩ͘͞ɻ

3. ໨࣍
σʔτήʔϜ ήʔϜཧ࿦#4*$ԋश͓ΑͼԋशΑΓ
σʔτήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
σʔτήʔϜʹ͓͚Δऑ‫׬‬શϕΠδΞϯ‫ߧۉ‬
ऑ‫׬‬શϕΠδΞϯ‫ߧۉ‬ͷఆٛ
ߦಈઓུͷ૊͕‫׬‬શʹ֬཰త
ஞ࣍‫ߧۉ‬ͷఆٛ
σʔτήʔϜͰ֬ೝ
֤‫ߧۉ‬ͷؔ܎·ͱΊ

4. ҎԼͷల։‫ܗ‬ήʔϜͰද‫͞ݱ‬ΕΔήʔϜΛσʔτήʔϜͱ‫͢ʹͱ͜Ϳݺ‬Δ
σʔτήʔϜ
P1
P2
P1
(2, 0)
(0, 0)
home
৘ใू߹
ϓϨΠϠʔ
P2
P2
(1, 3)
out
movie
m
shopping
s
movie shopping
(3, 1)
(0, 0)
ϓϨΠϠʔ
P1

5. ҎԼͷల։‫ܗ‬ήʔϜͰද‫͞ݱ‬ΕΔήʔϜΛσʔτήʔϜͱ‫͢ʹͱ͜Ϳݺ‬Δ
σʔτήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬Λ‫ٻ‬ΊΑ
෦෼ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬Λߟ͑Δͱ
ͱ Ͱ͋Δ
ɾ෦෼ήʔϜͰ ͕औΒΕΔͱ͢Ε͹ϓϨΠϠʔ͸ ΛબͿͷ͕࠷ద
ɾ෦෼ήʔϜͰ ͕औΒΕΔͱ͢Ε͹
ϓϨΠϠʔ͸ ΛબͿͷ͕࠷ద
Ώ͑ʹ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬͸
(movie, movie) (shopping, shopping)
(movie, movie) home
(shopping, shopping)
out
(home − movie, movie) (out − shopping, shopping)
σʔτήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
P1
P2
P1
(2, 0)
(0, 0)
home
σʔτήʔϜ
৘ใू߹
ϓϨΠϠʔ
P1
ϓϨΠϠʔ
P2
P2
(1, 3)
out
movie
m
shopping
s
movie shopping
(3, 1)
(0, 0)
1ʘ2
movie
shopping
movie shopping
෦෼ήʔϜ
෦෼ήʔϜ
P1
(2, 0)
home out
(1, 3)
P1
(2, 0)
home out
(3, 1)

6. ҎԼͷల։‫ܗ‬ήʔϜͰද‫͞ݱ‬ΕΔήʔϜΛσʔτήʔϜͱ‫͢ʹͱ͜Ϳݺ‬Δ
σʔτήʔϜʹ͓͚Δ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
‫׬‬શϕΠδΞϯ‫ߧۉ‬͸
ͨͩ͠
ͨͩ͠
φογϡ‫ͱߧۉ‬ಉ͡ઓུͷ૊Λ‫ؚ‬Έ
෦෼ήʔϜ‫׬‬શ‫Ͱߧۉ‬ഉআͰ͖ͨઓུͷ૊΋‫ؚ‬ΜͰ͠·͍ͬͯΔ
ஞ࣍‫ߧۉ‬Λ࢖ͬͯΈͯ͸ʁ
(home − movie, movie) μ1
1 μ1
2 μ2
(1,1,(r,1 − r)) r ≥
1
4
(home − shopping, movie) μ1
1 μ1
2 μ2
(1,1,(r,1 − r)) r ≥
1
4
(out − shopping, shopping) μ1
1 μ1
2 μ2
(1,1,(0,1))
σʔτήʔϜʹ͓͚Δऑ‫׬‬શϕΠδΞϯ‫ߧۉ‬
P1
P2
P1
(2, 0)
(0, 0)
home
σʔτήʔϜ
৘ใू߹
ϓϨΠϠʔ
P1
ϓϨΠϠʔ
P2
P2
(1, 3)
out
movie
m
shopping
s
movie shopping
(3, 1)
(0, 0)
μ1
1 = 1
μ1
2 = 1
r 1 − r

7. ऑ
‫׬‬શϕΠδΞϯ‫ߧۉ‬
‫׬‬શϕΠδΞϯ‫ߧۉ‬ઓུͷ૊ ͱ৴೦ͷମ‫ܥ‬ ͷ૊ Ͱ
͕ ͷ΋ͱͰஞ࣍߹ཧతͰ͋Γ
͕ ʹ੔߹తͰ͋Δ΋ͷΛऑ‫׬‬શϕΠδΞϯ‫͏͍ͱߧۉ‬
ߦಈઓུͷஞ࣍߹ཧੑల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
৴೦ͷମ‫ܥ‬ ͕༩͑ΒΕ͍ͯΔͱ͢Δ
͜ͷͱ͖
ઓུͷ૊ ͕
ϓϨΠϠʔ ͷ৘ใू߹ ʹ͓͍ͯҎԼΛຬͨ͢ͱ͖
͸৴೦ͷମ‫ܥ‬ ͷ΋ͱͰ৘ใू߹ ʹ͓͍ͯஞ࣍߹ཧతͰ͋Δͱ͍͏
৴೦ͷମ‫ͱܥ‬੔߹ੑల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
ઓུͷ૊ ͕༩͑ΒΕ͍ͯΔͱ͢Δ
͜ͷͱ͖
৴೦ͷମ‫ܥ‬ ͕
೚ҙͷϓϨΠϠʔ ͷ೚ҙͷ৘ใू߹ ʹ͓͍ͯ
Ͱ͋Ε͹
͢΂ͯͷ ͷ఺ ʹ͍ͭͯ
ͱͳΔͱ͖
͸ઓུͷ૊ ʹ͓͍ͯ੔߹తͰ͋Δͱ͍͏
b μ (b, μ)
b μ μ b
Γ μ
b = (b1
, ⋯, bn
) i ui
l
b μ ui
l
Hi
(ui
l, μ, (bi,ui
l, b−i,ui
l)) ≥ Hi
(ui
l, μ, (b′

i,ui
l, b−i,ui
l)), ∀b′

i,ui
l ∈ Bi,ui
l
Γ b = (b1
, ⋯, bn
)
μ i ui
l
Prob(ui
l |b) 0 ui
l x*
μ(x*) =
Prob(x*|b)
∑x∈ui
l
Prob(x|b)
μ b
ߦಈઓུʹΑͬͯ౸ୡՄೳͳ
‫ܦ‬࿏ʹ͓͚Δ߹ཧੑͷΈߟ͑Δ

8. ߦಈઓུͷ૊͕‫׬‬શʹ֬཰త
ߦಈઓུͷ૊ ͕‫׬‬શʹ֬཰తͰ͋Δͱ͸
͢΂ͯͷࢬʹ͍ͭͯਖ਼ͷ֬཰Λ༩͑Δͱ͖Λ͍͏
b = (b1
, ⋯, bn
)
ߦಈઓུͷ૊͕‫׬‬શʹ֬཰త
P1
P2
P1
(2, 0)
(0, 0)
home
৘ใू߹
ϓϨΠϠʔ
P1
ϓϨΠϠʔ
P2
P2
(1, 3)
out
movie
m
shopping
s
movie shopping
(3, 1)
(0, 0)
μ1
1 = 1
μ1
2 = 1
r 1 − r
P1
P2
P1
(2, 0)
(0, 0)
home
P2
(1, 3)
out
movie
m
shopping
s
movie shopping
(3, 1)
(0, 0)
μ1
1 = 1
μ1
2 = 1
r 1 − r
‫׬‬શʹ֬཰తͳߦಈઓུ
Ͱ͋Ε͹Ͳͷ৘ใू߹΋
ඞͣ౸ୡ͢Δ
‫׬‬શʹ֬཰తͰͳ͍ ‫׬‬શʹ֬཰తͰ͋Δ

9. ஞ࣍‫ߧۉ‬
ஞ࣍‫ߧۉ‬ઓུͷ૊ ͱ৴೦ͷମ‫ܥ‬ ͷ૊ ͕ஞ࣍‫͋Ͱߧۉ‬Δͱ͸
‫׬‬શʹ֬཰తͳߦಈઓུͷ૊ͱ৴೦ͷ
ମ‫ܥ‬ͷྻ ͕ଘࡏͯ͠
ҎԼΛຬͨ͢ɿ
ߦಈઓུͷஞ࣍߹ཧੑ৴೦ͷମ‫ܥ‬ ʹରͯ͠
ઓུͷ૊ ͕
ϓϨΠϠʔ ͷ৘ใू߹ ʹ͓͍ͯ
৴೦ͷମ‫ͱܥ‬੔߹ੑ֤ ʹ͍ͭͯ
৴೦ͷମ‫ܥ‬ ͸ߦಈઓུͷ૊ ͔Βಋ͔Ε
͕੒Γཱͭ
͜͜Ͱ
ߦಈઓུͷ૊ ͔Βಋ͔ΕΔͱ͸
ϕΠζͷϧʔϧΛ༻͍ͯ
ͱͳΔͱ͖Λ͍͏
b μ (b, μ)
{(bk, μk)}∞
k=1
μ b = (b1
, ⋯, bn
) i ui
l
Hi
(ui
l, μ, (bi,ui
l, b−i,ui
l)) ≥ Hi
(ui
l, μ, (b′

i,ui
l, b−i,ui
l)), ∀b′

i,ui
l ∈ Bi,ui
l
k = 1, 2, ⋯ μk bk = (b1
k , ⋯, bn
k )
(bk, μk) → (b, μ) k → ∞
bk = (b1
k , ⋯, bn
k )
μk(x*) =
Prob(x*|bk)
∑x∈ui
l
Prob(x|bk)
x* x**
μk(x*) =
1
3
1
3
+ 1
4
=
4
7
ui
l
1
3
y
1
4
5
12
ऑ‫׬‬શϕΠδΞϯ‫ͱߧۉ‬ಉ͡
‫׬‬શʹ֬཰తͳߦಈઓུʹΑͬͯ
͢΂ͯͷ‫ܦ‬࿏ʹ͓͚Δ߹ཧੑΛߟ͑Δ

10. σʔτήʔϜʹ͓͚Δஞ࣍‫ߧۉ‬Λ֬ೝ͢Δ
‫׬‬શϕΠδΞϯ‫ݕ͍ͯͭʹߧۉ‬౼
ͨͩ͠
ϓϨΠϠʔͷߦಈઓུ
ϓϨΠϠʔͷߦಈઓུ
ͨͩ͠
ͱ͢Δ͢Δͱ
͜ͷͱ͖
͜ͷߦಈઓུ͔Βಋ͔ΕΔ৴೦͸ ࣗ໌ͳ΋ͷ͸আ͘
‫͜ʹٯ‬ͷ৴೦ͷମ‫ ܥ‬
ͷ‫Ͱݩ‬͸
ϓϨΠϠʔͷߦಈઓུ ͸࠷ద
͜ͷߦಈઓུʹରͯ͠ϓϨΠϠʔͷߦಈઓུ ΋࠷ద
Ώ͑ʹ
ஞ࣍‫ ߧۉ‬
(home − movie, movie) μ1
1 μ1
2 μ2
(1,1,(r,1 − r)) r ≥
1
4
b1
k = ((1 − ϵ1
k , ϵ1
k ), (1 − ϵ′

1
k , ϵ′

1
k )) b2
k = ((1 − ϵ2
k , ϵ2
k ))
ϵ1
k → 0 ϵ′

1
k → 0 ϵ2
k → 0 k → ∞
b1
k = ((1 − ϵ1
k , ϵ1
k ), (1 − ϵ′

1
k , ϵ′

1
k )) → ((1,0), (1,0)) b2
k = ((1 − ϵ2
k , ϵ2
k )) → ((1,0))
r =
ϵ1
k (1 − ϵ′

1
k )
ϵ1
k (1 − ϵ′

1
k ) + ϵ1
k ϵ′

1
k
→ 1 (k → ∞)
μ1
1 μ1
2 μ2
(1,1,(1,0))
b2
= ((1,0))
((1,0), (1,0))
(home − movie, movie) μ1
1 μ1
2 μ2
(1,1,(1,0))
σʔτήʔϜͰ֬ೝ
P1
P2
P1
(2, 0)
(0, 0)
home
σʔτήʔϜ
৘ใू߹
ϓϨΠϠʔ
P1
ϓϨΠϠʔ
P2
P2
(1, 3)
out
movie
m
shopping
s
movie shopping
(3, 1)
(0, 0)
μ1
1 = 1
μ1
2 = 1
r 1 − r
ϵ1
k
ϵ′

1
k
ϵ2
k
ϵ2
k
ߦಈઓུͰ‫ه‬ड़͢Ε͹
b = (b1
, b2
) = (((1,0), (1,0)), ((1,0)))
1 0

11. σʔτήʔϜʹ͓͚Δஞ࣍‫ߧۉ‬Λ֬ೝ͢Δ
‫׬‬શϕΠδΞϯ‫ݕ͍ͯͭʹߧۉ‬౼
ͨͩ͠
ϓϨΠϠʔͷߦಈઓུ
ϓϨΠϠʔͷߦಈઓུ
ͨͩ͠
ͱ͢Δ͢Δͱ
͜ͷͱ͖
͜ͷߦಈઓུ͔Βಋ͔ΕΔ৴೦͸ ࣗ໌ͳ΋ͷ͸আ͘
‫͜ʹٯ‬ͷ৴೦ͷମ‫ ܥ‬
ͷ‫Ͱݩ‬͸
ϓϨΠϠʔͷߦಈઓུ ͸࠷దͰ͸ͳ͍
Ώ͑ʹ
ߦಈઓུͷ૊ ͸ஞ࣍‫Ͱߧۉ‬͸ͳ͍
(home − shopping, movie) μ1
1 μ1
2 μ2
(1,1,(r,1 − r)) r ≥
1
4
b1
k = ((1 − ϵ1
k , ϵ1
k ), (ϵ′

1
k ,1 − ϵ′

1
k )) b2
k = ((1 − ϵ2
k , ϵ2
k ))
ϵ1
k → 0 ϵ′

1
k → 0 ϵ2
k → 0 k → ∞
b1
k = ((1 − ϵ1
k , ϵ1
k ), (ϵ′

1
k ,1 − ϵ′

1
k )) → ((1,0), (0,1)) b2
k = ((1 − ϵ2
k , ϵ2
k )) → ((1,0))
r =
ϵ1
k ϵ′

1
k
ϵ1
k ϵ′

1
k + ϵ1
k (1 − ϵ′

1
k )
→ 0 (k → ∞)
μ1
1 μ1
2 μ2
(1,1,(0,1))
b2
= ((1,0))
(home − movie, movie)
σʔτήʔϜͰ֬ೝ
P1
P2
P1
(2, 0)
(0, 0)
home
σʔτήʔϜ
৘ใू߹
ϓϨΠϠʔ
P1
ϓϨΠϠʔ
P2
P2
(1, 3)
out
movie
m
shopping
s
movie shopping
(3, 1)
(0, 0)
μ1
1 = 1
μ1
2 = 1
r 1 − r
ϵ1
k
ϵ′

1
k
ϵ2
k
ϵ2
k
ߦಈઓུͰ‫ه‬ड़͢Ε͹
b = (b1
, b2
) = (((1,0), (0,1)), ((1,0)))
0 1

12. σʔτήʔϜʹ͓͚Δஞ࣍‫ߧۉ‬Λ֬ೝ͢Δ
‫׬‬શϕΠδΞϯ‫ݕ͍ͯͭʹߧۉ‬౼
ϓϨΠϠʔͷߦಈઓུ
ϓϨΠϠʔͷߦಈઓུ
ͨͩ͠
ͱ͢Δ͢Δͱ
͜ͷͱ͖
͜ͷߦಈઓུ͔Βಋ͔ΕΔ৴೦͸ ࣗ໌ͳ΋ͷ͸আ͘
‫͜ʹٯ‬ͷ৴೦ͷମ‫ ܥ‬
ͷ‫Ͱݩ‬͸
ϓϨΠϠʔͷߦಈઓུ ͸࠷దͰ͋Δ
͜ͷߦಈઓུʹରͯ͠ϓϨΠϠʔͷߦಈઓུ ΋࠷ద
Ώ͑ʹ
ஞ࣍‫ ߧۉ‬
(out − shopping, shopping) μ1
1 μ1
2 μ2
(1,1,(0,1))
b1
k = ((ϵ1
k ,1 − ϵ1
k ), (ϵ′

1
k ,1 − ϵ′

1
k )) b2
k = ((ϵ2
k ,1 − ϵ2
k ))
ϵ1
k → 0 ϵ′

1
k → 0 ϵ2
k → 0 k → ∞
b1
k = ((ϵ1
k ,1 − ϵ1
k ), (ϵ′

1
k ,1 − ϵ′

1
k )) → ((0,1), (0,1)) b2
k = ((ϵ2
k ,1 − ϵ2
k )) → ((0,1))
r =
(1 − ϵ1
k )ϵ′

1
k
(1 − ϵ1
k )ϵ′

1
k + (1 − ϵ1
k )(1 − ϵ′

1
k )
→ 0 (k → ∞)
μ1
1 μ1
2 μ2
(1,1,(0,1))
b2
= ((0,1))
((0,1), (0,1))
(out − shopping, shopping) μ1
1 μ1
2 μ2
(1,1,(0,1))
σʔτήʔϜͰ֬ೝ
P1
P2
P1
(2, 0)
(0, 0)
home
σʔτήʔϜ
৘ใू߹
ϓϨΠϠʔ
P1
ϓϨΠϠʔ
P2
P2
(1, 3)
out
movie
m
shopping
s
movie shopping
(3, 1)
(0, 0)
μ1
1 = 1
μ1
2 = 1
r 1 − r
ϵ1
k
ϵ′

1
k
ϵ2
k
ϵ2
k
ߦಈઓུͰ‫ه‬ड़͢Ε͹
b = (b1
, b2
) = (((0,1), (0,1)), ((0,1)))
0 1

13. ֤‫ߧۉ‬ͷ·ͱΊ
φογϡ‫ߧۉ‬
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
ऑ‫׬‬શϕΠδΞϯ‫ߧۉ‬
ஞ࣍‫ߧۉ‬