ゲーム理論 BASIC 演習74 補足 -コアの存在, 線型計画問題-

1. ήʔϜཧ࿦#4*$ԋशิ଍
ίΞͷଘࡏ ઢ‫ܭܕ‬ը໰୊

2. ਓήʔϜʹ͓͚ΔίΞͷଘࡏ৚݅
/ਓήʔϜʹ͓͚ΔίΞͷଘࡏ৚݅
ઢ‫ܭܕ‬ը໰୊Ͱߟ͑Δ
લճͷ໰୊
1V-1 1ZUIPOͷ਺ཧ࠷దԽύοέʔδ

3. ਖ਼‫ن‬Խ͞ΕͨਓήʔϜ ͰͷίΞ͕ඇۭͰ͋ΔͨΊͷඞཁे෼৚݅͸
ͭͷಛੑؔ਺‫ܗ‬ήʔϜ ͕ઓུతಉ౳Ͱ͋Δͱ͸
ਖ਼਺ ͱ࣮਺ ͕ଘࡏͯ͠ɺ೚ҙͷఏ‫ܞ‬ ʹରͯ͠ ͕੒Γཱͭ͜ͱͰ͋Δ
ಛੑؔ਺ ͕ຊ࣭తͰ͋Δͱ͸
ຊ࣭తͳಛੑؔ਺‫ܗ‬ήʔϜ ʹରͯ͠
·ͨ͸
Λຬͨ͢ઓུతಉ౳ͳήʔϜ ͕ଘࡏ͢Δ
͕੒Γཱͭͱ͖ Λਖ਼‫ن‬ԽήʔϜ
͕੒Γཱͭͱ͖ Λ ਖ਼‫ن‬ԽήʔϜ
ͱ͍͏
(N, v)
v({1,2}) + v({2,3}) + v({1,3}) ≤ 2v({1,2,3})
(N, ν) (N, ν′

)
α β1, ⋯, βn S ν′

(S) = αν(S) +
∑
i∈S
βi
v v(N)
n
∑
i=1
v(i)
(N, ν)
ν′

(i) = 0, i = 1,⋯ . n (1) ν′

(N) = 1, ν′

(i) = 0, i = 1,⋯, n (2)
(N, ν′

)
(N, ν′

)
(N, ν′

)
ਓήʔϜʹ͓͚ΔίΞͷଘࡏ৚݅

4. ίΞ͕ඇۭͱͳΔͨΊʹ͸ ҎԼΛຬͨ͢഑෼ ͕ଘࡏ͢Δͱ͖Ͱ͋Δɿ
ҎԼͷઢ‫ܭܕ‬ը໰୊Λߟ͑
࠷ద஋͕ Λ௒͑ͳ͍ͱ͖ ίΞ͕ඇۭͱͳΔɿ
x = (x1, ⋯, xn)
∑
i∈N
xi = v(N)
∑
i∈S
xi ≥ v(S), ∀S ⊂ N, S ≠ N, ∅
v(N)
min
x ∑
i∈N
xi
s . t .
∑
i∈S
xi ≥ v(S), ∀S ⊂ N, S ≠ N, ∅
/ਓήʔϜʹ͓͚ΔίΞͷଘࡏ৚݅
ͷ৔߹
ͱͯ͠
ྫ͑͹ ͱ͢Ε͹
੍໿৚݅ ఏ‫߹ܞ‬ཧੑ Λຬͨͭͭ͠
શମ߹ཧੑ Λຬͨ͢͜ͱ͕Մೳ
‫ʹٯ‬ ͷ৔߹͸
Ͳͷ Λখͯ͘͞͠΋੍໿৚݅Λຬͨ͞ͳ͘ͳΔͨΊ
શମ߹ཧੑ ͸ୡ੒Ͱ͖ͳ͍
min
x ∑
i∈N
xi v(N)
ϵ = v(N) − min
x ∑
i∈N
xi 0
x′

1 = x1 + ϵ
∑
i∈N
xi = v(N)
min
x ∑
i∈N
xi v(N)
xi
∑
i∈N
xi = v(N)

5. ҎԼͷಛੑؔ਺‫ܗ‬ήʔϜʹ͓͍ͯ ίΞ͕ଘࡏ͢Δ͔Ͳ͏͔൑அͤΑ
ίΞ͸ଘࡏ
ίΞ͸ଘࡏ
ίΞ͸ଘࡏ͠ͳ͍
v({1,2,3}) = 4 v({1,2}) = 2 v({1,3}) = 3 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
v({1,2,3}) = 4 v({1,2}) = 2 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
v({1,2,3}) = 4 v({1,2}) = 3 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
લճͷ໰୊

6. JNQPSUQVMQ
from fractions import Fraction
problem=pulp.LpProblem('74-(1)', pulp.LpMinimize)
x = pulp.LpVariable.dicts('x', ([1,2,3]), 0, None,'Continuous')
problem += x[1]+x[2]+x[3]
problem += x[1]+x[2]=2
problem += x[1]+x[3]=3
problem += x[2]+x[3]=2
problem += x[1]=0
problem += x[2]=0
problem += x[3]=0
problem.solve()
print('Optimal value:'+str(Fraction(problem.objective.value()).limit_denominator(10)))
print('x[1]:'+str(Fraction(x[1].value()).limit_denominator(10)))
print('x[2]:'+str(Fraction(x[2].value()).limit_denominator(10)))
print('x[3]:'+str(Fraction(x[3].value()).limit_denominator(10)))
v({1,2,3}) = 4 v({1,2}) = 2 v({1,3}) = 3 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
1V-1
Optimal value:7/2
x[1]:3/2
x[2]:1/2
x[3]:3/2
ίΞ͸ଘࡏ͢Δ
v({1,2,3}) = 4
7
2

7. JNQPSUQVMQ
from fractions import Fraction
problem=pulp.LpProblem(’74-(2)', pulp.LpMinimize)
x = pulp.LpVariable.dicts('x', ([1,2,3]), 0, None,'Continuous')
problem += x[1]+x[2]+x[3]
problem += x[1]+x[2]=2
problem += x[1]+x[3]=4
problem += x[2]+x[3]=2
problem += x[1]=0
problem += x[2]=0
problem += x[3]=0
problem.solve()
print('Optimal value:'+str(Fraction(problem.objective.value()).limit_denominator(10)))
print('x[1]:'+str(Fraction(x[1].value()).limit_denominator(10)))
print('x[2]:'+str(Fraction(x[2].value()).limit_denominator(10)))
print('x[3]:'+str(Fraction(x[3].value()).limit_denominator(10)))
v({1,2,3}) = 4 v({1,2}) = 2 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
1V-1
Optimal value:4
x[1]:2
x[2]:0
x[3]:2
ίΞ͸ଘࡏ͢Δ
v({1,2,3}) = 4 = 4

8. JNQPSUQVMQ
from fractions import Fraction
problem=pulp.LpProblem(’74-(3)', pulp.LpMinimize)
x = pulp.LpVariable.dicts('x', ([1,2,3]), 0, None,'Continuous')
problem += x[1]+x[2]+x[3]
problem += x[1]+x[2]=3
problem += x[1]+x[3]=4
problem += x[2]+x[3]=2
problem += x[1]=0
problem += x[2]=0
problem += x[3]=0
problem.solve()
print('Optimal value:'+str(Fraction(problem.objective.value()).limit_denominator(10)))
print('x[1]:'+str(Fraction(x[1].value()).limit_denominator(10)))
print('x[2]:'+str(Fraction(x[2].value()).limit_denominator(10)))
print('x[3]:'+str(Fraction(x[3].value()).limit_denominator(10)))
v({1,2,3}) = 4 v({1,2}) = 3 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
1V-1
Optimal value:9/2
x[1]:5/2
x[2]:1/2
x[3]:3/2
ίΞ͸ଘࡏ͠ͳ͍
v({1,2,3}) = 4
9
2

9. ήʔϜཧ࿦#4*$ԋशิ଍
ίΞͷଘࡏ ઢ‫ܭܕ‬ը໰୊