ゲーム理論 BASIC 演習100 -一対一マッチング -

1. ήʔϜཧ࿦#4*$ԋश
ҰରҰϚονϯά

2. ෮श #4*$ୈճʙճ
໰୊
ղ౴

3. Ϛονϯάͱ͸
ࠗ‫׆‬ύʔςΟͷΑ͏ʹஉੑάϧʔϓͱঁੑάϧʔϓ͕͋Γ
உੑͱঁੑͷϖΞΛ૊·ͤΔ͜ͱΛߟ͑Δ͜ΕΛϚονϯάͱ͍͏
֤உੑ͸ ঁੑʹର͠޷Έ બ޷ Λ΋ͪ ‫ঁʹٯ‬ੑ͸உੑʹର͠޷ΈΛ΋ͭ
Ұํͷάϧʔϓ͔͠બ޷Λ΋ͨͳ͍৔߹͸ׂ౰໰୊Ͱ͋Γ ผϞσϧ
બ޷Λ΋͍ͬͯΔάϧʔϓʹର͠ ద੾ͳϚονϯάΛ͍ͤͨ͞
͜͜Ͱ ద੾͞ʹ͍ͭͯ ҆ఆੑΛ༻͍Δ
҆ఆੑͱ͸ ֤ϖΞ͕ҳ୤͢ΔΠϯηϯςΟϒ ු‫ؾ‬ Λ͠ͳ͍ঢ়ଶͰ͋Δ

4. Ϟσϧ
உੑͷू߹Λ ঁੑͷू߹Λ ͱ͢Δ
ͱ͢Δ
ศٓతʹ உੑ͸ఴࣈ Λ෇༩͠ ঁੑ͸ఴࣈ Λ෇༩͢Δ
உੑ ͸ ্ʹબ޷ Λ΋ͪ
ঁੑ ͸ ্ʹબ޷ Λ΋ͭ
۩ମྫ
உੑʹ͍ͭͯ ͱॻ͍ͨ৔߹
͕΋ͬͱ΋๬·͍͠ ෇͖߹͍͍ͨ ͱߟ͓͑ͯΓ ࣍ʹ๬·͍͠ͷ͸
΍ͱ෇͖߹͏͘Β͍ͳΒಠΓͰ͍͍ͨ ͱղऍ͢Δ
M W
m = |M| 0 w = |W| 0
1,2,⋯, m m + 1,m + 2,⋯, m + w
i ∈ M W∪{i} ≿i (i = 1,2,⋯, m)
j ∈ W M∪{j} ≿j (j = m + 1,m + 2,⋯, m + w)
M = {1,2,3} W = {4,5,6,7}
≿1: 4 7 1 6 5

5. Ϟσϧ
͢΂ͯͷਓͷબ޷Λ Λฒ΂ͨ΋ͷΛ
ͱ͢Δ
͜ͷબ޷૊͕༩͑ΒΕͨ‫Ͱݩ‬ԿΒ͔ͷϖΞΛಋؔ͘਺Λ Ͱද͢
͜ͷ ͕Ϛονϯάͱͯ͠ຬͨ͢΂͖৚݅͸
ɾ֤ ʹ͍ͭͯ ͳ͓ ͸୭ͱ΋ϖΞΛ૊ΜͰͳ͍ঢ়ଶͰ͋Δ
ɾ֤ ʹ͍ͭͯ ͳ͓ ͸୭ͱ΋ϖΞΛ૊ΜͰͳ͍ঢ়ଶͰ͋Δ
ɾਓ͕ෳ਺ਓͱϖΞΛ૊Ή͜ͱ͸ͳ͍ ͭ·Γ ͱͳΔΑ͏ͳ͜ͱ͸ͳ͍
ɾஉੑ ͱঁੑ ͕ϖΞΛ૊Ή͜ͱ͸ ঁੑ ͱஉੑ ͕ϖΞͱͳΔ͜ͱΛҙຯ͢Δ
Ώ͑ʹ ͳΒ͹ Ͱ͋Δ
≿i (i = 1,2,⋯, m, m + 1,m + 2,⋯, m + w)
≿ = ( ≿1 , ≿2 , ⋯, ≿m , ≿m+1 , ≿m+2 , ⋯, ≿m+w )
μ : M ∪ W → M ∪ W
μ
i ∈ M μ(i) = j ∈ W ∪ {i} μ(i) = i
j ∈ W μ(j) = i ∈ M ∪ {j} μ(j) = j
μ(i) = j, μ(i′

) = j
i ∈ M j ∈ W j ∈ W i ∈ M
μ(i) = j μ(j) = i

6. ‫ݸ‬ਓ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
Ϛονϯά ͕ ͷ΋ͱͰ‫ݸ‬ਓ߹ཧతͰ͋Δͱ͸
͢΂ͯͷਓ ʹ͍ͭͯ ͕੒Γཱͭ
ϖΞΛ૊Μͩ૬ख͸ ʮಠΓͰ͍ΔʯΑΓ๬·͘͠ͳ͍૬खͱ͸ͳΒͳ͍ͱ͍͏͜ͱ
۩ମྫ
உੑʹ͍ͭͯ
உੑʹ͍ͭͯ
உੑʹ͍ͭͯ
ͱͳΓ ‫ݸ‬ਓ߹ཧੑΛຬͨ͢
μ ≿ = ( ≿1 , ≿2 , ⋯, ≿m , ≿m+1 , ≿m+2 , ⋯, ≿m+w )
i ∈ M ∪ W μ(i) ≿i i
M = {1,2,3} W = {4,5,6,7}
≿1: 4 7 1 6 5
≿2: 4 6 7 2 5
≿3: 7 4 5 6 3
μ = (μ(1), μ(2), μ(3), μ(4), μ(5), μ(6), μ(7)) = (7, 6, 4, 3, 5, 2, 1)
μ(1) = 7 ≿1 1 μ(2) = 6 ≿2 2 μ(3) = 4 ≿3 3 μ(4) = 3 ≿4 4 μ(5) = 5 ≿5 5 μ(6) = 2 ≿6 6 μ(7) = 1 ≿7 7
ঁੑʹ͍ͭͯ
ঁੑʹ͍ͭͯ
ঁੑʹ͍ͭͯ
ঁੑʹ͍ͭͯ
≿4: 2 1 3 4
≿5: 2 1 5 3
≿6: 1 2 3 6
≿7: 1 3 7 2

7. ҆ఆੑ
҆ఆੑ
Ϛονϯά ͷ΋ͱͰ ͋ΔஉঁͷϖΞ ͕
͔ͭ Λຬͨ͢ͱ͢Δ ͸‫޷ʹີݫ‬Ή͜ͱΛҙຯ͢Δ
͜ͷ৔߹ ͸͜ͷϚονϯάʹैΘͣ ҳ୤ͯ͠ϖΞΛ૊Μͩํ͕๬·͍͠
͜ΕΛબ޷ ͷ΋ͱͰ ͸ ΛϒϩοΫ͢Δͱ͍͏
Ϛονϯά ͕ ͷ΋ͱͰ҆ఆͰ͋Δͱ͸
͕‫ݸ‬ਓ߹ཧతͯ͋Δ
ͲͷΑ͏ͳϖΞ ʹΑͬͯ΋ϒϩοΫ͞Εͳ͍
ͱ͖Λ͍͏
͜ͷ҆ఆͳϚονϯάΛ‫ٻ‬ΊΔΞϧΰϦζϜ͕ଘࡏ͢Δ
μ (i, j)
j ≻i μ(i) i ≻j μ(j) ≻k
(i, j)
≿ (i, j) μ
μ ≿
μ
(i, j)

8. ޮ཰ੑ
ޮ཰ੑ
Ϛονϯά ͕ ͷ΋ͱͰޮ཰తͰ͋Δͱ͸ ҎԼͷΑ͏ͳϚονϯά ͕ଘࡏ͠ͳ͍ͱ͖Λ͍͏
໋୊҆ఆϚονϯάͳΒޮ཰ੑΛຬͨ͢
҆ఆϚονϯάͷํ͕ޮ཰ੑΑΓ΋‫͍ڧ‬৚݅Ͱ͋Δ
μ ≿ μ′

μ′

(i) ≿i μ(i) ∀i ∈ M ∪ W
μ′

(j) ≻j μ(j) ∃j ∈ M ∪ W

9. ‫هʹີݫ‬ड़͢Δ%ΞϧΰϦζϜ
%FGFSSFEDDFQUBODF BMHPSJUIN %ΞϧΰϦζϜ डೖอཹΞϧΰϦζϜͱ༁͞ΕΔ
உੑ ʹ͍ͭͯ ঁੑ ͕‫ڐ‬༰ՄೳͰ͋Δͱ͸ Λຬͨ͢ͱ͖Λ͍͏
ঁੑ ʹ͍ͭͯ உੑ ͕‫ڐ‬༰ՄೳͰ͋Δͱ͸ Λຬͨ͢ͱ͖Λ͍͏
உੑʹͱͬͯϓϩϙʔζՄೳͳঁੑͱ͸ ͜Ε·ͰৼΒΕ͓ͯΒ͔ͣͭ‫ڐ‬༰Մೳͳঁੑͷ͜ͱΛ͍͏
ࠓճ͸உੑଆ%ΞϧΰϦζϜ உੑଆ͔Βϓϩϙʔζ͢ΔΞϧΰϦζϜ Λઆ໌͢Δ͕
உੑͱঁੑΛೖΕସ͑Ε͹ ঁੑଆΞϧΰϦζϜΛಘΔ
i ∈ M j ∈ W j ≻i i
j ∈ W i ∈ M i ≻j j
ಠΓͰ͍ΔΑΓ෇͖߹͏ํ͕޷·͍͠ਓ

10. εςοϓ
֤உੑ͸ ‫ڐ‬༰ՄೳͳதͰ൪޷͖ͳঁੑʹϓϩϙʔζ͢Δ
‫ڐ‬༰Մೳͳঁੑ͕͍ͳ͍உੑ͸͜ͷ࣌఺ͰಠΓ਎͕֬ఆ͢Δ
ঁੑଆ͸ࣗ෼ʹϓϩϙʔζ͖ͯͨ͠உੑͷதͰ ‫ڐ‬༰ՄೳͳஉੑͷதͰ࠷΋޷͖ͳஉੑΛΩʔϓ͢Δ
Ωʔϓ͞Εͳ͔ͬͨଞͷஉੑ͸ৼΒΕΔ‫ڐ‬༰Մೳͳஉੑͷ୭͔Β΋ϓϩϙʔζ͞Εͳ͔ͬͨঁੑ͸ಠΓ਎ͷ
··Ͱ͋Δ
‫هʹີݫ‬ड़͢Δ%ΞϧΰϦζϜ

11. εςοϓ
εςοϓ ͰৼΒΕͨ͢΂ͯͷஉੑ͸ ͦͷͱ͖ϓϩϙʔζՄೳͳঁੑͷதͰ
൪޷͖ͳঁੑʹϓϩϙʔζ͢Δ
ϓϩϙʔζՄೳͳঁੑ͕͍ͳ͍உੑ͸ ͜ͷ࣌఺ͰಠΓ਎֬ఆ͢Δ
ঁੑଆͷରԠ
ɾεςοϓ ͰΩʔϓ͓ͯ͠Γ ͔ͭ͜ͷεςοϓͰ୭͔Β΋ϓϩϙʔζ͞Εͳ͔ͬͨঁੑ͸
ɹΩʔϓதͷஉੑΛΩʔϓ͠ଓ͚Δ
ɾεςοϓ ͰΩʔϓ͓ͯ͠Βͣ ͔ͭ͜ͷεςοϓͰ୭͔Β΋ϓϩϙʔζ͞Εͳ͔ͬͨঁੑ͸
ɹಠΓ਎ͷ··Ͱ͋Δ
ɾεςοϓ ͰஉੑΛΩʔϓ͓ͯ͠Γ ͔ͭ ਓͷ‫ڐ‬༰Մೳͳஉੑ͔Βϓϩϙʔζ͞Εͨঁੑ͸
ͦΕΒ ਓ͔Β൪޷͖ͳஉੑΛΩʔϓ͢ΔΩʔϓ͞Εͳ͔ͬͨஉੑ͸ৼΒΕΔ
·ͨ ͦ΋ͦ΋‫ڐ‬༰ՄೳͰͳ͍உੑ΋ৼΒΕΔ
ɾεςοϓ ͰஉੑΛΩʔϓ͓ͯ͠Βͣ ͔ͭ ਓͷ‫ڐ‬༰Մೳͳஉੑ͔Βϓϩϙʔζ͞Εͨঁੑ͸
ͦΕΒ ਓ͔Β൪޷͖ͳஉੑΛΩʔϓ͢ΔΩʔϓ͞Εͳ͔ͬͨஉੑ͸ৼΒΕΔ
·ͨ ͦ΋ͦ΋‫ڐ‬༰ՄೳͰͳ͍உੑ΋ৼΒΕΔ
s ≥ 2
s − 1
s − 1
s − 1
s − 1 n
n + 1
s − 1 n
n
‫هʹີݫ‬ड़͢Δ%ΞϧΰϦζϜ

12. ऴྃ৚݅
Ͳͷঁੑʹ΋ϓϩϙʔζ͢Δஉੑ͕͍ͳ͘ͳͬͨεςοϓͰΞϧΰϦζϜ͸ఀࢭ͢Δ
֤உੑ͸֤ঁੑʹ౓ͷΈϓϩϙʔζ͔͠Ͱ͖ͣ
ϓϩϙʔζՄೳͳঁੑ͸‫ݮ‬ΔҰํͳͷͰ ඞͣΞϧΰϦζϜ͸ఀࢭ͢Δ
͜ͷ࣌఺ͰΩʔϓ͍ͯ͠Δஉঁ͸ϖΞͱͳΓ ͦΕҎ֎͸ಠΓ਎ͱͳΔ
‫هʹີݫ‬ड़͢Δ%ΞϧΰϦζϜ

13. உੑͷάϧʔϓ ঁੑͷάϧʔϓ ͱ͢Δ
֤ਓͷબ޷͸ҎԼͷ௨ΓͰ͋Δ
உੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
ঁੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
M = {1,2,3} W = {4,5,6}
໰୊
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

14. உੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
εςοϓ
࠷΋޷·͍͠ঁੑʹϓϩϙʔζ͢Δ
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

15. உੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
ਓͷ‫ڐ‬༰Մೳͳஉੑ͔Βϓϩϙʔζ͞Εͨঁੑ͸
ͦΕΒ ਓ͔Β൪޷͖ͳஉੑΛΩʔϓ͢ΔΩʔϓ͞Εͳ͔ͬͨஉੑ͸ৼΒΕΔ
2
2
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

16. உੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
εςοϓ
εςοϓͰৼΒΕͨஉੑ͸ ͦͷͱ͖ϓϩϙʔζՄೳͳঁੑͷதͰ ൪޷͖ͳঁੑʹϓϩϙʔζ͢Δ
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

17. உੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
ɾεςοϓͰΩʔϓ͓ͯ͠Γ ͔ͭ͜ͷεςοϓͰ୭͔Β΋ϓϩϙʔζ͞Εͳ͔ͬͨঁੑ͸
ɹΩʔϓதͷஉੑΛΩʔϓ͠ଓ͚Δ
ɾεςοϓͰஉੑΛΩʔϓ͓ͯ͠Γ ͔ͭ‫ڐ‬༰Մೳͳஉੑ͔Βϓϩϙʔζ͞Εͨঁੑ͸
ͦΕΒਓ͔Β൪޷͖ͳஉੑΛΩʔϓ͢ΔΩʔϓ͞Εͳ͔ͬͨஉੑ͸ৼΒΕΔ
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

18. உੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
உੑ͸‫ڐ‬༰Մೳͳঁੑ͸͍ͳ͘ͳΓ ϓϩϙʔζΛ͠ͳ͍
Ͳͷঁੑʹ΋ϓϩϙʔζ͢Δஉੑ͕͍ͳ͘ͳͬͨεςοϓͰΞϧΰϦζϜ͸ఀࢭ͢Δ
Ώ͑ʹ உੑଆ%ΞϧΰϦζϜͰಘΒΕΔϚονϯά͸
μ = (μ(1), μ(2), μ(3), μ(4), μ(5), μ(6)) = (1, 5, 4, 3, 2, 6)
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

19. ঁੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
εςοϓ
࠷΋޷·͍͠உੑʹϓϩϙʔζ͢Δ
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

20. ঁੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
ਓͷ‫ڐ‬༰Մೳͳঁੑ͔Βϓϩϙʔζ͞Εͨஉੑ͸
ͦΕΒ ਓ͔Β൪޷͖ͳঁੑΛΩʔϓ͢ΔΩʔϓ͞Εͳ͔ͬͨঁੑ͸ৼΒΕΔ
2
2
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

21. ঁੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
εςοϓ
εςοϓͰৼΒΕͨঁੑ͸ ͦͷͱ͖ϓϩϙʔζՄೳͳஉੑͷதͰ ൪޷͖ͳஉੑʹϓϩϙʔζ͢Δ
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

22. ঁੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
உੑ͸ঁੑ͸‫ڐ‬༰ՄೳͰͳ͍ͷͰ ৼΒΕΔ
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

23. ঁੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
εςοϓ
εςοϓͰৼΒΕͨঁੑ͸ ͦͷͱ͖ϓϩϙʔζՄೳͳஉੑͷதͰ ൪޷͖ͳஉੑʹϓϩϙʔζ͢Δ
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

24. ঁੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
ɾεςοϓͰঁੑΛΩʔϓ͓ͯ͠Γ ͔ͭ‫ڐ‬༰Մೳͳঁੑ͔Βϓϩϙʔζ͞Εͨஉੑ͸
ͦΕΒਓ͔Β൪޷͖ͳঁੑΛΩʔϓ͢ΔΩʔϓ͞Εͳ͔ͬͨঁੑ͸ৼΒΕΔ
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

25. ঁੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
ঁੑ͸‫ڐ‬༰Մೳͳஉੑ͸͍ͳ͘ͳΓ ϓϩϙʔζΛ͠ͳ͍
Ͳͷஉੑʹ΋ϓϩϙʔζ͢Δঁੑ͕͍ͳ͘ͳͬͨεςοϓͰΞϧΰϦζϜ͸ఀࢭ͢Δ
Ώ͑ʹ ঁੑଆ%ΞϧΰϦζϜͰಘΒΕΔϚονϯά͸ μ = (μ(1), μ(2), μ(3), μ(4), μ(5), μ(6)) = (1, 5, 4, 3, 2, 6)
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

26. உੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯά
ঁੑଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯά
μ = (μ(1), μ(2), μ(3), μ(4), μ(5), μ(6)) = (1, 5, 4, 3, 2, 6)
μ = (μ(1), μ(2), μ(3), μ(4), μ(5), μ(6)) = (1, 5, 4, 3, 2, 6)
ղ౴
உੑ ≿1: 4 5 1 6
உੑ ≿2: 5 6 2 4
உੑ ≿3: 4 5 6 3
ঁੑ ≿4: 3 2 1 4
ঁੑ ≿5: 2 1 3 5
ঁੑ ≿6: 3 1 2 6

27. ήʔϜཧ࿦#4*$ԋश
ҰରҰϚονϯά