ゲーム理論 BASIC 演習103 -一対多マッチング- #ゲーム理論 #gametheory #数学

1. ήʔϜཧ࿦#4*$ԋश
ҰରଟϚονϯά

2. ෮श #4*$ୈճ ճ
໰୊
ղ౴

3. Ϟσϧ
ֶੜΛ‫ׂʹࣨڀݚ‬Γ౰ͯΔϞσϧΛߟ͑Δ
ֶੜͷू߹Λ ‫ࣨڀݚ‬ͷू߹Λ ͱ͢Δ
ͱ͢Δ
ศٓతʹ ֶੜ͸ఴࣈ Λ෇༩͠ ‫ࣨڀݚ‬͸ఴࣈ Λ෇༩͢Δ
ֶੜ ͸ ্ʹબ޷ Λ΋ͪ
‫ࣨڀݚ‬ ͸ ্ʹબ޷ Λ΋ͭ
۩ମྫ
ֶੜʹ͍ͭͯ ͱॻ͍ͨ৔߹
͕΋ͬͱ΋๬·͍͠ ॴଐ͍ͨ͠ ‫͓ͯ͑ߟͱࣨڀݚ‬Γ ࣍ʹ๬·͍͠‫ࣨڀݚ‬͸
΍ͷ‫ॴʹࣨڀݚ‬ଐ͢Δ͘Β͍ͳΒແॴଐ ‫࣮ݱ‬తͰ͸ͳ͍͕ Ͱ͍͍ͨ ͱղऍ͢Δ
‫͍ͯͭʹࣨڀݚ‬ ͱॻ͍ͨ৔߹
͸ॴଐ͍͕ͤͨ͞ ͸ॴଐͤͨ͘͞ͳ͍ 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
≿4: 1 2 4 3

4. Ϟσϧ
͢΂ͯͷֶੜ͓Αͼ‫ࣨڀݚ‬ͷબ޷Λ Λฒ΂ͨ΋ͷΛ
ͱ͢Δ
·ͨ ֤‫ࣨڀݚ‬ ͸ΩϟύγςΟ Λ΋ͪ ͸੔਺Ͱ Λຬͨ͢ͱ͢Δ
ͭ·Γ গͳ͘ͱ΋ਓ͸ॴଐͤ͞Δ͜ͱ͕Ͱ͖ ਓ·Ͱॴଐͤ͞Δ͜ͱ͕Ͱ͖Δ
͜ͷબ޷૊͕༩͑ΒΕͨ‫ֶͰݩ‬ੜͷॴଐઌΛಋؔ͘਺Λ Ͱද͢
͜ͷ ͕Ϛονϯάͱͯ͠ຬͨ͢΂͖৚݅͸
ɾֶ֤ੜ ʹ͍ͭͯ ͳ͓ ͸ॴଐ͍ͯ͠ͳ͍ঢ়ଶͰ͋Δ
ɾ‫߸ه‬ ʹΑΓ ʹॴଐ͢Δֶੜͷਓ਺Λද͢΋ͷͱ͢Δ͜Ε͕ Λຬͨ͢
͸୭΋ॴଐ͍ͯ͠ͳ͍ঢ়ଶͰ͋Δ
≿i (i = 1,2,⋯, m, m + 1,m + 2,⋯, m + w)
≿ = ( ≿1 , ≿2 , ⋯, ≿m , ≿m+1 , ≿m+2 , ⋯, ≿m+w )
j ∈ W cj cj cj ≥ 1
cj
μ : M → M ∪ W
μ
i ∈ M μ(i) = j ∈ W ∪ {i} μ(i) = i
|μ(j)| j ∈ W 0 ≤ |μ(j)| ≤ cj
|μ(j)| = 0
ֶ֤ੜΛͲ͜ʹׂΓ౰ͯΔ͔ͱ͍͏ؔ਺

5. ‫ݸ‬ਓ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
Ϛονϯά ͕ ͷ΋ͱͰ‫ݸ‬ਓ߹ཧతͰ͋Δͱ͸
Ͱ͋Δ͢΂ͯͷ ʹ͍ͭͯ ͔ͭ ͕੒Γཱͭ͜ͱͰ͋Δ
͸ֶੜ ͸‫ࣨڀݚ‬ ʹॴଐ͢Δ͜ͱʹͳΔ
͜ͷͱ͖ Ͱ͋Δ͜ͱ͸ ʹͱͬͯແॴଐͰ͋ΔΑΓ ʹॴଐ͢Δ͜ͱ͕๬·্͍͠
Ͱ͋Δ͜ͱ͸ ʹͱͬͯ Λ‫ڋ‬൱͢ΔΑΓॴଐͤͨ͞ํ͕๬·͍͠ͱ͍͏͜ͱͰ͋Δ
μ ≿ = ( ≿1 , ≿2 , ⋯, ≿m , ≿m+1 , ≿m+2 , ⋯, ≿m+w )
μ(i) = j i ∈ M, j ∈ W j ≻i i i ≻j j
μ(i) = j i j
j ≻i i i j
i ≻j j j i

6. ҆ఆੑ
҆ఆੑ
Ϛονϯά ͕ ͷ΋ͱͰ҆ఆͰ͋Δͱ͸
͕‫ݸ‬ਓ߹ཧతͯ͋Δ
Ͱ͋ΔΑ͏ͳ ʹ͍ͭͯߟ͑Δ
΋͠ ͳΒ͹ ͳΔ͢΂ͯͷ ʹ͍ͭͯ Ͱ͋Δ
΋͠ ͳΒ͹ Ͱ͋Δ
҆ఆੑΛຬͨ͢Ϛονϯά͸ ඞͣଘࡏ͠
ҰରҰͰͷ%ΞϧΰϦζϜͱಉ༷ͷΞϧΰϦζϜͰ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δ
μ ≿
μ
j ≻i μ(i) i ∈ M, j ∈ W
|μ(j)| = cj μ(k) = j k ∈ M k ≻j i
|μ(j)| cj j ≻j i ΑΓ๬·ֶ͍͠ੜ͕ॴଐ
ΩϟύγςΟ͸௒͑ͯͳ͍͕
͜ͷֶੜ͸ॴଐͤͨ͘͞ͳ͍
͸ׂΓ౰ͯΒΕͨ‫ࣨڀݚ‬ ΑΓ
๬·͍͠‫ࣨڀݚ‬ Λ‫ر‬๬
i μ(i)
j

7. ֶੜଆ%ΞϧΰϦζϜ
%FGFSSFEDDFQUBODF BMHPSJUIN %ΞϧΰϦζϜ डೖอཹΞϧΰϦζϜ
ֶੜ ʹ͍ͭͯ ‫ࣨڀݚ‬ ͕‫ڐ‬༰ՄೳͰ͋Δͱ͸ Λຬͨ͢ͱ͖Λ͍͏
‫ࣨڀݚ‬ ʹ͍ͭͯ ֶੜ ͕‫ڐ‬༰ՄೳͰ͋Δͱ͸ Λຬͨ͢ͱ͖Λ͍͏
ֶੜʹͱͬͯࢦఆՄೳͳ‫ͱࣨڀݚ‬͸ ͜Ε·ͰஅΒΕ͓ͯΒ͔ͣͭ‫ڐ‬༰Մೳͳ‫ࣨڀݚ‬ͷ͜ͱΛ͍͏
ֶੜଆ%ΞϧΰϦζϜ ֶੜଆ͔Βࢦఆ͢ΔΞϧΰϦζϜ Λઆ໌͢Δ
i ∈ M j ∈ W j ≻i i
j ∈ W i ∈ M i ≻j j

8. εςοϓ
ֶ֤ੜ͸ ‫ڐ‬༰ՄೳͳதͰ൪޷͖ͳ‫ࣨڀݚ‬Λࢦఆ͢Δ
‫ڐ‬༰Մೳͳ‫ֶ͍ͳ͍͕ࣨڀݚ‬ੜ͸͜ͷ࣌఺Ͱແॴଐ͕֬ఆ͢Δ
‫ࣨڀݚ‬ଆ͸ࢦఆֶ͖ͯͨ͠ੜͷதͰ ‫ڐ‬༰ՄೳͳֶੜͷதͰ্Ґ ਓ·ͰֶੜΛΩʔϓ͢Δ
ଞͷֶੜ͸அΒΕΔ
εςοϓ
εςοϓ ͰஅΒΕͨ͢΂ͯͷֶੜ͸ ͦͷͱ͖ࢦఆՄೳͳ‫ࣨڀݚ‬ͷதͰ ൪޷͖ͳ‫ࣨڀݚ‬Λࢦఆ͢Δ
ࢦఆՄೳͳ‫ֶ͍ͳ͍͕ࣨڀݚ‬ੜ͸ ͜ͷ࣌఺Ͱແॴଐ֬ఆ͢Δ
‫ࣨڀݚ‬ଆͷରԠ
ɾʮεςοϓ ͰΩʔϓֶͨ͠ੜʯͱʮεςοϓ Ͱࢦఆ͖ͯͨ͠‫ڐ‬༰Մೳͳֶੜʯͷத͔Β
্Ґ ਓ·ͰֶੜΛΩʔϓ͢ΔΩʔϓ͠ͳֶ͔ͬͨੜ͸͢΂ͯஅΔ
ऴྃ৚݅
Ͳͷ‫ʹࣨڀݚ‬΋ࢦఆ͢Δֶੜ͕͍ͳ͘ͳͬͨεςοϓͰΞϧΰϦζϜ͸ఀࢭ͢Δ
ֶ֤ੜ͸֤‫౓Ͱߴ࠷ʹࣨڀݚ‬ͷΈ͔͠ࢦఆͰ͖ͣ ࢦఆՄೳͳ‫ࣨڀݚ‬͸‫ݮ‬ΔҰํͳͷͰ
ඞͣΞϧΰϦζϜ͸ఀࢭ͢Δ
͜ͷ࣌఺ͰΩʔϓ͞ΕͯΔֶੜ͸ਖ਼ࣜʹॴଐ͢Δ͜ͱʹͳΔ
cj
s ≥ 2
s − 1
s − 1 s
cj
ֶੜଆ%ΞϧΰϦζϜ

9. ֶੜଆ%ΞϧΰϦζϜ͕ಋ͘Ϛονϯάͷੑ࣭
҆ఆੑ
ͲͷΑ͏ͳબ޷ ʹରͯ͠΋ ֶੜଆ%ΞϧΰϦζϜ͕ಋ͘Ϛονϯά͸҆ఆͰ͋Δ
ֶੜଆ଱ઓུੑ
ֶੜଆ%ΞϧΰϦζϜͰಘΒΕΔϚονϯά͸ ֶੜଆ଱ઓུੑΛຬͨ͢
ֶੜଆ࠷దੑ
೚ҙͷબ޷ ʹରͯ͠ ֶੜଆ%ΞϧΰϦζϜ͕ಋ͘ϚονϯάΛ Ͱද͢
ʹରͯ͠ͲͷΑ͏ͳ҆ఆϚονϯά ʹରͯ͠΋ શͯͷֶੜ ʹ͍ͭͯ ͕੒Γཱͪ
গͳ͘ͱ΋ਓͷֶੜ ʹ͍ͭͯ ͕੒Γཱͭ
≿
≿ μ
≿ μ′

≠ μ i μ(i) ≿i μ′

(i)
i μ(i) ≻i μ′

(i)

10. ֶੜଆ%ΞϧΰϦζϜʹΑΓ ҆ఆϚονϯάΛ‫ٻ‬ΊΑ
໰୊
ֶੜ ≿1: 8 9 10 1
ֶੜ ≿2: 10 9 8 2
ֶੜ ≿3: 8 10 9 3
‫ࣨڀݚ‬ ≿8: 3 2 6 4 5 7 1 8
‫ࣨڀݚ‬ ≿9: 3 1 6 5 7 4 2 9
ֶੜ ≿4: 8 10 9 4
c8 = 2
c9 = 2
ֶੜ ≿5: 8 9 10 5
ֶੜ ≿6: 9 8 10 6
ֶੜ ≿7: 10 8 9 7
c10 = 4
‫ࣨڀݚ‬ ≿10: 4 6 1 3 5 7 2 10

11. ղ౴
ֶੜ ≿1: 8 9 10 1
ֶੜ ≿2: 10 9 8 2
ֶੜ ≿3: 8 10 9 3
‫ࣨڀݚ‬ ≿8: 3 2 6 4 5 7 1 8
‫ࣨڀݚ‬ ≿9: 3 1 6 5 7 4 2 9
ֶੜ ≿4: 8 10 9 4
c8 = 2
c9 = 2
ֶੜ ≿5: 8 9 10 5
ֶੜ ≿6: 9 8 10 6
ֶੜ ≿7: 10 8 9 7
‫ࣨڀݚ‬ ≿10: 4 6 1 3 5 7 2 10
๬·͍͠‫ࣨڀݚ‬Λࢦఆ
c10 = 4

12. ղ౴
ֶੜ ≿1: 8 9 10 1
ֶੜ ≿2: 10 9 8 2
ֶੜ ≿3: 8 10 9 3
‫ࣨڀݚ‬ ≿8: 3 2 6 4 5 7 1 8
‫ࣨڀݚ‬ ≿9: 3 1 6 5 7 4 2 9
ֶੜ ≿4: 8 10 9 4
c8 = 2
c9 = 2
ֶੜ ≿5: 8 9 10 5
ֶੜ ≿6: 9 8 10 6
ֶੜ ≿7: 10 8 9 7
‫ࣨڀݚ‬ ≿10: 4 6 1 3 5 7 2 10
c10 = 4
ΩϟύγςΟ͕‫ݶ͢ڐ‬Γ
‫ڐ‬༰Մೳͳֶੜ͔Β্ҐΛΩʔϓ͢Δ

13. ղ౴
ֶੜ ≿1: 8 9 10 1
ֶੜ ≿2: 10 9 8 2
ֶੜ ≿3: 8 10 9 3
‫ࣨڀݚ‬ ≿8: 3 2 6 4 5 7 1 8
‫ࣨڀݚ‬ ≿9: 3 1 6 5 7 4 2 9
ֶੜ ≿4: 8 10 9 4
c8 = 2
c9 = 2
ֶੜ ≿5: 8 9 10 5
ֶੜ ≿6: 9 8 10 6
ֶੜ ≿7: 10 8 9 7
‫ࣨڀݚ‬ ≿10: 4 6 1 3 5 7 2 10
அΒΕֶͨੜ͸‫ڐ‬༰Մೳͳ‫͔ࣨڀݚ‬Β
΋ͬͱ๬·͍͠΋ͷΛࢦఆ
c10 = 4

14. ղ౴
ֶੜ ≿1: 8 9 10 1
ֶੜ ≿2: 10 9 8 2
ֶੜ ≿3: 8 10 9 3
‫ࣨڀݚ‬ ≿8: 3 2 6 4 5 7 1 8
‫ࣨڀݚ‬ ≿9: 3 1 6 5 7 4 2 9
ֶੜ ≿4: 8 10 9 4
c8 = 2
c9 = 2
ֶੜ ≿5: 8 9 10 5
ֶੜ ≿6: 9 8 10 6
ֶੜ ≿7: 10 8 9 7
‫ࣨڀݚ‬ ≿10: 4 6 1 3 5 7 2 10
c10 = 4
‫ࣨڀݚ‬͸ֶੜ ֶੜΛΩʔϓ

15. ղ౴
ֶੜ ≿1: 8 9 10 1
ֶੜ ≿2: 10 9 8 2
ֶੜ ≿3: 8 10 9 3
‫ࣨڀݚ‬ ≿8: 3 2 6 4 5 7 1 8
‫ࣨڀݚ‬ ≿9: 3 1 6 5 7 4 2 9
ֶੜ ≿4: 8 10 9 4
c8 = 2
c9 = 2
ֶੜ ≿5: 8 9 10 5
ֶੜ ≿6: 9 8 10 6
ֶੜ ≿7: 10 8 9 7
‫ࣨڀݚ‬ ≿10: 4 6 1 3 5 7 2 10
c10 = 4
அΒΕֶͨੜ͸‫ڐ‬༰Մೳͳ‫͔ࣨڀݚ‬Β
΋ͬͱ๬·͍͠΋ͷΛࢦఆ

16. ղ౴
ֶੜ ≿1: 8 9 10 1
ֶੜ ≿2: 10 9 8 2
ֶੜ ≿3: 8 10 9 3
‫ࣨڀݚ‬ ≿8: 3 2 6 4 5 7 1 8
‫ࣨڀݚ‬ ≿9: 3 1 6 5 7 4 2 9
ֶੜ ≿4: 8 10 9 4
c8 = 2
c9 = 2
ֶੜ ≿5: 8 9 10 5
ֶੜ ≿6: 9 8 10 6
ֶੜ ≿7: 10 8 9 7
‫ࣨڀݚ‬ ≿10: 4 6 1 3 5 7 2 10
c10 = 4
‫ࣨڀݚ‬͸ֶੜ΋‫ڐ‬༰ՄೳͰΩʔϓ

17. ղ౴
ֶੜ ≿1: 8 9 10 1
ֶੜ ≿2: 10 9 8 2
ֶੜ ≿3: 8 10 9 3
‫ࣨڀݚ‬ ≿8: 3 2 6 4 5 7 1 8
‫ࣨڀݚ‬ ≿9: 3 1 6 5 7 4 2 9
ֶੜ ≿4: 8 10 9 4
c8 = 2
c9 = 2
ֶੜ ≿5: 8 9 10 5
ֶੜ ≿6: 9 8 10 6
ֶੜ ≿7: 10 8 9 7
‫ࣨڀݚ‬ ≿10: 4 6 1 3 5 7 2 10
c10 = 4
ࢦఆ͢Δֶੜ͕͍ͳ͘ͳͬͨͷͰΞϧΰϦζϜऴྃ

18. ղ౴
ֶੜ ≿1: 8 9 10 1
ֶੜ ≿2: 10 9 8 2
ֶੜ ≿3: 8 10 9 3
‫ࣨڀݚ‬ ≿8: 3 2 6 4 5 7 1 8
‫ࣨڀݚ‬ ≿9: 3 1 6 5 7 4 2 9
ֶੜ ≿4: 8 10 9 4
c8 = 2
c9 = 2
ֶੜ ≿5: 8 9 10 5
ֶੜ ≿6: 9 8 10 6
ֶੜ ≿7: 10 8 9 7
‫ࣨڀݚ‬ ≿10: 4 6 1 3 5 7 2 10
c10 = 4
ֶੜଆ%ΞϧΰϦζϜͰಘΒΕΔϚονϯά͸
μ = (μ(1), μ(2), μ(3), μ(4), μ(5), μ(6), μ(7)) = (9, 10, 8, 8, 10, 9, 10)

19. ήʔϜཧ࿦#4*$ԋश
ҰରଟϚονϯά