投票ゲーム/拒否権をもつプレイヤーと独裁者

1. ήʔϜཧ࿦#4*$ԋश
౤ථήʔϜ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͱಠࡋऀ

2. ౤ථήʔϜʹ͓͚Δ‫ڋ‬൱‫ݖ‬ϓϨΠϠʔಠࡋऀ
໰୊
ղ౴

3. ౤ථήʔϜʹ͓͚Δ‫ڋ‬൱‫ݖ‬ϓϨΠϠʔಠࡋऀ
ಛੑؔ਺͕ ͷ஋ΛͱΓ ͱͳΔಛੑؔ਺‫ܗ‬ήʔϜ Λ୯७ήʔϜͱ͍͏
୯७ήʔϜͷ͏ͪ
ͳΒ͹
ͳΒ͹
ͷ̎ͭͷੑ࣭Λຬͨ͢΋ͷΛ ౤ථήʔϜͱ͍͏
ͱͳΔఏ‫ܞ‬Λউརఏ‫͍͍ͱܞ‬ ͱͳΔఏ‫ܞ‬Λഊ๺ఏ‫͏͍ͱܞ‬
͋ΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬ϓϨΠϠʔͱ͸ ͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔͷ͜ͱͰ͋Δ
͋ΔϓϨΠϠʔ͕ಠࡋऀͰ͋Δͱ͸ ͜ͷϓϨΠϠʔΛ‫ؚ‬Ήશͯͷఏ‫͕ܞ‬উརఏ‫ͳͱܞ‬Δͱ͖Λ͍͏
ˎϓϨΠϠʔ ͕ಠࡋऀͩͱ͢Δͱ ਓఏ‫͍͓ͯʹܞ‬ ͕੒ΓཱͭͨΊ ͱͳΔ
Ώ͑ʹ౤ථήʔϜʹ͓͍ͯ ಠࡋऀ͸ߴʑਓ͔͠ଘࡏ͠ͳ͍
౤ථήʔϜʹ͍ͭͯ͸zήʔϜཧ࿦#4*$ୈճγϟʔϓϨΠ஋Ԡ༻ɿ౤ථྗࢦ਺l΋ࢀর
v(N) = 1 (N, v)
S ⊆ T v(S) ≤ v(T)
v(S) = 1 v(NS) = 0
v(S) = 1 v(S) = 0
i v({i}) = 1 v(N{i}) = 0

4. ҎԼͷঢ়‫گ‬Λ౤ථήʔϜͱͯ͠ද‫ͤݱ‬Α·ͨ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ ಠࡋऀ͸͍Δ͔Λ֬ೝͤΑ
ܾٞ‫ݖ‬Λ΋ͭϓϨΠϠʔ͕ ͋ΔձࣾͷܾٞҊʹର͠ ౤ථ͢Δঢ়‫گ‬Λߟ͑Δ
͜ͷͱ͖ ա൒਺ϧʔϧΛ࠾༻͢Δ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
໰୊

5. ҎԼͷঢ়‫گ‬Λ౤ථήʔϜͱͯ͠ද‫ͤݱ‬Α·ͨ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ ಠࡋऀ͸͍Δ͔Λ֬ೝͤΑ
ܾٞ‫ݖ‬Λ΋ͭϓϨΠϠʔ͕ ͋ΔձࣾͷܾٞҊʹର͠ ౤ථ͢Δঢ়‫گ‬Λߟ͑Δ
͜ͷͱ͖ ա൒਺ϧʔϧΛ࠾༻͢Δ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Γ
͔ͭϓϨΠϠʔΛ‫ؚ‬Ήఏ‫͕ܞ‬ඞͣউརఏ‫ܞ‬ͷͨΊ ϓϨΠϠʔ͸ಠࡋऀͰ΋͋Δ
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 1 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3} {1}
ղ౴

6. ҎԼͷঢ়‫گ‬Λ౤ථήʔϜͱͯ͠ද‫ͤݱ‬Α·ͨ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ ಠࡋऀ͸͍Δ͔Λ֬ೝͤΑ
ܾٞ‫ݖ‬Λ΋ͭϓϨΠϠʔ͕ ͋ΔձࣾͷܾٞҊʹର͠ ౤ථ͢Δঢ়‫گ‬Λߟ͑Δ
͜ͷͱ͖ ա൒਺ϧʔϧΛ࠾༻͢Δ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
ಠࡋऀ͸ଘࡏ͠ͳ͍
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3}
ղ౴

7. ҎԼͷঢ়‫گ‬Λ౤ථήʔϜͱͯ͠ද‫ͤݱ‬Α·ͨ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ ಠࡋऀ͸͍Δ͔Λ֬ೝͤΑ
ܾٞ‫ݖ‬Λ΋ͭϓϨΠϠʔ͕ ͋ΔձࣾͷܾٞҊʹର͠ ౤ථ͢Δঢ়‫گ‬Λߟ͑Δ
͜ͷͱ͖ ա൒਺ϧʔϧΛ࠾༻͢Δ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͸͍ͳ͍ͷͰ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ͸ଘࡏ͠ͳ͍
·ͨ ಠࡋऀ΋͍ͳ͍
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3} {2,3}
ղ౴

8. ҎԼͷঢ়‫گ‬Λ౤ථήʔϜͱͯ͠ද‫ͤݱ‬Α·ͨ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ ಠࡋऀ͸͍Δ͔Λ֬ೝͤΑ
ܾٞ‫ݖ‬Λ΋ͭϓϨΠϠʔ͕ ͋ΔձࣾͷܾٞҊʹର͠ ౤ථ͢Δঢ়‫گ‬Λߟ͑Δ
͜ͷͱ͖ ա൒਺ϧʔϧΛ࠾༻͢Δ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Γ
͔ͭϓϨΠϠʔΛ‫ؚ‬Ήఏ‫͕ܞ‬ඞͣউརఏ‫ܞ‬ͷͨΊ ϓϨΠϠʔ͸ಠࡋऀͰ΋͋Δ
v({1,2,3,4}) = 1
v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 1 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4} {1}
ղ౴

9. ҎԼͷঢ়‫گ‬Λ౤ථήʔϜͱͯ͠ද‫ͤݱ‬Α·ͨ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ ಠࡋऀ͸͍Δ͔Λ֬ೝͤΑ
ܾٞ‫ݖ‬Λ΋ͭϓϨΠϠʔ͕ ͋ΔձࣾͷܾٞҊʹର͠ ౤ථ͢Δঢ়‫گ‬Λߟ͑Δ
͜ͷͱ͖ ա൒਺ϧʔϧΛ࠾༻͢Δ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
v({1,2,3,4}) = 1
v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}
ղ౴

10. ҎԼͷঢ়‫گ‬Λ౤ථήʔϜͱͯ͠ද‫ͤݱ‬Α·ͨ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ ಠࡋऀ͸͍Δ͔Λ֬ೝͤΑ
ܾٞ‫ݖ‬Λ΋ͭϓϨΠϠʔ͕ ͋ΔձࣾͷܾٞҊʹର͠ ౤ථ͢Δঢ়‫گ‬Λߟ͑Δ
͜ͷͱ͖ ա൒਺ϧʔϧΛ࠾༻͢Δ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͸͍ͳ͍ͷͰ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ͸ଘࡏ͠ͳ͍
·ͨ ಠࡋऀ΋͍ͳ͍
v({1,2,3,4}) = 1
v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 1
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 0 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}
ղ౴

11. ήʔϜཧ࿦#4*$ԋश
౤ථήʔϜ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͱಠࡋऀ

12. ಛੑؔ਺ͷੑ࣭
ఆٛɿ༏Ճ๏ੑ
ಛੑؔ਺ ͕༏Ճ๏ੑΛຬͨ͢ͱ͸
‫ަʹ͍ޓ‬ΘΒͳ͍೚ҙͷఏ‫ܞ‬ ͱ ʹରͯ͠
͕੒Γཱͭ͜ͱͰ͋Δ
v
S T S ∩ T = ∅
v(S ∪ T) ≥ v(S) + v(T)

13. ༏Ճ๏ੑΛຬͨ͢ήʔϜʹ͓͚ΔίΞ
ఆཧ
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ ͕༏Ճ๏ੑΛຬͨ͢ͱ͖
ίΞ͸
ఆٛɿఏ‫ܞ‬ ͷෆຬ
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ ഑෼ ͱఏ‫ܞ‬ ΛͱΔͱ͖
Λ഑෼ ʹର͢Δఏ‫ܞ‬ ͷෆຬͱ͍͏
ίΞʹ‫·ؚ‬ΕΔ഑෼͸ ͱͳΔͨΊ
͢΂ͯͷఏ‫ܞ‬ ʹ͍ͭͯෆຬ͕ͳ͍ঢ়ଶͱ΋‫͑ݴ‬Δ
(N, v) v
C(v) = {x ∈
𝒜
(v)|
∑
i∈S
xi ≥ v(S), ∀S ⊂ N, S ≠ ∅, N}
S
(N, v) x ∈
𝒜
(v) S ⊆ N
e(S, x) = v(S) −
∑
i∈S
xi x S
e(S, x) ≤ 0
S ⊆ N
ఏ‫߹ܞ‬ཧੑ

14. ҎԼͷঢ়‫گ‬Λਓͷ౤ථήʔϜͱͯ͠ද‫͠ݱ‬ ίΞΛ‫ٻ‬ΊΑ
ա൒਺ϧʔϧΛ࠾༻͠ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷථ਺Λ΋ͭͱ͢Δ
શථ਺͕Ͱ͋Γ ա൒਺͸Ҏ্Ͱ͋Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜ͸༏Ճ๏ੑΛຬͨ͢
഑෼Λ ͱ͢Δͱ ഑෼͓Αͼఏ‫߹ܞ‬ཧੑ͔Β
͜ΕΒͷ৚݅Λຬͨ͢ͷ͸
ˎϓϨΠϠʔ͸ಠࡋऀͰ͋Δ
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 1 v({2}) = 0 v({3}) = 0
(x1, x2, x3)
x1 + x2 + x3 = 1 x1 + x2 ≥ 1 x1 + x3 ≥ 1 x2 + x3 ≥ 0
x1 ≥ 1 x2 ≥ 0 x3 ≥ 0
(x1, x2, x3) = (1, 0, 0)
ղ౴