ゲーム理論NEXT 期待効用理論第7/8/9回 -期待効用定理の証明1/2/3-
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ゲーム理論NEXT 期待効用理論第7/8/9回 -期待効用定理の証明1/2/3-
- 2. ҎԼԬా ʹΑΔ ͏গ͠Ұൠతͳ ࠞ߹ू߹ ༗͘͡ݶ ࠞ߹ू߹্ͷબॱংʹؔͯͭ͠ͷެཧ දݱఆཧͱҰҙੑʢূ໌ʣ දݱఆཧͱҰҙੑʢূ໌ʣ ϦεΫճආ
- 3. ࣍ ͘͡Λͬͯද͢ݱΔظޮ༻Ծઆʢ෮शʣ Ͳ͏͍͏ू߹্ͷબॱংΛߟ͑Δ͔ʁʢ෮शʣ ̏ͭͷެཧʢ෮शʣ ظޮ༻ఆཧʢ෮शʣ ূ໌ ิ ิ ୯ௐੑ ิ ࿈ଓੑ ิ ಠཱੑެཧͷແࠩผ7FS ఆཧূ໌
- 4. ͘͡Λͬͯද͢ݱΔظޮ༻Ծઆ ظޮ༻Ծઆ ҙࢥܾఆओମෳͷۚ͘͡ ͷબʹ͓͍ͯ ظޮ༻ Λ࠷େʹ͢Δۚ͘͡Λબ͢Δ ͜͜Ͱ ՟ฎֹ ʹର͢Δޮ༻Λද͢ ͲͷΑ͏ͳબॱংͰ͋Ε ظޮ༻ͱͯ͠ද͖ͰݱΔͷ͔ʁ ͭ·Γ ͘͡ͷू߹ ʹରͯ͠ ҙࢥܾఆओମͷબॱং ͕ԿΒ͔ͷੑ࣭Λຬͨ͢ͱ͖ ҙͷۚ͘͡ ʹରͯ͠ ͘͡ͷू߹্ͷؔ ͕͋ͬͯ Ͱ͋Γ ͞Βʹ͜ͷ ظޮ༻ͷܗ Ͱද͞ΕΔͣ z = [x1, ⋯xs; p1, ⋯, ps] s ∑ k=1 pku(xk) u(xk) xk L ≻ z, z′  ∈ L U z ≻ z′  ⇔ U(z) U(z′  ) U U(z) = s ∑ k=1 pku(xk)
- 5. Ͳ͏͍͏ू߹্ͷબॱংΛߟ͑Δ͔ʁ ۚ͘͡ ͷબͱ͍͏ͷΛѻ͏ͨΊʹɾɾɾ ݁Ռʹର͢Δۚͷू߹ɿ ্ͷ֬શମ͔ΒͳΔू߹ʢ͘͡ͷू߹ʣɿ ͘͡ͷࠞ߹ɿ ͜ͷ ͷத͔Βҙͷ͘͡ ΛબΜͩͱ͖ʹ ҙͷ ʹର͠ ͱ͢Δͱ ۚ ʹର͢Δ֬Λද͢ ֬Ͱ ͕ൃੜ͢Δ͘͡ɿ ҙࢥܾఆओମ͘͡ͷू߹ ্ʹબॱং Λͭ z = [x1, ⋯xs; p1, ⋯, ps] X = {x1, ⋯, xs} X L = {(p1, ⋯, ps)|pk ≥ 0 ∀k = 1,⋯, s and s ∑ k=1 pk = 1} L P = (p1, ⋯, ps) Q = (q1, ⋯, qs) α(0 α 1) αP + (1 − α)Q = α(p1, ⋯, ps) + (1 − α)(q1, ⋯, qs) = (αp1 + (1 − α)q1, ⋯, αps + (1 − α)qs) αpk + (1 − α)qk xk xk Pk = ( 1∼ k−1 ⏞ 0,⋯ , k ⏞ 1, k+1∼ s ⏞ ⋯0 ) ∈ L L ≻
- 6. ͭͷެཧ ߹ཧੑ બॱং ͕ऑॱংͰ͋Δ ͜Ε ͕ਪҠੑͱඋੑΛͭ͜ͱͱಉ ಠཱੑ ҙͷ ʹରͯ͠ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕ଘࡏͯ͠ ≻ ≿ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R P, Q, R ∈ L P ≻ Q, Q ≻ R α, β(0 α, β 1) αP + (1 − α)R ≻ Q, Q ≻ βP + (1 − β)R గਖ਼
- 7. ظޮ༻ఆཧ ظޮ༻ఆཧ ݁Ռʹର͢Δۚͷू߹ ͱ͠ Λ ্ͷ͘͡ͷू߹ͱ͢Δ ্ͷબॱং ͕ެཧ Λຬͨ͢ͳΒ ͋Δ ্ͷ࣮ؔ ͕ଘࡏ͠ Ͱఆٛ͞ΕΔؔ બ Λද͢ݱΔ ʹٯ Ͱද͞ݱΕΔબ ެཧ Λຬͨ͢ ؔ ΛϑΥϯϊΠϚϯϞϧήϯγϡςϧϯޮ༻ؔ W/.ؔ ͱ ͼݺ ਖ਼ΞϑΟϯมΛআ͍ͯҰҙͰ͋Δ X L X L ≻ X u : X → ℜ U(P) = s ∑ k=1 pku(xk) U : L → ℜ ≻ P ≻ Q ⇔ U(P) U(Q) U : L → ℜ ≻ u u
- 8. ظޮ༻ఆཧɿূ໌ ิҙͷ ʹର͠ ิ ୯ௐੑ ҙͷ ʹର͠ ͳΒ ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ ิ ಠཱੑެཧͷແࠩผ7FS ҙͷ ʹର͠ Λຬͨ͢ ఆཧূ໌εςοϓҙͷ ʹରͯ͠ ͱͳΔ ͕Ұҙʹଘࡏ͢Δ͜ͷͱ͖ͷ Λ Ͱද͢ ఆཧূ໌εςοϓؔ Λ Ͱఆٛ͢Δ͜ͷ Λද͢ݱΔ ఆཧূ໌εςοϓҙͷ ʹରͯ͠ ఆཧূ໌εςοϓ͋Δؔ ͕ଘࡏͯ͠ ҙͷ ʹରͯ͠ ͱॻ͚Δ w w w P, Q ∈ L α ∈ (0,1) P ≻ Q ⇒ P ≻ αP + (1 − α)Q, αP + (1 − α)Q ≻ Q P, Q ∈ L α, β ∈ [0,1] P ≻ Q αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ α ≥ β αP + (1 − α)Q ∼ βP + (1 − β)Q ⇔ α = β αP + (1 − α)Q ≻ βP + (1 − β)Q ⇔ α β P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R P, Q, R ∈ L α ∈ [0,1] P ∼ Q ⇒ αP + (1 − α)R ∼ αQ + (1 − α)R P ∈ L P ∼ αP + (1 − α)P α ∈ [0,1] α αP U : L → ℜ U(P) = αP U ≻ P, Q ∈ L α ∈ [0,1] U(αP + (1 − α)Q) = αU(P) + (1 − α)U(Q) u : X → ℜ P ∈ L U(P) = s ∑ k=1 pku(xk) બॱং ͕ͭͷެཧΛຬͨ͢ͱ͖ ≻
- 9. ิ ิҙͷ ʹର͠ ΑΓಠཱੑͷެཧΑΓ ಉ༷ʹ P, Q ∈ L α ∈ (0,1) P ≻ Q ⇒ P ≻ αP + (1 − α)Q, αP + (1 − α)Q ≻ Q P ≻ Q (1 − α)P + αP ≻ (1 − α)Q + αP ⇔ P ≻ (1 − α)Q + αP αP + (1 − α)Q ≻ αQ + (1 − α)Q ⇔ αP + (1 − α)Q ≻ Q ಠཱੑͷެཧ ҙͷ ʹରͯ͠ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R
- 10. ิ ୯ௐੑ ิ ୯ௐੑ ҙͷ ʹର͠ ͳΒ ͱͳΔ͜ͱΛࣔ͢ ͷͱ͖ ͠ ͱ͢Ε Ͱ͋Γ ͜Ε ެཧ ऑॱং ͷඇରশੑʹໃ६ ಉ༷ʹ ཱ͠ͳ͍ Ώ͑ʹ ͷੑ࣭͔Β P, Q ∈ L α, β ∈ [0,1] P ≻ Q αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ α ≥ β αP + (1 − α)Q ∼ βP + (1 − β)Q ⇔ α = β αP + (1 − α)Q ≻ βP + (1 − β)Q ⇔ α β ⇐ α = β αP + (1 − α)Q = βP + (1 − β)Q αP + (1 − α)Q ≻ βP + (1 − β)Q αP + (1 − α)Q ≻ αP + (1 − α)Q ≻ βP + (1 − β)Q ≻ αP + (1 − α)Q ≻ αP + (1 − α)Q ∼ βP + (1 − β)Q ಠཱੑͷެཧ ҙͷ ʹରͯ͠ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R ඇରশੑ ʹ͍ͭͯ ͳΒ Ͱͳ͍Λຬͨ͢ x, y ∈ X x ≻ y y ≻ x Λ ্ͷऑॱংͱ͢Δ ͷݩ ʹରͯ͠ ͷ͍ͣΕ͔͕ඞͣΓཱͭ ≻ X X x y x ≻ y, x ∼ y, y ≻ x
- 11. ิ ୯ௐੑ ิ ୯ௐੑ ҙͷ ʹର͠ ͳΒ ͱͳΔ͜ͱΛࣔ͢ ͷͱ͖Λߟ͑Δ ͰิΑΓ ͜͜Ͱ ͱΈͳͤ ͷͱ͖ Γཱͭ ͷͱ͖ Ͱ͋Δ͜ͱʹҙͯ͠ ิΑΓ ͱͳΓ Γཱͭ P, Q ∈ L α, β ∈ [0,1] P ≻ Q αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ α ≥ β αP + (1 − α)Q ∼ βP + (1 − β)Q ⇔ α = β αP + (1 − α)Q ≻ βP + (1 − β)Q ⇔ α β ⇐ α β P ≻ Q αP + (1 − α)Q ≻ Q Q = 0P + (1 − 0)Q β = 0 ⇐ α β 0 α β 0 ⇔ 1 β α 0 αP + (1 − α)Q ≻ Q ⇒ αP + (1 − α)Q ≻ β α [αP + (1 − α)Q] + ( 1 − β α) Q = βP + (1 − β)Q ⇐ ิҙͷ ʹର͠ P, Q ∈ L α ∈ (0,1) P ≻ Q ⇒ P ≻ αP + (1 − α)Q, αP + (1 − α)Q ≻ Q
- 12. ิ ୯ௐੑ ิ ୯ௐੑ ҙͷ ʹର͠ ͳΒ ͱͳΔ͜ͱΛࣔ͢ ͷͱ͖ ͱ͢Δͱ ·ͨ Ͱ͋Γ ͜ͷͱ͖ͦΕͧΕ ·ͨ ͕Γཱ͕ͭ Ͱ͋Δ͜ͱʹໃ६ Ώ͑ʹ ࣮ͷେখؔʹ͍ͭͯ ͷ͍ͣΕ͔͕Γཱͭ͜ͱʹҙ P, Q ∈ L α, β ∈ [0,1] P ≻ Q αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ α ≥ β αP + (1 − α)Q ∼ βP + (1 − β)Q ⇔ α = β αP + (1 − α)Q ≻ βP + (1 − β)Q ⇔ α β ⇒ αP + (1 − α)Q ∼ βP + (1 − β)Q α ≠ β α β α β αP + (1 − α)Q ≻ βP + (1 − β)Q βP + (1 − β)Q ≻ αP + (1 − α)Q αP + (1 − α)Q ∼ βP + (1 − β)Q α = β x y, x = y, x y Λ ্ͷऑॱংͱ͢Δ ͷݩ ʹରͯ͠ ͷ͍ͣΕ͔͕ඞͣΓཱͭ ≻ X X x y x ≻ y, x ∼ y, y ≻ x
- 13. ิ ୯ௐੑ ิ ୯ௐੑ ҙͷ ʹର͠ ͳΒ ͱͳΔ͜ͱΛࣔ͢ ͷͱ͖ ͱ͢Δͱ ͕Γཱ͕ͭໃ६ ͱ͢Δͱ Ͱ͋Δ͕͜Εໃ६ Ώ͑ʹ ࣮ͷେখؔʹ͍ͭͯ ͷ͍ͣΕ͔͕Γཱͭ͜ͱʹҙ P, Q ∈ L α, β ∈ [0,1] P ≻ Q αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ α ≥ β αP + (1 − α)Q ∼ βP + (1 − β)Q ⇔ α = β αP + (1 − α)Q ≻ βP + (1 − β)Q ⇔ α β ⇒ αP + (1 − α)Q ≻ βP + (1 − β)Q α β βP + (1 − β)Q ≻ αP + (1 − α)Q α = β αP + (1 − α)Q ∼ βP + (1 − β)Q α β x y, x = y, x y Λ ্ͷऑॱংͱ͢Δ ͷݩ ʹରͯ͠ ͷ͍ͣΕ͔͕ඞͣΓཱͭ ≻ X X x y x ≻ y, x ∼ y, y ≻ x
- 14. ิ ୯ௐੑ ิ ୯ௐੑ ҙͷ ʹର͠ ͳΒ Ҏ্ΑΓ ·ͨ ·ͨ P, Q ∈ L α, β ∈ [0,1] P ≻ Q αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ α ≥ β αP + (1 − α)Q ∼ βP + (1 − β)Q ⇔ α = β αP + (1 − α)Q ≻ βP + (1 − β)Q ⇔ α β αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ αP + (1 − α)Q ∼ βP + (1 − β)Q αP + (1 − α)Q ≻ βP + (1 − β)Q ⇔ α = β α β ⇔ α ≥ β
- 15. ิ ࿈ଓੑ ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ Ͱ͋Δ߹ Ͱ͋Γ ҙͷ ʹ͍ͭͯ ୯ௐੑΑΓ ऑॱংͷੑ࣭ΑΓ Ώ͑ʹ ͕Ұҙͷ࣮Ͱ͋Δ Ͱ͋Δ߹ಉ༷ʹ ͕Ұҙͷ࣮Ͱ͋Δ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R P ∼ Q Q ∼ 1P + 0R α 1 1P + 0R ≻ αP + (1 − α)R Q ≻ αP + (1 − α)R μ = 1 Q ∼ R μ = 0 ิ ୯ௐੑ ҙͷ ʹର͠ ͳΒ P, Q ∈ L α, β ∈ [0,1] P ≻ Q αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ α ≥ β Λ ্ͷऑॱংͱ͢Δ ͷݩ ʹରͯ͠ ͔ͭ ͳΒ Ͱ͋Γ ͔ͭ ͳΒ Ͱ͋Δ ≻ X X x y z x ≻ y y ∼ z x ≻ z x ∼ y y ≻ z x ≻ z
- 16. ิ ࿈ଓੑ ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ Ͱ͋Δ߹ Λຬͨ͢ ҰҙͰ͋Δ ͳͥͳΒ Λຬͨ͢ผͷ࣮ ͕ଘࡏ͢Δͱ͢Δͱ ਪҠੑͱ୯ௐੑΑΓ Ͱ ͜ͷΑ͏ͳ࣮ ଘࡏ͢Δͷ͔ʁ࿈ଓੑͷެཧΛͬͯࣔ͢ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R P ≻ Q, Q ≻ R Q ∼ αP + (1 − α)R α Q ∼ α′  P + (1 − α′  )R α′  αP + (1 − α)R ∼ α′  P + (1 − α′  )R ⇒ α = α′  α ิ ୯ௐੑ ҙͷ ʹର͠ ͳΒ P, Q ∈ L α, β ∈ [0,1] P ≻ Q αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ α ≥ β
- 17. ิ ࿈ଓੑ ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ ࿈ଓੑͷެཧΛͬͯࣔ͢ Ͱ͋Δ͔Β ͋Δ࣮ ͕ଘࡏͯ͠ ਪҠੑ͔Β Ͱ͋Γ ୯ௐੑΑΓ Ͱ͋Δ ͜ͷͱ͖ ͱ͓͘ͱ J JJ JJJ ͷ͍ͣΕ͔͕Γཱͭ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R P ≻ Q, Q ≻ R α, β(0 α, β 1) αP + (1 − α)R ≻ Q, Q ≻ βP + (1 − β)R αP + (1 − α)R ≻ βP + (1 − β)R α β a0 = α, b0 = β, c = a0 + b0 2 cP + (1 − c)R ∼ Q cP + (1 − c)R ≻ Q Q ≻ cP + (1 − c)R ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕ଘࡏͯ͠ P, Q, R ∈ L P ≻ Q, Q ≻ R α, β(0 α, β 1) αP + (1 − α)R ≻ Q, Q ≻ βP + (1 − β)R ิ ୯ௐੑ ҙͷ ʹର͠ ͳΒ P, Q ∈ L α, β ∈ [0,1] P ≻ Q αP + (1 − α)Q ≿ βP + (1 − β)Q ⇔ α ≥ β
- 18. ิ ࿈ଓੑ ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ J JJ JJJ ͷ͍ͣΕ͔͕Γཱͭ ্هਤͩͱ JJJ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R cP + (1 − c)R ∼ Q cP + (1 − c)R ≻ Q Q ≻ cP + (1 − c)R Q ≻ cP + (1 − c)R P R Q ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕ଘࡏͯ͠ P, Q, R ∈ L P ≻ Q, Q ≻ R α, β(0 α, β 1) αP + (1 − α)R ≻ Q, Q ≻ βP + (1 − β)R αP + (1 − α)R βP + (1 − β)R β 1 − β α 1 − α α + β α + β 2 = c
- 19. ิ ࿈ଓੑ ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ J JJ JJJ ͷ͍ͣΕ͔͕Γཱͭ J ͷ߹ ͜ͷ࣮ ͕ଘࡏ͠ ͱͯ͠ JJ ͷ߹ ͱ͠ JJJ ͷ߹ ͱ͓͘ ͱ͓͚ ಉ༷ͷૢ࡞Λ܁Γฦ͢͜ͱʹΑΓ ΛߏͰ͖ ͱͰ͖Δ ͕ఆΊΔ࣮Λ ͱ͢Δͱ͖ Λূ໌͢Δ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R cP + (1 − c)R ∼ Q cP + (1 − c)R ≻ Q Q ≻ cP + (1 − c)R c μ = c Q ∼ μP + (1 − μ)R a1 = c, b1 = b0 a1 = a0, b1 = c c = a1 + b1 2 {an}∞ n=1 {bn}∞ n=1 anP + (1 − an)R ≻ Q Q ≻ bnP + (1 − bn)R, n = 1,2,⋯ |an − bn | → 0. (n → ∞) {an}∞ n=1 {bn}∞ n=1 μ Q ∼ μP + (1 − μ)R ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕ଘࡏͯ͠ P, Q, R ∈ L P ≻ Q, Q ≻ R α, β(0 α, β 1) αP + (1 − α)R ≻ Q, Q ≻ βP + (1 − β)R
- 20. ิ ࿈ଓੑ ͱͰ͖Δ ͕ఆΊΔ࣮Λ ͱ͢Δ ୯ௐੑΑΓ Ͱ͋Γ ඇ૿ՃྻͰԼʹ༗քͳͷͰ ಉ༷ʹ ඇݮগྻͰ্ʹ༗քͳͷͰ ·ͨ ·ͨ Ώ͑ʹ Ώ͑ʹ ͜ͷऩଋઌΛ ͱ͢Δ anP + (1 − an)R ≻ Q Q ≻ bnP + (1 − bn)R, n = 1,2,⋯ |an − bn | → 0. (n → ∞) {an}∞ n=1 {bn}∞ n=1 μ an bn, n = 1,2,⋯ a0 ≥ a1 ≥ ⋯ ≥ an bn ≥ ⋯ ≥ b1 ≥ b0 an an → a bn bn → b an − bn = an−1 − an−1 + bn−1 2 = an−1 − bn−1 2 an − bn = an−1 + bn−1 2 − bn−1 = an−1 − bn−1 2 an − bn = ( 1 2 ) n (a0 − b0) → 0. (n → ∞) an = bn + ( 1 2) n (a0 − b0) → b . (n → ∞) a = b μ
- 21. ิ ࿈ଓੑ ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ J JJ JJJ ͷ͍ͣΕ͔͕Γཱͭ ্هਤͩͱ JJJ ͜ͷͱ͖ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R cP + (1 − c)R ∼ Q cP + (1 − c)R ≻ Q Q ≻ cP + (1 − c)R Q ≻ cP + (1 − c)R a1 = a0, b1 = c P R Q a0P + (1 − a0)R b0P + (1 − b0)R b0 1 − b0 a0 1 − a0 a0 + b0 a0 + b0 2 = c
- 22. ิ ࿈ଓੑ ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ J JJ JJJ ͷ͍ͣΕ͔͕Γཱͭ ্هਤͩͱ JJ ͜ͷͱ͖ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R cP + (1 − c)R ∼ Q cP + (1 − c)R ≻ Q Q ≻ cP + (1 − c)R cP + (1 − c)R ≻ Q a2 = c, b2 = b1 P R Q a1P + (1 − a1)R b1P + (1 − b1)R a1 1 − a1 b1 1 − b1 a1 + b1 2 = c a1 + b1
- 23. ิ ࿈ଓੑ ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ ͱͰ͖Δ ͕ఆΊΔ࣮Λ ͱ͢Δͱ͖ Λূ໌͢Δ Ͱ͋ΔͱԾఆ͢Δ࿈ଓੑͷެཧΑΓ ͕ଘࡏͯ͠ ΑΓ ेେ͖ͳ ʹରͯ͠ Ͱ͋Δ ͷऔΓํΑΓ ਪҠੑ͔Β ୯ௐੑΑΓ ͱͳΓ ໃ६ ಉ༷ʹໃ६͕ੜ͡Δ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R anP + (1 − an)R ≻ Q Q ≻ bnP + (1 − bn)R, n = 1,2,⋯ |an − bn | → 0. (n → ∞) {an}∞ n=1 {bn}∞ n=1 μ Q ∼ μP + (1 − μ)R μP + (1 − μ)R ≻ Q 0 γ 1 γ[μP + (1 − μ)R] + (1 − γ)R = γμP + (1 − γμ)R ≻ Q μ γμ n bn γμ bn Q ≻ bnP + (1 − bn)R γμP + (1 − γμ)R ≻ bnP + (1 − bn)R γμ bn Q ≻ μP + (1 − μ)R ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕ଘࡏͯ͠ P, Q, R ∈ L P ≻ Q, Q ≻ R α, β(0 α, β 1) αP + (1 − α)R ≻ Q, Q ≻ βP + (1 − β)R
- 24. ิ ಠཱੑެཧͷແࠩผ7FS ิ ಠཱੑެཧͷແࠩผ7FS ҙͷ ʹର͠ Λຬͨ͢ ·ͣ ʹ͍ͭͯ Ͱ͋Ε ͱͳΔ͜ͱΛࣔ͢ ಠཱੑͷެཧΑΓ ΑΓ ਪҠੑ͔Β ΛಘΔ P, Q, R ∈ L α ∈ [0,1] P ∼ Q ⇒ αP + (1 − α)R ∼ αQ + (1 − α)R P, Q, R, S ∈ L P ≻ Q, R ≻ S α(0 α 1) αP + (1 − α)R ≻ αQ + (1 − α)S αP + (1 − α)R ≻ αQ + (1 − α)R (1 − α)R + αQ ≻ (1 − α)S + αQ αP + (1 − α)R ≻ αQ + (1 − α)S ಠཱੑͷެཧ ҙͷ ʹରͯ͠ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R
- 25. ิ ಠཱੑެཧͷແࠩผ7FS ิ ಠཱੑެཧͷແࠩผ7FS ҙͷ ʹର͠ Λຬͨ͢ ࣍ʹ ͳΒ ͱͳΔ͜ͱΛࣔ͢ Ͱͳ͍͜ͱΛࣔ͢ ΛԾఆͯ͠ໃ६Λࣔ͢ ΑΓ ઌ΄Ͳͷ͔ٞΒ ͔ͭ Ͱ͋Ε ͱͳΓ ͷඇରশੑʹໃ६ Ͱͳ͍͜ͱಉ༷ʹࣔͤΔ P, Q, R ∈ L α ∈ [0,1] P ∼ Q ⇒ αP + (1 − α)R ∼ αQ + (1 − α)R P ∼ Q α(0 α 1) P ∼ αP + (1 − α)Q P ≻ αP + (1 − α)Q αP + (1 − α)Q ≻ P P ≻ αP + (1 − α)Q P ∼ Q P ≻ αP + (1 − α)Q Q ≻ αP + (1 − α)Q P ≻ αP + (1 − α)Q Q ≻ αP + (1 − α)Q αP + (1 − α)Q ≻ α[αP + (1 − α)Q] + (1 − α)[αP + (1 − α)Q] = αP + (1 − α)Q ≻ αP + (1 − α)Q ≻ P ಠཱੑͷެཧ ҙͷ ʹରͯ͠ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R ʹ͍ͭͯ Ͱ͋Ε ͱͳΔ P, Q, R, S ∈ L P ≻ Q, R ≻ S α(0 α 1) αP + (1 − α)R ≻ αQ + (1 − α)S Λ ্ͷऑॱংͱ͢Δ ͷݩ ʹରͯ͠ ͷ͍ͣΕ͔͕ඞͣΓཱͭ ≻ X X x y x ≻ y, x ∼ y, y ≻ x Λ ্ͷऑॱংͱ͢Δ ͔ͭ ͳΒ Ͱ ͋Γ ͔ͭ ͳΒ Ͱ͋Δ ≻ X x ≻ y y ∼ z x ≻ z x ∼ y y ≻ z x ≻ z ඇରশੑ ʹ͍ͭͯ ͳΒ Ͱͳ͍Λຬͨ͢ x, y ∈ X x ≻ y y ≻ x
- 26. ิ ಠཱੑެཧͷແࠩผ7FS ิ ಠཱੑެཧͷແࠩผ7FS ҙͷ ʹର͠ Λຬͨ͢ ͯ͞ ิΛࣔ͢ ໌Β͔ʹΓཱͭͷͰ ͱ͢Δ ͳΒ ͷରশੑ ਪҠੑ ͓Αͼઌ΄Ͳࣔͨ͠ੑ࣭ΑΓ Ώ͑ʹ P, Q, R ∈ L α ∈ [0,1] P ∼ Q ⇒ αP + (1 − α)R ∼ αQ + (1 − α)R α = 0, α = 1 α ∈ (0,1) P ∼ R ∼ αP + (1 − α)R ∼ P, P ∼ Q, Q ∼ αQ + (1 − α)R αP + (1 − α)R ∼ αQ + (1 − α)R ಠཱੑͷެཧ ҙͷ ʹରͯ͠ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R ͳΒ ͱͳΔ P ∼ Q α(0 α 1) P ∼ αP + (1 − α)Q
- 27. ิ ಠཱੑެཧͷແࠩผ7FS ิ ಠཱੑެཧͷແࠩผ7FS ҙͷ ʹର͠ Λຬͨ͢ ࣍ʹ ͷ߹ ΛԾఆ͢Δ ͱಠཱੑެཧΑΓ ิΑΓ ͋Δ࣮ ͕ଘࡏͯ͠ Ұํ ͔ͭ Ͱ Ͱ͋Δ͔Β ಠཱੑެཧΑΓ ͭ·Γ Ͱ͋Δ͔Β Ͱ͋ΔಠཱੑެཧΑΓ ͱͳΓ ໃ६ P, Q, R ∈ L α ∈ [0,1] P ∼ Q ⇒ αP + (1 − α)R ∼ αQ + (1 − α)R P ≻ R αQ + (1 − α)R ≻ αP + (1 − α)R P ≻ R αP + (1 − α)R ≻ αR + (1 − α)R = R μ (0 μ 1) αP + (1 − α)R ∼ μ[αQ + (1 − α)R] + (1 − μ)R = μαQ + (1 − μα)R P ≻ R P ∼ Q Q ≻ R Q = (1 − μ)Q + μQ ≻ (1 − μ)R + μQ = μQ + (1 − μ)R Q ≻ μQ + (1 − μ)R P ∼ Q P ≻ μQ + (1 − μ)R αP + (1 − α)R ≻ α[μQ + (1 − μ)R] + (1 − α)R = αμQ + (1 − αμ)R ಠཱੑͷެཧ ҙͷ ʹରͯ͠ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R
- 28. ิ ಠཱੑެཧͷແࠩผ7FS ิ ಠཱੑެཧͷແࠩผ7FS ҙͷ ʹର͠ Λຬͨ͢ ࣍ʹ ͷ߹ ΛԾఆ͢Δ ͱಠཱੑެཧΑΓ ิΑΓ ͋Δ࣮ ͕ଘࡏͯ͠ ·ͨ Ͱ͋Δ͔Β ಠཱੑެཧΑΓ ͱ߹ΘͤΔͱ Ͱ͋Δ͔Β ಠཱੑެཧΑΓ ͱͳΓ ໃ६ Ώ͑ʹ P, Q, R ∈ L α ∈ [0,1] P ∼ Q ⇒ αP + (1 − α)R ∼ αQ + (1 − α)R P ≻ R αP + (1 − α)R ≻ αQ + (1 − α)R Q ≻ R αQ + (1 − α)R ≻ αR + (1 − α)R = R μ (0 μ 1) αQ + (1 − α)R ∼ μ[αP + (1 − α)R] + (1 − μ)R = μαP + (1 − μα)R P ≻ R P = (1 − μ)P + μP ≻ (1 − μ)R + μP = μP + (1 − μ)R Q ∼ P Q ≻ μP + (1 − μ)R αQ + (1 − α)R ≻ α[μP + (1 − μ)R] + (1 − α)R = αμP + (1 − αμ)R αP + (1 − α)R ∼ αQ + (1 − α)R ಠཱੑͷެཧ ҙͷ ʹରͯ͠ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R
- 29. ิ ಠཱੑެཧͷແࠩผ7FS ิ ಠཱੑެཧͷແࠩผ7FS ҙͷ ʹର͠ Λຬͨ͢ ࣍ʹ ͷ߹ ΛԾఆ͢Δ ͱಠཱੑެཧΑΓ ิΑΓ ͋Δ࣮ ͕ଘࡏͯ͠ Ͱ͋Δ͔Β ಠཱੑެཧΑΓ ಠཱੑެཧΑΓ ͱͳΓ ໃ६ P, Q, R ∈ L α ∈ [0,1] P ∼ Q ⇒ αP + (1 − α)R ∼ αQ + (1 − α)R R ≻ P αQ + (1 − α)R ≻ αP + (1 − α)R R ≻ Q R ≻ αQ + (1 − α)R μ (0 μ 1) αQ + (1 − α)R ∼ μ[αP + (1 − α)R] + (1 − μ)R = μαP + (1 − μα)R R ≻ Q μP + (1 − μ)R ≻ P( ∼ Q) α[μP + (1 − μ)R] + (1 − α)R = αμP + (1 − αμ)R ≻ αQ + (1 − α)R ಠཱੑͷެཧ ҙͷ ʹରͯ͠ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R
- 30. ิ ಠཱੑެཧͷແࠩผ7FS ิ ಠཱੑެཧͷແࠩผ7FS ҙͷ ʹର͠ Λຬͨ͢ ࣍ʹ ͷ߹ ΛԾఆ͢Δ ͱಠཱੑެཧΑΓ ิΑΓ ͋Δ࣮ ͕ଘࡏͯ͠ Ͱ͋Δ͔Β ಠཱੑެཧΑΓ ಠཱੑެཧΑΓ ͱͳΓ ໃ६ Ώ͑ʹ P, Q, R ∈ L α ∈ [0,1] P ∼ Q ⇒ αP + (1 − α)R ∼ αQ + (1 − α)R R ≻ P αP + (1 − α)R ≻ αQ + (1 − α)R R ≻ P R ≻ αP + (1 − α)R μ (0 μ 1) αP + (1 − α)R ∼ μ[αQ + (1 − α)R] + (1 − μ)R = μαQ + (1 − μα)R R ≻ Q μQ + (1 − μ)R ≻ Q( ∼ P) α[μQ + (1 − μ)R] + (1 − α)R = αμQ + (1 − αμ)R ≻ αP + (1 − α)R αP + (1 − α)R ∼ αQ + (1 − α)R ಠཱੑͷެཧ ҙͷ ʹରͯ͠ P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R ิ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕Ұҙʹଘࡏͯ͠ P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R μ(0 ≤ μ ≤ 1) Q ∼ μP + (1 − μ)R
- 33. ͱͳΓ ಉ༷ʹ Ͱ͋Δ͔Β ͕Γཱͭ ্هͰྫͳۃҰൠతʹ ͋Δ ʹ͍ͭͯ Ͱ͋Δͱ͢Δ͢Δͱ ͱͳΔ࠷ߴͷ͘͡ ͕ଘࡏ͠ ͔ͭ ͱͳΔ࠷ͷ͘͡ ͕ଘࡏ͢Δ ·ͨ ͕Γཱͭ P, Q ∈ L P ∼ Q u(xk) = c ∀k = 1,⋯, s U(P) = s ∑ k=1 pku(xk) = s ∑ k=1 pkc = c s ∑ k=1 pk = c U(Q) = c U(P) = U(Q) P, Q ∈ L P ≻ Q P ≿ R, ∀R ∈ L{P} P̄ R ≿ P, ∀R ∈ L{P} P P ≻ P
- 34. ༧ఆ ४උ αϯΫτϖςϧϒϧΫͷύϥυοΫε ظޮ༻Ծઆ ͭͷެཧͱظޮ༻ఆཧ ҎԼԬా ʹΑΔ ͏গ͠Ұൠతͳ ࠞ߹ू߹ ༗͘͡ݶ ࠞ߹ू߹্ͷબॱংʹؔͯͭ͠ͷެཧ දݱఆཧͱҰҙੑʢূ໌ʣ දݱఆཧͱҰҙੑʢূ໌ʣ ϦεΫճආ