Recommended
More Related Content
What's hot
What's hot (20)
Similar to ゲーム理論NEXT 期待効用理論第6回 -3つの公理と期待効用定理-
Similar to ゲーム理論NEXT 期待効用理論第6回 -3つの公理と期待効用定理- (20)
More from ssusere0a682
More from ssusere0a682 (20)
Recently uploaded
Recently uploaded (20)
ゲーム理論NEXT 期待効用理論第6回 -3つの公理と期待効用定理-
- 2. ҎԼԬా ʹΑΔ ͏গ͠Ұൠతͳ ࠞ߹ू߹ ༗͘͡ݶ ࠞ߹ू߹্ͷબॱংʹؔͯͭ͠ͷެཧ දݱఆཧͱҰҙੑʢূ໌ʣ දݱఆཧͱҰҙੑʢূ໌ʣ ϦεΫճආ
- 3. ࣍ ͘͡Λͬͯද͢ݱΔظޮ༻Ծઆ Ͳ͏͍͏ू߹্ͷબॱংΛߟ͑Δ͔ʁ ̏ͭͷެཧ ߹ཧੑ ಠཱੑ ࿈ଓੑ ظޮ༻ఆཧ
- 4. ͘͡Λͬͯද͢ݱΔظޮ༻Ծઆ ظޮ༻Ծઆ ҙࢥܾఆओମෳͷۚ͘͡ ͷબʹ͓͍ͯ ظޮ༻ Λ࠷େʹ͢Δۚ͘͡Λબ͢Δ ͜͜Ͱ ՟ฎֹ ʹର͢Δޮ༻Λද͢ ͲͷΑ͏ͳબॱংͰ͋Ε ظޮ༻ͱͯ͠ද͖ͰݱΔͷ͔ʁ ͭ·Γ ͘͡ͷू߹ ʹରͯ͠ ҙࢥܾఆओମ͕બॱং ͕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 ࣮ࡍ Ώ͑ʹ 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 αpk + (1 − α)qk ≥ 0, ∀k = 1,⋯, s s ∑ k=1 (αpk + (1 − α)qk) = α s ∑ k=1 pk + (1 − α) s ∑ k=1 qk = α + (1 − α) = 1 αP + (1 − α)Q ∈ L ू߹ ತू߹ L
- 6. 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 ≻
- 7. ͭ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
- 8. ߹ཧੑ ߹ཧੑ બॱং ͕ऑॱংͰ͋Δ ͜Ε ͕ਪҠੑͱඋੑΛͭ͜ͱͱಉ ४උ̍ࢀর ू߹ ্ͷೋ߲ؔ ͕ ඇରশੑͱ൱ఆͷਪҠੑΛຬͨ͢ͱ͖ऑॱংͱ͍͏ ඇରশੑ ʹ͍ͭͯ ͳΒ Ͱͳ͍Λຬͨ͢ͱ͖ ඇରশੑΛຬͨ͢ͱ͍͏ ൱ఆͷਪҠੑ ʹ͍ͭͯ Ͱͳ͔ͭ͘ Ͱͳ͍ͳΒ Ͱͳ͍Λຬͨ͢ͱ͖ ൱ఆͷਪҠੑΛຬͨ͢ͱ͍͏ ४උࢀর ࣍ͷੑ࣭Λຬͨ͢ ਪҠੑ ʹ͍ͭͯ ͔ͭ ͳΒ ͱͳΔ උੑ ͷҙͷݩ ʹରͯ͠ ·ͨ Ͱ͋Δ ≻ ≿ X R x, y ∈ X xRy yRx R x, y, z ∈ X xRy yRz xRz R ≿ x, y, z ∈ X x ≿ y y ≿ z x ≿ z X x y x ≿ y y ≿ x ߹ཧతͳҙࢥܾఆओମͳΒ͍࣋ͬͯΔͩΖ͏ʁ
- 9. ಠཱੑ ಠཱੑ ҙͷ ʹରͯ͠ ͱແؔͳ͘͡ ͕ࠞ͟ΔΑ͏ͳ߹Ͱ ͱ ͷͲͪΒΛબͿ͔ ͷબͰఆ·Δ ͘͡ ΛબͿҙࢥܾఆͷલʹ ผͷ͘͡ Λબ͢Δঢ়ଶ͕ ͷ֬Ͱൃੜ͢Δঢ়ͯ͠ͱگଊ͑ Ͱ Λબ͢Δঢ়ଶ͕͔ͨͬͳ͖ى߹ ͷ͘͡ΛબͿҙࢥܾఆ͕ߦΘΕΔ w w w w P, Q, R ∈ L α(0 α 1) P ≻ Q ⇒ αP + (1 − α)R ≻ αQ + (1 − α)R P, Q R αP + (1 − α)R αQ + (1 − α)R P, Q P, Q R 1 − α 1 − α R P, Q 1 − α α P Q R αP + (1 − α)R αQ + (1 − α)R ແؔ ͱແؔʹ, ୯ʹ ͷબͰఆ·Δϋζ R P, Q
- 10. ࿈ଓੑ ࿈ଓੑ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕ଘࡏͯ͠ ͜ͷެཧ͔Βಋ͔ΕΔ͜ͱ ʹର͠ બॱংͷҙຯͰ͍͋ͩʹ͋Δҙͷ͘͡ બॱংͷҙຯͰಉͳ͘͡ ͕ଘࡏ P, Q, R ∈ L P ≻ Q, Q ≻ R α, β(0 ≤ α, β ≤ 1) αP + (1 − α)R ≻ Q, Q ≻ βP + (1 − β)R P, R Q γP + (1 − γ)R 1 − β β P Q R ͷؒͷ͘͡ ͰඞͣදݱՄೳ ͷܗͷ͕͘͡࿈ଓతʹ P, R P, R γP + (1 − γ)R α 1 − α ͘͡Λࠞ߹
- 11. 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 ≻ U : L → ℜ ≻ u u
- 12. ༧ఆ ४උ αϯΫτϖςϧϒϧΫͷύϥυοΫε ظޮ༻Ծઆ ͭͷެཧͱظޮ༻ఆཧ ҎԼԬా ʹΑΔ ͏গ͠Ұൠతͳ ࠞ߹ू߹ ༗͘͡ݶ ࠞ߹ू߹্ͷબॱংʹؔͯͭ͠ͷެཧ දݱఆཧͱҰҙੑʢূ໌ʣ දݱఆཧͱҰҙੑʢূ໌ʣ ϦεΫճආ
- 13. ظޮ༻ఆཧɿ࠷ߴͷ͘͡ɾ࠷ͷ͘͡ ͯ͢ͷ ʹର͠ Ͱ͋Ε ͱ͢Ε 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
- 14. ظޮ༻ఆཧɿূ໌εςοϓ εςοϓҙͷ ʹର͠ εςοϓҙͷ ʹର͠ ͳΒ εςοϓҙͷ ʹରͯ͠ ͱͳΔ ͕Ұҙʹଘࡏ͢Δ ͜ͷͱ͖ͷ Λ Ͱද͢ ҙͷ ʹରͯ͠ ͳΒ ͋Δ࣮ ͕ଘࡏͯ͠ εςοϓؔ Λ Ͱఆٛ͢Δ͜ͷ Λද͢ݱΔ εςοϓҙͷ ʹରͯ͠ ͜͜Ͱ Λຬͨ͢ͱͯٞ͢͠Δ εςοϓ͋Δؔ ͕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 ∈ L P ∼ αP + (1 − α)P α ∈ [0,1] α αP P, Q, R ∈ L P ≿ Q, Q ≿ R, P ≻ R β(0 ≤ β ≤ 1) Q ∼ βP + (1 − β)R U : L → ℜ U(P) = αP U ≻ P, Q ∈ L α ∈ [0,1] U(αP + (1 − α)Q) = αU(P) + (1 − α)U(Q) P ∼ Q ⇒ αP + (1 − α)R ∼ αQ + (1 − α)R u : X → ℜ P ∈ L U(P) = s ∑ k=1 pku(xk) ઢܕੑ ࿈ଓੑ ಠཱੑ