論文紹介 Budget Management Strategies in Repeated Auctions (公開版)

࿦จ঺հ Budget Management Strategies in Repeated Auctions Balseiro, Santiago, et
al. "Budget Management Strategies in Repeated Auctions." Proceedings of the 26th International Conference on World Wide Web. 2017. https://dl.acm.org/doi/abs/10.1145/3038912.3052682 %4࿦จಡΈձ! ੢ྛ޹ 5BLBTIJ/JTIJCBZBTIJIBHJOP 1

֓ཁ w 888࠾୒ w Πϯλʔωοτ޿ࠂͷ܁Γฦ͠ΦʔΫγϣϯʹ͓͚Δ༧ࢉ؅ཧઓུʹ͍ͭͯ w ഑৴ϓϥοτϑΥʔϜ͕༧ࢉ؅ཧઓུΛఏڙ͢Δͱ͖ʹ w ͭͷઓུʹγεςϜۉߧͷଘࡏΛূ໌ w
͜ΕΒͷઓུ͕ചΓखͷརӹͱങ͍खͷޮ༻ͷτϨʔυΦϑʹͲΜͳӨڹ ͕͋Δͷ͔Λ໌Β͔ʹͨ͠ w ࣮ݧ w ϦεςΟϯά޿ࠂͷΦʔΫγϣϯͷ࣮σʔλͰ࣮ূΛߦͳͬͨ

ิ଍ͳͥۉߧͷ෼ੳΛ͢Δͷ͔ w ܦࡁֶͰ͸ࢢ৔ͷΞ΢τϓοτ͸ࢢ৔ͷϧʔϧɾϝΧχζϜʹΑͬͯมΘΔࣄ ͕஌ΒΕ͍ͯΔ w ചΓखͷརӹɺങ͍खͷޮ༻ɺ8FMGBSFʜ w ϓϨΠϠʔ͸ࣗ༝ʹ߹ཧతͳߦಈ͕ͱΕΔ͕ɺ৚͕݅ἧ͑͹ۉߧঢ়ଶʹ޲͏ w ۉߧ͸ϝΧχζϜ͕΋ͨΒ͢ӨڹΛಛ௃͚ͮͨ΋ͷͱ͍͑Δ
w ۉߧΛൺֱ͢ΔࣄͰ༷ʑͳ؍఺ʹ͓͍ͯϝΧχζϜͷ༏ྼ͕Θ͔Δ w ചΓख͕ಘ͔ങ͍ख͕ಘ͔ɺޮ཰ੑɺެฏੑɺӕΛ͍ͭͨํ͕ಘ͔ͳͲ w ϝΧχζϜͷྫΦʔΫγϣϯํࣜ w ࢢ৔͸݁ࠗࢢ৔΍ब৬ࢢ৔ͱ͍ͬͨՁ֨Λ͚ͭͳ͍औҾࢢ৔΋ؚΉ

*OUSPEVDUJPO

޿ࠂϓϥοτϑΥʔϜ͕޿ࠂओʹఏڙ͢Δ ༧ࢉ؅ཧػೳ w ೔༧ࢉΛઃఆ͓͚ͯ͠͹ͦΕҎ্࢖Θͳ͍ w ํࡦͷྫ w ࢖͍Ռͨͨ͠ΒࢭΊΔ͚ͩ w
֬཰తʹεϩοτϦϯάͯ͠೔͔͚ͯ࢖͏ w ΦʔΫγϣϯೖࡳֹۚΛԼ͛Δ উ཰ΛԼ͛Δ w ͜ΕΒ͸ϓϥοτϑΥʔϜ͓Αͼ޿ࠂओʹҟͳΔ݁ՌΛಋ͘ w طଘݚڀ͸༧ࢉ؅ཧͷ֬཰త࠷దԽͷςΫχοΫ΍ͦͷઃఆʹ͓͚ ΔήʔϜཧ࿦తͳۉߧͷ෼ੳͩͬͨ

޿ࠂϓϥοτϑΥʔϜ͕޿ࠂओʹఏڙ͢Δ ༧ࢉ؅ཧػೳ ଓ w Ұൠతͳ޿ࠂओޮ༻࠷େԽϞσϧʹ͓͍ͯ͸༧ࢉ੍໿ΑΓ΋ɺޮ༻ Λ࠷େʹ͢Δೖࡳʹؔ৺͕͋Δɻͭ·Γ༧ࢉΛ௚઀޿ࠂओ͕؅ཧ͢ Δɻ w ͔࣮͠͠ࡍʹ͸޿ࠂओ͸͜ͷΑ͏ͳήʔϜཧ࿦తͳۉߧઓུΛͱΔ ͷͰ͸ͳ͘ɺ͔ͳΓͷׂ߹ͷ޿ࠂओ͕ϓϥοτϑΥʔϜͷ༧ࢉ؅ཧ
ϝΧχζϜΛར༻͍ͯ͠Δ w ͱ͋Δ͕͜ͷ࿦จ͸ηΧϯυϓϥΠεΦʔΫγϣϯલఏͳͷͰ ϑΝʔετϓϥΠεΦʔΫγϣϯԽ͕ਐΜͩσΟεϓϨΠ޿ࠂͷྖҬ Ͱ΋ͦ͏ͳͷ͔͸ٙ໰

ຊ࿦จͷߩݙ w ޿ࠂϓϥοτϑΥʔϜ͕޿ࠂओʹ୅Θͬͯ༧ࢉ؅ཧϝΧχζϜΛఏ ڙͨ͠ͱ͖ͷγεςϜۉߧΛ෼ੳɾൺֱ͢Δ w ༧ࢉ؅ཧઓུΛݚڀ͢ΔͨΊͷ৽͍͠ϑϨʔϜϫʔΫͷఏҊ w ༧ࢉ؅ཧϝΧχζϜؒͷ༏ྼؔ܎ͷূ໌ w ͋Δ༧ࢉ؅ཧϝΧχζϜͷݩͰɺങ͍ख͕ͦΕͧΕ࠷΋߹ཧతͳ
ߦಈΛऔͬͨͱ͖ͷΞ΢τϓοτ͕ϝΧχζϜؒͰͲ͏͕ࠩ͋Δ ͷ͔Λ໌Β͔ʹͨ͠ͷʹ૬౰͢Δ w ཧ࿦ʹΑͬͯಋ͔Εͨ༧ࢉ؅ཧઓུͷ૬ରతͳύϑΥʔϚϯεΛ࣮ ࡍͷσʔλͰ࣮ূ

ઃఆ w ങ͍ख͸ਖ਼௚ʹධՁֹΛਃࠂ͢Δ w ങ͍ख͸༧ࢉ੍໿Λ׬શʹຬͨ͢૯ࢧग़ʹͷΈڵຯ͕͋Δ w ചΓख͸ങ͍खͷ୅ߦͰ༧ࢉ؅ཧϝΧχζϜΛ࣮ߦ͢Δ

܁Γฦ͠ΦʔΫγϣϯઃఆ w ਓͷങ͍ख w ചΓख͸ ճɺػձίετ ͰࡒΛചΔ w ങ͍ख ͸༧ࢉ
Λͭ w ങ͍ख ͷධՁֹ ͷ୆͸ ͱ͠ɺධՁֹ͸֬཰ີ౓෼෍ ͔ΒಠཱʹҾ͘ɻ ͜ͷ෼෍ͷྦྷੵؔ਺Λ ɺ ͱ͢Δɻ w ͷԾఆʹ͍͔ͭ͘ͷ৚݅Λ͓͘ SFHVMBS w WJSUVBMWBMVFGVODUJPO ͕୯ௐ૿Ճ w ചΓखʹͱͬͯͷ࠷ద஋ Λղ͘ͱ͖ʹͰͯ͘Δ n T c > 0 i Bi i x χi fi Fi ¯ Fi (x) := 1 − Fi (x) fi ψi (x) := x − ¯ Fi (x) f(x) p = argmaxv v ⋅ (1 − F(v))

༧ࢉ؅ཧϝΧχζϜͷఆࣜԽ w ങ͍खͷධՁֹۭؒ w ങ͍ख ͷϓϥοτϑΥʔϜύϥϝʔλ w Λύϥϝʔλۭؒ
w ༧ࢉ؅ཧϝΧχζϜ͸ׂΓ౰ͯϧʔϧ ͱ੥ٻϧʔϧ ͰදΘ͢ w ͸֤ങ͍ख΁ͷׂΓ౰ͯ֬཰ w w ͷݩͰͷങ͍ख ͷظ଴ࢧग़ w ͷݩͰͷങ͍ख ͷ֫ಘՁ஋ χ = χ1 × χ2 × ⋯ × χn i 𝒮i ∈ [s i , ¯ si ] 𝒮 = 𝒮1 × 𝒮2 × ⋯ × 𝒮n Q M Q : χ × 𝒮 → Δn Δn M : χ × 𝒮 → ℝn s i Gi (s) = T𝔼x [Mi (x; s)] s i Vi (s) = T𝔼x [xi Qi (x; s)]

γεςϜۉߧ w ങ͍खશһͷϓϥοτϑΥʔϜύϥϝʔλͷ૊߹ͤ w w ֤ങ͍खͷύϥϝʔλ ͸ങ͍खͷظ଴ޮ༻ Λ࠷େʹͭͭ͠ɺظ ଴ࢧग़ΛͦΕҎ্૿΍ͤͳ͍஋
w γεςϜۉߧʹ͓͚Δརӹͱޮ༻ͷൺֱͷҙٛ w ͋Δ༧ࢉ؅ཧϝΧχζϜͷݩͰɺങ͍ख͕ͦΕͧΕ࠷΋߹ཧతͳߦಈΛऔͬͨ ͱ͖ͷചΓखͷརӹɾങ͍खͷޮ༻ΛϝΧχζϜؒͰൺֱ͢Δ͜ͱʹ૬౰͢Δ s* = (s* i )n i=1 ∈ 𝒮 s Vi (s) − Gi (s)

.FDIBOJTNT

ݕ౼͢Δ༧ࢉ؅ཧϝΧχζϜ ࠷௿མࡳՁ֨༗ΓηΧϯυϓϥΠεΦʔΫγϣϯΛগ͠վม͢Δɻ ϓϥοτϑΥʔϜ͸ങ͍खຖͷ༧ࢉ؅ཧϝΧχζϜͷύϥϝʔλTΛܾΊΔɻT͕େ͖͍ͱࢧग़͕ݮΔɻ #JE4IBEJOH 4 ೖࡳֹۚΛׂΓҾ͘ɻׂΓҾֹ͍ͨۚͰΦʔΫγϣϯΛߦͳ͏ɻ ങ͍खຖʹׂΓҾ͘౓߹͍Λઃఆ͢Δ .VMUJQMJDBUJWF#PPTUJOH .# উར৚݅ʹೋҐՁ֨ͱͷՁ֨ࠩΛ෇༩͢Δ
ങ͍खຖʹՁ֨ࠩͷେ͖͞Λઃఆ͢Δ 3FTFSWF1SJDJOH 3 ങ͍खຖʹϑϩΞϓϥΠεΛઃఆ͢Δ 5ISFTIPMEJOH 5 ങ͍खຖʹΦʔΫγϣϯࢀՃᮢ஋ͱͳΔධՁֹΛઃఆ͢Δ 5ISPUUMJOH 50 ങ͍खຖʹΦʔΫγϣϯࢀՃ֬཰Λઃఆ͢Δ 1SPpU0QUJNBM 10 རӹ࠷େԽ ϕϯνϚʔΫ 6UJMJUZ0QUJNBM 60 ޿ࠂओޮ༻࠷େԽ ϕϯνϚʔΫ

༧ࢉ؅ཧϝΧχζϜͷఆٛ 5BCMF ׂΓ౰ͯϧʔϧ2 ͱ ੥ٻϧʔϧ.

γεςϜۉߧ͕ଘࡏ͢Δ͜ͱͷূ໌ w ങ͍खͷධՁ͕ങ͍खຖʹҟͳΔઃఆ BTZNNFUSJDTFUUJOH w l#JETIBEJOH 4 NVMUJQMJDBUJWFCPPTUJOH .#
UISFTIPMEJOH 5 BOEUISPUUMJOH 50 BENJUBTZTUFNFRVJMJCSJVNz w 3FTFSWFQSJDJOH 3 BENJUTBTZTUFNFRVJMJCSJVNXIFOUIF SBOHFTPGSFTFSWFQSJDFTBSFTVJUBCMZSFTUSJDUFE BOECJE EJTUSJCVUJPOTIBWFTUSJDUMZJODSFBTJOH('3Tz w 3 ʹ͍ͭͯ͸3FTFSWFQSJDFͱධՁֹͷ෼෍ʹ੍໿͕͋Δ w ূ໌͸ϑϧόʔδϣϯ<>Λࢀর <>#BMTFJSP 43 ",JN ..BIEJBO BOE7.JSSPLOJ#VEHFUNBOBHFNFOUTUSBUFHJFTJO SFQFBUFEBVDUJPOT 0DUPCFS "WBJMBCMFBU443/IUUQTTTSODPN BCTUSBDU

%PNJOBODF3FMBUJPOTJO UIFTZNNFUSJDTFUUJOH

ิ଍ࢧ഑ؔ܎ %PNJOBODF3FMBUJPOT w ઓུͷൺֱؔ܎ w ͋Δઓུ"͕૬खͷઓུʹ͔͔ΘΒͣઓུ#ΑΓ΋େ͖ͳརಘΛ΋ͨΒ ͢৔߹ɺઓུ#͸ઓུ"ʹࢧ഑͞Ε͍ͯΔͱ͍͏ w नਓͷδϨϯϚͷྫ w
φογϡۉߧ w ࣗനઓུ͕໧ൿઓུΛࢧ഑͍ͯ͠Δˠ߹ཧతͳनਓ͸ࣗന͢Δ w ৗʹ"Λ΍͓͚ͬͯ͹͍͍ͳΒ"͕ࢧ഑ઓུ w ηΧϯυϓϥΠεΦʔΫγϣϯͰ͸ධՁֹͰೖࡳ͢Δͷ͕ऑࢧ഑ઓུ

*OUIFTZNNFUSJDTFUUJOH w શͯͷങ͍ख͕ಉ͡ධՁؔ਺ͱ༧ࢉΛ΋ͭύλʔϯ w ֤ങ͍खͷϓϥοτϑΥʔϜύϥϝʔλ͸ಉ͡ʹͳΔ w ͜ͷͱ͖།ҰͷTZNNFUSJDFRVJMJCSJVNΛ࣋ͪɺඇઢܗ૬ิੑ৚ ݅ΛΈͨ͢ w TZNNFUSJDFRVJMJCSJVNˠ͢΂ͯͷϓϨΠϠʔ͕ಉ͡ઓུΛ
ͱͬͨͱ͖ͷۉߧ w ۉߧͷϢχʔΫੑ͸ϝΧχζϜؒͷൺֱΛ΍Γ΍͘͢͢Δ

ઓུಉ࢜ͷൺֱ w ࢧ഑ؔ܎ͷهड़ʹ ͱʚΛ࢖͍ͬͯ͘Α w ༧ࢉ؅ཧϝΧχζϜ ͕ Λࢧ഑͢Δͱ͖ɺ ͸
ͷγεςϜۉߧΑΓ΋େ͖ͳ໨తՁ஋Λಋ͘ w ͜ΕΛ w ඇࢧ഑ؔ܎ʹ͋Δ৔߹͸ʚ w ൺֱΛਤࣔͰ͖ΔΑ͏ʹੑೳۂઢΛಋೖ͢Δˠ w ͷੑೳۂઢ͕ৗʹ ͷ্ʹ͋ΔͳΒ w ަࠩ͢Δ৔߹͸ઃఆʹΑͬͯ༏ྼؔ܎͕มΘΔͷͰ ≥ (Qπ, Mπ) (Qπ′ , Mπ′ ) (Qπ, Mπ) (Qπ′ , Mπ′ ) π ≥ π′ π π′ π ≥ π′ π||π′

ചΓखͷརӹ؍఺Ͱͷൺֱ w ఆཧ w 1SPpU0QUJNBM3FTFSWF1SJDJOH≥5ISFTIPMEJOH≥ 5ISPUUMJOH≥*'#JE4IBEJOH≥6UJMJUZ0QUJNBM w 3FTFSWF1SJDJOH≥6.VMUJ#PPTUJOH≥6#JE4IBEJOH w .VMU#PPTUJOHcc5ISFTIPMEJOH.VMUJ#PPTUJOHcc
5ISPUUMJOH w ചΓख͸*NQຖʹػձίετΛ࣋ͭͷͰগͳ͍ൢച਺ͰചΓखͷ ༧ࢉΛ࢖͍Ռ͢ํ͕རӹ͕େ͖͘ͳΔ w *NQͷࢧ෷ֹ͍ۚΛߴ͘Ͱ͖Δ3FTFSWF1SJDJOH͕ޮՌత

ചΓखͷརӹ؍఺Ͱͷൺֱ ଓ͖ w ൢച*NQ਺Λ ͱ͢ΔͱചΓखͷརӹ ͸ ͳͷͰফԽֹۚͱ*NQ਺ ͕ಉ͡Ͱ͋Ε͹Ͳͷઓུ΋ಉ݁͡Ռʹͳ Δ w
આ໌͸ແ͍͚Ͳ6UJMJUZ0QUJNBM͸ೖࡳஊ ߹ঢ়ଶʹͳ͍ͬͯΔɻશһ͕ݩͷ 3FTFSWF1SJDF ͜ͷ࿦จͰ͸ػձί ετ ΪϦΪϦͰೖࡳ͢ΔͱചΓखͷར ӹ͸ͳ͘ͳΔ I P = nG − cI

ങ͍खͷޮ༻؍఺Ͱͷൺֱ w ޮ༻ w ఆཧ w 6UJMJUZ0QUJNBM≥#JE4IBEJOH≥5ISFTIPMEJOH≥3FTFSWF1SJDJOH 1SPpU0QUJNBM w
#JE4IBEJOH≥6.VMUJ#PPTUJOH≥63FTFSWF1SJDJOH w 5ISPUUMJOHcc5ISFTIPMEJOH5ISPUUMJOHcc.VMU#PPTUJOH w ചΓखͷརӹͷ؍఺͔ΒΈͨ༏Ґੑͱٯస͍ͯ͠ΔɻചΓखͷརӹ͸ങ͍ खͷޮ༻Λ٘ਜ਼ʹ͢Δ͔Βࣗવɻ w #JE4IBEJOH͸5ISFTIPMEJOHΑΓ΋ߴ͍ޮ༻Λ΋ͨΒ͢͜ͱ͔Β*NQͷ ਺͕ଟ͍ͱޮ༻͸্͕Δ U = nV − nG

ങ͍खͷޮ༻؍఺Ͱͷൺֱ ଓ͖ w #JE4IBEJOH͕ϕϯνϚʔΫʹ͍ۙ w l#JETIBEJOHJTLOPXOUPCFBO PQUJNBMCFTUSFTQPOTFBOEB TUSBUFHJDCVZFSJTCFUUFSP⒎ VOJMBUFSBMMZEFWJBUJOHUPCJE TIBEJOHXIFOFWFSQPTTJCMFz
w #JE4IBEJOHͷ ͸҆͘ങ ͑Δ͚Ͳ͋͑ͯ༧ࢉফԽͷͨΊʹߴ ͘ೖࡳ͍ͯ͠Δঢ়ଶ G > 1500

&NQJSJDBM4UVEZ

γεςϜۉߧΛٻΊΔखଓ͖ ࣮σʔλ͔Βಋ͍ͨۉߧ͕ཧ࿦తͳ݁ՌͱҰக͢ΔࣄΛ֬ೝ͢ΔͨΊ

࣮σʔλͰͷ֬ೝ w ࣮ࡍͷೖࡳϩάΛೖࡳֹ< >ʹϦεέʔϧͯ͠ಗ໊σʔλΛ ࡞ͬͨɻೖࡳ਺্ҐࣾͷࢀՃͨ͠ΦʔΫγϣϯΛར༻ɺػ ձίετDͱͨ͠ɻ w ඇ߆ଋ༧ࢉ ࢖͍͖ͬͯ΋༨Βͤͯ΋ྑ͍
ʹֻ͚Δׂ߹ Λม͑ͳ͕ΒੑೳۂઢΛ࡞੒ w ݁Ռ͸ϑϧόʔδϣϯͷ࿦จΛࢀরͷ͜ͱ B1 , ⋯, B5 γ ∈ [0,1]

࣮σʔλͰͷ֬ೝ

8FMGBSFʹ͍ͭͯൺֱ GVMMWFSTJPO͔Β ʜަ׵Ͱੜ·ΕͨՁ஋ʹ૬౰͢Δ W(s) = n ∑ i=1 Vi (s)
− cI(s)

݁Ռ w 'JHVSFʹࣔ݁͢Ռ͸ཧ࿦తʹূ໌ͨ݁͠Ռͱ౳͘͠ͳͬͨ w ചΓखࢹ఺ w 3FTFSWF1SJDF͕࠷ྑ #JE4IBEJOH͕࠷ѱ w ങ͍खࢹ఺
w #JE4IBEJOH͕࠷ྑ 3FTFSWF1SJDF͕࠷ѱ w 8FMGBSFࢹ఺ w #JE4IBEJOH͕࠷ྑ 3FTFSWF1SJDF͕࠷ѱ

ײ૝ w ۉߧ෼ੳʹΑΔΞυςΫϩδοΫͷൺֱ͸ॳΊͯಡΜͩͷͰ໘ന ͔ͬͨ w 441͕3FTFSWF1SJDFPQUJNJ[BUJPOʹྗΛೖΕΔͷɺ8FMGBSF૿ ͑ͳ͍͔Βηί͍Αͳͱࢥ͍͚ͬͯͨͲ͋ͬͯͨ w ୯ҰͷϓϥοτϑΥʔϜ಺Ͱ׬݁͢ΔΦʔΫγϣϯͳΒͱ΋͔͘ ଟஈΦʔΫγϣϯͰ#JE4IBEJOH͢ΔͱޙଓΦʔΫγϣϯͰෛ͚
΍͘͢ͳΔ͔ΒσΟεϓϨΠ޿ࠂͩͱ͜ͷ࿦จͷ݁Ռͱ͸มΘΔ ؾ͕ͨ͠

論文紹介 Budget Management Strategies in Repeated Auctions (公開版)

論文紹介 Budget Management Strategies in Repeated Auctions (公開版)

Takashi Nishibayashi

More Decks by Takashi Nishibayashi

Other Decks in Science

Featured

Transcript

࿦จ঺հ Budget Management Strategies in Repeated Auctions Balseiro, Santiago, et

֓ཁ w 888࠾୒ w Πϯλʔωοτ޿ࠂͷ܁Γฦ͠ΦʔΫγϣϯʹ͓͚Δ༧ࢉ؅ཧઓུʹ͍ͭͯ w ഑৴ϓϥοτϑΥʔϜ͕༧ࢉ؅ཧઓུΛఏڙ͢Δͱ͖ʹ w ͭͷઓུʹγεςϜۉߧͷଘࡏΛূ໌ w

*OUSPEVDUJPO

޿ࠂϓϥοτϑΥʔϜ͕޿ࠂओʹఏڙ͢Δ ༧ࢉ؅ཧػೳ w ೔༧ࢉΛઃఆ͓͚ͯ͠͹ͦΕҎ্࢖Θͳ͍ w ํࡦͷྫ w ࢖͍Ռͨͨ͠ΒࢭΊΔ͚ͩ w

ઃఆ w ങ͍ख͸ਖ਼௚ʹධՁֹΛਃࠂ͢Δ w ങ͍ख͸༧ࢉ੍໿Λ׬શʹຬͨ͢૯ࢧग़ʹͷΈڵຯ͕͋Δ w ചΓख͸ങ͍खͷ୅ߦͰ༧ࢉ؅ཧϝΧχζϜΛ࣮ߦ͢Δ

.PEFM

܁Γฦ͠ΦʔΫγϣϯઃఆ w ਓͷങ͍ख w ചΓख͸ ճɺػձίετ ͰࡒΛചΔ w ങ͍ख ͸༧ࢉ

༧ࢉ؅ཧϝΧχζϜͷఆࣜԽ w ങ͍खͷධՁֹۭؒ w ങ͍ख ͷϓϥοτϑΥʔϜύϥϝʔλ w Λύϥϝʔλۭؒ

γεςϜۉߧ w ങ͍खશһͷϓϥοτϑΥʔϜύϥϝʔλͷ૊߹ͤ w w ֤ങ͍खͷύϥϝʔλ ͸ങ͍खͷظ଴ޮ༻ Λ࠷େʹͭͭ͠ɺظ ଴ࢧग़ΛͦΕҎ্૿΍ͤͳ͍஋

.FDIBOJTNT

༧ࢉ؅ཧϝΧχζϜͷఆٛ 5BCMF ׂΓ౰ͯϧʔϧ2 ͱ ੥ٻϧʔϧ.

γεςϜۉߧ͕ଘࡏ͢Δ͜ͱͷূ໌ w ങ͍खͷධՁ͕ങ͍खຖʹҟͳΔઃఆ BTZNNFUSJDTFUUJOH w l#JETIBEJOH 4 NVMUJQMJDBUJWFCPPTUJOH .#

%PNJOBODF3FMBUJPOTJO UIFTZNNFUSJDTFUUJOH

ิ଍ࢧ഑ؔ܎ %PNJOBODF3FMBUJPOT w ઓུͷൺֱؔ܎ w ͋Δઓུ"͕૬खͷઓུʹ͔͔ΘΒͣઓུ#ΑΓ΋େ͖ͳརಘΛ΋ͨΒ ͢৔߹ɺઓུ#͸ઓུ"ʹࢧ഑͞Ε͍ͯΔͱ͍͏ w नਓͷδϨϯϚͷྫ w

*OUIFTZNNFUSJDTFUUJOH w શͯͷങ͍ख͕ಉ͡ධՁؔ਺ͱ༧ࢉΛ΋ͭύλʔϯ w ֤ങ͍खͷϓϥοτϑΥʔϜύϥϝʔλ͸ಉ͡ʹͳΔ w ͜ͷͱ͖།ҰͷTZNNFUSJDFRVJMJCSJVNΛ࣋ͪɺඇઢܗ૬ิੑ৚ ݅ΛΈͨ͢ w TZNNFUSJDFRVJMJCSJVNˠ͢΂ͯͷϓϨΠϠʔ͕ಉ͡ઓུΛ

ઓུಉ࢜ͷൺֱ w ࢧ഑ؔ܎ͷهड़ʹ ͱʚΛ࢖͍ͬͯ͘Α w ༧ࢉ؅ཧϝΧχζϜ ͕ Λࢧ഑͢Δͱ͖ɺ ͸

ചΓखͷརӹ؍఺Ͱͷൺֱ w ఆཧ w 1SPpU0QUJNBM3FTFSWF1SJDJOH≥5ISFTIPMEJOH≥ 5ISPUUMJOH≥*'#JE4IBEJOH≥6UJMJUZ0QUJNBM w 3FTFSWF1SJDJOH≥6.VMUJ#PPTUJOH≥6#JE4IBEJOH w .VMU#PPTUJOHcc5ISFTIPMEJOH.VMUJ#PPTUJOHcc

ചΓखͷརӹ؍఺Ͱͷൺֱ ଓ͖ w ൢചNQ਺Λ ͱ͢ΔͱചΓखͷརӹ ͸ ͳͷͰফԽֹۚͱNQ਺ ͕ಉ͡Ͱ͋Ε͹Ͳͷઓུ΋ಉ݁͡Ռʹͳ Δ w

ങ͍खͷޮ༻؍఺Ͱͷൺֱ w ޮ༻ w ఆཧ w 6UJMJUZ0QUJNBM≥#JE4IBEJOH≥5ISFTIPMEJOH≥3FTFSWF1SJDJOH 1SPpU0QUJNBM w

ങ͍खͷޮ༻؍఺Ͱͷൺֱ ଓ͖ w #JE4IBEJOH͕ϕϯνϚʔΫʹ͍ۙ w l#JETIBEJOHJTLOPXOUPCFBO PQUJNBMCFTUSFTQPOTFBOEB TUSBUFHJDCVZFSJTCFUUFSP⒎ VOJMBUFSBMMZEFWJBUJOHUPCJE TIBEJOHXIFOFWFSQPTTJCMFz

&NQJSJDBM4UVEZ

γεςϜۉߧΛٻΊΔखଓ͖ ࣮σʔλ͔Βಋ͍ͨۉߧ͕ཧ࿦తͳ݁ՌͱҰக͢ΔࣄΛ֬ೝ͢ΔͨΊ

࣮σʔλͰͷ֬ೝ w ࣮ࡍͷೖࡳϩάΛೖࡳֹ< >ʹϦεέʔϧͯ͠ಗ໊σʔλΛ ࡞ͬͨɻೖࡳ਺্ҐࣾͷࢀՃͨ͠ΦʔΫγϣϯΛར༻ɺػ ձίετDͱͨ͠ɻ w ඇ߆ଋ༧ࢉ ࢖͍͖ͬͯ΋༨Βͤͯ΋ྑ͍

࣮σʔλͰͷ֬ೝ

8FMGBSFʹ͍ͭͯൺֱ GVMMWFSTJPO͔Β ʜަ׵Ͱੜ·ΕͨՁ஋ʹ૬౰͢Δ W(s) = n ∑ i=1 Vi (s)

݁Ռ w 'JHVSFʹࣔ݁͢Ռ͸ཧ࿦తʹূ໌ͨ݁͠Ռͱ౳͘͠ͳͬͨ w ചΓखࢹ఺ w 3FTFSWF1SJDF͕࠷ྑ #JE4IBEJOH͕࠷ѱ w ങ͍खࢹ఺