࿦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

.PEFM

܁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

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&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