経営課題にAIを!ビジネスインフォマティクス研究会 (第14回)
JSAI Special Interest Group on Business Informatics (SIG-BI #14)

‫ن‬ൣϞσϧʹ‫ͮ͘ج‬ஊ߹ҡ࣋ϝΧχζϜͷ‫ڀݚ‬
Analisys of collusion model in application of norm game modelॾ౻ल޾ 1 ɹ૔‫ڮ‬અ໵ 1
Hideyuki MOROFUJII1 ,Setsuya KURAHASHI111ஜ೾େֶେֶӃϏδωεՊֶ‫ڀݚ‬ՊGraduate School of Business Sciences, University of Tsukuba

Abstract: ެ‫ڞ‬ௐୡʹ͓͚Δஊ߹͕ɼࣾձతʹ໰୊ͱ͞Ε͍ͯΔɽAxelrod ͸ɼ‫ن‬ൣΛʮॴ༩ͷࣾձతഎ‫͍͓ͯʹܠ‬ɼ‫ݸ‬ਓ͕Ұൠతʹ͋Δछͷํ๏Ͱߦಈ͠ɼͦͷΑ͏ͳํ๏ͰͷߦಈΛଵΔͷΛ໨ܸ͞Εͨͱ͖ʹ͸ॲേ͞ΕΔɼͦͷఔ౓ʹґଘͯ͠ଘࡏ͢Δ΋ͷʯͱఆٛͯ͠ɼ‫ن‬ൣͷҡ࣋ʹ͸ʮϝλ‫ن‬ൣʯ͕ඞཁͰ͋Δ͜ͱΛࣔͨ͠ɽ͜ͷϞσϧΛԉ༻ͯ͠ஊ߹ϞσϧΛߏங͠ɼγϛϡϨʔγϣϯΛ࣮ߦͯ͠ɼஊ߹ͷ੒ཱʹ͸ϝλ‫ن‬ൣ·Ͱ͸ඞཁͰ͸ͳ͘ɼൺֱత༰қʹҡ࣋͠ಘΔ͜ͱΛࣔͨ͠ɽ1͸͡Ίʹެ‫ڞ‬ௐୡʹ͓͚Δஊ߹͕ɼࣾձతʹ໰୊ͱ͞Ε͍ͯΔɽஊ߹Λ๷ࢭ͢ΔͨΊʹ͸ɼͦΕ͕ҡ࣋͞ΕΔϝΧχζϜΛ෼ੳ͢Δඞཁ͕͋Δɽຊ‫Ͱڀݚ‬͸ɼஊ߹ͷҡ࣋ʹ͸ɼ‫ྗ੍ڧ‬ʢ‫ن‬ൣʣ͕ඞཁͰ͋ΔͱͷԾઆΛཱͯɼ‫ن‬ൣ͕ஊ߹ͷҡ࣋ʹՌͨ͢໾ׂΛ෼ੳ͢Δɽ2ؔ࿈‫ڀݚ‬ஊ߹͸ɼඇެࣜ૊৫Ͱ͋Γɼ‫ݸ‬ʑͷΤʔδΣϯτͷߦಈ͕ஊ߹Λ੒ཱͤ͞Δ΋ͷͱߟ͑Δͱɼެࣜ૊৫Λ૊৫͢Δͱ͖ʹඞཁͳτοϓμ΢ϯ‫ܕ‬ͷΞϓϩʔνͰ͸ͳ͘ɼϘτϜΞοϓ‫ܕ‬ͷΞϓϩʔν͕ඞཁͱͳΔɽϘτϜΞοϓ‫ܕ‬ͷΞϓϩʔνʹ͸ ABMʢAgentBasedModeling) ͕͋Δ͕ɼ͜ΕʹΑΓ‫ن‬ൣΛ‫ͨ͠ڀݚ‬΋ͷͱ͠
ͯɼAxelrod[3] ͷ‫͕͋ڀݚ‬Δɽ·ͨɼೖࡳ͸ҰछͷΦʔΫγϣϯͰ͋Δ͕ɼΦʔΫγϣϯʹ͓͚Δஊ߹Λ‫ͨ͠ڀݚ‬΋ͷͱͯ͠ɼMcAfee,
R.P. and McMillan, J. [2] ͕͋Δɽ

2.1‫ن‬ൣϞσϧ

Axelrod[3] ͸ɼϞσϧԽͱγϛϡϨʔγϣϯʢ100 ճࢼߦʣͷ݁Ռ͔Βɼ‫ن‬ൣͷҡ࣋ʹ͸ʮϝλ‫ن‬ൣʯ͕ඞཁͰ͋Δ͜ͱΛࣔͨ͠ɽ˙Axelrod ͷϞσϧΛਤͰࣔ͢ͱɼਤ 1 ͷͱ͓ΓͰ͋Δɽ
Axelrod ͷϞσϧͱγϛϡϨʔγϣϯͷ݁Ռʹର͠ɼ‫ن‬ൣͷ਺ཧϞσϧΛల։͢Δͱͱ΋ʹγϛϡϨʔγϣϯΛ࠶ࢼߦͯ͠ɼࢼߦճ਺Λ૿΍͢͜ͱʹΑΓϝλ‫ن‬ൣΛҎͯͯ͠΋ඞͣ͠΋‫ن‬ൣ͕ҡ࣋͞Εͳ͍͜ͱΛࣔ͠ਤ 1: Axelrod ͷ‫ن‬ൣϞσϧ [3]ͨͷ͕ɼJose Manuel Galan and Luis R. Izquierdo[1]Ͱ͋Δɽ

2.2ΦʔΫγϣϯཧ࿦

McAfee, R.P. and McMillan, J. [2] ͸ɼஊ߹Λ StrongCartels ͱ WeakCartels ʹ෼͚ͯ෼ੳΛߦ͍ͬͯΔɽStrongCartels Ͱ͸ɼஊ߹αʔΫϧ಺Ͱͷརಘͷ෼഑͕ߦΘΕΔɽҰํͰɼWeakCartels Ͱ͸ɼརಘͷ෼഑͸ߦΘΕͣɼམࡳऀΛॱ൪ʹܾఆͨ͠ΓɼܾΊΒΕͨՁ֨Ͱೖࡳͯ͠ɼམࡳऀͷܾఆ͸ൃ஫ऀଆͷϧʔϧʢ͘͡Ҿ͖౳ʣʹ೚ͤͨΓ͢Δ͢Δ΋ͷͱ͞ΕΔɽ·ͨɼ͜ͷ෼ੳͰ͸ɼஊ߹Λҡ࣋͢ΔͨΊʹ͸ԿΒ͔ͷ‫͕ྗ੍ڧ‬ඞཁͱ͞Ε͍ͯΔ͕ɼͦͷ͋Γํʹ͍ͭͯ͸ɼઌߦ‫ڀݚ‬ΛҾ༻ཱͯͪ͠ೖΒͳ͍͜ͱͱ͍ͯ͠Δɽͦͯ͠ɼ෼ੳͷओ୊͸ɼஊ߹ͷҡ࣋ͱೖࡳऀͷ࣋ͭকདྷׂҾ཰ͷؒͷؔ܎ͱஊ߹ʹର͢ΔചΓखʢൃ஫ऀʣଆͷରࡦͱͳ͍ͬͯΔɽ

経営課題にAIを!ビジネスインフォマティクス研究会 (第14回)
JSAI Special Interest Group on Business Informatics (SIG-BI #14)

ͳ͓ɼMcAfee, R.P. and McMillan, J. [2] ͕ઌߦ‫ݚ‬‫͍ͯ͛͋ͯ͠ͱڀ‬Δจ‫ݙ‬͸ɼҰൠతͳෳ਺ੜ࢈ऀͱෳ਺ফඅऀͷҰൠతࢢ৔Λର৅ͱͨ͠෼ੳͱͳ͍ͬͯΔɽ

3త҆ఆ৚݅ (ESS:Evolutionary Stable State) Λࢉఆ͢Δɽ͜ΕʹΑΓɼஊ߹γεςϜͷ҆ఆ৚݅Λ෼ੳ͢Δɽ͜͜Ͱ͸ɼStrong Cartels ʹ͍ͭͯ෼ੳ͢Δɽஊ߹ϞσϧͰͷ Payoﬀ-equation ͸ɼҎԼͷͱ͓ΓʹఆࣜԽ͞ΕΔɽஊ߹ͷ਺ཧϞσϧ‫ن‬ൣͷ਺ཧϞσϧΛɼஊ߹ͷ਺ཧϞσϧʹվมͯ͠ɼ‫ྗ੍ڧ‬ʢ௄േʣ͕ஊ߹ʢ‫ن‬ൣʣͷҡ࣋ʹՌͨ͢໾ׂΛ෼ੳ͢Δɽ͜͜Ͱͷओ୊͸ɼҰൠࢢ৔Ͱ͸ͳ͘ɼ1 ͷൃ஫ऀ (buyer) ͱෳ਺ͷೖࡳऀ (seller) ͷ‫ݶ‬ΒΕͨ‫ڥ؀‬ͷதͰɼseller ؒͷஊ߹ (‫ن‬ൣ) ͷҡ࣋ʹ‫ྗ੍ڧ‬ʢ௄േʣ͕Ռͨ͢໾ׂͰ͋Δɽஊ߹Ͱͷ‫ྗ੍ڧ‬ʢ௄േʣ͕Կ͔͕໰୊Ͱ͋Δ͕ɼྫ͑͹ɼެ‫Ͱۀࣄڞ‬͸ଟॏԼ੥͚ߏ଄͕͋Δ͜ͱ͕ࢦఠ͞Ε͍ͯΔɽStrongCartels Ͱͷརಘͷ෼഑ͷҰ‫ܗ‬ଶͱͯ͠Լ੥͚ʹೖΔ͜ͱʹΑΔརӹ෼഑͕ߟ͑ΒΕɼ‫ڧ‬੍ྗʢ௄േʣͷҰ‫ܗ‬ଶͱͯͦ͠ͷԼ੥͚͔Β֎͞ΕΔ͜ͱʹΑΔඃ֐͕ߟ͑ΒΕΔɽWeakCartels ʹ͍ͭͯ͸ɼྫ͑͹‫ݐ‬ઃ‫͚͓ʹۀ‬Δ‫ۀ‬քஂମ͕͋Δ͜ͱʹΑΔ૬‫ޓ‬ැॿؔ܎͕͋ΓɼैΘͳ͍͜ͱʹΑΔඃ֐͕ߟ͑ΒΕΔɽஊ߹Ϟσϧ΁ͷվมʹ͋ͨΓɼ3 ͭͷ఺Λมߋ͢ΔɽୈҰʹɼஊ߹ʢ‫ڠ‬ௐʣʹΑΔརಘ R(Reward) ͷಋೖͰ
͋ΔɽAxelrod ͷ‫ن‬ൣϞσϧʹ͓͍ͯ͸ɼR ͸ 0 ͱ͞Ε͍ͯΔɽೖࡳʹ͓͍ͯ͸ɼೖࡳʹΑΓউརऀʹͳΔͱ‫ܖ‬໿͕੒ཱͯ͠རӹΛ֬อ͢ΔલఏΛ͓͍ͯɼR > 0ͱ͢Δɽୈೋʹɼཪ੾ΓʹΑΔपғͷΤʔδΣϯτʹ
༩͑Δ H Λ֎͢ɽAxelrod ͷ‫ن‬ൣϞσϧ͸ɼࣾձʹ͓͚Δެ‫ͯ͠ͱࡒڞ‬ͷ‫ن‬ൣͷ෼ੳ͕ओ୊Ͱ͋Γɼެ‫ࡒڞ‬ͷࡡऔʹΑΔपғ΁ͷඃ֐͕૝ఆ͞Ε͍ͯΔ͕ɼຊ‫ݚ‬‫͍͓ͯʹڀ‬͸ɼཪ੾Γʢஊ߹ഁΓʣʹΑΔଞͷΤʔδΣϯτʹର͢Δ௚઀ඃ֐͕૝ఆ͠ʹ͍͘͜ͱʹΑΔɽୈࡾʹɼೖࡳʹ͓͍ͯ͸ɼೖࡳ݁Ռ͕ެද͞ΕΔ͜ͱ͔Βɼཪ੾Γʢஊ߹ഁΓʣ͸‫ج‬ຊతʹൃ‫͞ݟ‬ΕΔ΋ͷͱߟ͑ΒΕΔ͜ͱ͔Βɼൃ‫ ཰ݟ‬S ͷཁૉΛ֎͢ɽϞσϧͷ֓ཁΛਤࣔ͢Δͱɼਤ 2 ͷͱ͓ΓͰ͋Δɽ

P ayof fi = Defi1 ∗ T + Defi2 ∗+j=1
j̸=i͜ΕʹΑΓɼ·ͣஊ߹ͷ਺ཧϞσϧΛߏஙͯ͠ਐԽ

nn!
P unij ∗ E +P unji ∗ P(1)

j=1
j̸=i

R Λ n Ͱআ͍ͯ͠Δͷ͸ɼஊ߹ࢀՃऀͰ R Λ෼഑͢ΔͨΊͰ͋Δɽ͜͜ͰɼAxelrod ͷ‫ن‬ൣϞσϧͷઃఆΛ౿ऻͯ͠ɼTɿཪ੾Γརಘ (Temptation)ɼEɿ௄േίετ (Enforcement)ɼPɿ௄േඃ֐ (Punishment) ͱ͠ɼAxelrod ͷ‫ن‬ൣϞσϧͱಉ͘͡ɼT = 3, E = −2, P =−9, n = 20 ͱ͢Δɽ⎧
1 If agent i def ects⎪⎪⎪bi⎪⎪⎪
P rob(Defi1 ≡ 1) = n⎪⎪!
⎪⎪⎪bi⎪⎨
i=1
Defi1 =⎪
0 If agent i cooperates
⎪
⎪
⎪bi
⎪
⎪
⎪
P rob(Defi1 ≡ 0) = 1 − n
⎪
⎪!
⎪⎪
⎪
bi
⎩
n
&i=1bi ͸ɼdetect ͢ΔೖࡳࢀՃऀͷਓ਺ͷ‫ظ‬଴஋Ͱ͋i=1Γɼ‫ڝ‬૪ೖࡳऀʢஊ߹ഁΓΛ͢Δऀʣ͔ΒϥϯμϜʹ
མࡳऀ͕બ୒͞ΕΔͨΊͰ͋Δɽ
⎧1 If agent i cooperates
⎪
⎪
⎪
n
⎪
'
⎪
⎪
⎪
Prob(Def≡1)=(1 − bi )
⎪i2
⎨
i=1
Defi2 =⎪
0 If agent i def ects
⎪
⎪n
⎪
'⎪
⎪
⎪
⎪
Prob(Def≡0)=1−(1 − bi )i2⎩n'i=1

ਤ 2: ஊ߹Ϟσϧn!

Ri=1(1 − bi ) ͱͳ͍ͬͯΔͷ͸ɼೖࡳࢀՃऀ͕શһஊ߹ʹࢀՃ͢Δඞཁ͕͋ΔͨΊͰ͋Δɽ⎧⎪
1 If agent i punishes agent j⎪⎪⎨
P rob(Defij ≡ 1) = bj ∗ vi
P unij =⎪0Ifagent i does not punish agent j⎪⎪⎩
P rob(Defij ≡ 0) = 1 − bj ∗ viൃ‫཰ݟ‬Λද͢ (bj /2) Λ֎͍ͯ͠Δɽ͜Ε͸ɼൃ‫཰ݟ‬S ͕Ұ༷෼෍Ͱ͋Δ͜ͱ͔Βɼ0 ͔Β bj ·ͰͷؒͰ͋Δ‫ظ‬଴஋Λ‫͍ͯ͠ࢉܭ‬Δ΋ͷͰ͋Δ͕ɼൃ‫ؔʹ཰ݟ‬ΘΒͣ௄േΛߦ͏͜ͱͱ͢ΔͷͰɼ౰֘‫ظ‬଴஋Λ֎͢ɽ͕ͨͬͯ͠ɼΤʔδΣϯτ i ͷ 1 ϥ΢ϯυͰͷ‫ظ‬଴རಘ͸ɼ࣍Ͱද͞ΕΔɽ経営課題にAIを!ビジネスインフォマティクス研究会 (第14回)
JSAI Special Interest Group on Business Informatics (SIG-BI #14)

Exp(payof fi ) = T ∗bin!

bi͜͜Ͱ͸ɼγϛϡϨʔγϣϯΛ࣮ߦͯ͠ɼͲͪΒͷESS ͕ͲͷΑ͏ʹग़‫͢ݱ‬Δͷ͔ɼ͋Δ͍͸ग़‫͍ͳ͠ݱ‬ͷ͔ɼ࣮ࡍͷ‫ڍ‬ಈΛ‫͢ূݕ‬ΔɽnR '+ ∗ (1 − bi ) + En i=1(2)i=1∗ vi ∗n!
j=1
j̸=i

bj + P ∗ bi ∗n!

4.1

vj

j=1
j̸=i

͜͜ʹ͓͍ͯɼΤʔδΣϯτͷ bi Λʮେ୾͞ʯɼ viΛʮ෮़౓ʯͱͯ͠ɼͦΕͧΕ࿈ଓతͰ͋Δͱ͢ΔɽmΛॴ༩ͷूஂϺʹ͓͚Δ೚ҙͷ (ઓུΛม͑Δ mutant)ΤʔδΣϯτͱͯ͠ɼͦͷઓུΛͦΕͧΕ bm , vm ͱ͢ΔɽI Λूஂ Θ ͰͷΤʔδΣϯτ m Λআ͍ͨΤʔδΣϯτͷू߹ͱ͢Δɽ͜ͷ͜ͱ͔Βɼeq(3) ‫ ͼٴ‬eq(4) ͕ ESS ঢ়ଶʹ͍ΔΤʔδΣϯτͷඞཁ৚݅ͱͳΔɽ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩∂Exp(payof fm )
⎛=

∂bm

∂bm
∂Exp(payof fm )⎜OR ⎜⎝bm = 1 AN D⎜⎜⎝

OR bm = 0 AN D

⎛

=

⎜OR ⎜
⎝vm = 1 AN D
⎛⎜⎜⎝

≤

∂bm

∂vmOR vm = 0 AN D

∂Exp(payof fi )∂vm∂Exp(payof fm )∂vm∂Exp(payof fm )∂vm∀i ∈ I≥∂bm∂Exp(payof fm )⎛∂Exp(payof fm )∂Exp(payof fi )∂Exp(payof fi )∂bm∂Exp(payof fi )
∂bm

⎞
⎟

∀i ∈ I ⎟⎠∀i ∈ I⎞⎟⎟⎠∀i ∈ I≥≤∂Exp(payof fi )∂vm∂Exp(payof fi )∂vm⎞⎟∀i ∈ I ⎟⎠∀i ∈ I⎞⎟⎟⎠⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭∀m ∈ Θ(3)∀m ∈ Θ(4)ͳ͓ɼbi ͕φογϡ‫͋Ͱߧۉ‬ΔͨΊͷඞཁ৚݅͸ɼҎԼͷͱ͓ΓͱͳΔɽ⎧ ∂Exp(payof fm )⎫=0⎪⎪⎪⎪∂bm()⎬⎨∂Exp(payof fm )OR bm = 1 AN D≥
0∀m ∈ Θ∂bm() ⎪⎪⎪⎩ OR b = 0 AN D ∂Exp(payof fm ) ≤ 0 ⎪⎭m∂bm(5)
Appendix ʹ͓͍ͯɼeq(3) ‫ ͼٴ‬eq(3) ͔ΒɼESS ͷඞཁ৚͕݅શͯͷΤʔδΣϯτ͕ಉ͡ઓུΛ࣋ͭ͜ͱͰ͋Γɼ-bi = 0, vi ̸= 0 ∀ibi = 1, vi = 0 ∀i

͕ ESS ͷඞཁ৚݅Ͱ͋Δ͜ͱ͕ূ໌͞ΕΔɽ

4γϛϡϨʔγϣϯஊ߹ͷ਺ཧϞσϧͷ෼ੳ݁Ռͱͯ͠ɼஊ߹͕੒ཱ͢
Δ bi = 0 ∀i ͱஊ߹่͕յ͢Δ bi = 1 ∀i ͷ૬൓͢Δೋͭͷঢ়ଶ͕ ESS Ͱ͋Δ͜ͱ͕ಋग़͞ΕͨɽγϛϡϨʔγϣϯ‫ͱڥ؀‬ιʔείʔυγϛϡϨʔγϣϯ‫ڥ؀‬͸ɼNetLogo1 Λར༻ͨ͠ɽ·ͣɼήʔϜͷ‫ج‬ຊతͳྲྀΕ͸ɼgameʢϞσϧͷ࣮ߦʣɼcalcStandardDeviationʢඪ४ภࠩ‫ࢉܭ‬ʣɼevolutionʢརಘʹΑΔ౫ଡͱઓུͷਐԽʣͰ͋Δɽ
ͦͯ͠ɼgame ͸ɼdetect cooperate ΛΤʔδΣϯτ਺‫܁‬Γฦ͢΋ͷͱͳ͓ͬͯΓɼdetect cooperate ͷιʔείʔυ͸ɼιʔείʔυ 1 ͷͱ͓ΓͰ͋Δɽιʔείʔυ 1: game
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3839;ཪ੾Δ͔‫ڠ‬ௐ͢Δ͔
to defect cooperate
ask turtles[;֬཰B rate ʹΑΓ‫ڠ‬ௐ͢Δ(‫ڠ‬ௐ͠ͳ͚Ε͹ཪ੾Γʣifelse random float 1 >= B rate [;‫ڠ‬ௐή Ϝ;‫ڠ‬ௐλʔτϧͷϥϕϧ෇͚set label "cooperate"][;ཪ੾Γή Ϝ;ཪ੾Γλʔτϧͷϥϕϧ෇͚set label "detect"];རಘ‫ࢉܭ‬;ม਺’cooperate_turtles’ʹʢஊ߹ʣ‫ྗڠ‬ΤʔδΣϯτΛ୅ೖ
let cooperate turtles turtles with [label = "cooperate"];ม਺’defect_turtles’ʹཪ੾Γʢஊ߹ഁΓʣΤʔδΣϯτΛ୅ೖlet detect turtles turtles with [label = "detect"]
if any? cooperate turtles [;શһ͕ʢஊ߹ʣ‫͔ྗڠ‬ΛνΣοΫ
if count cooperate turtles = number [;શһʢஊ߹ʣ‫͋Ͱྗڠ‬Ε͹ɼஊ߹རಘR ΛΤʔδΣϯτͰ෼͚߹͏ɽ
ask parents [ set utility utility + (R / number)]]if any? detect turtles [;ཪ੾ΓΤʔδΣϯτͷத͔Βɼམࡳऀ (উརऀ)ΤʔδΣϯτΛϥϯμϜʹܾఆͯ͠མࡳऀ͸རӹT ΛಘΔ
ask one of detect turtles [ set utility utility +T]]
ask turtles[;‫ن‬ൣή Ϝ
if label = "detect"[let group j other parents;ࣗ෼Λturtle i ʹ୅ೖ
let turtle i turtle who;ཪ੾Γ‫ن‬ൣή Ϝnorm game defection turtle i group j

1 https://ccl.northwestern.edu/netlogo/

経営課題にAIを!ビジネスインフォマティクス研究会 (第14回)
JSAI Special Interest Group on Business Informatics (SIG-BI #14)
40414243444546
47484950
515253544.2]]
endto norm game defection [turtle i group j];ཪ੾Γ‫ن‬ൣή Ϝask group j [;ൃ‫ऀݟ‬ͷ௄േ֬཰V ʹΑΓ௄േΛߦ͏
ifelse V rate > random float 1 [ask turtle i [set utility utility + P];ൃ‫’ऀݟ‬j’͸௄േίετ E Λෛ͏
set utility utility + E]
endఏ৚݅ͱͯ͠ R > 0 Ͱ͋ΔͷͰɼbi = 1, vi = 0 ∀i ͸ɼφογϡ‫ͳʹߧۉ‬Βͳ͍ɽ͕ͨͬͯ͠ɼஊ߹Ϟσϧͷ͏ͪ StrongCartels ʹ͓͍ͯ͸ɼϝλ‫ن‬ൣΛඞཁͱ͢Δ͜ͱͳ͘ɼஊ߹‫ن‬ൣ͕ҡ࣋͞ΕΔ͜ͱͱͳΔ6·ͱΊ‫ن‬ൣϞσϧΛԉ༻ͯ͠ஊ߹ϞσϧΛఏࣔ͢Δͱͱ΋ʹγϛϡϨʔγϣϯΛ࣮ߦͯ͠ɼStrongCartels Ͱ͸ɼ௄േͷଘࡏʹΑΓஊ߹͕༰қʹҡ࣋͞ΕΔ͜ͱΛࣔͨ͠ɽࠓ‫ޙ‬͸ɼWeakcartels ͷ৔߹ͷ෼ੳ͕ඞཁͱ͞ΕΔɽγϛϡϨʔγϣϯ݁ՌStrongCartels
100 ճࢼߦͱ 10000 ճࢼߦͷ݁ՌΛࣔ͢ɽ݁Ռͱͯ͠͸ɼbi = 0, vi ̸= 0 ∀i ͷ݁ՌͷΈ͕ग़‫͢ݱ‬Δɽਤ 3: StrongCartels Ͱͷ 100 ճࢼߦͷ݁Ռ

Appendix A: ূ໌ A͢΂ͯͷΤʔδΣϯτ͕ಉ͡ઓུΛͱΔ͜ͱ͕ ESSͷඞཁ৚݅Ͱ͋Δ͜ͱΛূ໌͢Δɽ

Exp(P ayof fi ) = Exp(P ayof fj )∀i, j ∈ I(m ∈/ I); ∀m ∈ Θ; ∀bm , vm
F = Exp(P ayof fi ) − Exp(P ayof fj ) = 0∀i, j ∈ I; ∀m ∈ Θ; ∀bm , vmbi − bj
F =T∗n!
bm +bk⎛ ⎛k=1
k̸=m⎜ ⎜+E ⎝vi ⎝bm +⎛ ⎛⎜ ⎜+P ⎝bi ⎝vm +=0ਤ 4: StrongCartels Ͱͷ 10000 ճࢼߦͷ݁Ռ5ߟ࡯਺ཧϞσϧʹΑΔ෼ੳͰ͸ɼೋͭͷ ESS ͷඞཁ৚͕݅ಋग़͞ΕͨɽҰํͰɼಉ͡ϞσϧΛγϛϡϨʔγϣϯͰ‫͢ূݕ‬Δͱɼbi = 0, vi ̸= 0 ∀i ͷΈ͕ग़‫͢ݱ‬Δɽ͜ͷ‫ݪ‬Ҽ͸ɼESS ͷୈ 2 ඞཁ৚͕݅φογϡ‫ߧۉ‬ͷඞཁ৚݅Λຬͨ͞ͳ͍ͨΊͰ͋Δɽvi = 0 ∀i ͷ৔߹ʹ eq(5) ʹ͓͍ͯ bi = 1 ∀i ͕੒ཱ͢ΔͨΊͷ RΛࢉఆ͢ΔͱɼAppendix ͷ eq(B.4) ΑΓɼR ≤ 0 =T ∗ (n − 1)
s.t. bi = 1 ∀i ͱͳΓɼϞσϧͷલn ∗ bi ∗ (1 − bi )n−1n!

k=1k̸=i,mn!

k=1
k̸=i,m⎞⎛⎟⎜bk ⎠ − vj ⎝bm +⎞⎛⎟
⎜v k ⎠ − b j ⎝v m +n!

k=1
k̸=j,mn!

k=1k̸=j,m∀i, j ∈ I; ∀m ∈ Θ; ∀bm , vm⎞⎞⎟⎟bk ⎠⎠⎞⎞⎟⎟v k ⎠⎠͜͜ͰɼR ʹؔ͢Δ߲͕ফ͑Δͷ͸ɼஊ߹ʢ‫ڠ‬ௐʣརಘ͸ɼશһ͕‫ڠ‬ௐ͢Δඞཁ͕͋ΔͨΊͰ͋Δɽ

∂F= 0 ∀i, j ∈ I; ∀m ∈ Θ; ∀bm ∈ (0, 1)∂vmP∗ (bi − bj) = 0 ∀i, j ∈ I; ∀m ∈ Θ; ∀bm ∈ (0, 1)
2bi = bj ∀i, j ∈ I; ∀m ∈ Θ ⇒ bi = bj ∀i, j ∈ Θ∂F=0∀i, j ∈ I; ∀m ∈ Θ; ∀bm ∈ (0, 1)∂bmbi − bj−T ∗+ E ∗ bm ∗ (vi − vj) = 0n!
2(bm +bk )
k=1
k̸=m

∀i, j ∈ I; ∀m ∈ Θ; ∀bm ∈ (0, 1)

bi = bj ⇒E ∗ bm ∗ (vi − vj) = 0 ∀i, j ∈ I; ∀m ∈ Θ; ∀bm ∈ (0, 1)
vi = vj ∀i, j ∈ I; ∀m ∈ Θ ⇒ vi = vj ∀i, j ∈ Θ経営課題にAIを!ビジネスインフォマティクス研究会 (第14回)
JSAI Special Interest Group on Business Informatics (SIG-BI #14)

͕ͨͬͯ͠ɼશͯͷΤʔδΣϯτ͕ಉ͡ઓུΛ࣋ͭ͜ͱ͕ ESS ͷඞཁ৚݅Ͱ͋Δ͜ͱ͕ূ໌͞Εͨɽ

Appendix B: ূ໌ BΤʔδΣϯτͷ ESS ͷඞཁ৚݅Ͱ͋ΔઓུΛಋग़͢Δɽ4∂Exp(payof fm ) 44T ∗ (n − 1) ∗ B
R=− ∗ (1 − B)n−1
42∂bm(n∗B)nV =0T ∗BRn−1≥−− ∗ (1 − B)(n ∗ B)2n(B.4)
4∂Exp(payof fi ) 44=4∂bm
V =0
⇒ {eq(3)} ⇒ B = 1
͕ͨͬͯ͠ɼ

{eq(2)} ⇒Exp(payof fi ) = T ∗bibi + bm +n!

bjR∗ (1 − bi ) ∗ (1 − bm ) ∗nn'j=1
j̸=i,m+E ∗ vi ∗ (bm +(1 − bj )n!

bj )

j=1
j̸=i,m+P ∗ bi ∗ (vm +n!

vj )

j=1
j̸=i,mҎԼͰ͸ɼbi = B vi = V ∀i ∈ Θ ͱ͓͘ɽ∂Exp(payof fm )T ∗ (n − 1) ∗ B=∂bm(n ∗ B)2R
n−1− ∗ (1 − B)
+ (n − 1) ∗ V ∗ Pn∂Exp(payof fi )T ∗B=−∂bm(n ∗ B)2
R− ∗ (1 − B)n−1 + E ∗ Vn∂Exp(payof fm )= E ∗ (n − 1) ∗ B∂vm∂Exp(payof fi )=P ∗Bi ̸= m∂vm

bi = 0, vi ̸= 0 ∀ibi = 1, vi = 0 ∀i

͕ ESS ͷඞཁ৚݅Ͱ͋Δ͜ͱ͕ূ໌͞Εͨɽj=1
j̸=i,m+-ࢀߟจ‫ݙ‬[1] Jose Manuel Galan and Luis R. Izquierdo. Appearances can be deceiving: Lessons learned
re-implementing axelrod ʟs ʟevolutionary approach to norms ʟ.
Journal of Artificial
Societies and Social Simulation, Vol. 8, p.
http://jasss.soc.surrey.ac.uk/8/3/2.html, 2005.
[2] McAfee P. and J. McMillan. Bidding Rings. American Economic Review, Vol. 82, pp. 579–599, 1992.

i ̸= mi ̸= mi ̸= m(B.1)

4∂Exp(payof fm ) 44
R= − + (n − 1) ∗ V ∗ P
4∂bm
n
B=04
R
∂Exp(payof fi ) 44≤− +E∗V =4n∂bm

B=0(B.2)⇒ {eq(3)}4 ⇒B=04∂Exp(payof fm ) 44
∂Exp(payof fi ) 44=0=0=44∂vm∂vm
B=0B=0⇒ {eq(4)} ⇒ V = ∀x(0 ≤ x ≤ 1) ⇒ {eq(B.4)} ⇒ V ̸= 044∂Exp(payof fm ) 44
∂Exp(payof fi ) 44
≤44∂vm∂vm
B̸=0B̸=0⇒ {eq(4)} ⇒ V = 0(B.3)[3] RobertAxelrodʢࣉ໺ོ༤ʢ‫؂‬༁ʣ). ‫ن‬ൣͷଅਐ.
ʮ‫ڠ‬ௐͱରཱͷՊֶʯ(The Complexity of Cooperation. Princeton University Press.(1997)). μΠΞϞϯυࣾ, 2003.