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今からはじめるボードゲーム解析
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Kei Hirata
July 03, 2017
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1.1k
今からはじめるボードゲーム解析
組み合わせゲームの解析を通じて、様々なゲームの特性や最善手について話します。
Kei Hirata
July 03, 2017
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Transcript
ࠓ͔Β͡ΊΔϘʔυήʔϜղੳ ฏాܓ
ࠓ͢͜ͱ Έ߹ΘͤήʔϜͱʁ جຊతͳٕ๏ ؼ݁ྨʹ͍ͭͯ ہ໘ͷ
ࢀߟॻ੶ ߹ͤήʔϜཧೖ ʵউརͷํఔࣜʵ M.H.Albert (ஶ), R.J.Nowakowski (ஶ), D.Wolfe (ஶ),
ล ࣏೭ (༁) 2011/9/21 ڞཱग़൛
Έ߹ΘͤήʔϜͱʁ
Έ߹ΘͤήʔϜͱʁ ਓͷରہऀ͕ަޓʹखΛଧͪɺͲͪΒ͔ͷରہऀ͕نଇ ʹैͬͨखΛଧͭ͜ͱ͕ग़དྷͳ͘ͳΔ·Ͱଓ͘ήʔϜ ! શใήʔϜͷہ໘ ঢ়ଶ ͕શͯެ։͞Ε͍ͯΔ ! ֬ఆ
αΠίϩͳͲ ۮવʹࠨӈ͞ΕΔཁૉ͕ͳ͍ ! ༗ݶ༗ݶͷखʹΑͬͯඞͣήʔϜ͕ऴྃ͢Δ 㲈ೋਓ༗ݶ֬ఆશใήʔϜ
Έ߹ΘͤήʔϜͷྫυϛφϦϯά υϛφϦϯάͷϧʔϧ ! ަޓʹ൫໘ʹυϛϊṛΛஔ ! ઌख ࠨ ॎɺޙख ӈ
ԣ ! ઌʹஔ͚ͳ͘ͳͬͨରہऀ͕ෛ͚ ‣ ਖ਼نܗήʔϜ
υϛφϦϯάͷਐΈํ ࠨ ࠨ ӈ ӈ ࠨ ࠨ ӈ ӈ
ࠨ ӈ ࠨ ࠨͷউͪ
υϛφϦϯάͷجຊઓུ ࠨ ӈ ࠨ ࠨ ӈ ࠨ͕༗རͳख ӈ͕༗རͳख ޙͰࣗͷυϛϊṛΛஔ͚Δ༷ʹ֬อ͢Δͷ͕ྑ͍ख
ෆภήʔϜͱඇෆภήʔϜ શہ໘Ͱɺࠨͱӈͷબࢶ͕ಉ͡ήʔϜΛෆภήʔϜͱݺͿ ྫʣରہऀ͕υϛϊṛΛͲͪΒʹ͓͍ͯྑ͍߹ ΫϥϜ ରہऀʹΑͬͯબࢶ͕ҟͳΔ߹ඇෆภήʔϜͱݺͿ υϛφϦϯά
جຊతͳٕ๏
ཉுΓઓུ ήʔϜͷہ໘ʹؔ࿈͚ΒΕͨ͋ΔྔΛɺ ࠷େ·ͨ࠷খʹ͢ΔखΛଧͭ͜ͱ ྫʣυϛφϦϯάͰͷஔՄೳͳϚεͷ֬อɺ ɹɹޒฒͰͷฒͼͱΛ૿͢खͳͲ ܭࢉ͕༰қͳͨΊɺ؆୯ʹήʔϜʹద༻Ͱ͖Δ ҰํͰෳࡶͳήʔϜͷ߹ɺظతʹෆརͳબͱͳΔ͜ͱ͋Δ
બࢶաଟ ૬खͷબࢶΛ࠷େԽ ہ໘ΛෳࡶԽ ͤ͞Δઓུ w ߟྀ͖͢બࢶΛ૿͢͜ͱͰɺ૬खͷѱ͍ԠखΛ༠͏͜ͱ͕త w ෆརͳ൫໘ʹ͓͍ͯɺ૬खͷஅͷޡΓΛૂ͏ ҰํͰɺࣗͷબࢶΛ࠷େԽ͠ɺ૬खͷબࢶΛ࠷খԽ͢Δઓུ
͕ϓϩάϥϜͰ༻͍ΒΕΔ͜ͱଟ͍
ਅࣅઓུ ΫϥϜͰͷਅࣅઓུͷྫ 2n*2mͷ൫໘Ͱৗʹӈ͕উͭ ήʔϜ͕ରশੑΛ࣋ͭ߹ɺ ޙख͕ઌखͷखΛਅࣅͯଧͭඞউ๏ 2n*(2m-1)ͷ൫໘ͰɺυϛϊṛΛ தԝʹஔ͘͜ͱͰৗʹࠨ͕উͭ ӈ͕ࣗہ໘ͰͷΓखΛۮʹอͭ͜ͱͰɺ ৗʹࣗͷख൪ͰήʔϜΛऴہͤ͞ΔΑ͏ཧ͢Δ
ήʔϜͷح۰ੑ
ਅࣅઓུήʔϜͷح۰ੑ ήʔϜͷح۰ੑͱɺ ήʔϜͷ͋Δྔ͕ح͔ۮ͔ͱ͍͏ੑ࣭ͷ͜ͱ ଟ͘ͷήʔϜͰɺ͋Δྔͷح۰ੑ͕উഊΛܾఆ͢ΔॏཁͳཁૉͱͳΔ ;ͨࢁͰͷχϜʹ͓͚Δح۰ੑͱਅࣅઓུ ৗʹ2*n ʹอͭ͜ͱͰޙखউͪͷہ໘ͱͳΔ
ઓུഈआ ૬ख͕ήʔϜͷඞউ๏Λ࣋ͭͱԾఆ͢Δͱ͖ɺ ήʔϜʹΑͬͯͦͷઓུΛഈआ͢Δ͜ͱͰউརͰ͖Δ A B B A A νϣϯϓͰͷରہྫ
νϣϯϓઓུഈआʹΑͬͯɺઌखඞউ͕ূ໌Ͱ͖Δ
ઓུഈआνϣϯϓͷඇߏূ໌ νϣϯϓ͕ޙखඞউͩͱԾఆ͢Δ ઌख͕ӈ্۱Λଧͭͱ͢Δͱɺޙख ඞউͱͳΔϚεYʹखΛଧͭ ઌख͕ϚεYʹखΛଧͭͱɺͰͷہ໘ ͱಉ݁͡ՌʹͳΔͨΊɺઌख͕ඞউͱ
ͳΔ ΑͬͯɺͷԾఆޡΓͱͳΔ ࠨ ӈ ઌख͕ӈ্ʹखΛଧͬͨ߹ɺ ޙखͷશखઌखΛؚΜͩ݁ՌͱͳΔ ➡ ઌखޙखͷઓུഈआ͕ग़དྷΔ ۩ମతͳखͷఏ͕ࣔͳ͘ɺήʔϜͷੑ࣭͔Βઌख/ޙखඞউΛূ໌ = ඇߏূ໌
ήʔϜͷʮੳʯՄೳ͔ʁ ͜Ε·Ͱͷٕ๏ɺήʔϜͷ͋Δྔੑ࣭ʹ͢Δ͜ͱͰ ہ໘Λ༗རʹਐΊΔ͜ͱΛతͱ͍ͯͨ͠ அ͕ൺֱత༰қͳࢦඪʹͯ͠ɺଥͳબΛߦ͏͜ͱ ࣮ફత͕ͩɺ࠷ળखͱݶΒͳ͍ ہ໘ʹରͯͯ͢͠ͷબࢶͷ݁ՌΛੳ͠ɺ ࠷ળखઌखޙखඞউΛΔ͜ͱͰ͖Δ͔ʁ
ؼ݁ྨʹ͍ͭͯ
ήʔϜ ͋Δہ໘Ͱͷ֤ରہऀͷ࣋ͭબࢶɺήʔϜͰදݱͰ͖Δ ࠨબࢶ ӈબࢶ
ήʔϜͷԆ ήʔϜͷࢬऴہ·ͰԆ͢͜ͱ͕ग़དྷΔ ࠨબࢶ ӈબࢶ
ήʔϜͱہ໘ͷׂ ࠨ ӈ ࠨ ήʔϜہ໘Λׂͨ͠෦ήʔϜʹରͯ͠ద༻Ͱ͖Δ ෦ήʔϜͰख൪͕ަޓʹͳΔݶΒͳ͍ ήʔϜͰɺہ໘ͷख൪ߟྀ͠ͳ͍
Έ߹ΘͤήʔϜͷجຊఆཧ "ͱ#ͷରہ͢ΔήʔϜG͕༩͑ΒΕͯɺ"Λઌख൪ͱ͢Δ ͱ͖ɺGઌख൪ͷ"͕ඞͣউ͔ͭɺޙख൪ͷ#͕ඞͣউͭ ͔ͷͲͪΒ͔Ͱ͋Γɺͦͷ྆ํͱ͍͏͜ͱͳ͍ɻ ؼೲతʹɺ"͕खΛଧͬͨޙͷશہ໘ɺ#͕ઌख൪Ͱউ͔ͭɺ "͕ޙख൪Ͱউ͔ͭͷͲͪΒ͔ɻ "͕উͭબࢶ͕ҰͭͰ͋Εɺ"͜ΕΛબ͠ඞͣউͭɻ ·ͨɺ"͕উͭબࢶ͕Ұͭͳ͚Εɺ#͕ඞͣউͭ
جຊఆཧͷྫ ࠨબࢶ ӈબࢶ ࠨબࢶ ӈબࢶ A͕ඞͣউͭ B͕ඞͣউͭ
ΑΓେ͖ͳہ໘ʹ͍ͭͯɺήʔϜΛ৳͢͜ͱͰ ࠶ؼతʹઌखޙखউͪΛஅ͢Δ͜ͱ͕ग़དྷΔ
ؼ݁ྨ جຊఆཧࠨӈͲͪΒ͕ઌख൪Ͱಉ༷ʹΓཱͭɻ ؼ݁ྨ ໊শ ఆٛ N ϑΝδʔ ઌख൪ /FYU
͕ඞͣউͭ P ྵ ޙख൪ 1SFWJPVT ͕ඞͣউͭ L ਖ਼ ͲͪΒ͕ઌख൪Ͱɺࠨ -FGU ͕ඞͣউͭ R ෛ ͲͪΒ͕ઌख൪Ͱɺӈ 3JHIU ͕ඞͣউͭ ͭ·ΓࠨPSӈ͕ઌख൪ͷͱ͖ɺઌखPSޙखඞউͱͳΔ ͜ͷͱ͖ɺہ໘ύλʔϯͷؼ݁ྨʹྨ͞ΕΔ
ؼ݁ྨͷྫ Nઌख͕উͭ Pޙख͕উͭ Lࠨ͕উͭ Rӈ͕উͭ
ہ໘ͷબࢶͱؼ݁ྨ ہ໘GͷࠨӈબࢶΛGL,GRͱ͢Δͱ͖ɺ Gͷؼ݁ྨҎԼ͔ΒٻΊΒΕΔɻ ͋ΔGR 㱨R ∪P ͯ͢ͷGR
㱨R ∪ P ͋ΔGL ∈ L ∪ P N L ͯ͢ͷGL ∈ R ∪ N R P ͜ͷϧʔϧΛ༻͍Δ͜ͱͰɺ ήʔϜ͔Βہ໘ͷؼ݁ྨΛٻΊΔ͜ͱ͕Ͱ͖Δ
ήʔϜͷؼ݁ྨ ήʔϜΛऴہ·Ͱ୳ࡧ͢Δɻ
ήʔϜͷؼ݁ྨ P P P P P P P
P P P P P P P P P P P P P ऴΛPʹஔ͖͑Δ ऴඞͣPͱͳΔ
ήʔϜͷؼ݁ྨ P P P P P P P
P P P P P P P P P P P P P N N N N N N N N N P L L P N ֤બࢶͷؼ݁ྨΛͱʹɺલہ໘ͷؼ݁ྨΛٻΊΔ
ήʔϜͱؼ݁ྨʹΑΔੳ ήʔϜͱؼ݁ྨΛ༻͍Δ͜ͱͰɺ͋Δہ໘ʹ͓͚Δ ઌखޙखඞউ͜ͷͨΊͷ࠷ળखΛΔ͜ͱ͕Ͱ͖ͨ ͨͩ͠ɺҰൠʹہ໘ͷେ͖͞ʹԠͯ͡બࢶരൃతʹ૿͠ɺ ݁Ռͱͯ͠ήʔϜͷܭࢉࠔͱͳΔɻ ͜Εʹର͠ɺେ͖ͳہ໘Λখ͞ͳہ໘ͷͱଊ͑Δ͜ͱͰɺ ؼ݁ྨͷಋग़Λߦ͑Δ͜ͱ͕͋Δɻ
ہ໘ͷ
ہ໘ͷ ෳͷখ͞ͳہ໘ͷΈ߹Θ͔ͤΒͳΔہ໘Λɺ ͦΕͧΕͷখ͞ͳہ໘ͷͱΈͳ͢ɻ + +
ہ໘ͷ O͔ΒͳΔہ໘ɺ໌Β͔ʹࠨʹͱͬͯ༗རͳہ໘ͱͳΔ ͜ͷΑ͏ͳ༗ར͞Λہ໘ͷͱ͢Δ ࠨʹͱͬͯ1ख༗ར : 1 ࠨʹͱͬͯ2ख༗ར
: 2 ӈʹͱͬͯ1ख༗ར : -1 ӈʹͱͬͯ2ख༗ར : -2 or or
ہ໘ͷ ͦΕͧΕͷہ໘ͷʹ͍ͭͯɺؼ݁ྨΛߟ͑Δ + L R +
+ = = P + L R + + = = L ಉ͡ؼ݁ྨΛ࣋ͭہ໘͔ΒͳΔͰɺ ͦΕͧΕͷʹΑͬͯؼ݁ྨ͕ҟͳΔ
ͱͳΔہ໘ ؼ݁ྨPΛ࣋ͭہ໘ɺඞͣΛ࣋ͭɻ͜ͷͨΊɺ ہ໘Z ∈ Pͱͨ͠ͱ͖ɺҙͷہ໘Gʹ͓͍ͯ G + Zͷؼ݁ྨඞͣGͷؼ݁ྨͱಉ͡ͱͳΔ +
G Z ͲͪΒͷରہऀɺZʹखΛଧͨͳ͍ݶΓG+Zͷ ؼ݁Gʹۙ͘ͳΔɻ Ұํͷରہऀ͕ZʹखΛଧͬͨ߹ɺ͏Ұํ ͕ZʹޙखඞউͷԠखΛଧͭ͜ͱͰGͷؼ݁ʹม Խى͜Βͳ͍
ہ໘ͷʹ͓͚Δؼ݁ྨ L P R N L L L ?
L·ͨN P L P R N R ? R R R·ͨN N L·ͨN N R·ͨN ? ͭͷہ໘ͷ͕࣋ͭؼ݁ྨͷରԠҎԼͷ௨ΓͱͳΔ
·ͱΊ w Έ߹ΘͤήʔϜͷհ w දྫͱͯ͠υϛφϦϯά w ܦݧతʹಘΒΕΔٕज़ઓུͷհ w ؼ݁ྨͱήʔϜʹ͍ͭͯͷղઆ
w ͋Δہ໘ͷઌखޙखউ࠷ળखͷղੳ w Λ༻͍ͯہ໘Λׂ w ෦తʹʹؼ݁ྨ͕ద༻ग़དྷΔ͜ͱΛࣔͨ͠
None