Kelly Criterion iyo Roulette
Kelly criterion wuxuu u sheegaa sharadlaha faa'iido leh inta uu sharadsanayo. Roulette wuxuu ku siinayaa jawaab cad: waxba.
Sharadka Kelly = faa'iido / odds. Faa'iido taban, qaaciddadu waxay soo celisaa jajab taban, taasoo macnaheedu yahay ha sharadsan. Habkani weli wuu waxtar leeyahay si loo fahmo sababta ciyaartoyda eexdu sharadyadooda sidaas u cabbireen, iyo sababta qof kasta oo kale uu u cabbiro miisaaniyad ahaan.
Habka
John Kelly (1956) wuxuu weydiiyay sida khamaarle faa'iido la yaqaan leh uu u cabbiro sharadyada si uu u kordhiyo koritaanka muddada dheer ee raasamaalka ciyaartalacagRaasamaalka ciyaartaWadarta lacagta loo qoondeeyay ciyaarta. Maaraynta raasamaalka ciyaarta waa go'aaminta inta laga khatarinayo sharad kasta iyo fadhi kasta.Akhri dhammaan →. Jawaabtu, sharad bixiya b ilaa 1 oo leh itimaalxisaabItimaalFursadda natiijo, 0 ilaa 1 ama 0% ilaa 100%. Hal lambar: 1/37 = 2.70%.Akhri dhammaan → guul p iyo itimaal khasaare q = 1 − p, waa inuu sharadsado jajabka f* = (bp − q) / b ee raasamaalka. f* togan macnaheedu waa sharadso jajabkaas; eber ama taban macnaheedu waa ha sharadsan. Kelly wuxuu ka shaqeeyay Bell Labs, warqaddiisa “A new interpretation of information rate” waxay masalada dhigtay aragtida macluumaadka ee Claude Shannon: khamaarle talooyin ku hela khad buuq leh, odds caddaalad ah, wuxuu raasamaalka ku kordhin karaa ugu badnaan heerka uu khadku macluumaad qaado. 1961 Leo Breiman wuxuu caddeeyay in sharadka Kelly, muddada dheer, uu ka horreeyo istaraatiijiyad kasta oo asal ahaan ka duwan oo uu gaaro yool la dejiyay qiyaastii waqtiga ugu gaaban.
Sharadka Kelly ee roulette
CasaankasharadCasaanSharad lacag isle'eg ah oo ku saabsan 18-ka lambar ee casaanka ah. Faransiis: rouge.Akhri dhammaan →: b = 1, p = 18/37, q = 19/37; f* = (18/37 − 19/37)/1 = −1/37. Hal lambarsharadHal lambarHal lambar oo keliya, wuxuu bixiyaa 35:1.Akhri dhammaan →: b = 35, p = 1/37, q = 36/37; f* = (35/37 − 36/37)/35 = −1/1295. Sharad kasta oo caadi ah wuxuu bixiyaa jajab Kelly oo taban, sababtoo ah sharad kastaa wuxuu leeyahay faa'iido taban, sharadka ugu fiican ee koritaankuna waa eber. Sharadka Kelly ee roulette waa hab rasmi ah oo lagu yiraahdo waxa bogga faa'iidada kasiinadu sheego: ma jiro cabbir uu sharad qiime la filayo oo taban leh ku koriyo raasamaal.
Halka Kelly dhab ahaan ku shaqeeyay
Ciyaartoyda eexdu waxay lahaayeen faa'iido iyo dhibaato cabbir. Jagger, García-Pelayo-yada iyo Eudaemons dhammaantood waxay go'aamin jireen inta ay ku sharadsanayaan faa'iidadooda la cabbiray; sharad yar wuxuu khasaariyaa faa'iidada, sharad aad u badanna wuxuu khatar u geliyaa burbur ka hor inta faa'iidadu muuqan. GodqalabGodMid ka mid ah 37-ka qol ee lambarrada leh ee rotor-ka. Sidoo kale: slot, cup.Akhri dhammaan → ku dhaca 1 ka mid ah 30, Kelly wuxuu leeyahay 0.57% raasamaalka wareeg kasta lambarkaas: $10,000, qiyaastii $57 wareeggii, koritaan caadi ah oo qiyaastii 0.054% wareeggii ah, ama ×1.7 1,000 wareeg gudahood. Celceliska faa'iidadu waa labanlaab taas, 0.11% wareeggii, laakiin Kelly wuxuu beegsadaa raasamaalka caadiga ah, ma aha celceliska, kaas oo dhowr taxane oo nasiib leh ay bararaan. Nus-Kelly waa doorashada caadiga ah ee dhabta ah sababtoo ah qiyaasta faa'iidadu ma hubo.
Caadi iyo celcelis
Farqiga u dhexeeya raasamaalka caadiga ah iyo kan celceliska ah wuu ballaaran yahay. Qaado god ku dhaca 1 jeer 25 kiiba, faa'iido 44% ah, tirada Thorp u bixiyo qalabkiisa. Kelly wuxuu ku sharadsadaa 1.26% raasamaalka. Kadib 1,000 wareeg raasamaalka caadiga ahi wuxuu koray qiyaastii 11.5 jeer, halka celceliska, oo ay kor u qaadeen dhowr taxane oo aad nasiib u leh, uu koray qiyaastii 249 jeer. Sharad ka badan Kelly wuxuu kor u qaadaa celceliska wuxuuna hoos u dhigaa natiijada caadiga ah: laba jeer sharadka Kelly raasamaalka caadiga ahi wuxuu kaliya noqdaa labanlaab, laba iyo bar jeer sharadka Kelly wuu yaraadaa, inkastoo sharad kastaa weli leeyahay faa'iido 44% ah. Sharad ka yar wax yar ayuu ku kacaa: nus sharadka wuxuu hayaa qiyaastii saddex meelood afar meel koritaanka.
Marka faa'iidadu tahay male
Kelly wuxuu u qaataa in faa'iidada la yaqaan. Miiska waa la qiyaasaa, qiyaastuna inta badan way ka sarreysaa. Tusaalaha Epstein, oo ku dhisan 44% ee Thorp, waa lambar ku dhacay 49 jeer 1,225 wareeg oo giraangir laba-ebergiraangirEberHal godka cagaaran ee aan daboolin sharad lacag isle'eg ah, dersin ama tiir; isha faa'iidada kasiinada. Sidoo kale: single zero, 0.Akhri dhammaan → ah: 1 jeer 25 kiiba, faa'iido 44% ah, sharad Kelly oo 1.26% ah. Laakiin giraangir laba-eber ah oo caddaalad ahi waxay leedahay 38 lambar, kan ugu mashquulsan wuxuu gaaraa 49 ama ka badan 1,225 wareeg qiyaastii hal jeer sagaal kiiba. Lambarka ugu mashquulsan diiwaan kasta sidoo kale waa kan nasiibku ugu caawiyay, sidaas darteed xataa god dhab ahaan eex leh wuxuu diiwaanka uga muuqdaa si ka fiican sida uu yahay: inkaarta guulaystaha. Haddii godkaasi dhab ahaan eex lahayn, sharadka Kelly waa sharad kaliya oo celcelis ahaan khasaara 5.26%. Jawaabaha caadiga ahi waa in la sharadsado jajab Kelly ah oo la sii tiriyo ilaa faa'iidadu ka badbaado diiwaan cusub (eeg Sida Loo Ogaado Eexda Giraangirta Roulette).
Dad faa'iido leh
Ogaanshaha faa'iidadu la mid maaha isticmaalkeeda si fiican. Haghani iyo Dewey (2016) waxay 61 qof mid kasta siiyeen $25 si ay nus saac ugu sharadsadaan lacag bir ah oo loo sheegay inay madax ku soo dhacdo 60% waqtiga, iyadoo guulaha lagu xaddiday $250 (ciyaartoydu xadka waxay ogaadeen oo keliya markay gaareen). Kelly wuxuu leeyahay sharadso 20% raasamaalka madaxa mar kasta, qorayaashuna waxay qiyaaseen in sharadyo joogto ah oo 10–20% ah ay gaari lahaayeen xadka qiyaastii 95% waqtiga. Kaliya 21% ciyaartoyda ayaa gaaray. 28% way burbureen, 18 ka mid ah 61 waxay wax walba ku sharadsadeen hal tuuris, qiyaastii saddex meelood meel waxay ku dhammeeyeen wax ka hooseeya $25 ay ku bilaabeen, celceliska lacag-bixintunalacagLacag-bixinWaxa sharad guuleystay soo celiyo, oo lagu sheego “X ilaa 1” oo lagu daro sharadka la soo celiyay.Akhri dhammaan → wuxuu ahaa $91. Kaliya shan ka mid ah 61 ayaa maqlay Kelly criterion. Faa'iido leh, cabbirka sharadyadu ayaa go'aamiya inaad haysato; faa'iido la'aan, sida giraangir caddaalad ah, cabbir ma caawiyo.
Waxa la isticmaalo halkii
Faa'iido la'aan, cabbirku waa su'aal miisaaniyadeed, ma aha su'aal koritaan: sharad = inta aad diyaar u tahay inaad ku bixiso wareeg kasta, oo laga soo saaray miisaaniyadda saacadda iyo xawaaraha miiska (bogga Maaraynta Raasamaalka Ciyaarta Roulette). Waxa Kelly ku biiriyay ciyaartoyga caadiga ah waa caddaynta jawaabtiisa: eber. Wax kasta oo aad ka sarraysiiso eber waa qiimaha ciyaarta, adigaa doortay, ma aha mid qaacido kasta hagaajisay.
Qoraal ku saabsan Kelly jajab ah iyo “maaraynta lacagta”
Nidaamyada loo iibiyo “Kelly roulette” ee ku taliya boqolley go'an oo raasamaalka sharad kasta waa sharad joogto ahistaraatiijiSharad joogto ahIsla qadarka la sharadsado wareeg kasta. Saldhigga kordhin kasta lala barbardhigo.Akhri dhammaan → oo sharadkiisu yaraanayo. Waxay khasaaraan 2.70% wadarta sharadyadalacagWadarta sharadyadaWadarta lacagta la sharadsaday fadhi ama muddo, oo lagu tiriyo guulaha dib loo sharadsaday. Sidoo kale: handle, action.Akhri dhammaan →, sababtoo ah sharadku wuu yaraadaa marka aad khasaarto, waxay burburka ka dhigaan mid gaabis ah halkii ay ka dhigi lahaayeen mid aan suurtagal ahayn. Waa hab macquul ah oo miisaaniyad loogu dheereeyo; ma aha Kelly, kaas oo odhan lahaa jooji.
Qaacido
Su'aalo
Kaliya faa'iido la cabbiray oo ka timid giraangir eex lehshaybaarGiraangir eex lehGiraangir ay cilladaheeda jireed ka dhigaan godad qaarkood kuwo ka itimaal badan 1/37. Waa dhif, waxaana lagu ogaan karaa oo keliya kumanaan wareeg oo la diiwaangeliyay.Akhri dhammaan → ama qalab waqti-cabbir. Ma aha giraangir caddaalad ah.
Sababtoo ah qiyaasta faa'iidadu khalad bay leedahay, sharad aad u badanna wuxuu ka qaalisan yahay sharad yar. Nus-Kelly wuxuu bixiyaa saddex meelood afar meel koritaanka iyadoo kala-duwanaanshuhuxisaabKala-duwanaanshoCabbir muujinaya sida ballaaran ee natiijooyinku ugu kala firdhsan yihiin qiimaha la filayo agtiisa. Sare hal lambar, hoose sharadyada lacag isle'eg.Akhri dhammaan → yahay afar meelood meel.
Maya. Kelly wuxuu hagaajiyaa koritaanka marka faa'iido jirto. Maaraynta raasamaalku waxay xakameysaa kharashka marka aysan jirin.
Madadaalo Xisaabeed
Marka hore adigu isku day. Xal kasta waa qarsoon yahay; fur marka aad diyaar u tahay.
Qalabka saadaasha ee Thorp iyo Shannon wuxuu siin jiray qiyaastii 44% faa'iido. Hal lambar oo la saadaaliyay, Kelly wuxuu leeyahay sharadso 1.26% raasamaalka ciyaartalacagRaasamaalka ciyaartaWadarta lacagta loo qoondeeyay ciyaarta. Maaraynta raasamaalka ciyaarta waa go'aaminta inta laga khatarinayo sharad kasta iyo fadhi kasta.Akhri dhammaan →. Ka warran haddii isla 44%-kaas faa'iidada ahi uu yimaado hal sharad oo ku kala fidsan shan lambar oo isku xiga?
Muuji xalka
Jawaab Markaas Kelly wuxuu leeyahay 7.1% raasamaalka, raasamaalkuna wuxuu u koraa ku dhawaad lix jeer ka dhakhso badan: qiyaastii 1.25 milyan laab 1,000 sharad gudahood, marka la barbardhigo qiyaastii 11.5 laab.
Hal lambarsharadHal lambarHal lambar oo keliya, wuxuu bixiyaa 35:1.Akhri dhammaan → oo leh 44% faa'iido wuxuu guulaystaa fursad 0.04, maadaama 36 × 0.04 = 1.44. Sharadka Kelly waa faa'iidada loo qaybiyay odds-ka: 0.44/35 = 1.26%. Korriinka caadiga ah ee sharadkiiba waa 0.04 × ln(1 + 35 × 0.0126) + 0.96 × ln(1 − 0.0126) ≈ 0.00244, sidaa darteed 1,000 sharad waxay raasamaalka ku dhuftaan qiyaastii e^2.44 ≈ 11.5.
Hal sharad ku kala fidi shan lambar, sharadkuna wuxuu guulaystaa fursad 0.20 wuxuuna bixiyaa 36/5 halkii 1, taas oo ah 6.2 ilaa 1: weli 44% faa'iido ah, maadaama 0.2 × 7.2 = 1.44. Sharadka Kelly wuxuu noqdaa 0.44/6.2 = 7.10%, korriinka sharadkiibana 0.01404, qiyaastii 5.75 jeer intaas. 1,000 sharad gudahood taasi waa e^14.04, qiyaastii 1.25 milyan.
Faa'iidadu waa isku mid; isbeddelladu waa ka yar yihiin. Sharad guulaysta hal mar shantii wuxuu qaadi karaa qayb ka weyn raasamaalka iyadoo aan khatar burbur la gelin, korriinka Kelly-na taas ayuu abaal-mariyaa. Thorp (1984) tusaalahan wuxuu u adeegsadaa inuu muujiyo sababta ciyaartoyda saadaashu ugu sharadsadaan qaybogiraangirQaybQaanso isku xiran oo godad ah oo giraangirta ku taal, oo ka duwan koox ku taal shaxda sharadka.Akhri dhammaan →. Dhab ahaan shan derissharadDerisLambar iyo labada god ee dhinac kasta uga yaal giraangirta, shan chip oo hal lambar ah.Akhri dhammaan → waxay leeyihiin faa'iido wax yar ka hoosaysa lambarka keliya ee ugu fiican (eeg Ciyaarta Faa'iido-raadinta ee Roulette).
Ciyaartoy ayaa helay laba godqalabGodMid ka mid ah 37-ka qol ee lambarrada leh ee rotor-ka. Sidoo kale: slot, cup.Akhri dhammaan → oo isku xiga oo mid kastaa soo baxo 1 jeer 25-kii halkii 1 jeer 37-kii. Ma inuu sharadsado splitsharadSplitLaba lambar oo isku xiga, wuxuu bixiyaa 17:1.Akhri dhammaan →-ka u dhexeeya 17 ilaa 1, mise lambar kasta hal lambar ahaan? Intee ayuu Kelly leeyahay, raasamaalkuna intee dhakhso ah ayuu u koraa marka la barbardhigo hal god oo noocaas ah?
Muuji xalka
Jawaab Farqi ma samayso: 2.59% raasamaalka split-ka, ama 1.29% lambar kasta. Raasamaalka caadiga ahi wuxuu koraa qiyaastii 155 jeer 1,000 wareeg gudahood, marka la barbardhigo 11.5 jeer hal god.
Split-ku wuxuu guulaystaa itimaalxisaabItimaalFursadda natiijo, 0 ilaa 1 ama 0% ilaa 100%. Hal lambar: 1/37 = 2.70%.Akhri dhammaan → 2/25 = 0.08 wuxuuna bixiyaa 17 ilaa 1, sidaa darteed wuxuu celiyaa 18 × 0.08 = 1.44 unuglacagUnugSharadka saldhigga ah ee nidaam ama ciyaartoy ku shaqeeyo. Kordhinnada waxaa lagu sharxaa unugyo si ay ugu shaqeeyaan miis kasta.Akhri dhammaan → kasta oo la sharadsaday: faa'iido 44% ah, la mid ah hal god oo eex leh. Sharadka Kelly waa faa'iidada loo qaybiyay odds-ka: 0.44/17 = 2.59%.
Laba sharad oo hal lambar ah oo 1.29% midkiiba waxay bixiyaan natiijooyin isku mid ah. Marka mid ka mid ah lambarradu soo baxo, hal chiplacagChipCalaamad u taagan qadar go'an, oo la dul dhigo shaxda sharadka si sharad loo sameeyo. Sidoo kale: check.Akhri dhammaan → wuxuu helaa 35 ilaa 1 kan kalena waa lumaa, saafi 34 × 1.29% = 44% raasamaalka, sida split-ku bixiyo; marka midkoodna soo bixin, labaduba waa lumaan.
Korriinka wareeggiiba waa 0.00505, marka la barbardhigo 0.00244 hal god, sidaa darteed 1,000 wareeg gudahood raasamaalka caadiga ahi wuxuu koraa e^5.046 ≈ 155 jeer halkii e^2.44 ≈ 11.5 jeer. Laba lambar oo wanaagsan waxay aad uga qiimo badan yihiin laba jeer mid, taasina waa sababta ciyaartoyda eexdu ugu sharadsadaan kooxo lambarro ah. Thorp (1984) wuxuu sameeyaa isla barbardhiggaas.
Kaftan yar
Sharadka roulette ee ugu fiican koritaanka waa tiro qof walba horay u yaqaan.
Ilaha
- Kelly, J. L. “A new interpretation of information rate.”
- Breiman, L. “Optimal gambling systems for favorable games.”
- Thorp, E. O. “Optimal gambling systems for favorable games.”
- Thorp, E. O. The Mathematics of Gambling.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.
- Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Haghani, V. & Dewey, R. “Rational decision-making under uncertainty: observed betting patterns on a biased coin.”