Kriter Kelly ek roulet
Kriter Kelly dir enn parier ki ena enn lavantaz komie pou mize. Pou roulet li donn enn repons kler: nanye.
Miz Kelly = lavantaz / kot. Avek enn lavantaz negatif, formil-la donn enn fraksion negatif, ki vedir pa parye. Kriter-la res itil pou konpran kifer bann zwer ki ti eksplwat defo ti swazir zot gro miz koumsa, ek kifer tou lezot dimounn bizin swazir zot miz dapre zot bidze.
Kriter-la
John Kelly (1956) ti demande kouma enn zwer ki konn so lavantaz bizin swazir so gro miz pou fer so kapital zwelarzanKapital zweTotal larzan ki ou met de kote pou zwe. Zesion kapital zwe vedir deside komie ladan ou riske par pari ek par sesion.Lir tou → grandi le plis posib lor enn long dire. Repons-la, pou enn pari ki pey b kont 1 avek probabilitematematikProbabiliteSans enn rezilta, depi 0 ziska 1 ouswa depi 0% ziska 100%. Enn nimero: 1/37 = 2.70%.Lir tou → gagne p ek probabilite perdi q = 1 − p, se mize fraksion f* = (bp − q) / b kapital-la. Enn f* pozitif vedir parye sa fraksion-la; zero ouswa negatif vedir pa parye. Kelly ti travay kot Bell Labs, ek so lartik, “A new interpretation of information rate”, ti met problem-la dan teori lenformasion Claude Shannon: enn zwer ki gagn tuyo lor enn lign ki ena lebrwi, avek kot zis, kapav fer so kapital grandi o-maximom a vites ki lign-la transport lenformasion. An 1961 Leo Breiman ti prouve ki pari Kelly, lor enn long dire, fini an avans lor ninport ki stratezi vremem diferan ek ariv enn bi fixe dan preske pli kourt letan posib.
Pari Kelly dan roulet
Pou rouzpariRouzPari peman egal lor 18 nimero rouz. An franse: rouge.Lir tou →: b = 1, p = 18/37, q = 19/37; f* = (18/37 − 19/37)/1 = −1/37. Pou enn pari lor enn sel nimeropariPari lor enn sel nimeroEnn sel nimero, peye 35:1.Lir tou →: b = 35, p = 1/37, q = 36/37; f* = (35/37 − 36/37)/35 = −1/1295. Sakenn pari standar donn enn fraksion Kelly negatif, akoz sakenn pari ena enn lavantaz negatif, ek miz optimal pou kroisans se zero. Pari Kelly lor roulet se enn fason formel pou dir seki paz lavantaz kazino dir: pena okenn gro miz kot enn pari avek lesperans negatif fer enn kapital grandi.
Kot Kelly ti vremem servi
Bann zwer ki ti eksplwat defo ti ena enn lavantaz ek enn problem gro miz. Jagger, fami García-Pelayo ek bann Eudaemons ti tou bizin deside komie pou parye lor lavantaz ki zot ti mezire, ek parye tro tipti gaspiy lavantaz-la, alor ki parye tro gro riske ruin avan ki lavantaz-la paret. Pou enn kazekipmanKazEnn bann 37 konpartiman avek nimero lor rotor. Osi: slot, cup.Lir tou → ki sorti 1 lor 30, Kelly dir 0.57% kapital zwe par tour lor sa nimero-la: avek $10,000, apepre $57 par tour, pou enn kroisans tipik apepre 0.054% par tour, ouswa ×1.7 lor 1,000 tour. Gin mwayen se de fwa sa, 0.11% par tour, me Kelly vize kapital tipik, pa kapital mwayen, ki enn pe seri ere gonfle. Mwatie Kelly se swa pratik abitiel akoz lestimasion lavantaz-la pa sir.
Tipik ek mwayen
Diferans ant kapital tipik ek kapital mwayen larz. Pran enn kaz ki sorti 1 fwa lor 25, enn lavantaz 44%, sif ki Thorp donn pou so aparey. Kelly mize 1.26% kapital-la lor li. Apre 1,000 tour, kapital tipik inn grandi apepre 11.5 fwa, alor ki mwayenn-la, ki enn pe seri byen ere tir anler, inn grandi apepre 249 fwa. Parye plis ki Kelly ogmant mwayenn-la ek bes rezilta tipik: a de fwa miz Kelly, kapital tipik zis double, ek a de fwa edmi miz Kelly li retresi, mem si sakenn pari ankor ena enn lavantaz 44%. Parye mwins pa kout boukou: lamwatie miz-la gard apepre trwa kar kroisans-la.
Kan lavantaz-la zis enn devinet
Kelly sipoze ki ou konn lavantaz-la. Lor latab li estime, ek lestimasion-la zeneralman tro for. Legzanp Epstein, lor baz 44% Thorp, se enn nimero ki ti sorti 49 fwa lor 1,225 tour enn rou doub zerorouZeroSel kaz ver ki okenn pari peman egal, douzenn ouswa kolonn pa kouver; sours lavantaz kazino. Osi: enn sel zero, 0.Lir tou →: 1 fwa lor 25, enn lavantaz 44%, enn miz Kelly 1.26%. Me enn rou doub zero zis ena 38 nimero, ek sa enn ki sorti plis parmi zot ariv 49 ouswa plis lor 1,225 tour apepre enn fwa lor nef. Nimero ki sorti plis dan ninport ki rezistr se osi sa enn ki lasans inn ed plis, alor mem enn kaz ki vremem ena defo paret pli bon dan rezistr ki li ete vremem: malediksion gagnan. Si sa kaz-la an realite pena defo, miz Kelly zis enn pari ki perdi 5.26% an mwayenn. Bann repons abitiel se parye enn fraksion Kelly ek kontign konte ziska ki lavantaz-la tini dan enn nouvo rezistr (get Kouma detekte defo enn rou roulet).
Dimounn ki ena enn lavantaz
Konn lavantaz-la pa parey kouma servi li byen. Haghani ek Dewey (2016) ti donn 61 dimounn $25 sakenn pou parye pandan enn demi-ler lor enn pies ki zot ti dir zot tom lor fas 60% letan, avek gin limite a $250 (zwer ti aprann limit-la zis kan zot ti ariv ladan). Kelly dir parye 20% kapital-la lor fas sak fwa, ek loter-la estime ki bann pari stab 10–20% ti pou ariv limit-la apepre 95% letan. Zis 21% zwer ti ariv laba. 28% ti fini fayit, 18 lor 61 ti parye tou lor enn sel lans, apepre enn tier ti fini anba $25 ki zot ti koumanse avek, ek pemanlarzanPemanSeki enn pari gagnan rapporte, ekrir kouma “X kont 1” anplis miz ki retourne.Lir tou → mwayen ti $91. Zis sink lor 61 ti deza tann koz kriter Kelly. Avek enn lavantaz, gro miz-la deside si ou gard li; san lavantaz, kouma lor enn rou zis, okenn gro miz pa ede.
Ki pou servi alaplas
San lavantaz, swazir gro miz se enn kestion bidze, pa enn kestion kroisans: miz = seki ou dakor depanse par tour, kalkile depi bidze par ler ek rit latab (paz zesion kapital zwe). Kontribision Kelly pou zwer ordiner se kler so repons: zero. Tou seki ou mize depase zero se pri pou zwe, ki ou swazir ou mem, pa okenn formil ki optimiz li.
Enn not lor Kelly fraksionel ek “zesion larzan”
Bann sistem ki vann kouma “Kelly pou roulet” ki rekomand enn pourantaz fix kapital-la par pari se miz fixstrateziMiz fixParye mem montan sak tour. Referans ki ou konpar sak progresion avek li.Lir tou → avek enn miz ki retresi. Zot perdi 2.70% total paryelarzanTotal paryeTotal larzan ki ou parye lor enn sesion ouswa enn peryod, ek bann gin ki ou reparye ladan. Osi: handle, action.Lir tou → ek, akoz miz-la retresi kan ou perdi, zot fer ruin vinn dousman olie ki zot fer li pa probab. Se enn fason rezonab pou fer enn bidze dire; se pa Kelly, ki ti pou dir aret.
Formil
Kestion
Zis avek enn lavantaz mezire lor enn rou ki ena defolaboratwarRou ki ena defoEnn rou kot bann defo fizik fer serten kaz pli probab ki 1/37. Rar, ou kapav detekte li zis avek plizier milye tour ki finn note.Lir tou → ouswa avek enn aparey kronometraz. Pa lor enn rou zis.
Akoz lestimasion lavantaz-la ena erer, ek parye tro gro kout pli ser ki parye tro tipti. Mwatie Kelly donn trwa kar kroisans-la pou enn kar varyansmatematikVaryansEnn mezir lor kouma bann rezilta dispers otour valer atandi. For pou pari lor enn sel nimero, ba pou pari peman egal.Lir tou →.
Non. Kelly optimiz kroisans kan ena enn lavantaz. Zesion kapital zwe kontrol depans kan pena okenn.
Lamizman ar matematik
Esey rezoud zot ou mem avan. Sakenn solision kase; ouver li kan ou pare.
Aparey prediksion Thorp ek Shannon ti donn enn lavantaz apepre 44%. Lor enn sel nimero predi, Kelly dir parye 1.26% kapital zwe-lalarzanKapital zweTotal larzan ki ou met de kote pou zwe. Zesion kapital zwe vedir deside komie ladan ou riske par pari ek par sesion.Lir tou →. Ki arive si mem lavantaz 44%-la ti vini lor enn pari partaze lor sink nimero vwazin?
Montre solision-la
Repons Lerla Kelly dir 7.1% kapital zwe-la, ek kapital-la grandi preske sis fwa pli vit: apepre 1.25 milion fwa lor 1,000 pari, kont apepre 11.5 fwa.
Enn pari lor enn sel nimeropariPari lor enn sel nimeroEnn sel nimero, peye 35:1.Lir tou → avek enn lavantaz 44% gagne avek sans 0.04, akoz 36 × 0.04 = 1.44. Miz Kelly se lavantaz-la divize par kot-la: 0.44/35 = 1.26%. Kroisans tipik par pari se 0.04 × ln(1 + 35 × 0.0126) + 0.96 × ln(1 − 0.0126) ≈ 0.00244, alor 1,000 pari miltipliy kapital-la par apepre e^2.44 ≈ 11.5.
Partaz enn miz lor sink nimero ek pari-la gagne avek sans 0.20 ek peye 36/5 pou 1, vedir 6.2 kont 1: touzour enn lavantaz 44%, akoz 0.2 × 7.2 = 1.44. Miz Kelly vinn 0.44/6.2 = 7.10%, ek kroisans par pari 0.01404, apepre 5.75 fwa plis. Lor 1,000 pari sa fer e^14.04, apepre 1.25 milion.
Lavantaz-la parey; bann balansman pli tipti. Enn pari ki gagn enn fwa lor sink kapav pran enn pli gro par kapital zwe-la san riske ruin, ek kroisans Kelly rekonpans sa. Thorp (1984) servi sa egzanp-la pou montre kifer bann zwer prediksion parye lor sekterrouSekterEnn ark kaz ki swiv lor rou-la, pa enn group lor tapi pari.Lir tou →. Dan pratik sink vwazinpariVwazinEnn nimero ek de kaz lor sak kote li lor rou-la, sink zeton lor enn sel nimero.Lir tou → ena enn lavantaz enpe pli ba ki sel meyer nimero (get Advantage Play).
Enn zwer finn trouv de kazekipmanKazEnn bann 37 konpartiman avek nimero lor rotor. Osi: slot, cup.Lir tou → vwazin ki sakenn sorti 1 fwa lor 25 olie 1 lor 37. Eski li bizin met enn pari lor de nimeropariPari lor de nimeroDe nimero kot-a-kot, peye 17:1.Lir tou → ant zot a 17 kont 1, ouswa sakenn nimero tousel? Komie Kelly dir, ek komie vit kapital-la grandi kompare ar enn sel kaz koumsa?
Montre solision-la
Repons Pa fer okenn diferans: 2.59% kapital zwe-la lor pari lor de nimero, ouswa 1.29% lor sakenn nimero. Kapital tipik grandi apepre 155 fwa lor 1,000 tour, kont 11.5 fwa avek enn sel kaz.
Pari lor de nimero gagne avek probabilitematematikProbabiliteSans enn rezilta, depi 0 ziska 1 ouswa depi 0% ziska 100%. Enn nimero: 1/37 = 2.70%.Lir tou → 2/25 = 0.08 ek peye 17 kont 1, alor li retourn 18 × 0.08 = 1.44 par initelarzanIniteMiz baz ki enn sistem ouswa enn zwer servi. Bann progresion dekrir an inite pou ki zot aplik lor nenport ki latab.Lir tou → mize: enn lavantaz 44%, parey kouma enn kaz ki ena defo. Miz Kelly se lavantaz-la divize par kot-la: 0.44/17 = 2.59%.
De pari lor enn sel nimero 1.29% sakenn donn egzakteman mem rezilta. Kan enn de nimero-la sorti, enn zetonlarzanZetonEnn zeton ki reprezant enn montan fix, ki ou poze lor tapi pari pou fer enn pari. Osi: check.Lir tou → gagn 35 kont 1 ek lot-la perdi, enn net 34 × 1.29% = 44% kapital zwe-la, parey kouma pari lor de nimero peye; kan ni enn ni lot pa sorti, toulede perdi.
Kroisans par tour se 0.00505, kont 0.00244 pou enn kaz, alor lor 1,000 tour kapital tipik grandi e^5.046 ≈ 155 fwa olie e^2.44 ≈ 11.5 fwa. De bon nimero vo boukou plis ki de fwa enn, ek se pou sa ki bann zwer defo parye lor group nimero. Thorp (1984) fer mem konparezon-la.
Enpe riye
Miz roulet ki optimiz kroisans se enn nonb ki tou dimounn deza konn.
Sours
- 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, 2em edision.
- Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2em edision.
- 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.”