Kelly rul ɛn rulɛt
Di Kelly rul de tɛl pɔsin we gɛt advantej ɔmɔs fɔ bɛt. Fɔ rulɛt, di ansa klia: natin.
Kelly bɛt = advantej / ɔds. If di advantej nɛgativ, di fɔmula de gi nɛgativ pat, ɛn dat min se nɔ bɛt. Di rul stil gud fɔ ɔndastand wetin mek pleya dɛn we de ple wil we gɛt bayas bin pik dɛn bɛt mɔni so, ɛn wetin mek ɔlman ɔda fɔ pik bɛt mɔni frɔm dɛn bɔjit.
Di rul
John Kelly (1956) bin aks aw gambula we no in advantej fɔ pik in bɛt mɔni so dat in gambul mɔnimɔniGambul mɔniƆl di mɔni we yu dɔn put asay fɔ ple. Fɔ manej yu gambul mɔni min fɔ disayd ɔmɔs pan am yu go risk fɔ ɛni bɛt ɛn fɔ ɛni sɛshɔn.Rid ɔl → go gro pas ɔl if i ple lɔng tɛm. Fɔ bɛt we de pe b to 1, wit prɔbabilitimatematiksPrɔbabilitiDi chans fɔ wan rizɔlt, frɔm 0 to 1 ɔ frɔm 0% to 100%. Wan nɔmba: 1/37 = 2.70%.Rid ɔl → p fɔ win ɛn prɔbabiliti q = 1 − p fɔ lɔs, di ansa na fɔ bɛt di pat f* = (bp − q) / b pan di gambul mɔni. If f* pɔsitiv, bɛt dat pat; if i na ziro ɔ nɛgativ, nɔ bɛt. Kelly bin de wok na Bell Labs, ɛn in pepa, “A new interpretation of information rate”, bin put di prɔblɛm insay Claude Shannon in tiori fɔ infɔmeshɔn: gambula we de gɛt tip ova lain we gɛt nɔys, wit fɛr ɔds, kin mek in gambul mɔni gro, bɔt i nɔ go pas di rɛt we di lain de kɛr infɔmeshɔn. Na 1961, Leo Breiman bin pruv se Kelly bɛt, if yu ple lɔng tɛm, de dɔn ɔp pas ɛni stratɛji we rili difrɛn, ɛn i de rich gol we yu sɛt lɛk insay di shɔtis tɛm.
Kelly bɛt na rulɛt
Fɔ rɛdbɛtRɛdIvin-mɔni bɛt pan di 18 rɛd nɔmba. Na Frɛnch: rouge.Rid ɔl →: b = 1, p = 18/37, q = 19/37; f* = (18/37 − 19/37)/1 = −1/37. Fɔ bɛt pan wan nɔmbabɛtBɛt pan wan nɔmbaWan nɔmba nɔmɔ, i de pe 35:1.Rid ɔl →: b = 35, p = 1/37, q = 36/37; f* = (35/37 − 36/37)/35 = −1/1295. Ɛni standad bɛt de gi nɛgativ Kelly pat, bikɔs ɛni bɛt gɛt nɛgativ advantej, ɛn di bɛt mɔni we de mek mɔni gro pas ɔl na ziro. Kelly bɛt pan rulɛt na wan fɔmal we fɔ se wetin di pej fɔ kasino advantej de se: no bɛt mɔni nɔ de we go mek gambul mɔni gro wit bɛt we in ɛkspɛkteshɔn nɛgativ.
Usay Kelly bin wok
Pleya dɛn we bin de ple wil we gɛt bayas bin gɛt advantej, ɛn prɔblɛm fɔ pik ɔmɔs fɔ bɛt. Jagger, di García-Pelayo dɛn ɛn di Eudaemons ɔl bin fɔ disayd ɔmɔs fɔ bɛt pan di advantej we dɛn mɛzhɔ, ɛn if yu bɛt tu smɔl yu de wes di advantej, bɔt if yu bɛt tu bɔku yu kin lɔs ɔl bifo di advantej sho. Fɔ pɔkitikwipmɛntPɔkitWan pan di 37 smɔl ples dɛn wit nɔmba pan di rotɔ. Dɛn kin kɔl am bak: slot, kɔp.Rid ɔl → we de kam 1 tɛm insay 30, Kelly se bɛt 0.57% pan di gambul mɔni ɛni spin pan dat nɔmba: wit $10,000, dat na lɛk $57 fɔ ɛni spin, ɛn di nɔmal gro na lɛk 0.054% fɔ ɛni spin, ɔ ×1.7 ova 1,000 spin. Di avrej gro na tu tɛm dat, 0.11% fɔ ɛni spin, bɔt Kelly de aim fɔ di nɔmal gambul mɔni, nɔto di avrej wan, we sɔm lɔki strik dɛn de pof ɔp. Af-Kelly na di tin we pipul kin yuz fɔ tru bikɔs dɛn nɔ shɔ bɔt di advantej we dɛn ɛstimet.
Nɔmal ɛn avrej
Di gap bitwin di nɔmal gambul mɔni ɛn di avrej wan wayd. Tek pɔkit we de kam 1 tɛm insay 25, advantej fɔ 44%, di nɔmba we Thorp gi fɔ in mashin. Kelly de bɛt 1.26% pan di gambul mɔni pan am. Afta 1,000 spin, di nɔmal gambul mɔni dɔn gro lɛk 11.5 tɛm, bɔt di avrej, we sɔm rili lɔki strik dɛn pul ɔp, dɔn gro lɛk 249 tɛm. If yu bɛt mɔ pas Kelly, di avrej de go ɔp ɛn di nɔmal rizɔlt de go dɔŋ: wit tu tɛm di Kelly bɛt, di nɔmal gambul mɔni de jɔs dɔbul, ɛn wit tu ɛn af tɛm di Kelly bɛt i de shɔnk, ilɛk ɛni bɛt stil gɛt 44% advantej. If yu bɛt smɔl pas Kelly, yu nɔ lɔs bɔku: af di bɛt mɔni de kip lɛk tri kwata pan di gro.
Wen di advantej na gɛs
Kelly de tek am se yu no di advantej. Na di tebul, yu de ɛstimet am, ɛn di ɛstimet kin ay pas aw i rili de. Epstein in ɛgzampul, we i bil pan Thorp in 44%, na nɔmba we kam 49 tɛm insay 1,225 spin pan wil wit dɔbul zirowilZiroDi wan grin pɔkit we nɔ ivin-mɔni, dɔzin ɔ kɔlɔm bɛt nɔ kɔba; na usay di kasino advantej kɔmɔt. Dɛn kin kɔl am bak: wan ziro, 0.Rid ɔl →: 1 tɛm insay 25, advantej fɔ 44%, Kelly bɛt fɔ 1.26%. Bɔt fɛr wil wit dɔbul ziro gɛt 38 nɔmba, ɛn di wan we kam pas ɔl de rich 49 ɔ mɔ insay 1,225 spin lɛk wan tɛm insay nayn. Di nɔmba we kam pas ɔl na ɛni rɛkɔd na bak di wan we lɔk dɔn ɛp pas ɔl, so ivin pɔkit we rili gɛt bayas de luk bɛtɛ na di rɛkɔd pas aw i rili de: dis na di winɛ in kɔs. If dat pɔkit nɔ gɛt bayas fɔ tru, di Kelly bɛt na jɔs bɛt we de lɔs 5.26% pan avrej. Di ansa we pipul kin gi na fɔ bɛt pat pan Kelly ɛn fɔ kɔntinyu fɔ kɔnt te di advantej sho bak na nyu rɛkɔd (luk “Aw fɔ fɛn bayas na rulɛt wil”).
Pipul we gɛt advantej
Fɔ no di advantej nɔto di sem tin as fɔ yuz am fayn. Haghani ɛn Dewey (2016) bin gi 61 pipul $25 ɛni wan fɔ bɛt fɔ af awa pan kɔyn we dɛn tɛl dɛn se i de kam ed 60% pan di tɛm, ɛn di win mɔni nɔ fɔ pas $250 (di pleya dɛn bin jɔs no bɔt di limit wen dɛn rich am). Kelly se bɛt 20% pan di gambul mɔni pan ed ɛni tɛm, ɛn di raita dɛn ɛstimet se stedi bɛt fɔ 10–20% go rich di limit lɛk 95% pan di tɛm. Na jɔs 21% pan di pleya dɛn rich de. 28% lɔs ɔl, 18 pan di 61 bɛt ɔltin pan wan flip, lɛk wan tɔd dɔn wit smɔl pas di $25 we dɛn bigin wit, ɛn di avrej pemɔniPeWetin bɛt we win de gi yu, dɛn de tɔk am as “X to 1” ɔn tɔp di bɛt mɔni we kam bak.Rid ɔl → na $91. Na jɔs fayv pan di 61 bin dɔn yɛri bɔt di Kelly rul. If yu gɛt advantej, di sayz fɔ di bɛt dɛn de disayd if yu go kip am; if yu nɔ gɛt advantej, lɛk na fɛr wil, no sayz nɔ de ɛp.
Wetin fɔ yuz instɛd
If yu nɔ gɛt advantej, fɔ pik bɛt mɔni na bɔjit kwɛstyɔn, nɔto gro kwɛstyɔn: bɛt mɔni = wetin yu gri fɔ spɛn fɔ ɛni spin, we kɔmɔt frɔm di bɔjit fɔ ɛni awa ɛn aw di tebul de go kwik (luk di pej fɔ aw fɔ manej yu gambul mɔni). Wetin Kelly de gi di nɔmal pleya na in ansa we klia: ziro. Ɛni bɛt mɔni we pas ziro na di prays fɔ ple, we yu yɛsɛf pik, nɔto wan fɔmula.
Smɔl tin bɔt pat-Kelly ɛn “mɔni manejmɛnt”
Sistɛm dɛn we dɛn de sɛl as “Kelly fɔ rulɛt” ɛn we se bɛt wan fiks pasɛnt pan di gambul mɔni ɛni tɛm na bɛt di sem mɔni ɔltɛmstratɛjiBɛt di sem mɔni ɔltɛmFɔ bɛt di sem mɔni ɛni spin. Na dis ɛni prɔgrɛshɔn de kɔmpia wit.Rid ɔl → wit bɛt mɔni we de shɔnk. Dɛn de lɔs 2.70% pan ɔl di mɔni we yu bɛtmɔniƆl di bɛt mɔniƆl di mɔni we yu bɛt na wan sɛshɔn ɔ wan tɛm, wit di win mɔni we yu bɛt bak. Dɛn kin kɔl am bak: handle, action.Rid ɔl → ɛn, bikɔs di bɛt mɔni de shɔnk as yu de lɔs, dɛn de mek lɔs ɔl kam slo, nɔto se i nɔ go kam. I na fayn we fɔ mek bɔjit las; i nɔto Kelly, we go se mek yu stɔp.
Fɔmula
Kwɛstyɔn dɛn
Na jɔs wit advantej we yu mɛzhɔ frɔm wil we gɛt bayaslabWil we gɛt bayasWil we gɛt pɔblɛm na in bɔdi we mek sɔm pɔkit kin kam pas 1/37. I nɔ kɔmɔn, ɛn yu kin fɛn am nɔmɔ wit tawzin tawzin spin we yu dɔn rayt dɔn.Rid ɔl → ɔ mashin fɔ tɛm. Nɔto pan fɛr wil.
Bikɔs di ɛstimet fɔ di advantej gɛt mistek, ɛn fɔ bɛt tu bɔku de kɔst pas fɔ bɛt tu smɔl. Af-Kelly de gi tri kwata pan di gro wit wan kwata pan di variansmatematiksVariansWan mɛzhɔ fɔ aw di rizɔlt dɛn sprɛd rawnd di valyu we fɔ ɛkspɛkt. I ay fɔ bɛt pan wan nɔmba, i lɔ fɔ ivin-mɔni bɛt.Rid ɔl →.
Nɔ. Kelly de mek gro bɛtɛ pas ɔl wen advantej de. Fɔ manej yu gambul mɔni de kɔntrol wetin yu de spɛn wen advantej nɔ de.
Ple wit Mat
Tray dɛn yusɛf fɔs. Ɛni ansa de ayd; opin am we yu rɛdi.
Thorp ɛn Shannon dɛn mashin fɔ si bifo bin gi lɛk 44% advantej. Pan wan nɔmba we di mashin pik, Kelly se bɛt 1.26% pan di gambul mɔnimɔniGambul mɔniƆl di mɔni we yu dɔn put asay fɔ ple. Fɔ manej yu gambul mɔni min fɔ disayd ɔmɔs pan am yu go risk fɔ ɛni bɛt ɛn fɔ ɛni sɛshɔn.Rid ɔl →. Bɔt if di sem 44% advantej kam pan wan bɛt we yu sprɛd oba fayv nɔmba we de nia wan ɔda nko?
Sho di ansa
Ansa Dɔn Kelly se 7.1% pan di gambul mɔni, ɛn di gambul mɔni de gro nia siks tɛm mɔ kwik: lɛk 1.25 milyɔn tɛm oba 1,000 bɛt, ɛn di ɔda wan na lɛk 11.5 tɛm.
Bɛt pan wan nɔmbabɛtBɛt pan wan nɔmbaWan nɔmba nɔmɔ, i de pe 35:1.Rid ɔl → wit 44% advantej de win wit chans 0.04, bikɔs 36 × 0.04 = 1.44. Kelly in bɛt na di advantej sheb bay di ɔds: 0.44/35 = 1.26%. Di nɔmal gro pan ɛni bɛt na 0.04 × ln(1 + 35 × 0.0126) + 0.96 × ln(1 − 0.0126) ≈ 0.00244, so 1,000 bɛt de mɔltiplay di gambul mɔni bay lɛk e^2.44 ≈ 11.5.
Sprɛd wan bɛt oba fayv nɔmba ɛn di bɛt de win wit chans 0.20 ɛn i de pe 36/5 fɔ 1, dat na 6.2 to 1: na stil 44% advantej, bikɔs 0.2 × 7.2 = 1.44. Kelly in bɛt tɔn 0.44/6.2 = 7.10%, ɛn di gro pan ɛni bɛt 0.01404, lɛk 5.75 tɛm mɔ. Oba 1,000 bɛt dat na e^14.04, lɛk 1.25 milyɔn.
Di advantej na di sem; di ɔp ɛn dɔŋ smɔl pas. Bɛt we de win wan tɛm insay fayv kin kɛr big pat pan di gambul mɔni ɛn i nɔ de put yu na denja fɔ lɔs ɔl, ɛn Kelly in gro de pe fɔ dat. Thorp (1984) yuz di ɛgzampul fɔ sho wetin mek pleya dɛn we de si bifo de bɛt pan sɛktawilSɛktaPɔkit dɛn we de fala wan ɔda na wan pat fɔ di wil, nɔto grup na di bɛt tebul.Rid ɔl →. Fɔ rial, fayv nɛbabɛtNɛba dɛnWan nɔmba ɛn di tu pɔkit dɛn na ɛni sayd fɔ am na di wil, fayv chip pan wan nɔmba ɛni wan.Rid ɔl → gɛt advantej we smɔl smɔl pas di wan bɛst nɔmba (luk Advantage Play).
Pleya dɔn fɛn tu pɔkitikwipmɛntPɔkitWan pan di 37 smɔl ples dɛn wit nɔmba pan di rotɔ. Dɛn kin kɔl am bak: slot, kɔp.Rid ɔl → we de nia wan ɔda, ɛn ɛni wan de kam 1 tɛm insay 25 nɔto 1 insay 37. I fɔ bɛt di splitbɛtSplitTu nɔmba we de nia wan ɔda, i de pe 17:1.Rid ɔl → bitwin dɛn bay 17 to 1, ɔ ɛni nɔmba wan wan? Ɔmɔs Kelly se, ɛn aw kwik di gambul mɔni de gro if yu kɔmpia am wit wan pɔkit lɛk dat?
Sho di ansa
Ansa I nɔ mek difrɛns: 2.59% pan di gambul mɔni pan di split, ɔ 1.29% pan ɛni nɔmba. Di nɔmal gambul mɔni de gro lɛk 155 tɛm insay 1,000 spin, ɛn wit wan pɔkit na 11.5 tɛm.
Di split de win wit prɔbabilitimatematiksPrɔbabilitiDi chans fɔ wan rizɔlt, frɔm 0 to 1 ɔ frɔm 0% to 100%. Wan nɔmba: 1/37 = 2.70%.Rid ɔl → 2/25 = 0.08 ɛn i de pe 17 to 1, so i de gi bak 18 × 0.08 = 1.44 fɔ ɛni yunitmɔniYunitDi bes bɛt mɔni we sistɛm ɔ pleya de wok wit. Dɛn de ɛksplen prɔgrɛshɔn wit yunit so dat i go wok na ɛni tebul.Rid ɔl → we yu bɛt: advantej fɔ 44%, di sem lɛk wan pɔkit we gɛt bayas. Kelly in bɛt na di advantej sheb bay di ɔds: 0.44/17 = 2.59%.
Tu bɛt pan wan nɔmba fɔ 1.29% ɛni wan de gi di sem rizɔlt dɛn rayt rayt. We ɛni wan pan di tu nɔmba kam, wan chipmɔniChipTokɛn we min wan sɛt mɔni; yu put am na di bɛt tebul fɔ mek bɛt. Dɛn kin kɔl am bak: check.Rid ɔl → de win 35 to 1 ɛn di ɔda wan lɔs, nɛt na 34 × 1.29% = 44% pan di gambul mɔni, lɛk aw di split de pe; we nɔn pan dɛn nɔ kam, ɔl tu de lɔs.
Di gro pan ɛni spin na 0.00505, ɛn fɔ wan pɔkit na 0.00244, so oba 1,000 spin di nɔmal gambul mɔni de gro e^5.046 ≈ 155 tɛm, nɔto e^2.44 ≈ 11.5 tɛm. Tu gud nɔmba valyu fa pas tu tɛm wan, ɛn na dat mek pleya dɛn we de yuz bayas de bɛt grup fɔ nɔmba. Thorp (1984) du di sem kɔmpia.
Smɔl jok fɔ ɛz di maynd
Di rulɛt bɛt mɔni we de mek mɔni gro pas ɔl na nɔmba we ɔlman dɔn no.
Usay wi gɛt am frɔm
- 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.”