Kelly Criterion na Rulet
Lo bilong Kelly i tokim wanpela pleia i gat edj hamas mani em i ken putim long bet. Long rulet em i givim klia bekim: nogat samting.
Mani bilong bet long Kelly = edj / ods. Sapos edj i nogut (i stap aninit long siro), formula i givim hap namba i nogut, na dispela i min olsem: no ken putim bet. Tasol lo i helpim yu long save bilong wanem ol pleia bilong bias i bin putim ol bet long sais ol i bin mekim. Em i soim tu bilong wanem olgeta narapela i mas makim sais bilong bet long baset bilong ol.
Lo bilong Kelly
John Kelly (1956) i askim: sapos wanpela gembla i save edj bilong em, em i mas putim bet long wanem sais bilong mani bilong pilaimaniMani bilong pilaiOlgeta mani yu putim long sait bilong pilai. Lukautim mani bilong pilai em i makim hamas long en yu redi long lusim long wanpela bet na long wanpela sesen.Ritim olgeta → i ken kamap bikpela tru long longpela taim? Bekim em i olsem. Bilong bet i pe b long 1, wantaim sansmatematikSansSans bilong wanpela resal, long 0 i go 1 o 0% i go 100%. Wanpela namba: 1/37 = 2.70%.Ritim olgeta → p bilong win na sans q = 1 − p bilong lus, putim hap f* = (bp − q) / b bilong mani bilong pilai. Sapos f* i antap long siro, betim dispela hap. Sapos em i siro o i stap aninit long siro, no ken putim bet. Kelly i bin wok long Bell Labs. Pepa bilong em, “A new interpretation of information rate”, i putim dispela askim insait long tiori bilong infomesen bilong Claude Shannon. Wanpela gembla i kisim tok hait long wanpela lain i gat nois, long ods i stret. Mani bilong pilai bilong em i ken kamap bikpela, tasol i no kwik moa long hamas infomesen dispela lain i karim. Long 1961 Leo Breiman i soim klia olsem: long longpela taim, bet bilong Kelly i save winim olgeta narapela strateji i narakain tru, na i kisim wanpela mak klostu long sotpela taim tru.
Bet bilong Kelly long rulet
Bilong retbetRetBet wan-long-wan long 18 namba ret. Tok Frans: rouge.Ritim olgeta →: b = 1, p = 18/37, q = 19/37; f* = (18/37 − 19/37)/1 = −1/37. Bilong bet long wanpela nambabetWanpela nambaWanpela namba tasol, pe em i 35:1.Ritim olgeta →: b = 35, p = 1/37, q = 36/37; f* = (35/37 − 36/37)/35 = −1/1295. Olgeta nambawan bet i givim hap bilong Kelly aninit long siro, long wanem olgeta bet i gat edj i nogut. Olsem na mani bilong bet i gutpela tru bilong kamap bikpela em i siro. Bet bilong Kelly long rulet i tok wankain olsem pes bilong edj bilong kasino: i no gat wanpela sais we bet i gat EV aninit long siro i ken mekim mani bilong pilai i kamap bikpela.
We lo bilong Kelly i bin wok
Ol pleia bilong bias i bin gat edj, na hevi bilong makim sais bilong bet. Jagger, ol García-Pelayo na ol Eudaemon olgeta i mas makim hamas mani long betim long edj ol i bin mesarim. Sapos yu betim liklik tumas, yu lusim edj. Sapos yu betim bikpela tumas, yu ken lusim olgeta mani bipo edj i kamap klia. Bilong wanpela hultulHulWanpela long ol 37 hul i gat namba long rota. Narapela nem: slot, cup.Ritim olgeta → i kamap 1 long 30, Kelly i tok: 0.57% bilong mani bilong pilai long wanwan tanim long dispela namba. Wantaim $10,000, em klostu $57 long wanpela tanim. Mani i save kamap bikpela klostu 0.054% long wanpela tanim, o ×1.7 long 1,000 tanim. Avarej win em tupela taim moa, 0.11% long wanpela tanim. Tasol Kelly i lukluk long mani bilong pilai i save kamap planti taim, i no long avarej, long wanem sampela ran i gat gutpela luk tumas i mekim avarej i go antap. Hap-Kelly em ol i save makim long wok tru, long wanem save long edj i no klia olgeta.
Wanem i save kamap na avarej
Namel long mani bilong pilai i save kamap planti taim na avarej i gat bikpela spes. Kisim wanpela hul i kamap 1 taim long 25, edj bilong 44%. Em namba Thorp i givim bilong masin bilong em. Kelly i putim 1.26% bilong mani bilong pilai long en. Bihain long 1,000 tanim, mani i save kamap planti taim i bin kamap bikpela klostu 11.5 taim. Tasol avarej, we sampela ran i gat gutpela luk tru i pulim i go antap, i bin kamap bikpela klostu 249 taim. Sapos yu betim moa long Kelly, avarej i go antap na risalt i save kamap planti taim i go daun. Long tupela taim mani bilong Kelly, mani i save kamap planti taim i kamap dabel tasol. Long tupela na hap taim mani bilong Kelly em i go daun, maski olgeta bet i gat edj bilong 44% yet. Sapos yu betim liklik moa, yu no lusim planti: hap mani bilong bet i holim klostu tripela hap long fopela bilong kamap bikpela.
Taim edj em i tingting tasol
Kelly i ting olsem yu save edj stret. Long tebol yu ges long en, na ges i save i bikpela tumas. Eksampel bilong Epstein, i kamap long 44% bilong Thorp, em wanpela namba i kamap 49 taim long 1,225 tanim bilong wil i gat dabel sirowilSiroWanpela grinpela hul we i no gat bet wan-long-wan, dosen o kolam i karamapim; em i as bilong edj bilong kasino. Narapela nem: wanpela siro, 0.Ritim olgeta →: 1 taim long 25, edj bilong 44%, mani bilong Kelly 1.26%. Tasol wanpela stretpela wil i gat dabel siro i gat 38 namba. Namba i kamap moa long olgeta i save kisim 49 o moa long 1,225 tanim klostu wanpela taim long nainpela. Na namba i kamap moa long olgeta long wanpela rekod em tu namba we gutpela luk i helpim moa. Olsem na maski wanpela hul i gat bias tru, em i luk gutpela moa long rekod long tru: ol i kolim dispela “bagarap bilong husat i win”. Sapos dispela hul i no gat bias tru, mani bilong Kelly em wanpela bet tasol i save lus 5.26% long avarej. Ol pasin bilong stretim em: betim hap bilong Kelly tasol, na kaunim yet inap edj i stap long wanpela nupela rekod (lukim pes bilong painim bias bilong wil).
Ol manmeri i gat edj
Save long edj em i no wankain olsem yusim gut. Haghani na Dewey (2016) i givim $25 long wanwan bilong 61 manmeri bilong betim long hap aua long wanpela koin. Ol i tokim ol olsem koin i save kamap het 60% bilong taim. Win i gat mak $250 (ol pleia i save long mak taim ol i kisim tasol). Kelly i tok: betim 20% bilong mani bilong pilai long het long olgeta taim. Ol raita i ting olsem ol bet i stap wankain long 10–20% bai kisim mak klostu 95% bilong taim. Tasol 21% tasol bilong ol pleia i kisim. 28% i lusim olgeta mani. 18 long 61 i betim olgeta mani long wanpela tromoi. Klostu wanpela hap long tripela i pinis aninit long $25 ol i bin stat long en. Avarej pemaniPeMani bet i win i givim yu, ol i tok olsem “X long 1”, antap long mani bilong bet i kam bek.Ritim olgeta → em $91. Faivpela tasol long 61 i bin harim pinis long lo bilong Kelly. Sapos yu gat edj, sais bilong ol bet i makim sapos yu holim o nogat. Sapos yu no gat edj, olsem long stretpela wil, i no gat sais i ken helpim.
Wanem samting long yusim
Sapos yu no gat edj, sais bilong bet em askim bilong baset, i no askim bilong kamap bikpela. Mani bilong bet = hamas yu laik spendim long wanpela tanim. Yu wokim long baset bilong wanpela aua na hamas tanim tebol i save mekim (lukim pes bilong lukautim mani bilong pilai). Samting Kelly i givim long nambawan pleia em klia bekim bilong en: siro. Olgeta mani yu betim antap long siro em pe bilong pilai. Yu yet i makim, i no wanpela formula i makim.
Liklik tok long hap Kelly na “lukautim mani”
Ol sistem ol i salim olsem “Kelly bilong rulet” i tok yu mas betim wanpela pesen bilong mani bilong pilai i no senis long wanwan bet. Ol dispela em bet wankain oltaimstratejiBet wankain oltaimBetim wankain mani long olgeta tanim. Olgeta progresen ol i skelim wantaim dispela.Ritim olgeta → wantaim mani bilong bet i go daun. Ol i lusim 2.70% bilong olgeta mani yu betimmaniOlgeta mani yu betimOlgeta mani yu betim long wanpela sesen o long wanpela taim, wantaim mani yu win na yu betim gen. Narapela nem: handle, action.Ritim olgeta →. Long wanem mani bilong bet i go daun taim yu lus, ol i mekim lus bilong olgeta mani i kam isi isi, tasol i no mekim i no ken kamap. Em gutpela rot bilong mekim baset i stap longpela taim. Tasol em i no Kelly. Kelly bai tok: stop.
Fomula
Ol askim
Sapos yu gat edj yu mesarim long wil i gat biaslaboratoriWil i gat biasWil we sampela bagarap long bodi bilong en i mekim sampela hul i kamap moa long 1/37. Em i no planti, na yu inap painim tasol wantaim planti tausen tanim yu raitim.Ritim olgeta → o long masin bilong kaunim taim tasol. I no long stretpela wil.
Long wanem ges long edj i gat rong, na betim bikpela tumas i bagarapim moa long betim liklik tumas. Hap-Kelly i givim tripela hap long fopela bilong kamap bikpela, wantaim wanpela hap long fopela bilong variansmatematikVariansMak bilong hamas ol resal i kalap nabaut long ekspektet velyu. I bikpela long bet long wanpela namba, i liklik long bet wan-long-wan.Ritim olgeta →.
Nogat. Kelly i mekim kamap bikpela i gutpela tru taim yu gat edj. Lukautim mani bilong pilai i bosim spending taim yu no gat edj.
Amamas wantaim Matematik
Traim ol dispela yu yet pastaim. Bekim bilong wanwan i hait; opim taim yu redi.
Masin bilong Thorp na Shannon bilong tok profet i givim edj klostu 44%. Long wanpela namba ol i tok profet long en, Kelly i tok yu mas betim 1.26% bilong mani bilong pilaimaniMani bilong pilaiOlgeta mani yu putim long sait bilong pilai. Lukautim mani bilong pilai em i makim hamas long en yu redi long lusim long wanpela bet na long wanpela sesen.Ritim olgeta →. Olsem wanem sapos wankain edj 44% i kam long wanpela bet i tilim long faivpela namba i stap klostu?
Soim bekim
Bekim Orait Kelly i tok 7.1% bilong mani bilong pilai, na mani bilong pilai i kamap bikpela klostu sikis taim hariap moa: klostu 1.25 milion taim long 1,000 bet, na klostu 11.5 taim long narapela rot.
Bet long wanpela nambabetWanpela nambaWanpela namba tasol, pe em i 35:1.Ritim olgeta → i gat edj 44% i win wantaim sans 0.04, long wanem 36 × 0.04 = 1.44. Bet bilong Kelly em edj tilim long ods: 0.44/35 = 1.26%. Nomol kamap bikpela long wanwan bet em 0.04 × ln(1 + 35 × 0.0126) + 0.96 × ln(1 − 0.0126) ≈ 0.00244, olsem na 1,000 bet i taimim mani bilong pilai long klostu e^2.44 ≈ 11.5.
Tilim wanpela bet long faivpela namba na bet i win wantaim sans 0.20 na i pe 36/5 bilong 1, em 6.2 long 1: em i gat yet edj 44%, long wanem 0.2 × 7.2 = 1.44. Bet bilong Kelly i kamap 0.44/6.2 = 7.10%, na kamap bikpela long wanwan bet 0.01404, klostu 5.75 taim moa. Long 1,000 bet em e^14.04, klostu 1.25 milion.
Edj em wankain; ol senis antap na daun i liklik moa. Bet i win wanpela taim long faiv inap karim bikpela hap bilong mani bilong pilai na i no gat hevi long lusim olgeta, na kamap bikpela bilong Kelly i givim pe long dispela. Thorp (1984) i yusim eksampel long soim bilong wanem ol pleia i tok profet i betim ol seksenwilSeksenWanpela hap bilong wil we ol hul i stap wan bihain long narapela, i narakain long wanpela grup long laiaut.Ritim olgeta →. Long wok tru, faivpela namba klostubetOl namba klostuWanpela namba na tupela hul long tupela sait bilong en long wil, faivpela sip long wan wan namba.Ritim olgeta → i gat edj liklik daunbilo long wanpela namba i nambawan (lukim Pilai bilong Kisim Edj long Rulet).
Wanpela pleia i painim tupela hultulHulWanpela long ol 37 hul i gat namba long rota. Narapela nem: slot, cup.Ritim olgeta → i stap klostu, na wanwan i kamap 1 taim long 25, i no 1 long 37. Em i mas betim bet long tupela nambabetTupela nambaTupela namba i stap klostu, pe em i 17:1.Ritim olgeta → namel long tupela long 17 long 1, o wanwan namba olsem bet long wanpela namba? Kelly i tok hamas, na mani bilong pilai i kamap bikpela hariap olsem wanem sapos yu skelim wantaim wanpela kain hul?
Soim bekim
Bekim I no senisim wanpela samting: 2.59% bilong mani bilong pilai long tupela namba, o 1.29% long wanwan namba. Nomol mani bilong pilai i kamap bikpela klostu 155 taim long 1,000 tanim, na 11.5 taim wantaim wanpela hul.
Bet long tupela namba i win wantaim sansmatematikSansSans bilong wanpela resal, long 0 i go 1 o 0% i go 100%. Wanpela namba: 1/37 = 2.70%.Ritim olgeta → 2/25 = 0.08 na i pe 17 long 1, olsem na em i givim bek 18 × 0.08 = 1.44 long wanwan yunitmaniYunitMani bilong bet bilong stat we wanpela sistem o wanpela pleia i wok long en. Ol i stori long ol progresen long yunit, olsem na ol i ken wok long wanem tebol.Ritim olgeta → yu betim: edj bilong 44%, wankain olsem wanpela hul i gat bias. Bet bilong Kelly em edj tilim long ods: 0.44/17 = 2.59%.
Tupela bet long wanpela namba bilong 1.29% wanwan i givim stret wankain risal. Taim wanpela namba i kamap, wanpela sipmaniSipSamting i makim wanpela namba mani i no save senis, yu putim long laiaut bilong mekim bet. Narapela nem: check.Ritim olgeta → i win 35 long 1 na narapela i lus, win tru bilong 34 × 1.29% = 44% bilong mani bilong pilai, olsem bet long tupela namba i pe; taim nogat wanpela i kamap, tupela i lus.
Kamap bikpela long wanwan tanim em 0.00505, na 0.00244 long wanpela hul, olsem na long 1,000 tanim nomol mani bilong pilai i kamap bikpela e^5.046 ≈ 155 taim, i no e^2.44 ≈ 11.5 taim. Tupela gutpela namba i winim tru tupela taim wanpela, na dispela em as na ol pleia bilong bias i betim ol grup namba. Thorp (1984) i wokim wankain skelim.
Liklik lap
Mani bilong bet i gutpela tru long rulet em wanpela namba olgeta i save pinis.
Ol As bilong Save
- 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.”