Kelly criterion ne roulette
Kelly criterion kyerɛ obi a ɔto bɛt a ɔwɔ mfasoɔ sika dodoɔ a ɔnto. Ma roulette no, ne mmuaeɛ da hɔ pefee: hwee.
Kelly bɛt sika = mfasoɔ / odds. Sɛ mfasoɔ no yɛ bɔne a, nhyehyɛe no ma nkyɛmu a ɛyɛ bɔne, a ɛkyerɛ sɛ nto bɛt. Nanso criterion no da so ho wɔ mfasoɔ sɛ ɛbɛma wote aseɛ deɛ enti a agorɔfoɔ a wɔde ntwahonan animhwɛ dii dwuma paw bɛt kɛseɛ sɛdeɛ wɔyɛeɛ no, ne deɛ enti a ɛsɛ sɛ obiara foforɔ de sika a wayi asie paw bɛt kɛseɛ.
Criterion no
John Kelly (1956) bisaa sɛdeɛ ɛsɛ sɛ sika agorɔfoɔ a onim ne mfasoɔ paw bɛt kɛseɛ na ama ne agorɔ dwetiresikaAgorɔ dwetireSika a woayi asie nyinaa ama agorɔ. Agorɔ dwetire so hwɛ ne sɛ wobɛsi gyinaeɛ wɔ dodoɔ a wode bɛto asiane mu wɔ bɛt biara ne agorɔ bere biara mu.Kenkan ne nyinaa → anyini sen biara wɔ bere tenten mu. Mmuaeɛ no, ma bɛt a ɛtua b tia 1 a ne nkonim hokwanakontaabuHokwanHokwan a biribi bɛba, firi 0 kɔsi 1 anaa 0% kɔsi 100%. Nɔma baako: 1/37 = 2.70%.Kenkan ne nyinaa → yɛ p na n'ahwereɛ hokwan yɛ q = 1 − p no, ne sɛ wode agorɔ dwetire no nkyɛmu f* = (bp − q) / b bɛto. Sɛ f* yɛ papa a, to saa nkyɛmu no; sɛ ɛyɛ hwee anaa bɔne a, nto bɛt. Kelly yɛɛ adwuma wɔ Bell Labs, na ne nsɛm a ɛyɛ “A new interpretation of information rate” no de asɛm no too Claude Shannon nsɛm ho nimdeɛ mu: sika agorɔfoɔ a ɔnya nsɛm wɔ telefon ahama a ɛyɛ dede so, wɔ odds a ɛyɛ pɛ so no, betumi ama agorɔ dwetire no anyini wɔ ahoɔhare a ɛsen biara a ɛne nsɛm a ahama no soa no yɛ pɛ. Wɔ afe 1961 mu no, Leo Breiman kyerɛɛ sɛ Kelly bɛt, wɔ bere tenten mu no, ba awieeɛ wɔ anim sen ɔkwan biara a ɛnte sɛ ɛno, na ɛduru botaeɛ a wɔasi so wɔ bere a ɛtia sen biara bɛyɛ mu.
Kelly bɛt wɔ roulette mu
Ma kɔkɔɔbɛtKɔkɔɔSika pɛ bɛt wɔ nɔma kɔkɔɔ 18 no so. Franse kasa mu: rouge.Kenkan ne nyinaa →: b = 1, p = 18/37, q = 19/37; f* = (18/37 − 19/37)/1 = −1/37. Ma nɔma baako bɛtbɛtNɔma baako bɛtNɔma baako pɛ, ɛtua 35:1.Kenkan ne nyinaa →: b = 35, p = 1/37, q = 36/37; f* = (35/37 − 36/37)/35 = −1/1295. Bɛt biara a ɛtaa ba ma Kelly nkyɛmu a ɛyɛ bɔne, ɛfiri sɛ bɛt biara mfasoɔ yɛ bɔne, na bɛt sika a ɛma nkɔsoɔ a ɛye sen biara ne hwee. Kelly bɛt wɔ roulette so yɛ ɔkwan a ɛfata a wɔfa so ka deɛ kasino mfasoɔ kratafa no ka: kɛseɛ biara nni hɔ a bɛt a ne boɔ a wɔhwɛ kwan yɛ bɔne bɛma agorɔ dwetire anyini.
Baabi a Kelly fa ho ampa
Agorɔfoɔ a wɔde ntwahonan animhwɛ dii dwuma no wɔ mfasoɔ ne bɛt kɛseɛ ho ɔhaw. Jagger, García-Pelayo abusua no ne Eudaemons no nyinaa sii gyinaeɛ wɔ sika dodoɔ a wɔbɛto wɔ mfasoɔ a wɔasusu no so, na sɛ wode kakra dodo to a, mfasoɔ no yera, na sɛ wode pii dodo to a, ebia agorɔ dwetire bɛsa ansa na mfasoɔ no ada adi. Ma tokurunneɛma a wɔde diTokuruRotor no so tokuru 37 a nɔma wom no mu baako. Wɔfrɛ no nso slot, cup.Kenkan ne nyinaa → a ɛba 30 mu 1 no, Kelly ka sɛ to 0.57% wɔ agorɔ dwetire so wɔ kyinhyia biara mu wɔ saa nɔma no so: sɛ wowɔ $10,000 a, bɛyɛ $57 kyinhyia biara, a ɛma nkɔsoɔ a ɛtaa ba bɛyɛ 0.054% wɔ kyinhyia biara mu, anaa ×1.7 wɔ kyinhyia 1,000 mu. Mfinimfini mfasoɔ no yɛ mmɔho mmienu wɔ saa so, 0.11% kyinhyia biara, nanso Kelly botaeɛ ne agorɔ dwetire a ɛtaa ba no, ɛnyɛ mfinimfini deɛ no a ntoatoasoɔ kakraa bi a akyiresɛm wom ma ɛkɔ soro no. Kelly fa ne deɛ nnipa taa paw wɔ deɛ wɔyɛ ampa mu ɛfiri sɛ mfasoɔ a wɔasusu no nni pintinn.
Deɛ ɛtaa ba ne mfinimfini
Ntam a ɛda agorɔ dwetire a ɛtaa ba ne mfinimfini deɛ no ntam no trɛ. Fa tokuru a ɛba mpɛn 1 wɔ 25 mu, mfasoɔ 44%, nɔma a Thorp de ma ne afidie no. Kelly de 1.26% wɔ agorɔ dwetire so to so. Kyinhyia 1,000 akyi no, agorɔ dwetire a ɛtaa ba no anyini bɛyɛ mmɔho 11.5, na mfinimfini no, a ntoatoasoɔ kakraa bi a akyiresɛm kɛseɛ wom twe kɔ soro no, anyini bɛyɛ mmɔho 249. Sɛ wode sen Kelly to a, mfinimfini no kɔ soro na deɛ ɛtaa ba no siane: sɛ wode Kelly bɛt sika mmɔho mmienu to a, agorɔ dwetire a ɛtaa ba no yɛ mmɔho mmienu pɛ, na sɛ ɛduru Kelly bɛt sika mmɔho mmienu ne fa a, ɛtew, ɛwom sɛ bɛt biara da so wɔ mfasoɔ 44% de. Sɛ wode kakra to a, ɛmmɔ ka kɛseɛ: bɛt sika no fa ma nkɔsoɔ no bɛyɛ nkyɛmu ɛnan mu mmiɛnsa.
Bere a mfasoɔ no yɛ nsusuiɛ
Kelly gye di sɛ wonim mfasoɔ no. Wɔ ɛpono no ho no, wɔsusu, na nsusuiɛ no taa kɔ soro dodo. Epstein nhwɛsoɔ, a ɛgyina Thorp 44% no so no, yɛ nɔma a ɛbaa mpɛn 49 wɔ kyinhyia 1,225 mu wɔ ntwahonan a hweentwahonanHweeTokuru ahabammono baako a sika pɛ bɛt, dumienu kuo bɛt anaa santen tee bɛt biara nkata so; kasino mfasoɔ fibea. Wɔfrɛ no nso hwee baako, 0.Kenkan ne nyinaa → mmienu wɔ so: mpɛn 1 wɔ 25 mu, mfasoɔ 44%, Kelly bɛt sika 1.26%. Nanso ntwahonan a hwee mmienu wɔ so a ɛyɛ pɛ wɔ nɔma 38, na emu deɛ ɛba pii sen biara duru 49 anaa deɛ ɛboro saa wɔ kyinhyia 1,225 mu bɛyɛ mpɛn baako wɔ nkron mu. Nɔma a ɛba pii sen biara wɔ nkrataa biara mu nso ne deɛ akyiresɛm aboa no sen biara, ɛno nti tokuru a ɛyɛ animhwɛ ampa mpo te sɛ deɛ ɛye sen sɛdeɛ ɛte ampa wɔ nkrataa no mu: nkonimdifoɔ nnome (winner’s curse). Sɛ saa tokuru no nyɛ animhwɛ ampa a, Kelly bɛt sika no yɛ bɛt a ɛhwere 5.26% wɔ mfinimfini mu kɛkɛ. Mmuaeɛ a ɛtaa ba ne sɛ wobɛto Kelly nkyɛmu bi na woakɔ so akan kɔsi sɛ mfasoɔ no bɛgyina nkrataa foforɔ mu (hwɛ Sɛdeɛ wobɛhu roulette ntwahonan animhwɛ).
Nnipa a wɔwɔ mfasoɔ
Sɛ wonim mfasoɔ no a, ɛnkyerɛ sɛ wode bɛdi dwuma yie. Haghani ne Dewey (2016) maa nnipa 61 $25 biara sɛ wɔmfa nto bɛt fa dɔnhwere wɔ sika kɔɔbere bi a wɔka kyerɛɛ wɔn sɛ ti ba bere no 60% mu so, na nkonim sika no ano yɛ $250 (agorɔfoɔ no tee ano no ho asɛm bere a wɔduruu so nko ara). Kelly ka sɛ to 20% wɔ agorɔ dwetire so wɔ ti so bere biara, na nhomanimfoɔ no susu sɛ bɛt a ɛyɛ 10–20% daa bɛduru ano no so bɛyɛ bere no 95% mu. Agorɔfoɔ no 21% pɛ na wɔduruu hɔ. 28% sika saeɛ, 61 no mu 18 de wɔn sika nyinaa too kɔɔbere baako so, bɛyɛ nkyɛmu mmiɛnsa mu baako wiee wɔ $25 a wɔde hyɛɛ aseɛ no ase, na mfinimfini akatuasikaAkatuaDeɛ bɛt a edi nkonim de ma, a wɔka no sɛ “X tia 1” ka bɛt sika a ɛsan ba no ho.Kenkan ne nyinaa → no yɛ $91. 61 no mu baanum pɛ na na wɔate Kelly criterion ho asɛm. Sɛ wowɔ mfasoɔ a, bɛt no kɛseɛ na ɛkyerɛ sɛ wobɛkora so anaa; sɛ wonni bi a, te sɛ wɔ ntwahonan a ɛyɛ pɛ so no, kɛseɛ biara mmoa.
Deɛ wode bɛdi dwuma mmom
Sɛ wonni mfasoɔ a, bɛt kɛseɛ yɛ sika a woayi asie ho asɛm, ɛnyɛ nkɔsoɔ ho asɛm: bɛt sika = deɛ wopɛ sɛ wode bɛsɛe wɔ kyinhyia biara mu, a ɛfiri dɔnhwere sika a woayi asie ne ɛpono no ahoɔhare mu (agorɔ dwetire so hwɛ kratafa). Deɛ Kelly de ma agorɔfoɔ a ɔyɛ daa daa no ne ne mmuaeɛ a ɛda hɔ pefee: hwee. Biribiara a wode to sen hwee no yɛ agorɔ no boɔ, a wo ara na wopaw, ɛnyɛ nhyehyɛe bi na ɛyɛ no papa.
Asɛm tiawa bi fa Kelly nkyɛmu ne “sika so hwɛ” ho
Nhyehyɛe a wɔtɔn no sɛ “Kelly ma roulette” a ɛkamfo ɔha nkyɛmu pintinn bi wɔ agorɔ dwetire so wɔ bɛt biara mu kyerɛ no yɛ sika pɛ a wɔto daaɔkwanSika pɛ a wɔto daaSika dodoɔ koro no ara a wode to bɛt kyinhyia biara. Ɛno ne gyinapɛn a wɔde bɛt nsakraeɛ ntoatoasoɔ biara toto ho.Kenkan ne nyinaa → a bɛt sika no retew. Ɛhwere 2.70% wɔ bɛt sika nyinaasikaBɛt sika nyinaaSika nyinaa a wɔde to bɛt wɔ agorɔ bere anaa bere bi mu, a nkonim sika a wɔsan de to ka ho. Wɔfrɛ no nso handle, action.Kenkan ne nyinaa → so, na esiane sɛ bɛt sika no tew bere a worehwere nti, ɛma agorɔ dwetire no sa brɛoo, na ɛnyɛ sɛ ɛrensa. Ɛyɛ ɔkwan pa a wobɛfa so ama sika a woayi asie atena hɔ kyɛ; ɛnyɛ Kelly, a anka ɛbɛka sɛ gyae.
Akontaabu kwan
Nsɛmmisa
Sɛ wowɔ mfasoɔ a wɔasusu firi ntwahonan a ɛyɛ animhwɛnhwehwɛmu danNtwahonan a ɛyɛ animhwɛNtwahonan a ne honam so mfomsoɔ ma ntokuru bi hokwan boro 1/37. Ɛnnsɛ ho pii, na wobɛtumi ahunu no ɛnkyɛ sɛ wode kyinhyia mpempem a woakyerɛw ahwɛ.Kenkan ne nyinaa → anaa bere a wobu afidie so nko ara. Ɛnyɛ wɔ ntwahonan a ɛyɛ pɛ so.
Ɛfiri sɛ mfasoɔ a wɔasusu no wɔ mfomsoɔ, na sɛ wode pii dodo to a, ne ka sen sɛ wode kakra dodo to. Kelly fa ma nkɔsoɔ no nkyɛmu ɛnan mu mmiɛnsa wɔ nsakraeɛakontaabuNsakraeɛNsusuiɛ a ɛkyerɛ sɛdeɛ nsɛm a ɛfiri mu ba no trɛ wɔ boɔ a wɔhwɛ kwan ho. Ɛso ma nɔma baako bɛt, ɛsua ma sika pɛ bɛt.Kenkan ne nyinaa → no nkyɛmu ɛnan mu baako so.
Dabi. Kelly ma nkɔsoɔ yɛ papa sen biara bere a mfasoɔ wɔ hɔ. Agorɔ dwetire so hwɛ hwɛ sika a wosɛe so bere a mfasoɔ biara nni hɔ.
Akontaabu ho anigyeɛ
Di kan yɛ eyinom ara wo ho. Wɔakata mmuaeɛ biara so; sɛ woasiesie wo ho a, bue no.
Thorp ne Shannon nkɔmhyɛ afidie no maa mfasoɔ bɛyɛ 44%. Wɔ nɔma baako a wɔahyɛ ho nkɔm so no, Kelly ka sɛ to agorɔ dwetiresikaAgorɔ dwetireSika a woayi asie nyinaa ama agorɔ. Agorɔ dwetire so hwɛ ne sɛ wobɛsi gyinaeɛ wɔ dodoɔ a wode bɛto asiane mu wɔ bɛt biara ne agorɔ bere biara mu.Kenkan ne nyinaa → no mu 1.26%. Na sɛ mfasoɔ 44% korɔ no ara baa bɛt baako a wɔakyekyɛ wɔ nɔma enum a ɛbɛn wɔn ho so a ɛ?
Kyerɛ mmuaeɛ no
Mmuaeɛ Afei Kelly ka sɛ agorɔ dwetire no mu 7.1%, na agorɔ dwetire no nyini ntɛm bɛyɛ mmɔho nsia: bɛyɛ mmɔho ɔpepem 1.25 wɔ bɛt 1,000 mu, sɛ wode toto bɛyɛ mmɔho 11.5 ho a.
Nɔma baako bɛtbɛtNɔma baako bɛtNɔma baako pɛ, ɛtua 35:1.Kenkan ne nyinaa → a ne mfasoɔ yɛ 44% di nkonim wɔ hokwan 0.04 mu, ɛfiri sɛ 36 × 0.04 = 1.44. Kelly bɛt sika yɛ mfasoɔ no a wɔde odds no akyekyɛ: 0.44/35 = 1.26%. Nkɔsoɔ a ɛtaa ba wɔ bɛt biara so yɛ 0.04 × ln(1 + 35 × 0.0126) + 0.96 × ln(1 − 0.0126) ≈ 0.00244, enti bɛt 1,000 bɔ agorɔ dwetire no ho bɛyɛ e^2.44 ≈ 11.5.
Kyekyɛ bɛt sika baako wɔ nɔma enum so na bɛt no di nkonim wɔ hokwan 0.20 mu na ɛtua 1 ma 36/5, kyerɛ sɛ 6.2 tia 1: ɛda so yɛ mfasoɔ 44%, ɛfiri sɛ 0.2 × 7.2 = 1.44. Kelly bɛt sika no bɛyɛ 0.44/6.2 = 7.10%, na nkɔsoɔ wɔ bɛt biara so 0.01404, bɛyɛ mmɔho 5.75. Wɔ bɛt 1,000 mu no, ɛyɛ e^14.04, bɛyɛ ɔpepem 1.25.
Mfasoɔ no yɛ pɛ; nsakraeɛ no sua. Bɛt a ɛdi nkonim mprɛ baako wɔ enum mu tumi soa agorɔ dwetire no kyɛfa kɛseɛ a asiane sɛ agorɔ dwetire bɛsa nni mu, na Kelly nkɔsoɔ tua saa ho ka. Thorp (1984) de nhwɛsoɔ no kyerɛ deɛ enti a nkɔmhyɛ agorɔfoɔ to bɛt wɔ afantwahonanƆfaNtokuru a ɛtoa so wɔ ntwahonan no so, a ɛnyɛ kuo a ɛwɔ bɛt nhyehyɛe no so.Kenkan ne nyinaa → so. Wɔ adwuma mu no, fipam nɔmabɛtFipam nɔmaNɔma bi ne ntokuru mmienu a ɛwɔ ne nkyɛn biara wɔ ntwahonan no so, nɔma baako bɛt agorɔ sika enum.Kenkan ne nyinaa → enum mfasoɔ sua kakra sen nɔma baako a ɛyɛ papa sen biara (hwɛ Advantage Play).
Agorɔfoɔ bi ahu ntokurunneɛma a wɔde diTokuruRotor no so tokuru 37 a nɔma wom no mu baako. Wɔfrɛ no nso slot, cup.Kenkan ne nyinaa → mmienu a ɛbɛn wɔn ho a biara ba mprɛ 1 wɔ 25 mu mmom sen 1 wɔ 37 mu. Ɛsɛ sɛ ɔto nɔma mmienu bɛtbɛtNɔma mmienu bɛtNɔma mmienu a ɛbɛn wɔn ho, ɛtua 17:1.Kenkan ne nyinaa → wɔ wɔn ntam wɔ 17 tia 1 so, anaa nɔma biara sɛ nɔma baako bɛt? Kelly ka dodoɔ sɛn, na agorɔ dwetire no nyini ntɛm sɛn sɛ wode toto tokuru baako a ɛte saa ho a?
Kyerɛ mmuaeɛ no
Mmuaeɛ Ɛnyɛ nsonsonoeɛ biara: agorɔ dwetire no mu 2.59% wɔ nɔma mmienu bɛt no so, anaa 1.29% wɔ nɔma biara so. Agorɔ dwetire a ɛtaa ba nyini bɛyɛ mmɔho 155 wɔ kyinhyia 1,000 mu, sɛ wode toto mmɔho 11.5 a ɛne tokuru baako ho a.
Nɔma mmienu bɛt no di nkonim wɔ hokwanakontaabuHokwanHokwan a biribi bɛba, firi 0 kɔsi 1 anaa 0% kɔsi 100%. Nɔma baako: 1/37 = 2.70%.Kenkan ne nyinaa → 2/25 = 0.08 mu na ɛtua 17 tia 1, enti ɛsan de 18 × 0.08 = 1.44 ba wɔ kyɛfasikaKyɛfaBɛt sika a ɛwɔ ase a nhyehyɛe anaa agorɔfoɔ de yɛ adwuma. Wɔka bɛt nsakraeɛ ntoatoasoɔ ho asɛm wɔ kyɛfa mu sɛdeɛ ɛbɛyɛ a ɛbɛfata ɛpono biara.Kenkan ne nyinaa → biara a wɔto so: mfasoɔ 44%, ɛne tokuru baako a ɛyɛ animhwɛ yɛ pɛ. Kelly bɛt sika yɛ mfasoɔ no a wɔde odds no akyekyɛ: 0.44/17 = 2.59%.
Nɔma baako bɛt mmienu a biara yɛ 1.29% ma nsunsuansoɔ korɔ no ara pɛpɛɛpɛ. Sɛ nɔma biara ba a, agorɔ sikasikaAgorɔ sikaAdeɛ a ɛgyina hɔ ma sika dodoɔ bi a ɛnsesa, a wode to bɛt nhyehyɛe no so de to bɛt. Wɔfrɛ no nso check.Kenkan ne nyinaa → baako di nkonim 35 tia 1 na ɔfoforɔ no hwere, ankasa 34 × 1.29% = agorɔ dwetire no mu 44%, sɛdeɛ nɔma mmienu bɛt no tua; sɛ emu biara amma a, ne mmienu nyinaa hwere.
Nkɔsoɔ wɔ kyinhyia biara so yɛ 0.00505, sɛ wode toto 0.00244 a ɛne tokuru baako ho a, enti wɔ kyinhyia 1,000 mu no, agorɔ dwetire a ɛtaa ba nyini e^5.046 ≈ mmɔho 155 mmom sen e^2.44 ≈ mmɔho 11.5. Nɔma pa mmienu som bo kɛseɛ sen baako mmɔho mmienu, na ɛno nti na animhwɛ agorɔfoɔ to bɛt wɔ nɔma akuo so. Thorp (1984) yɛ ntotohoɔ korɔ no ara.
Serew kakra
Roulette bɛt sika a ɛma nkɔsoɔ a ɛye sen biara yɛ nɔma a obiara nim dada.
Nsɛm fibea
- 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.”