Kelly Criterion na Roulette
Kelly criterion na-agwa onye ịgba nzọ nwere uru ego ole ọ ga-etinye. Maka roulette, ọ na-enye azịza doro anya: ọ dịghị ihe ọ bụla.
Ego nzọ Kelly = uru / odds. Mgbe uru dị n’okpuru efu, usoro mgbakọ ahụ na-enye akụkụ dị n’okpuru efu, nke pụtara na agbala nzọ. Criterion a ka bara uru maka ịghọta ihe mere ndị na-egwu mmasị wiil ji tụọ nha nzọ ha otu ha si tụọ ya, na ihe mere onye ọ bụla ọzọ ji kwesị itu nha site na mmefu ego ya.
Criterion ahụ
John Kelly (1956) jụrụ otu onye ịgba nzọ nwere uru a maara ga-esi tụọ nha nzọ ya iji mee ka ego egwuregwuegoEgo egwuregwuNgụkọta ego e debere maka igwu egwu. Njikwa ego egwuregwu bụ ikpebi ole n’ime ya a ga-etinye n’ihe egwu kwa nzọ na kwa oge egwuregwu.Gụọ ya niile → too nke kachasị n’ogologo oge. Azịza ya, maka nzọ na-akwụ b ruo 1 nwere oheremgbakọOhereOhere nke otu nsonaazụ, site na 0 ruo 1 ma ọ bụ 0% ruo 100%. Otu nọmba: 1/37 = 2.70%.Gụọ ya niile → mmeri p na ohere mfu q = 1 − p, bụ itinye akụkụ f* = (bp − q) / b nke ego egwuregwu. f* dị elu karịa efu pụtara gbaa akụkụ ahụ; efu ma ọ bụ n’okpuru efu pụtara agbala nzọ. Kelly rụrụ ọrụ na Bell Labs, akwụkwọ ya, “A new interpretation of information rate”, tinyere nsogbu ahụ n’ime ozizi ozi nke Claude Shannon: onye ịgba nzọ na-enweta ndụmọdụ site n’ahịrị nwere mkpọtụ, na odds ziri ezi, nwere ike ime ka ego egwuregwu too, n’ọnọdụ kacha mma, naanị n’ọsọ ozi ahịrị ahụ na-ebu. N’afọ 1961, Leo Breiman gosipụtara na nzọ Kelly, n’ogologo oge, na-emecha n’ihu atụmatụ ọ bụla dị iche n’ezie, ma na-eru ebumnuche a tọrọ n’ihe dị ka oge kacha nso.
Nzọ Kelly na roulette
Maka uhienzọUhieNzọ ego hara nhata na nọmba uhie 18. French: rouge.Gụọ ya niile →: b = 1, p = 18/37, q = 19/37; f* = (18/37 − 19/37)/1 = −1/37. Maka straight upnzọStraight upOtu nọmba, na-akwụ 35:1.Gụọ ya niile →: b = 35, p = 1/37, q = 36/37; f* = (35/37 − 36/37)/35 = −1/1295. Nzọ ọ bụla a na-agbakarị na-enye akụkụ Kelly dị n’okpuru efu, n’ihi na nzọ ọ bụla nwere uru dị n’okpuru efu, ego nzọ kachasị mma maka uto bụkwa efu. Nzọ Kelly na roulette bụ ụzọ e ji ekwu n’usoro ihe ibe uru ụlọ na-ekwu: ọ dịghị nha nzọ nwere atụmanya n’okpuru efu ga-eme ka ego egwuregwu too.
Ebe Kelly rụrụ ọrụ n’ezie
Ndị na-egwu mmasị wiil nwere uru na nsogbu nha nzọ. Jagger, ndị García-Pelayo na ndị Eudaemons niile ga-ekpebi ego ole ha ga-agba n’uru ha tụrụ, ebe ịgba nzọ dị obere na-emefu uru ahụ, ịgba nzọ karịrị akarị na-etinyekwa n’ihe egwu nke ọdịda tupu uru ahụ apụta. Maka ogherengwa ọrụOghereOtu n’ime ebe 37 nwere nọmba dị na rotor. A na-akpọkwa ya: slot, cup.Gụọ ya niile → na-akụ 1 ugboro n’ime 30, Kelly na-ekwu 0.57% nke ego egwuregwu kwa ntụgharị na nọmba ahụ: ya na $10,000, ihe dị ka $57 kwa ntụgharị, maka uto a na-ahụkarị nke ihe dị ka 0.054% kwa ntụgharị, ma ọ bụ ×1.7 n’ime ntụgharị 1,000. Uru nkezi bụ okpukpu abụọ nke ahụ, 0.11% kwa ntụgharị, mana Kelly na-elekwasị anya n’ego egwuregwu a na-ahụkarị, ọ bụghị nke nkezi, nke usoro chi ọma ole na ole na-eme ka ọ zaa. Ọkara Kelly bụ nhọrọ a na-ahọrọkarị n’ọrụ n’ihi na atụmatụ uru ahụ ejighị n’aka.
Nke a na-ahụkarị na nke nkezi
Ọdịiche dị n’etiti ego egwuregwu a na-ahụkarị na nke nkezi sara mbara. Were oghere na-akụ 1 ugboro n’ime 25, uru 44%, ọnụọgụ Thorp nyere maka ngwaọrụ ya. Kelly na-etinye 1.26% nke ego egwuregwu na ya. Mgbe ntụgharị 1,000 gasịrị, ego egwuregwu a na-ahụkarị etoola ihe dị ka okpukpu 11.5, ebe nkezi, nke usoro chi ọma ole na ole dọlitere, etoola ihe dị ka okpukpu 249. Ịgba karịa Kelly na-ebuli nkezi ma na-ebelata nsonaazụ a na-ahụkarị: na okpukpu abụọ nke ego nzọ Kelly, ego egwuregwu a na-ahụkarị na-amụba naanị okpukpu abụọ, ma n’okpukpu abụọ na ọkara nke ego nzọ Kelly ọ na-ebelata, n’agbanyeghị na nzọ ọ bụla ka nwere uru 44%. Ịgba obere na-efu obere ihe: ọkara ego nzọ na-echekwa ihe dị ka akụkụ atọ n’ime anọ nke uto ahụ.
Mgbe uru bụ nkọ
Kelly na-ewere na a maara uru ahụ. Na tebụl, a na-atụ ya, atụmatụ ahụ na-adịkarịkwa elu karịa. Ihe atụ Epstein, nke e wuru n’elu 44% nke Thorp, bụ nọmba kụrụ ugboro 49 n’ime ntụgharị 1,225 nke wiil efuwiilEfuOtu oghere akwụkwọ ndụ nke nzọ ego hara nhata, dozen ma ọ bụ column ọ bụla na-ekpuchighị; isi mmalite uru ụlọ. A na-akpọkwa ya: single zero, 0.Gụọ ya niile → abụọ: 1 ugboro n’ime 25, uru 44%, ego nzọ Kelly nke 1.26%. Mana wiil efu abụọ ziri ezi nwere nọmba 38, nke kachasị ọrụ n’ime ha na-eru 49 ma ọ bụ karịa n’ime ntụgharị 1,225 ihe dị ka otu ugboro n’ime itoolu. Nọmba kachasị ọrụ n’ihe ndekọ ọ bụla bụkwa nke chi ọma nyeere aka nke ukwuu, ya mere ọbụna oghere nwere mmasị n’ezie na-adị mma karịa ka ọ dị n’ihe ndekọ: ọbụbụ ọnụ onye meriri (winner’s curse). Ọ bụrụ na oghere ahụ enweghị mmasị n’ezie, ego nzọ Kelly bụ naanị nzọ na-atụfu 5.26% na nkezi. Azịza a na-enyekarị bụ ịgba akụkụ nke Kelly na ịga n’ihu ịgụ ruo mgbe uru ahụ lanarịrị ihe ndekọ ọhụrụ (lee Otu E Si Achọpụta Mmasị Wiil Roulette).
Ndị nwere uru
Ịmara uru abụghị otu ihe ahụ dịka iji ya mee nke ọma. Haghani na Dewey (2016) nyere mmadụ 61 $25 onye ọ bụla ka ha gbaa nzọ ọkara elekere na mkpụrụ ego a gwara ha na ọ na-ada isi 60% nke oge, ebe mmeri nwere oke $250 (ndị egwuregwu matara oke ahụ naanị mgbe ha ruru ya). Kelly na-ekwu gbaa 20% nke ego egwuregwu na isi oge ọ bụla, ndị dere ya na-atụkwa anya na nzọ kwụsiri ike nke 10–20% ga-eru oke ahụ ihe dị ka 95% nke oge. Naanị 21% nke ndị egwuregwu ruru ebe ahụ. 28% gwụrụ ego, 18 n’ime 61 gbara ihe niile n’otu ịtụ, ihe dị ka otu ụzọ n’ụzọ atọ mechara n’okpuru $25 ha ji bido, ụgwọegoỤgwọIhe nzọ meriri na-eweghachi, a na-ekwu ya dị ka “X ruo 1” na mgbakwunye ego nzọ e weghachiri.Gụọ ya niile → nkezi bụkwa $91. Naanị ise n’ime 61 nụtụrụla maka Kelly criterion. Mgbe ị nwere uru, nha nzọ na-ekpebi ma ị ga-echekwa ya; mgbe ị nweghị, dịka na wiil ziri ezi, ọ dịghị nha na-enyere aka.
Ihe ị ga-eji kama
Mgbe enweghị uru, nha nzọ bụ ajụjụ mmefu ego, ọ bụghị ajụjụ uto: ego nzọ = ihe ị dị njikere imefu kwa ntụgharị, nke e si na mmefu ego kwa elekere na ọsọ tebụl nweta (ibe Njikwa Ego Egwuregwu na Roulette). Onyinye Kelly nye onye egwuregwu nkịtị bụ ịdị nkọwa nke azịza ya: efu. Ihe ọ bụla ị gbara karịa efu bụ ọnụ ahịa nke igwu egwu, nke gị onwe gị họọrọ, ọ bụghị nke usoro mgbakọ ọ bụla mere ka ọ kacha mma.
Okwu banyere akụkụ Kelly na “njikwa ego”
Usoro a na-ere dịka “Kelly maka roulette” nke na-atụ aro pasentị a kapịrị ọnụ nke ego egwuregwu kwa nzọ bụ nzọ otu ọnụatụmatụNzọ otu ọnụItinye otu ego ahụ na ntụgharị ọ bụla. Ndabere e ji atụnyere usoro mmụba nzọ ọ bụla.Gụọ ya niile → nwere ego nzọ na-ebelata. Ha na-atụfu 2.70% nke ngụkọta nzọegoNgụkọta nzọNgụkọta ego e tinyere na nzọ n’otu oge egwuregwu ma ọ bụ otu oge, gụnyere mmeri a gbara nzọ ọzọ. A na-akpọkwa ya: handle, action.Gụọ ya niile → ma, n’ihi na ego nzọ na-ebelata ka ị na-atụfu, ha na-eme ka ọdịda bịa nwayọọ kama ime ka ọ ghara ime. Ọ bụ ụzọ kwesịrị ekwesị iji mee ka mmefu ego dịgide; ọ bụghị Kelly, nke ga-ekwu ka ị kwụsị.
Fọmụla
Ajụjụ
Naanị mgbe e nwere uru a tụrụ site na wiil nwere mmasịụlọ nnwaleWiil nwere mmasịWiil nke ntụpọ anụ ahụ ya na-eme ka ụfọdụ oghere nwee ohere karịa 1/37. Ọ dị ụkọ, a pụkwara ịchọpụta ya naanị site na puku kwuru puku ntụgharị e dekọrọ.Gụọ ya niile → ma ọ bụ ngwaọrụ na-atụ oge. Ọ bụghị na wiil ziri ezi.
N’ihi na atụmatụ uru ahụ nwere njehie, ịgba nzọ karịrị akarị na-efukwa karịa ịgba nzọ dị obere. Ọkara Kelly na-enye akụkụ atọ n’ime anọ nke uto n’otu ụzọ n’ụzọ anọ nke mgbasamgbakọMgbasaIhe ọtụtụ na-egosi otu nsonaazụ si gbasaa gburugburu uru a tụrụ anya. Ọ dị elu maka straight up, dị ala maka nzọ ego hara nhata.Gụọ ya niile →.
Mba. Kelly na-eme ka uto kacha mma mgbe e nwere uru. Njikwa ego egwuregwu na-achịkwa mmefu mgbe enweghị uru ọ bụla.
Egwuregwu Mgbakọ
Buru ụzọ nwaa ha n’onwe gị. A pịachiri azịza ọ bụla; meghee ya mgbe ị dị njikere.
Ngwaọrụ amụma Thorp na Shannon nyere uru dị ka 44%. N’otu nọmba e buru amụma ya, Kelly na-ekwu ka a gbaa nzọ 1.26% nke ego egwuregwuegoEgo egwuregwuNgụkọta ego e debere maka igwu egwu. Njikwa ego egwuregwu bụ ikpebi ole n’ime ya a ga-etinye n’ihe egwu kwa nzọ na kwa oge egwuregwu.Gụọ ya niile →. Ọ bụrụ na otu uru 44% ahụ bịara n’otu nzọ a gbasara n’elu nọmba ise dị n’akụkụ ibe ha?
Gosi azịza ya
Azịza Mgbe ahụ Kelly na-ekwu 7.1% nke ego egwuregwu, ego egwuregwu ahụ na-etokwa ihe fọrọ nke nta ka ọ bụrụ ugboro isii ngwa ngwa: ihe dị ka okpukpu nde 1.25 n’ime nzọ 1,000, megide ihe dị ka okpukpu 11.5.
Straight upnzọStraight upOtu nọmba, na-akwụ 35:1.Gụọ ya niile → nwere uru 44% na-emeri na ohere 0.04, ebe 36 × 0.04 = 1.44. Ego nzọ Kelly bụ uru ahụ kewara site na odds: 0.44/35 = 1.26%. Uto a na-ahụkarị kwa nzọ bụ 0.04 × ln(1 + 35 × 0.0126) + 0.96 × ln(1 − 0.0126) ≈ 0.00244, ya mere nzọ 1,000 na-amụba ego egwuregwu ahụ ihe dị ka e^2.44 ≈ 11.5.
Gbasaa otu ego nzọ n’elu nọmba ise, nzọ ahụ na-emeri na ohere 0.20 ma na-akwụ 36/5 maka 1, ya bụ 6.2 ruo 1: ka bụ uru 44%, ebe 0.2 × 7.2 = 1.44. Ego nzọ Kelly na-aghọ 0.44/6.2 = 7.10%, uto kwa nzọ 0.01404, ihe dị ka ugboro 5.75 karịa. N’ime nzọ 1,000 nke ahụ bụ e^14.04, ihe dị ka nde 1.25.
Uru ahụ bụ otu; mgbago na mgbada dị obere. Nzọ na-emeri otu ugboro n’ime ise nwere ike iburu òkè ka ukwuu nke ego egwuregwu n’etinyeghị ọdịda n’ihe egwu, uto Kelly na-akwụkwa ụgwọ maka nke ahụ. Thorp (1984) na-eji ihe atụ ahụ gosi ihe mere ndị na-egwu amụma ji agba nzọ na sectorwiilSectorAkụkụ oghere na-esochi ibe ha na wiil, nke dị iche na otu ìgwè na mbadamba nzọ.Gụọ ya niile →. N’omume, ndị agbata obinzọNdị agbata obiOtu nọmba na oghere abụọ n’akụkụ ọ bụla ya na wiil, chip straight up ise.Gụọ ya niile → ise nwere uru dị ntakịrị karịa otu nọmba kacha mma (lee Egwuregwu Uru na Roulette).
Onye egwuregwu achọtala ogherengwa ọrụOghereOtu n’ime ebe 37 nwere nọmba dị na rotor. A na-akpọkwa ya: slot, cup.Gụọ ya niile → abụọ dị n’akụkụ ibe ha nke ọ bụla na-apụta 1 ugboro n’ime 25 kama 1 n’ime 37. Ha kwesịrị ịgba nzọ splitnzọSplitNọmba abụọ dị n’akụkụ ibe ha, na-akwụ 17:1.Gụọ ya niile → n’etiti ha na 17 ruo 1, ka ọ bụ nọmba nke ọ bụla straight up? Ole ka Kelly na-ekwu, ma kedu ka ego egwuregwu si eto ngwa ngwa ma e jiri ya tụnyere otu oghere dị otu a?
Gosi azịza ya
Azịza Ọ dịghị ọdịiche: 2.59% nke ego egwuregwu na split ahụ, ma ọ bụ 1.29% na nọmba nke ọ bụla. Ego egwuregwu a na-ahụkarị na-eto ihe dị ka ugboro 155 n’ime ntụgharị 1,000, megide ugboro 11.5 na otu oghere.
Split ahụ na-emeri na oheremgbakọOhereOhere nke otu nsonaazụ, site na 0 ruo 1 ma ọ bụ 0% ruo 100%. Otu nọmba: 1/37 = 2.70%.Gụọ ya niile → 2/25 = 0.08 ma na-akwụ 17 ruo 1, ya mere ọ na-eweghachi 18 × 0.08 = 1.44 kwa unitegoUnitEgo nzọ ntọala nke otu usoro ma ọ bụ onye egwuregwu na-eji arụ ọrụ. A na-akọwa usoro mmụba nzọ na unit ka ha metụta tebụl ọ bụla.Gụọ ya niile → e tinyere: uru 44%, otu ihe ahụ dị ka otu oghere nwere mmasị. Ego nzọ Kelly bụ uru ahụ kewara site na odds: 0.44/17 = 2.59%.
Nzọ straight up abụọ nke 1.29% n’otu n’otu na-enye kpọmkwem otu nsonaazụ ahụ. Mgbe nọmba nke ọ bụla pụtara, otu chipegoChipIhe nnọchi anya ego a kara aka, a na-etinye na mbadamba nzọ iji gbaa nzọ. A na-akpọkwa ya: check.Gụọ ya niile → na-emeri 35 ruo 1, nke ọzọ efuo, ngụkọta 34 × 1.29% = 44% nke ego egwuregwu, dịka split ahụ na-akwụ; mgbe ọ nweghị nke pụtara, ha abụọ efuo.
Uto kwa ntụgharị bụ 0.00505, megide 0.00244 maka otu oghere, ya mere n’ime ntụgharị 1,000 ego egwuregwu a na-ahụkarị na-eto e^5.046 ≈ ugboro 155 kama e^2.44 ≈ ugboro 11.5. Nọmba ọma abụọ bara uru karịa okpukpu abụọ nke otu, nke a bụ ihe mere ndị na-egwu mmasị wiil ji agba nzọ n’ìgwè nọmba. Thorp (1984) na-arụ otu ntụnyere ahụ.
Ihe ọchị
Ego nzọ roulette kachasị mma maka uto bụ ọnụọgụ onye ọ bụla maralarị.
Isi Mmalite
- 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.”