Kelly Criterion aur Roulette
Kelly criterion edge wale bettor ko batata hai ki kitna stake lagaaye. Roulette ke liye iska jawab saaf hai: kuch nahi.
Kelly stake = edge / odds. Negative edge par formula negative fraction deta hai, jiska matlab hai bet mat lagaiye. Yeh criterion phir bhi yeh samajhne mein kaam aata hai ki bias players ne bets ka size waise kyun rakha, aur baaki sab ko budget ke hisaab se size kyun rakhna chahiye.
Criterion
John Kelly (1956) ne poocha ki jaane-maane edge wala gambler bankrollpaisaBankrollKhelne ke liye alag rakhi gayi kul raqam. Bankroll management ka matlab hai tay karna ki har bet aur har session mein iska kitna hissa risk karna hai.Poora padhein → ki lambi growth maximum karne ke liye bets ka size kaise rakhe. Jawab, aisi bet ke liye jo b to 1 deti hai, jeet ki probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → p aur haar ki probability q = 1 − p ke saath, yeh hai ki bankroll ka f* = (bp − q) / b hissa lagaaiye. Positive f* ka matlab woh hissa lagaaiye; zero ya negative ka matlab bet mat lagaaiye. Kelly Bell Labs mein kaam karte the, aur unke paper “A new interpretation of information rate” ne is sawaal ko Claude Shannon ki information theory mein rakha: jo gambler noisy line par tips paata hai, fair odds par, woh bankroll ko zyada se zyada usi rate se badha sakta hai jis rate se line information le jaati hai. 1961 mein Leo Breiman ne saabit kiya ki Kelly betting lambe samay mein kisi bhi asal mein alag strategy se aage nikalti hai aur tay lakshya tak lagbhag sabse kam samay mein pahunchti hai.
Roulette mein Kelly betting
RedbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → ke liye: b = 1, p = 18/37, q = 19/37; f* = (18/37 − 19/37)/1 = −1/37. Straight-upbetStraight-upEk single number, payout 35:1.Poora padhein → ke liye: b = 35, p = 1/37, q = 36/37; f* = (35/37 − 36/37)/35 = −1/1295. Har standard bet negative Kelly fraction deti hai, kyunki har bet ka edge negative hai, aur growth ke liye sabse achha stake zero hai. Roulette par Kelly betting wahi baat formal tareeke se kehti hai jo house-edge page kehta hai: aisa koi size nahi jis par negative-expectation bet bankroll badhaaye.
Kelly kahan sach mein laagu hua
Bias players ke paas edge bhi tha aur size ki samasya bhi. Jagger, García-Pelayos aur Eudaemons sabko tay karna pada ki apne maape hue fayde par kitna lagaayein, aur kam lagaane se edge bekaar jaata hai jabki zyada lagaane se edge dikhne se pehle ruin ka khatra hai. 30 mein 1 baar aane wale pocketsaamanPocketRotor ke 37 numbered khaanon mein se ek. Doosre naam: slot, cup.Poora padhein → ke liye Kelly kehta hai har spin par us number par bankroll ka 0.57%: $10,000 ke saath, lagbhag $57 ek spin, aam growth lagbhag 0.054% har spin, yaani 1,000 spins mein ×1.7. Average fayda isse dugna hai, 0.11% ek spin, lekin Kelly aam (typical) bankroll ko nishana banata hai, average ko nahi, jise kuch lucky runs phula dete hain. Half-Kelly aam practical choice hai kyunki edge ka andaaza pakka nahi hota.
Typical aur average
Typical bankroll aur average bankroll ke beech ka fark bada hai. Ek pocket lijiye jo 25 mein 1 baar aata hai, 44% ka edge, jo Thorp apne device ke liye batate hain. Kelly us par bankroll ka 1.26% lagaata hai. 1,000 spins ke baad typical bankroll lagbhag 11.5 guna badhta hai, jabki average, kuch bahut lucky runs ki wajah se upar khinchkar, lagbhag 249 guna. Kelly se zyada lagaane par average badhta hai aur typical result ghatta hai: Kelly stake ke dugne par typical bankroll sirf double hota hai, aur Kelly stake ke dhai gune par woh sikud jaata hai, halaanki har bet par ab bhi 44% edge hai. Kam lagaane ki keemat kam hai: aadha stake growth ka lagbhag teen-chauthai bachaata hai.
Jab edge sirf andaaza ho
Kelly maanta hai ki edge pata hai. Table par uska andaaza lagaaya jaata hai, aur andaaza aksar zyada hota hai. Epstein ka udaharan, Thorp ke 44% par bana, ek number hai jo double-zerowheelZeroAkela green pocket jise koi even-money, dozen ya column bet cover nahi karta; house edge ka source. Doosre naam: single zero, 0.Poora padhein → wheel ke 1,225 spins mein 49 baar aaya: 25 mein 1 baar, 44% edge, Kelly stake 1.26%. Lekin fair double-zero wheel par 38 numbers hain, aur unmein sabse zyada aane wala 1,225 spins mein lagbhag nau mein ek baar 49 ya usse zyada tak pahunchta hai. Kisi bhi record ka sabse busy number wahi hota hai jise luck ne sabse zyada madad di, isliye sach mein biased pocket bhi record mein asal se behtar dikhta hai: winner’s curse. Agar woh pocket asal mein unbiased hai, toh Kelly stake bas ek bet hai jo average mein 5.26% haarti hai. Aam jawab hain: Kelly ka ek hissa lagaaiye aur tab tak ginte rahiye jab tak edge naye record mein bhi tika rahe (How to Detect Roulette Wheel Bias dekhiye).
Edge wale log
Edge jaanna aur use achhe se use karna ek baat nahi. Haghani aur Dewey (2016) ne 61 logon ko har ek ko $25 diye taaki woh aadhe ghante ek sikke par bet lagaayein, jiske baare mein bataya gaya ki woh 60% baar heads aata hai, aur jeet ki seema $250 thi (players ko seema ka pata tabhi chala jab woh wahan pahunche). Kelly kehta hai har baar heads par bankroll ka 20% lagaaiye, aur lekhakon ka andaaza hai ki 10–20% ki lagatar bets lagbhag 95% baar seema tak pahunchtin. Sirf 21% players wahan pahunche. 28% kangaal ho gaye, 61 mein se 18 ne ek hi flip par sab laga diya, lagbhag ek-tihaai ne shuruaati $25 se kam par khatam kiya, aur average payoutpaisaPayoutJeetne wala bet kya lautata hai, “X to 1” ke roop mein bataya jaata hai, wapas mile stake ke upar.Poora padhein → $91 tha. 61 mein se sirf paanch ne Kelly criterion ka naam suna tha. Edge ho toh bets ka size tay karta hai ki aap use bacha paate hain ya nahi; edge na ho, jaise fair wheel par, toh koi size madad nahi karta.
Iske bajaye kya use karein
Edge ke bina size budget ka sawaal hai, growth ka nahi: stake = jitna aap har spin par kharch karne ko taiyaar hain, ghante ke budget aur table ki raftaar se nikaala hua (bankroll management page). Aam player ke liye Kelly ka yogdaan uske jawab ki saafgoi hai: zero. Zero se upar jo bhi aap lagaate hain woh khelne ki keemat hai, jo aap chunte hain, kisi formula se optimize nahi hoti.
Fractional Kelly aur “money management” par ek note
“Kelly for roulette” ke naam se bike systems jo har bet par bankroll ka fixed percentage sujhaate hain, woh sikudte stake ke saath flat bettingstrategyFlat bettingHar spin par same raqam lagana. Har progression ko isi baseline se compare kiya jaata hai.Poora padhein → hi hain. Woh turnoverpaisaTurnoverSession ya muddat mein lagayi gayi kul raqam, dobara lagayi gayi jeet milakar. Doosre naam: handle, action.Poora padhein → ka 2.70% haarte hain aur, kyunki haarne par stake sikudta hai, ruin ko unlikely nahi balki dheema bana dete hain. Budget ko lamba chalaane ka yeh theek tareeka hai; yeh Kelly nahi hai, jo rukne ko kehta.
Formula
Sawaal
Sirf tab jab biased wheellabBiased wheelAisa wheel jiske physical defects kuch pockets ko 1/37 se zyada likely bana dete hain. Bahut kam milta hai, aur hazaron record kiye gaye spins se hi pakda ja sakta hai.Poora padhein → ya timing device se maapa hua edge ho. Fair wheel par nahi.
Kyunki edge ke andaaze mein galti hoti hai, aur zyada lagaana kam lagaane se mehenga padta hai. Half-Kelly growth ka teen-chauthai deta hai, varianceganitVarianceYeh naap ki results expected value ke aas-paas kitna faile hue hain. Straight-up ke liye zyada, even-money bets ke liye kam.Poora padhein → ke ek-chauthai par.
Nahi. Kelly edge hone par growth optimize karta hai. Bankroll management edge na hone par kharch control karta hai.
Maths ke saath mazaa
Pehle khud try kijiye. Har solution chhupa hua hai; jab taiyaar hon tab kholiye.
Thorp aur Shannon ke prediction device ne lagbhag 44% edge diya. Ek predicted number par Kelly kehta hai bankrollpaisaBankrollKhelne ke liye alag rakhi gayi kul raqam. Bankroll management ka matlab hai tay karna ki har bet aur har session mein iska kitna hissa risk karna hai.Poora padhein → ka 1.26% lagaiye. Agar wahi 44% edge paanch padosi numbers par phaili ek bet par mile to?
Solution dikhaiye
Jawab Tab Kelly bankroll ka 7.1% kehta hai, aur bankroll lagbhag chhe guna tezi se badhta hai: 1,000 bets mein lagbhag 1.25 million guna, jabki pehle lagbhag 11.5 guna.
44% edge wala straight-upbetStraight-upEk single number, payout 35:1.Poora padhein → 0.04 chance se jeetta hai, kyunki 36 × 0.04 = 1.44. Kelly ka stake hai edge ko odds se bhaag: 0.44/35 = 1.26%. Per bet typical growth hai 0.04 × ln(1 + 35 × 0.0126) + 0.96 × ln(1 − 0.0126) ≈ 0.00244, isliye 1,000 bets bankroll ko lagbhag e^2.44 ≈ 11.5 se guna karti hain.
Ek stake ko paanch numbers par phailaiye to bet 0.20 chance se jeetti hai aur 1 ke liye 36/5 deti hai, yaani 6.2 to 1: edge phir bhi 44%, kyunki 0.2 × 7.2 = 1.44. Kelly ka stake 0.44/6.2 = 7.10% ho jaata hai, aur per bet growth 0.01404, lagbhag 5.75 guna zyada. 1,000 bets mein yeh e^14.04 hai, lagbhag 1.25 million.
Edge wahi hai; utaar-chadhaav chhote hain. Jo bet paanch mein ek baar jeetti hai woh barbaadi ke risk ke bina bankroll ka bada hissa utha sakti hai, aur Kelly ki growth iska inaam deti hai. Thorp (1984) is example se dikhate hain ki prediction players sectorswheelSectorWheel par pockets ka lagatar ek hissa (arc), layout ke kisi group se alag cheez.Poora padhein → par kyun bet lagate hain. Practice mein paanch neighborsbetNeighborsWheel par ek number aur uske dono taraf ke do-do pockets, paanch straight-up chips.Poora padhein → ka edge akele sabse achhe number se thoda kam hota hai (Advantage Play dekhiye).
Ek khiladi ko do padosi pocketssaamanPocketRotor ke 37 numbered khaanon mein se ek. Doosre naam: slot, cup.Poora padhein → mile hain jo har ek 37 mein 1 ki jagah 25 mein 1 baar hit karte hain. Kya unhe dono ke beech splitbetSplitDo saath wale numbers, payout 17:1.Poora padhein → 17 to 1 par lagana chahiye, ya har number straight-up? Kelly kitna kehta hai, aur ek aise pocket ke muqable bankroll kitni tezi se badhta hai?
Solution dikhaiye
Jawab Koi fark nahi padta: split par bankroll ka 2.59%, ya har number par 1.29%. Typical bankroll 1,000 spins mein lagbhag 155 guna badhta hai, jabki ek pocket ke saath 11.5 guna.
Split 2/25 = 0.08 probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → se jeetta hai aur 17 to 1 deta hai, isliye lagaye gaye har unitpaisaUnitBase stake jismein koi system ya player kaam karta hai. Progressions units mein batayi jaati hain taaki kisi bhi table par laagu hon.Poora padhein → par 18 × 0.08 = 1.44 lautata hai: 44% edge, ek biased pocket jitna hi. Kelly ka stake hai edge ko odds se bhaag: 0.44/17 = 2.59%.
1.29% ki do straight-up bets bilkul wahi results deti hain. Jab koi bhi number aata hai, ek chippaisaChipEk fixed raqam ka token, jise bet lagane ke liye layout par rakha jaata hai. Doosra naam: check.Poora padhein → 35 to 1 jeetti hai aur doosri haar jaati hai, net 34 × 1.29% = bankroll ka 44%, jitna split deta hai; jab koi nahi aata, dono haar jaati hain.
Per spin growth 0.00505 hai, ek pocket ke 0.00244 ke muqable, isliye 1,000 spins mein typical bankroll e^2.44 ≈ 11.5 guna ki jagah e^5.046 ≈ 155 guna badhta hai. Do achhe numbers ek ke dugne se kahin zyada keemti hain, isiliye bias players numbers ke groups par bet lagate hain. Thorp (1984) yahi tulna karte hain.
Thodi hansi
Roulette ka growth-optimal stake ek aisa number hai jo sabko pehle se pata hai.
Sources
- 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.”