Di Kelly criterion an roulette
Di Kelly criterion tell a gambla weh have a edge ow much fi stake. Fi roulette it give a clear ansa: nutten.
Kelly stake = edge / odds. Wen di edge negative, di formula give back a negative fraction, an dat mean nuh bet. Di criterion still useful fi undastand wah mek bias playa size dem bet di way dem did do it, an wah mek evrybaddy else fi size by budget.
Di criterion
John Kelly (1956) did ask ow a gambla weh know im edge fi size im bet fi mek di bankrollmoney mattaBankrollAll di money yuh set aside fi play. Fi manage yuh bankroll mean yuh decide ow much a it yuh a go risk pon evry bet an evry session.Read di whole ting → grow di most inna di long run. Fi a bet weh pay b to 1, wid win probabilitymathsProbabilityDi chance fi a result, fram 0 to 1 or 0% to 100%. One numba: 1/37 = 2.70%.Read di whole ting → p an loss probability q = 1 − p, di ansa a fi stake di fraction f* = (bp − q) / b a di bankroll. If f* positive, bet dat fraction; if it zero or negative, nuh bet. Kelly did work at Bell Labs, an im paper, “A new interpretation of information rate”, put di problem inna Claude Shannon theory a information: a gambla weh get tip ova a noisy line, at fair odds, can grow di bankroll at bes only as fast as di line carry information. Inna 1961 Leo Breiman prove seh Kelly bettin, inna di long run, finish ahead a any strategy weh different fi true, an reach a set goal inna bout di shortes time.
Kelly bettin inna roulette
Fi redbet demRedEven-money bet pon di 18 red numba. Inna French: rouge.Read di whole ting →: b = 1, p = 18/37, q = 19/37; f* = (18/37 − 19/37)/1 = −1/37. Fi a straight upbet demStraight upOne single numba, pay 35:1.Read di whole ting →: b = 35, p = 1/37, q = 36/37; f* = (35/37 − 36/37)/35 = −1/1295. Evry standard bet give a negative Kelly fraction, cause evry bet have a negative edge, an di stake weh grow di bankroll bes a zero. Kelly bettin pon roulette a jus a formal way fi seh wah di house-edge page seh: nuh size nuh deh weh a bet wid negative expectation can grow a bankroll.
Weh Kelly did apply
Bias playa did have a edge an a sizin problem. Jagger, di García-Pelayo dem an di Eudaemons all haffi decide ow much fi bet pon di advantage dem measure, an if yuh bet too likkle yuh waste di edge, while if yuh bet too much yuh risk ruin bifo di edge show. Fi a pocketgearPocketOne a di 37 compartment wid numba pon di rotor. Nex name: slot, cup.Read di whole ting → weh hit 1 outta 30, Kelly seh 0.57% a di bankroll evry spin pon dat numba: wid $10,000, dat a bout $57 a spin, fi a typical growth a bout 0.054% evry spin, or ×1.7 ova 1,000 spin. Di average gain a double dat, 0.11% a spin, but Kelly aim fi di typical bankroll, nuh di average one, weh a few lucky run blow up. Half-Kelly a di usual practical choice cause di edge estimate nuh sure.
Typical an average
Di gap between di typical bankroll an di average one wide. Tek a pocket weh hit 1 time outta 25, a edge a 44%, di figure Thorp give fi im device. Kelly stake 1.26% a di bankroll pon it. Afta 1,000 spin di typical bankroll grow bout 11.5 time, while di average, weh a few very lucky run pull up, grow bout 249 time. If yuh bet more dan Kelly, di average go up an di typical result go dung: at two time di Kelly stake di typical bankroll only double, an by two an a half time di Kelly stake it shrink, even doh evry bet still carry a 44% edge. If yuh bet less, it cost likkle: half di stake keep bout tree quarta a di growth.
Wen di edge a guess
Kelly tek it seh yuh know di edge. At di table yuh haffi estimate it, an di estimate usually too high. Epstein example, built pon Thorp 44%, a one numba weh hit 49 time inna 1,225 spin pon a double-zeropon di wheelZeroDi one green pocket weh no even-money, dozen or column bet nuh cover; a weh di house edge come fram. Nex name: single zero, 0.Read di whole ting → wheel: 1 time outta 25, a 44% edge, a Kelly stake a 1.26%. But a fair double-zero wheel have 38 numba, an di busies one a dem reach 49 or more inna 1,225 spin bout one time outta nine. Di busies numba inna any record a also di one weh luck help di most, suh even a pocket weh bias fi true look betta inna di record dan it really be: dat a di winna curse. If dat pocket nuh bias at all, di Kelly stake a jus a bet weh lose 5.26% pon average. Di usual ansa dem a fi bet a fraction a Kelly an fi keep countin till di edge survive a fresh record (look pon How to Detect Roulette Wheel Bias).
People weh have a edge
Fi know di edge nuh di same as fi use it good. Haghani an Dewey (2016) give 61 people $25 each fi bet fi half a hour pon a coin weh dem did tell dem come up heads 60% a di time, wid di winnins cap at $250 (di playa dem only find out bout di cap wen dem reach it). Kelly seh bet 20% a di bankroll pon heads evry time, an di writa dem estimate seh steady bet a 10–20% woulda reach di cap bout 95% a di time. Only 21% a di playa dem reach deh. 28% go broke, 18 outta di 61 bet evryting pon one flip, bout a third finish unda di $25 dem start wid, an di average payoutmoney mattaPayoutWah a bet weh win gi yuh back, weh dem quote as “X to 1” on top a di stake weh come back.Read di whole ting → did a $91. Only five outta di 61 did eva hear bout di Kelly criterion. Wen yuh have a edge, di size a di bet dem decide if yuh keep it; widout one, like at a fair wheel, nuh size cyaan help.
Wah fi use instead
Widout a edge, sizin a budget question, nuh a growth question: stake = wah yuh willin fi spend evry spin, work out fram di budget fi di hour an how fast di table go (bankroll management page). Wah Kelly give di ordinary playa a di clear ansa: zero. Evryting yuh stake ova zero a di price fi play, weh yuh choose yuhself, nuh weh any formula optimize.
A note pon fractional Kelly an “money management”
System weh people sell as “Kelly fi roulette” weh tell yuh fi bet a fix percentage a di bankroll pon evry bet a jus flat bettinstrategy demFlat bettinFi bet di same amount evry spin. Di baseline weh dem compare evry progression gainst.Read di whole ting → wid a stake weh a shrink. Dem lose 2.70% a turnovermoney mattaTurnoverAll di money yuh bet ova a session or a period, countin di winnins yuh bet again (all di money yuh bet, roun afta roun). Nex name: handle, action.Read di whole ting → an, cause di stake shrink as yuh lose, dem mek ruin come slow, nuh mek it unlikely. It a decent way fi mek a budget last; it nuh Kelly, cause Kelly woulda seh stop.
Formula
Question dem
Only wid a edge weh yuh measure fram a bias wheellabBias wheelA wheel weh have physical defect weh mek some pocket more likely dan 1/37. It rare, an yuh can only find it out wid thousands a spin weh yuh write dung.Read di whole ting → or a timin device. Nuh pon a fair wheel.
Cause di edge estimate have error inna it, an fi bet too much cost more dan fi bet too likkle. Half-Kelly give tree quarta a di growth fi a quarta a di variancemathsVarianceA measure a ow wide di result dem spread roun di expected value. High fi straight up, low fi even-money bet.Read di whole ting →.
No. Kelly optimize growth wen yuh have a edge. Bankroll management control spendin wen yuh nuh have none.
Fun wid maths
Try dem yuhself fus. Evry ansa fold up; open it wen yuh ready.
Thorp an Shannon prediction device did give bout a 44% edge. Pon a single numba weh it predict, Kelly seh fi bet 1.26% a di bankrollmoney mattaBankrollAll di money yuh set aside fi play. Fi manage yuh bankroll mean yuh decide ow much a it yuh a go risk pon evry bet an evry session.Read di whole ting →. Wah if di same 44% edge did come pon one bet spread ova five numba weh side by side?
Show di ansa
Ansa Den Kelly seh 7.1% a di bankroll, an di bankroll grow almost six time as fast: bout 1.25 million time ova 1,000 bet, gainst bout 11.5 time.
A straight upbet demStraight upOne single numba, pay 35:1.Read di whole ting → wid a 44% edge win wid chance 0.04, since 36 × 0.04 = 1.44. Kelly stake a di edge divide by di odds: 0.44/35 = 1.26%. Di typical growth pon each bet a 0.04 × ln(1 + 35 × 0.0126) + 0.96 × ln(1 − 0.0126) ≈ 0.00244, suh 1,000 bet multiply di bankroll by bout e^2.44 ≈ 11.5.
Spread one stake ova five numba an di bet win wid chance 0.20 an pay 36/5 fi 1, dat a 6.2 to 1: still a 44% edge, since 0.2 × 7.2 = 1.44. Kelly stake turn 0.44/6.2 = 7.10%, an di growth pon each bet 0.01404, bout 5.75 time as much. Ova 1,000 bet dat a e^14.04, bout 1.25 million.
Di edge a di same; di swing dem smalla. A bet weh win one time outta five can carry a bigga share a di bankroll widout risk a ruin, an Kelly growth reward dat. Thorp (1984) use di example fi show why prediction playa bet sectorpon di wheelSectorA piece a di wheel weh di pocket dem follow one anodda roun di arc, nuh a group pon di betting layout.Read di whole ting →. Inna practice five neighbourbet demNeighboursA numba an di two pocket pon evry side a it pon di wheel, five straight-up chip.Read di whole ting → carry a likkle lowa edge dan di single bes numba (look pon Advantage Play).
A playa find two pocketgearPocketOne a di 37 compartment wid numba pon di rotor. Nex name: slot, cup.Read di whole ting → side by side weh each one hit 1 time outta 25 instead a 1 outta 37. Im fi bet di splitbet demSplitTwo numba weh deh side by side, pay 17:1.Read di whole ting → between dem at 17 to 1, or each numba straight up? Ow much Kelly seh, an ow fast di bankroll grow compare to one pocket like dat?
Show di ansa
Ansa It nuh mek no difference: 2.59% a di bankroll pon di split, or 1.29% pon each numba. Di typical bankroll grow bout 155 time inna 1,000 spin, gainst 11.5 time wid one pocket.
Di split win wid probabilitymathsProbabilityDi chance fi a result, fram 0 to 1 or 0% to 100%. One numba: 1/37 = 2.70%.Read di whole ting → 2/25 = 0.08 an pay 17 to 1, suh it return 18 × 0.08 = 1.44 fi each unitmoney mattaUnitDi base stake weh a system or a playa work wid. Dem describe progression inna unit suh dem work at any table.Read di whole ting → yuh stake: a edge a 44%, di same as one bias pocket. Kelly stake a di edge divide by di odds: 0.44/17 = 2.59%.
Two straight-up bet a 1.29% each give exactly di same result dem. Wen eida numba come up, one chipmoney mattaChipA token weh stand fi a fix amount, weh yuh put pon di betting layout fi mek a bet. Nex name: check.Read di whole ting → win 35 to 1 an di odda one lose, a net a 34 × 1.29% = 44% a di bankroll, same as di split pay; wen neida one come, both a dem lose.
Di growth pon each spin a 0.00505, gainst 0.00244 fi one pocket, suh ova 1,000 spin di typical bankroll grow e^5.046 ≈ 155 time instead a e^2.44 ≈ 11.5 time. Two good numba worth far more dan two time one, an dat a why bias playa bet group a numba. Thorp (1984) work di same comparison.
Sumting fi laugh
Di roulette stake weh grow di bankroll bes a one numba evrybaddy done know.
Source dem
- 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.”