Alugen n Kelly d rulit
Alugen n Kelly yeqqar i umurar yesɛan lmaziyya acḥal ara yerhen. I rulit, yettak-d tiririt i d-ibanen: ula d acemma.
Rrehn n Kelly = lmaziyya / assaɣ. S lmaziyya ddaw n yilem, tasemselt tettak-d aḥric ddaw n yilem, i yebɣan ad yini ur rehhen ara. Alugen-a yenfeɛ ula d tura akken ad nefhem amek i ttwaseggmen rrhun ɣer yimuraren n umyal, d wayɣer wiyaḍ akk ilaq ad seggmen s tedrimt.
Alugen
John Kelly (1956) yesteqsa amek ilaq umurar s lmaziyya yettwassnen ad yesseggem rrhun akken ad yessimɣur timerna n ugerrujidrimenAgerrujAkk idrimen yettwaǧǧan i wurar. Asefrek n ugerruj d tagrut: acḥal seg-s ara tesrem i yal rrehn d yal taɣimit.Ɣer akk → deg uɣawas aɣezfan. Tiririt, i rrehn ixellṣen b ɣef 1 s taprubabilittusnaktTaprubabilitAzal n yiwen ugmuḍ ad d-yeḍru, seg 0 ɣer 1 neɣ seg 0% ɣer 100%. Yiwen wuṭṭun: 1/37 = 2,70%.Ɣer akk → n rrbeḥ p d taprubabilit n lexsara q = 1 − p, d arhen n uḥric f* = (bp − q) / b n ugerruj. f* sennig n yilem yebɣa ad yini rhen aḥric-nni; ilem neɣ ddaw n yilem yebɣa ad yini ur rehhen ara. Kelly yella yettxeddim deg Bell Labs, yerna amagrad-is, «A new interpretation of information rate», yessers ugur deg tmuɣli n telɣut n Claude Shannon: amurar i yettawin tilɣa s ujerriḍ n tilifun yesɛan ahuzzu, s wassaɣ n lḥeqq, yezmer ad yessimɣur agerruj, ma yella akka, s tazrit n telɣut i yettawi ujerriḍ. Deg 1961 Leo Breiman yesbeggen belli arhen n Kelly, deg uɣawas aɣezfan, yettfakk s rrbeḥ ugar n yal tastratijit yemgaraden s tidet, yettaweḍ ɣer yiswi yettwasbedden deg wakud awezlan akk qrib.
Arhen n Kelly deg rulit
I uzeggaɣrrehnAzeggaɣRrehn s lexlaṣ imsawi ɣef 18 n wuṭṭunen izeggaɣen. S tefransist: rouge.Ɣer akk →: b = 1, p = 18/37, q = 19/37; f* = (18/37 − 19/37)/1 = −1/37. I yiwen wuṭṭunrrehnYiwen wuṭṭunYiwen wuṭṭun kan, ixelleṣ 35:1.Ɣer akk →: b = 35, p = 1/37, q = 36/37; f* = (35/37 − 36/37)/35 = −1/1295. Yal rrehn amagnu yettak-d aḥric n Kelly ddaw n yilem, acku yal rrehn yesɛa lmaziyya ddaw n yilem, yerna rrehn yifen i tmerna d ilem. Arhen n Kelly ɣef rulit d abrid unṣib n wayen yeqqar usebter n lmaziyya n lkazinu: ulac tiddi ideg rrehn s wazal yettwaṛǧan ddaw n yilem yessimɣur agerruj.
Anida i yettwasnas Kelly
Imuraren n umyal sɛan lmaziyya d wugur n tiddi n rrhun. Jagger, García-Pelayo d Eudaemons, akk ilaq ad ferren acḥal ara rehnen ɣef lmaziyya-nsen yettwaktalen, yerna arhen s drus yesṛuḥuy lmaziyya, arhen s waṭas yezmer ad yawi ɣer uɣelluy uqbel ad d-tban lmaziyya. I texxamtibenkanTaxxamtYiwet seg 37 n texxamin s wuṭṭunen ɣef uqerru yettezzin. Daɣen: slot, cup.Ɣer akk → i d-yettbanen 1 seg 30, Kelly yeqqar 0,57% n ugerruj i tuzzya ɣef wuṭṭun-nni: s $10 000, qrib $57 i tuzzya, i tmerna s umata n qrib 0,054% i tuzzya, neɣ ×1,7 deg 1 000 n tuzzyiwin. Rrbeḥ alemmas d sin n tikkal waya, 0,11% i tuzzya, maca Kelly yettnadi agerruj s umata, mačči alemmas, i ssimɣuren kra n yizrar n zzher. Azgen n Kelly d afran n leqdic s umata acku aqadar n lmaziyya ur yettwaḍman ara.
S umata d ulemmas
Ameccaq gar ugerruj s umata d ugerruj alemmas d ameqqran. Ṭṭfet taxxamt i d-yettbanen 1 seg 25, lmaziyya n 44%, amḍan i d-yefka Thorp i yibenk-is. Kelly yettrehin 1,26% n ugerruj fell-as. Deffir 1 000 n tuzzyiwin, agerruj s umata yemɣer qrib 11,5 n tikkal, ma d alemmas, i d-ssalin kra n yizrar n zzher meqqren, yemɣer qrib 249 n tikkal. Arhen ugar n Kelly yessali alemmas, yessenqas agmuḍ s umata: s sin n tikkal rrehn n Kelly, agerruj s umata yettnerni kan s sin, yerna s sin u azgen n tikkal rrehn n Kelly yettenqas, ɣas akken yal rrehn yesɛa lmaziyya n 44%. Arhen s drus ur yettkellif ara aṭas: azgen n rrehn yeḥrez qrib kraḍ n rrbuɛ n tmerna.
Mi ara tili lmaziyya d ayen yettwaɣilen
Kelly yeɣil belli lmaziyya tettwassen. Ɣef ṭṭabla tettwaqedder, yerna aqadar s umata d ameqqran aṭas. Amedya n Epstein, yebnan ɣef 44% n Thorp, d uṭṭun i d-yeffɣen 49 n tikkal deg 1 225 n tuzzyiwin n uḍebsi s sin yilmawenaḍebsiIlemTaxxamt tazegzawt tasuft ur yeṭṭif ula d yiwen n rrehn s lexlaṣ imsawi, n ddzina neɣ n tgejdit; d aɣbalu n lmaziyya n lkazinu. Daɣen: yiwen yilem, 0.Ɣer akk →: 1 seg 25, lmaziyya n 44%, rrehn n Kelly n 1,26%. Maca aḍebsi imsawi s sin yilmawen yesɛa 38 n wuṭṭunen, yerna win yeffɣen ugar seg-sen yettaweḍ ɣer 49 neɣ ugar deg 1 225 n tuzzyiwin qrib yiwet n tikkelt seg tẓa. Uṭṭun yeffɣen ugar deg yal asekles d win daɣen i iɛawen zzher ugar, ihi ula d taxxamt s umyal n tidet tettban-d deg uɣmis yelha ugar n wayen tella: d ddeɛwessu n win yerbḥen. Ma yella taxxamt-nni s tidet war amyal, rrehn n Kelly d rrehn kan i yettxesren 5,26% s ulemmas. Tiririyin yettwassnen d arhen n uḥric n Kelly, d ukemmel n usiḍen alamma lmaziyya teṣmed i uɣmis amaynut (walit Amek ara d-naf amyal n uḍebsi).
Imdanen s lmaziyya
Tamussni n lmaziyya mačči d yiwet d useqdec-is s wudem yelhan. Haghani d Dewey (2016) fkan i 61 n yimdanen $25 i yal yiwen akken ad rehnen azgen n tsaɛett ɣef uṣurdi i sen-qqaren yettak-d aqerru 60% n wakud, s yirbaḥen yettwaḥeddden ɣer $250 (imuraren ẓran s talast mi kan wwḍen-tt). Kelly yeqqar ad yettwarhen 20% n ugerruj ɣef uqerru yal tikkelt, yerna imeskaren qedren belli rrhun yettɣimin n 10–20% ad wwḍen ɣer talast qrib 95% n wakud. 21% kan n yimuraren i yewwḍen. 28% xesren kullec, 18 seg 61 rehnen kullec ɣef yiwen uḍeqqer, qrib tulut fukken ddaw $25 i bdan, yerna alemmas n lexlaṣidrimenLexlaṣAyen i d-yettak rrehn i yerbḥen, yettwanna «X ɣef 1» nnig rrehn i d-yettuɣalen.Ɣer akk → yella $91. Semmus kan seg 61 i yeslan s ulugen n Kelly. S lmaziyya, tiddi n rrhun i yettafren ma ad tt-teḥrezem; war-s, am ɣef uḍebsi imsawi, ulac tiddi i iɛawnen.
Ayen ara tesseqdcem deg umkan-is
War lmaziyya, tiddi n rrhun d asteqsi n tedrimt, mačči n tmerna: adrim n rrehn = ayen i tebɣam ad tesrufem i yal tuzzya, yettwakkes seg tedrimt n tsaɛett d tazrit n ṭṭabla (asebter n usefrek n ugerruj). Ayen i d-yefka Kelly i umurar n yal ass d tafat n tririt-is: ilem. Yal ayen ara terhnem nnig yilem d ssuma n wurar, i tferrem s yiman-nwen, ur t-yesseggem ula d yiwet n tsemselt.
Tamawt ɣef uḥric n Kelly d «usefrek n yedrimen»
Inagrawen i ttzenzen am «Kelly i rulit» i d-yessumuren afmiḍi yettwasbedden n ugerruj i yal rrehn d rrehn ur nettbeddiltastratijitRrehn ur nettbeddilRrehn n yiwen wazal deg yal tuzzya. D aswir ɣer wacu i yettwaserwas yal anagraw n usnerni.Ɣer akk → s udrim yettenqasen. Ttxesren 2,70% n umḍan n rrhun meṛṛaidrimenAmḍan n rrhun meṛṛaTamerna n yidrimen yettwarhen deg yiwet n tɣimit neɣ deg yiwen wakud, s rrbeḥ i yettuɣalen yettwarhen. Daɣen: handle, action.Ɣer akk →, yerna acku adrim yettenqas mi ara texsrem, rrent aɣelluy yetteddu s ttawil, mačči d ayen yettwafen qlil. D abrid yeɛqel akken ad teṭṭul tedrimt; mačči d Kelly, i yeqqaren ad tḥebsem.
Tasemselt
Isteqsiyen
Kan s lmaziyya yettwaktalen seg uḍebsi s umyaltanaritAḍebsi s umyalAḍebsi i deg ara xedmen yixeṣṣaṣen-is n lǧessa kra n texxamin ad d-ffɣent ugar n 1/37. D ayen yettwaxeṣṣren, ur yettwaf ara ala s yigiman n tuzzyiwin yettwarun.Ɣer akk → neɣ seg yibenk n uktazal. Mačči ɣef uḍebsi imsawi.
Acku aqadar n lmaziyya yesɛa tuccḍa, yerna arhen s waṭas yettkellif ugar n urhen s drus. Azgen n Kelly yettak-d kraḍ n rrbuɛ n tmerna s rubeɛ n tavaryanstusnaktTavaryansAktazal n wamek i d-ttwazzaren igmaḍ i tama n wazal yettwaṛǧan. Meqqret i rrehn ɣef yiwen wuṭṭun, meẓẓiyet i rrhun s lexlaṣ imsawi.Ɣer akk →.
Ala. Kelly yesseggem tamerna ma tella lmaziyya. Asefrek n ugerruj yettḥakam deg usrufu ma ulac-itt.
Urar s tusnakt
Ɛerḍet-tent s yiman-nwen qbel. Yal tifrat tettwaffer; ldit-tt mi ara tilim twejdem.
Allal n tifin uqbel n Thorp d Shannon yettak-d qrib lmaziyya n 44%. Ɣef yiwen wuṭṭun yettwafen zik, Kelly yenna ad trehnem 1,26% n ugerrujidrimenAgerrujAkk idrimen yettwaǧǧan i wurar. Asefrek n ugerruj d tagrut: acḥal seg-s ara tesrem i yal rrehn d yal taɣimit.Ɣer akk →. Ma yella lmaziyya-nni n 44% d-tusa ɣef yiwen n rrehn yettwabḍan ɣef semmus n wuṭṭunen yemlalen?
Sken-d tifrat
Tiririt Kelly yenna dɣa 7,1% n ugerruj, u agerruj yettimɣur qrib sḍis n tikkal s tazzla ugar: qrib 1,25 n yimelyunen n tikkal deg 1 000 n rrhun, mgal qrib 11,5 n tikkal.
Rrehn ɣef yiwen wuṭṭunrrehnYiwen wuṭṭunYiwen wuṭṭun kan, ixelleṣ 35:1.Ɣer akk → s lmaziyya n 44% yerbeḥ s tprubabilit n 0,04, imi 36 × 0,04 = 1,44. Rrehn n Kelly d lmaziyya yettwabḍan ɣef wassaɣ: 0,44/35 = 1,26%. Timɣuri tamagnut i yal rrehn d 0,04 × ln(1 + 35 × 0,0126) + 0,96 × ln(1 − 0,0126) ≈ 0,00244, ihi 1 000 n rrhun sgeten agerruj s qrib e^2,44 ≈ 11,5.
Bḍut yiwen n rrehn ɣef semmus n wuṭṭunen, rrehn ad yerbeḥ s tprubabilit n 0,20 u ad ixelleṣ 36/5 i 1, d 6,2 ɣef 1: mazal lmaziyya n 44%, imi 0,2 × 7,2 = 1,44. Rrehn n Kelly yuɣal d 0,44/6,2 = 7,10%, u timɣuri i yal rrehn 0,01404, qrib 5,75 n tikkal ugar. Deg 1 000 n rrhun d e^14,04, qrib 1,25 n yimelyunen.
Lmaziyya d yiwet; ahuzzu d amecṭuḥ. Rrehn i yettrebḥen yiwet n tikkelt deg semmus yezmer ad yawi tasgunt tameqqrant ugar n ugerruj war lxeṭṛ n uɣelluy, u timɣuri n Kelly tessenfaɛ aya. Thorp (1984) yesseqdac amedya-a akken ad yesken acuɣer imuraren n tifin uqbel ttrehinen ɣef tigezminaḍebsiTagzemtAḥric n texxamin yemmeḍfaren ɣef uḍebsi, mačči d tagrumma ɣef tfelwit.Ɣer akk →. S tidet, semmus n lǧiranrrehnLǧiranYiwen wuṭṭun d snat n texxamin ɣef yal tama-s ɣef uḍebsi, semmus n yijiṭunen ɣef yiwen wuṭṭun.Ɣer akk → sɛan lmaziyya drus cwiṭ ɣef wuṭṭun yelhan akk iman-is (walit Urar n lmaziyya).
Amurar yufa-d snat n texxaminibenkanTaxxamtYiwet seg 37 n texxamin s wuṭṭunen ɣef uqerru yettezzin. Daɣen: slot, cup.Ɣer akk → yemlalen, yal yiwet tettak-d 1 n tikkelt deg 25 deg wadeg n 1 deg 37. Ilaq ad yerhen ɣef sin wuṭṭunenrrehnSin wuṭṭunenSin wuṭṭunen yemyuḍanen, ixelleṣ 17:1.Ɣer akk → gar-asent s 17 ɣef 1, neɣ ɣef yal uṭṭun iman-is? Acḥal i yenna Kelly, d s wacḥal n tazzla i yettimɣur ugerruj mgal yiwet n texxamt am ta?
Sken-d tifrat
Tiririt Ulac amgired: 2,59% n ugerruj ɣef sin wuṭṭunen, neɣ 1,29% ɣef yal uṭṭun. Agerruj amagnu yettimɣur qrib 155 n tikkal deg 1 000 n tuzzyiwin, mgal 11,5 n tikkal s yiwet n texxamt.
Rrehn ɣef sin wuṭṭunen yerbeḥ s tprubabilittusnaktTaprubabilitAzal n yiwen ugmuḍ ad d-yeḍru, seg 0 ɣer 1 neɣ seg 0% ɣer 100%. Yiwen wuṭṭun: 1/37 = 2,70%.Ɣer akk → n 2/25 = 0,08 u ixelleṣ 17 ɣef 1, ihi yerra-d 18 × 0,08 = 1,44 i yal aferdisidrimenAferdisRrehn agejdan s wacu i yettxeddem unagraw neɣ amurar. Inagrawen n usnerni ttwaglamen s yiferdisen akken ad ilin i yal ṭṭabla.Ɣer akk → yettwarehnen: lmaziyya n 44%, am yiwet n texxamt s umyal. Rrehn n Kelly d lmaziyya yettwabḍan ɣef wassaɣ: 0,44/17 = 2,59%.
Sin n rrhun ɣef yiwen wuṭṭun n 1,29% yal yiwen ttak-d igmaḍ-nni s tidet. Mi ara d-yeffeɣ yiwen seg wuṭṭunen-nni, yiwen ujiṭunidrimenAjiṭunTamatart i d-yemmalen azal yettwasbedden, yettwasersen ɣef tfelwit akken ad yili rrehn. Daɣen: check.Ɣer akk → yerbeḥ 35 ɣef 1 u wayeḍ yexser, amatu ačuran n 34 × 1,29% = 44% n ugerruj, akken ixelleṣ rrehn ɣef sin wuṭṭunen; mi ara ulac yiwen, xesren i sin.
Timɣuri i yal tuzzya d 0,00505, mgal 0,00244 i yiwet n texxamt, ihi deg 1 000 n tuzzyiwin agerruj amagnu yettimɣur e^5,046 ≈ 155 n tikkal deg wadeg n e^2,44 ≈ 11,5 n tikkal. Sin wuṭṭunen yelhan sɛan azal yugaren aṭas snat n tikkal n yiwen, d ayagi i yerran imuraren n umyal ttrehinen ɣef yigrawen n wuṭṭunen. Thorp (1984) yexdem aserwes-a yiwen.
Taḍsa i usgunfu
Rrehn n rulit yifen i tmerna d amḍan i yessen yal yiwen yakan.
Iɣbula
- 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.”