Statistical modelling of gambling probabilities

Transcription

1 Ttle Statstcal modellng of gamblng pobabltes Autho(s) Lo, Su-yan, Vcto.; 老 瑞 欣 Ctaton Issued Date 992 URL Rghts The autho etans all popetay ghts, (such as patent ghts) and the ght to use n futue woks.

2 STATISTICAL MODELLING OF GAMBLING PROBABILITIES Vcto Su Yan Lo A thess pesented towads the degee of Docto of Phlosophy Unvesty of Hong Kong June 992

3 Ths thess s submtted to the Unvesty of Hong Kong fo the awad of the degee of Docto of Phlosophy. The whole wok has been undetaken afte egstaton fo the degee and has not been pevously ncluded n a thess dssetaton o epot submtted to t h s o any o t h e n s t t u t o n f o a degee, dploma o othe qualfcaton- Vcto S.Y. Lo

4 ACKNOWLEDGEMENTS The subject supevso, D. matte J. of ths thess Bacon-Shone, Lectue was of suggested the by my Depatment of Statstcs.Decto of Socal Scences Reseach Cente, Faculty Dean of Socal Scences, Unvesty of Hong Kong and my nfomal advso, D. K. Busche Lectue o f t h e School o f Economcs, Unvesty o f Hong Kong. I must t a k e t h s oppotunty t o e x p e s s my deep g a t t u d e t o both o f t h e m f o t h e concen, suppot, contnuous gudance and enlghtenment thoughout t h e e n t e peod o f study. The good deas, stmulaton, p a t e n t coectons and encouagement made by m y supevso w e e ndspensable t o t h s t h e s s. I am a l s o g a t e f u l t o D. W.K. L and D. Y.C. Kuk f o t h e advce. Moeve, I would l k e t o thank D. K. Busche, P o f e s s o s M.M. Al and R. Quandt f o t h e data, and P o f e s s o Junj Shba f o h s h e l p n acqung t h e Japanese data.

5 Abstact o f thess enttled "STATISTICAL MODELLING OF GAMBLING PROBABILITIES" submtted by Vcto Su Yan Lo fo the degee of Docto of Phlosophy at the Unvesty of Hong Kong n June 992 Economsts and psychologsts have long shown Inteest n acetack bettng as a souce fo nvestgatng atttudes to sk and effcency of makets. Most of the pevous studes have concluded that bettos undebet favoute hoses and ovebet longshots. Hence, the bettos may be sk-loves. A dffeent phenomenon n Hong Kong acetack bettng was also epoted. Howeve, to date the statstcal methods have been faly udmentay and thus, thee s oom fo mpoved analyss. In ths thess, we popose and apply new statstcal methodology to study the bettng maket usng a lage amount of data obtaned n dffeent acetacks. A smple model s developed fo ths pupose. The conclusons fo the wn bet n dffeent acetacks ae consstent wth the pevous studes. Apat fom analysng the smple wn bet, we extend ou model to nclude moe complcated bets such as the exacta.

6 Usng wn bet factons alone, models fo pedctng odeng pobabltes based on dffeent dstbutons (exponental, nomal assumptons and of gamma) unlng ae tme compaed empcally and theoetcally. Patcula vaance and coelaton stuctues of unnng tmes ae also consdeed. As most of these models ae dffcult to apply n pactce, a smplfcaton of these sophstcated models s poposed. The smplfed models ae then appled to a bettng stategy (D.Z s system} n dffeent a c e t a c k s. Results ndcate t h a t t h e e s a n mpovement ove t h e ognal s t a t e g y.

7 To my Dad who neve loses because he neve bets and my gandmom who s a wnne because she ases me I

8 Ths thess s submtted to the Unvesty of Hong Kong fo the awad of the degee of Docto of Phlosophy, The whole wok has been undetaken afte egstaton fo the degee and has not been pevously ncluded n a thess, dssetaton o epot submtted to ths o any othe nsttuton fo a degee, dploma o qualfcaton. Vcto S.Y. Lo othe

9 CONTENTS Page Chapte Intoducton LI Backgound ].2 Racetack bettng makets ;.3 Reseach aeas - L4 Data souces Chapte 2 Lteatue evew 2. Intoducton Rsk behavou and maket effcency Estmaton of outcome pobabltes Optmal bettng Chapte 3 Modellng the wnnng pobabltes 3. Intoducton A s method of analysng wn bet data Classes of multnomal logt models 33 3A 3.5 Empcal analyses n the U.S. H-K-, Japan and Shangha Compason w t h t h e othe models Relatonshp between t h e b e t a and pool s z e s Monng l n e odds and e a l e odds 68

10 3.9 Conclusons Chapte 4 72 Analyses of moe complcated bets 4. Intoducton Descpton of some poposed models Extended logt model fo moe complcated bets Empcal analyss fo exacta bet Empcal analyss fo tfecta bet Empcal analyss fo qunella bet Empcal analyss fo double bet and double qunella Conclusons 04 Chapte 5 Detaled compason between the Havlle and Heney models 5. Intoducton The motvaton Condtonal logstc analyss fo the Havlle and Heney models Theoetcal nvestgaton of the Havlle and Heney models Concluson 23 Chapte 6 Extensons of the Heney model 6. Intoducton The selected model 25

11 6-3 Appoxmaton fomulas 3 6A 33 Chapte 7 A ecommended smple model 7. Intoducton Smlaty between the two models Some smulaton esults Empcal analyss usng fxed lambda and tan Matchng between the Heney model and exacta bet f Facton Othe appoxmatons fo the Heney model Compason of pobablty estmatons usng a closeness measue Concluson 64 Chapte 8 The Sten model n Japan 8. Intoducton Falue of the Heney model n Japan Fttng the Sten model A smple appoxmaton of the Sten model A theoetcal esult fo the compason between the Havlle and Sten models 8.6 Concluson Chapte 9 Study of bettng stategy 9. Intoducton 93

12 9.2 Descpton of D.Z's system Applcatons of D.Z's system Specal consdeaton n Japan Recommendatons 24 Chapte 0 Oveall conclusons 0. Bettng behavou Complcated bets analyses Models fo pedctng odeng pobabltes Bettng stategy 28 Appendx A Estmated paametes fo A-class and L-class models 220 Appendx B Gaussan-Hemte ntegatons fo complcated pobabltes 226 Appendx C Devaton of appoxmaton fo pobabltes unde extended model 228 Appendx D Devatons of second-ode appoxmatons fo the Heney model Bblogaphy

13 CHAPTER ONE INTRODUCTION. Backgound Gamblng behavou s one topc n decson unde sk. It s elated to at least two Economsts outcomes ae (e.g. Psychologsts nvestgate Kahneman academc felds - Economcs and Psychology. nteested n fndng possble Gethe S c Plott & Mathematcal the behavoual S c Tvesky (979 (979) Psychologsts, pocess (e.g. easons fo the Wetzman (965)). and on the Luce othe (965 & hand, 977), S c 983) and Tvesky & Kahneman 974 Se 98)). Nowadays, Statstcans can wok w t h Psychologsts and Economsts n studyng t h e human decson pocess and explanng vaous phenomena wth s ta t s t c s. Fo nstance, subjectve pobablty and objectve pobablty estmaton c a n be teated as s t a t s t c a l poblems. Racetack bettng has long b e e n ecognsed b y Economsts and Psychologsts as a souce of nfomaton f o nvestgatng atttudes t o sk and e f f c e n c y of makets. Racetacks and s e c u t e s makets have many chaactestcs n common. One k e y d f f e e n c e s t h e complexty; t h e acetack s e a l l y a sequence of makets t h a t ae e l a t v e l y smple and shot-lved.

14 To bettos, of couse, t s moe nteestng to pedct the poft. Bettos usng a bettng system attempt to fnd the maket neffcences and bet on such hoses when they have postve expected poft. To date statstcal methods have been faly udmentay. We have poposed mpoved methods and used these to analyse hose-acng data fom dffeent countes ncludng the U.S., Hong Kong, and Japan n bette ways..2 Racetack Bettng Makets Befoe ntoducng ou study, t s helpful to befly descbe the acetack bettng makets. Usually, fo each ace, bettos have a choce of bettng types. Fo example, wn bet means a betto wll have a postve etun f the hose he bets on wns place b e t means a b e t t o w l l have a p o s t v e e t u n f t h e h o s e h e chooses f n s h e s f s t o second; b e t t n g on t w o h o s e s t o e x a c t a p a y s o n l y w h e n t h e t w o h o s e s f n s h f s t and second n e x a c t ode, e t c. Cwe adopt t h e temnology o f t h e U.S. acetacks). Sepaate pools o f money ae kept f o each bet type. The numbe o f h o s e s n a a c e u s u a l l y a n g e s f o m 6 t o 2 n t h e U.S., 3 t o 4 n Hong Kong and 6 t o 8 n Japan. The smplest b e t t y p e s t h e w n bet. Unde t h e pamutuel b e t t n g system (o t h e totalze system), t h e odds (.e. pofts pe dolla b e t t o a

15 successful betto) o f the w n bet a e detemned f o m the bets made by the publc as follows : In a patcula ace, let Unde the pamutuel system, 0 = (-t) W - X X ^ =odds on hose. s calculated as follows =,2,..., whee t = the tack take, n = numbe o f hoses n t h e a c e, = t h e w n b e t amount on ho se, and W = t o t a l w n b e t amount on a l l h o s e s = S X The popoton o f money b e t on a h o s ex / W s called t h e w n b e t f p a c t o n. Some e s e a c h e s have suggested t h a t t h e w n b e t f a c t o n c a n b e used as a n e s t m a t e o f t h e t u e wnnng pobablty o f a h o s e. One eason s t h a t unde t h e assumpton of p o f t m a x m z a t o n o f bettos, t h e e x p e c t e d e t u n s f o a l l t h e hoses n t h e a c e a e t h e same and t c a n b e e a s l y s h o w n t h a t t h e bet f a c t o n s equal t o t h e t u e w n n n g pobablty. The t a c k take, t s t h e commsson pad t o t h e Jockey club. Some p a t o f t h e take may b e pad t o t h e Govenment n t h e f om o f t a x. T h e s e tack takes a e t y p c a l l y n t h e a n g e o f 5 t o 25 %. It s k n o w n t h a t n t h e U.S. and Hong Kong, t h e t a c k t a k e s f o t h e w n b e t a e about but n Japan, t h e t a c k t a k e f o t h e w n b e t Is appoxmately 26 %. Moe complcated b e t s (usually c a l l e d e x o t c

16 bets) wll have a hghe tack take. In addton, thee s a cost called Beakage n the U.S. Ths addtonal cost s due to the fact that all etuns pe dolla bet (.e. odds plus one) ae ounded down to the neaest fve o ten cents n the U.S. and thus t s not substantal. All changng wn odds ae shown clealy on a bg sceen befoe a ace stats. In Hong Kong, qunella odds ae also vsble. Howeve, fo moe complcated bets (e.g. tfecta - bettng on the hoses fnshng fst, second and thd n coect ode), bettos cannot obseve the odds. Anothe bettng system s the fxed-odds system. It Is countes such as Btan, Austala, Italy and used n Fance. The bookmakes set fxed odds that the bettos can place a bet. But the odds offeed change ove tme. Ths allows the bookmakes to balance the books so as to ensue themselves of a poft egadless of whch hose wns. In ths case, we have no explct estmates of the wnnng pobabltes. Howeve, ths system wll not be elated to the analyses n ths thess snce the bettng systems of ou data sets ae all pamutuel,.3 Reseach Aeas Ou study can be dvded nto thee pats. Fo each pat, we have absobed some knowledge fom the pevously publshed wok.

17 .3. Wnnng pobablty modellng Al (977) and Asch, Malkel 8«Quandt (982) concluded t h a t t h e b e t t o s behave a s sk-loves based on t h e smple analyses. Ths s because a favoute-longshot bas w a s obseved n t h e w n bet. Absence o f t h e bas w a s also epoted n Busche and Hall (988) usng a d f f e e n t data s e t. We have analysed Al^s data s e t by f t t n g a c l a s s o f multnomal logt model and f n a l l y a smple model s suggested. Ths smple model s f t t e d n othe data sets. Ou e s u l t s a e qute consstent w t h t h e e s u l t s o f pevous studes. Ths p a t s epoted n Chapte t h e e..3.2 A n a l y s e s of complcated bets and estmaton of complcated pobabltes Followng t h e smple model obtaned n t h e pevous pat o f ou study, an extenson of t h a t model allows u s t o analyse moe complcated b e t s l k e e x a c t a, t f e c t a and qunella. Estmaton o f moe complcated pobabltes can be based on models poposed by Havlle (973), Heney (98) and Ste n (990). These models a e t e d on d f f e e n t data s e t s. Results n some acetacks suggest t h a t t h e Havlle model s t h e w o s t one but t s most commonly used because of ts smplcty. Moe theo et ca l e s u l t s a e obtaned empcal esults and some f o t h e compasons between d f f e e n t models. These a e pesented In detal n Chapte fou and f v e. Extensons o f t h e Heney model a e dscussed n Chapte s x.

18 Moeove, a smple model s poposed to appoxmate the Heney model n Chapte seven. The Sten model s found to f t well n Japan and ths esult togethe wth ts appoxmaton s dscussed n Chapte eght,.3.3 Study of a bettng system Hausch, Zemba Sc Rubnsten (98) and Hausch & Zemba (985) suggest a system known as D.Z's system. They clam that the system s moe opeatonal and easonable than some tadtonal bettng stateges (e.g. Epsten (977) Isaacs Wlls (964)) whch usually (953) Rosne (975) and assume p e f e c t knowledge of wnnng pobabltes. Fo t h e estmaton o f complcated pobabltes, t h e y u s e t h e model poposed b y Havlle (973). The system optmses t h e b e t amounts f o place and show b e t s usng a nonlnea pogammng model. We have appled D.Z's system n ou data s e t s and ecommend u s n g t h e model poposed n Chapte s e v e n and eght f o estmatng t h e complcated pobabltes. Ths p a t o f t h e study s epoted n Chapte n n e. P a t one (Chapte thee) manly deals w t h a n economc poblem o f s k p e f e e n c e b y f t t n g s t a t s t c a l models w h l e pat t w o (Chaptes f o u, f v e, s x,seven and eght) s concened w t h t h e s t a t s t c a l dstbutons o f unnng t m e s o f hoses a s w e l l a s t h e ac cua ces of d f f e e n t b e t t y p e s. The l a s t p a t (Chapte nne) s c e t a n l y a n nteestng aea f o t h e geneal publc (bettos) a s w e l l a s f o academcs such a s s ta ts tc a n s, opeatonal eseaches

19 and economsts..4 Data souces We have obtaned some hose-acng data sets fom vaous acetacks n the U.S., H.K. and Japan. The detals ae as follows : Race t a c k () A l ' s d a t a data avalable yea no.of aces : (U.S.) Saatoga wn R o o sev e I t wn Yonkes wn ( ) Quand t ' s data Meadow ands : (U.S.) w n p a c e, s h o w, 84 70S ex a e t a, t f e c t a, double bet A t l a n t c Cty wn ( ) Hong Kong d a t a Shat n & wn,place, Happy V a l l e y qunella, 8-90 >4000 d ou b l e b e t s, double qunella, t fe c t a (v) J a p a n e s e data (v) Dffeent w n,place, a c e t a c k s a u n e l l a Old Chnese a c e t a c k data Sh a n gh a :

20 Moeove, the above acetacks have dffeent specal popetes. Fo examples, the pool szes In H.K. ae extemely lage elatve to all the othe tacks; the wn bet tack take n Japan s appoxmately 26% but t s only about 7-87«e l s e w h e e.

21 CHAPTER TWO L I T E R A T U R E REVIEW 2. Intoducton In ths chapte, we wll dscuss some pevous elevant studes. Pevous studes about hose-acng have been publshed n jounals elated to a wde ange of subjects - Statstcs, Pobablty, Mathematcs, Management Scence, Economcs, Busness, Fnance and Psychology. It s mpotant to pont out that the bettng systems beng used n Noth Ameca and Hong Kong ae pamutuel systems. Howeve, n Btan and Austala, both pamutuel system and fxed-odds systems ae beng used. (Fo defntons of these two systems, see Chapte one.) In Japan to ou knowledge, only p a m u t u e l s y s t e m s u s e d and n o e l e v a n t l t e a t u e s f o u n d Rsk behavou and maket e f f c e n c y 2.2. U n t e d S t a t e s and Hong Kong : B e t s a t t h e U.S. and H.K a c e t a c k s a e p e f o m e d unde a pamutuel s y s t e m t h a t c a l c u l a t e s and automatcally updates b e t t n g odds b a s e d o n a l l w a g e s. Gffth (949), eseache n the f e l d a of psychologst, s pobably the fst h o s e - b e t t n g behavou. He c l a s s f e s

22 hs data by odds and compaes the numbe of wnnes n an odds-goup (Wn ) multpled by tack-odds一phs one (+0 ) w t h t h e h h numbe o f h o s e s n t h a t goup. Ths s equvalent t o compang w n n n g f e q u e n c e s and subjectve wnnng pobabltes. Ths s b e c a u s e f t h e tack take s gnoed and t h e subjectve wnnng p o b a b l t e s a e good es t m a t e s o f t u e wnnng pobabltes, t h e n Wn (+0 ) ^ Numbe o f hoses n t h e goup h. He concludes t h a t t h e h h fnal odds chances o f ae, wnnng. t h e ch a n c es o f Anothe hose-acng on t h e aveage, accuate e f l e c t o n s o f t h e But t h e e s a systematc undeevaluaton o f f a v o t e s and oveevaluaton o f t h o s e o f longshots. psychologst, data McGlothln (956) analyses s e t. He also c l a s s f e s t h e d a t a b y t h e n f n d s out t h e e x p e c t e d e t u n ( wt h tack t a k e and coected). a odds lage and Beakage The e x p e c t e d e t u n s l o w when odds a e hghe (Ths e s u l t s m o e s g n f c a n t n t h e l a s t a c e f o e a c h a c e day). Hence, h e concludes t h a t t h e goup o f b e t t o s s sk-takng, n geneal. Two s t a t s t c a n s, Hoel elablty o f subjectve Se Falln (974) demonstate t h e e s t m a t e s usng h o s e - a c n g data. They s e p a a t e t h e a c e s z e s f s t and t h e n u s e Ch-squae t e s t s t o test t h e pecson espect good to the but subjectve o f s ubj e c t v e pobabltes o f wnnng w t h obseved f e q u e n c e s o f wnnng. The e s u l t s a e t h e e s a s l g h t t e n d e n c y t o ovepedct a t l o w e pobabltes. The e s u l t s smla t o t h e conclusons 0

23 of othe eseaches. Al (977) uses a vey lage data hose-acng. He uses the odds of hoses to set to analyse fnd the bet factons (whch ae not avalable n hs data set). He agues that hoses should be classfed by favoute poston (favoutsm) because only one hose fo each goup exsts n each ace. He woks out the aveage bet facton and obseved fequency of wnnng fa each favoute-goup. He assumes that each hose caes a tue wnnng pobablty tt. and defnes the wn bet facton fo each hose as the subjectve pobablty Smple Z-tests ( ae pobabltes a e hghe & then 之 0 and 2 7^= 2 used to show that ) subjectve (lowe) than obseved fequences f o longshots (favoutes). We call t h s knd o f analyss Al's table. Detals o f ths method w l l b e dscussed n Chapte two. A theoetcal explanaton f o t h s e s u l t s povded n h s pape but f o t h e case of t w o hoses only (based on some assumptons). In addton, t h e u t l t y f u n c t o n appoach and delta-measue a e used t o show t h a t bettos ae "sk-loves". Lastly, Al concludes t h a t t h e sk behavou s elated t o captal held by studyng t h e d f f e e n c e o f behavou between bettng on t h e l a s t a c e n a meetng day and othe aces. Anothe eseache, Snyde e f f c e n c y o f t h e hose-acng (978a,b) maket. tested An e f f c e n t the maket

24 means a pefectly compettve maket whee pces eflect all avalable nfomaton. Snyde tes the weak fom test (whethe knowledge about the used to ean (whethe any subjectve an above odds assgned by bettos can be aveage etun) and stong fom test specal goup of people can outpefom the othes n eanng poft). Fo the weak fom test, Snyde analyses sx dffeent sets of data and shows ate of etun gaphcally and sometmes statstcally that the deceases when the odds ncease. Ths fndng s consstent wth Al (977) s. In h s stong f o m t e s t, Snyde shows t h a t nealy all of the e x p e t s ' odds dveged moe f o m an unbased pedcton than dd the geneal bettng publc's pamutuel odds (o subjectve pobabltes of wnnng). Fnally, between the h e concludes t h a t w h l e s g n f c a n t subjectve dffeences and empcal pobabltes of wnnng f o patcula odds-goups o f hoses e x s t, t h e s e d f f e e n c e s ae n o t so lage a s t o exceed t h e pce o f bettng - t h e tack take, Al (979) t e s t s t h e equalty o f etuns between t w o smple b e t s (palay and double bet) o f unknown but dentcal wnnng pobabltes. He concludes t h a t s g n f c a n t l y d f f e e n t and thus they the ae s a n mplcaton of an e f f c e n t maket. 2 t w o etuns a e not "equally pced" whch

25 Fglewsk (979) uses a multnomal logt model to elate the obseved fequency of wnnng to the handcappes' nfomaton and fnal odds. He concludes that do contan consdeable nfomaton dscount almost t. do not all of dscount accuately the the handcappes' but that the Howeve, off-tack subjectve handcappe advce tack bettng odds systems nfomaton as as do on-tack bettos. Regadng Snyde's analyss, Vannebo (980) comments that the skewness of ate of etun s also mpotant n addton to mean and standad devaton of ate of etun. He suggests that the based expected etun s a esult nheent n atonal behavou towads sk and wll be ncued even when the wageng maket s "effcent". In fact, the mpotance of hghe moments was dscussed by some pevous eseaches (e.g. S.Tsang (972)), Losey and Talbott (980) e-analyse Snyde^s data and conclude that bettos tustng the handcappe ae not only unable to get above aveage etun but they may also get etuns lowe than the aveage. The esult s smla to and stonge than Fglewsk's. Asch, Malkel & Quandt (982) analyse the data set smla way to Al. The esults ae also smla bettos ovebet on longshots and undebet behavou s elated favotes to captal held. Moeove, estmated ates of etun 3 to they n a Al's,.e. and sk show that ae lowe f odds ae hghe. Ths esult

26 s smla to Snyde(978). Futhe, they show that the fnal pamutuel odds ae bette than monng-lne odds n estmatng the pobablty of wnnng. Asch, Malkel & Quandt (984,86) use the multnomal logt model to f t the wnnng fequency on vaous handcappng Infomaton. They conclude that poftable stateges fo wn bets cannot be devsed on the bass of maket data. Quandt (986) "Equlbum" theoetcal of fst ceates hs own hose bettng. Then he goes on defnton of to pove some popetes such as the maket cannot be n equlbum f the bet factons and the tue wnnng pobabltes ae the same. Fnally, he concludes that the objectve pobabltes of wnnng ae geate than the subjectve pobabltes of wth the evese beng tue fo wnnng longshots, fo favotes s a natual consequence of equng equlbum to hold n the bettng maket. Howeve, most mpotantly, hs esults ae stongly based on the assumpton of sk-lovng bettos Asch and Quandt (987) analyse exacta and daly double data collected fom Meadowlands n 984. Fo exacta data, they fst ft a lnea egesson of classfed wnnng fequences on subjectve pobabltes (unde the classfcaton by odds). But the assumptons fo lnea egesson ae not satsfed. Then they estmate the objectve pobabltes of success n the j-exacta as follows: 4 ⑴

27 Substtute n the above egesson equaton usng the actual subjectve pobablty of each hose and estmate the coespondng objectve pobablty; (2) Apply the Havlle (973) fomula (dscussed late) to estmate the coespondng objectve pobablty of success h^. Lastly, obtan an estmate of the mplct subjectve pobablty of success fo the j-exacta by substtutng ths * and solve fo s. Also, t h e y compute t h e dect estmate of t h e subjectve pobablty (.e. t h e amounts b e t on each possble e x a c t a combnaton dvded by t h e total exacta pool). Then, they f t a egesson of and * ntecept) ae on and f n d t h a t t h e t w o paametes (slope s g n f c a ntl y d f f e e n t f o m one and z e o * espectvely. Moeove, p a y o f f s deved f o m s^and s " a e compaed and t h e y conclude t h a t t h e p a y o f f s deved f o m a e sgnfcantly hghe. Thus, t h e y "thnk t h e e s nsde nfomaton. All of t h e above a e based on t h e assumpton t h a t Havlle,s f omula s appopate whch s, In f a c t, questonable. The second pat o f t h e pape s analyss o f daly double bets. They compae t h e p a y o f f s t o daly double b e t and palay and conclude s t a t s t c a l l y t h a t t s moe p o f t a b l e t o b e t on t h e daly double. They suggest a possble eason f o t h s f n d n g s t h a t t h e e s nsde nfomaton on t h e daly double bet. Thale and Zemba (988) povde a good summay, dscusson and comments on most pevous Amecan studes n t h s f e l d. They comment t h a t t h e favoute-longshot bas s pobably due t o some 5

28 dffeent easons whch they suggest athe than sk-takng behavou. Fo example, bettos mght oveestmate the chance that the long shots wll wn, bettos may deve utlty smply fom holdng a tcket on a longshot and t s moe fun to pck a long shot to wn than a favoute, etc.. They conclude that modellng bettng behavou s complcated. Bettos' behavou seems to depend on numeous factos such as how they have done n eale aces, and whch bets wll yeld the best stoes afte the fact. Busche & Hall (988 analyse a Hong Kong data s e t usng Al^s t a b l e and conclude t h a t Hong Kong bettos do not undebet f avoutes and ovebet longshots o vce vesa. Ths s vey nteestng and t h e easons f o t h s a e unde nvestgaton. In addton, Busche & Hall (988) dscuss t h e classfcaton poblem. In ode t o t e s t t h e accuacy o f b e t f a c t o n s, t h e pevous e s e a c h e s have t o aange t h e data n such a w a y t h a t t h e hoses w t h n e a c h goup ae "smla" so t h a t t h e wnnng fequency (o estmated t u e pobablty) of each hose-goup can b e computed. As dscussed above, pevous eseaches usually c l a s s f y t h e data n one o f t h e t w o w a y s by b e t f a c t o n s (o odds) o by favoute poston. Howeve, ethe method of c l a s s f y n g data d e f n e s t h e w n odds gven t h e bettng odds. Thee may b e an eo n measung t h e t u e wnnng fequences of epesentatve hoses. Hence, t Is b e t t e t o f n d a method of analyss w h c h s n o t a f f e c t e d b y t h e claussfcato poblem. 6

29 2.2.2 Btan Dowe (976 nvestgated Btsh hose-acng data. s y s t e m u s e d s f x e d - o d d s. We have t w o foecast The types of pces ( o d d s ) p c e (FP, f o e c a s t odds estmated b e f o e s t a t a c e ) and s t a t n g pce (SP, t h e f n a l odds made b y of the bookmakes). D o w e f n d s t h a t t h e coelaton b e t w e e n f o e c a s t p c e and wnnng fequency pce not sgnfcantly less than that between statng and wnnng fequency Then h e concludes t h a t t h e e s nsde wll s nfomaton because f nsde nf o ma t o n no exsts, n o t b e avalable t o t h e publc and only t h e f n a l SP t can eveal t. Howeve, Cafts (985) agues t h a t t h e t e s t u s e d b y Dowe s n o t appopate because a betto can b e t on odds much lage t h a n the fnal statng odds f he gets nsde Inf omaton and t h e e f o e h e can e a n much moe p o f t t h a n t h e aveage. Cafts u s e s h s o w n t e s t s t o conclude t h a t t h e Btsh f x e d - o d d s enables poftable abtage at pces dffeent f o m system SP I.e. p o f t a b l e nsde bettng not avalable t o SP b e t t o s e x s t s. Howeve, t h e s e t w o s t u d es on nsde n f o m a t o n and maket e f f c e n c y only apply t o t h e Btsh f x e d - o d d s s y s t e m but not t o pamutuel system n t h e U.S. o Hong Kong, Heney (985) f s t f n d s t h a t t h e a t e o f e t u n Is lowe If 7

30 the SP odds a e hghe (Ths e s u l t s, agan, s m l a to Snyde (978)). He explans t h s b y makng a n nteestng hypothess that Q=fq, w h e e Q=subjectve estmate o f losng pobablty by the betto, q s the t u e chance o f losng and - f s the f a c t o n dscounted of hs loss. He shows gaphcally and statstcally that but less than L Howeve, he only tes out of data (whch ae not lage). Q =f q h h f s vey close to ths model on two sets Also, f : f o a l l hoses h n a ace then n- = S Q = f 2 q = f (n-) h h h h whee n = no. of hoses n the ace, whch mples that f s constaned to be one Austala : Lke Btan, both bettng system the fxed-odds system and the pamutuel ae used n Austala. Tuckwell (983) analyses odds data fom Sydney and Melboune aces n 974. He obtans a sgnfcant pecentage loss of quadatc can but between sample bettos and the statng pce odds. He dscoves that f gambles bet on hoses begnnng elatonshp "fm" whch sgnfcantly obtan pofts. As the amount ae to be favotes, then they of money of gambles s lage, the oveall magn favoute hoses wll not shot-odds at the taken on ths goup of bookmakes fo the be below the aveage" magn. Howeve, on a 8

31 p o gounds, he expects that the bookmakes should have moe expetse n assessng wnnng chances. Because o f the above fndng, Tuckwell suggests that the only possble explanaton s the exstence of nsde nfomaton may ethe channels of communcaton "nsde" nfomaton. Ths be the consequence of mpefect and/o delbeate ntefeence n ace-outcomes, whch can occu n vaous ways. Fnally, he tests the weak fom and sem-stong fom effcency and fnds that both ae neffcent. Bd, McCae and Beggs (987) analyse data of fom the Melboune acetack n a smla way subjectve vesus objectve pobabltes favotsm). They also analyse the to Al's f o subjectve bookmakes5 odds each wok (.e. level of pobabltes deved fom dffeent odds shown at dffeent tme befoe the ace stats. The esult s smla to Al's. They ague that t would be unwse f o bookmakes to gnoe the bettos* the bettng behavou when pobabltes mpled by the assessment as adjustng bookmakes expessed the odds. by Hence, the odds soon come t o e f l e c t t h e assessment o f bettos. Next, elatonshp they ty an nteestng analyss : between aveage a t e o f e tu n w t h they t e s t the vaances and s k e w n e s s o f t h e etun by f t t n g a egesson COLS, GLS and then SURE ae ted). The vaance tem has a 9 sgnfcant postve

32 c o e f f c e n t w h c h suggests that bettos demand a hghe expected etun to compensate them fo acceptng a hghe vaances n the etun dstbuton. sgnfcantly The c o e f f c e n t o f the skewness tem negatve whch suggests that bettos ae w l l n g s to accept a lowe expected etun n ode to nvest on a hose that offes a l o w pobablty of hgh e t u n. That means, bettos dslke vaance but they lke skewness The authos ty to gve some explanatons fo ths sk-takng behavou. In summay, ths pape concentates on povdng an explanaton of bettos behavou w t h n t h e famewok o f expected u t l t y theoy. Howeve, t h e y note t h a t expected u t l t y models based on t h e assumptons o f goal dected behavou, atonalty and optmalty, have been e x t e n s v e l y ct c z e d n t h e psychologcal lteatue on judgment and choce behavou. Thus, t h e y menton that e c e n t appoaches on human decson may b e moe appopate othe e.g. pospect t h e o y poposed b y Kahneman and Tvesky (979). 2.3 E s t m a t o n o f o u t c o m e p o b a b l t e s Havlle appoxmaton) (973) that ntoduces makes t an assumpton possble t o obtan Havlle pobabltes assocated w t h any complete outcomes n t e m s o f o n l y t h e wnnng pobabltes. 20

33 Let p [ ] b e t h e p o b a b l t y t h a t h o s e s,.. k 2 k l' 2 f n s h fst secod. kth, e s p e c t v e l y, w h e e k ^ n, t h e no. k of hoses, t h e n Havlle's fomula s : M W. y W p [ y k = vwvy wv.-u whee Then = - pj^], etc. k=2, pobablty of p [, = p [ ]p [ ]/(l-p [ ]). Ths 2 X 2 2 X hoses ^ S c fnshng fst elates and the second, espectvely, to the own wnnng pobabltes. Applyng the assumpton to analyse hs data, that bettos ovebet on longshots and Havlle suggests undebet Futhe, although the obseved fequency of second on favotes. and thd place fnshes ae n easonable accod wth the theoetcal long-un fequences, thee oveestmate the seems to be somethng of a tendency to chances of a second o thd place fnsh fo hoses wth hgh theoetcal pobabltes of such fnshes and to undeestmate the chances of those wth low theoetcal pobabltes. Ths fndng may be explanable. Fnally, he compaes (based on hs fomula) wth aveage expected payoff pe dolla the aveage actual payoff pe dolla. The esults show that some of them have sgnfcant dffeences and thus hs assumpton may 2 not hold.

34 Plackett (974) ases the f o l l o w n g "geneal" queston : I f p s the p o b a b l t y that appeas f s t, s the pobablty appeas f s t and j appeas second and p jk ae smlaly defned, how many "levels" of these pobabltes ae equed know all pobabltes hose-acng stuaton. all pemutatons. Cetanly, one case but hs dscusson Is fo the geneal He suggests maxmum smple s wth to to use a logstc model to fnd out the "level" equed. He then apples the method to two vey examples. Howeve, the hoses ae usually dffeent n dffeent aces and thus the estmaton method mentoned n ths pape s not useful n ou hose-acng data sets. Although Havlle egadng the Zemba tue queston & Rubnsten and apply (whch wll the pape, of Havlle's but they (973) not condtonal have a fm concluson ndependence, Hausch, (98) nevetheless assume ths elaton s t to the well-known optmal stategy of bettng be dscussed they n Chapte eght). also poduce pape, slghtly of does.e. bettos In the fst pat of esults smla to the fst pat undebet favotes on wn bet ovebet favotes on place and show bet, etc. Havlle's f o m u l a s c t c z e d b y Heney (98). He develops anothe theoy and woks out anothe appoxmaton fomula f o t h e pobabltes t h a t t h e j t h hose f n s h e s kth, e t c. f a n Independent nomal dstbuton s assumed f o t h e mnng t m e o f each hose. Hs method f s t detemnes estmates o f t h e mean unng tmes f o m 22

35 the the w n n n g pobabltes ( f known o estmable). Then, based on these estmated mean unnng tmes, we can obtan the appoxmate pobabltes f o any complcated bet. Hs esult s qute nteestng but he has not povded any empcal suppot w t h hose-acng data. We w l l dscuss Heney's model n detal n Chapte fou. McCullogh 8 c Z j l (986) gve a "dect" test fo the Havlle fomula by usng the Austalan data. The data set has a popety that the payoff pe dolla to a show bet s ndependent of whch othe hoses n the ace also show and thus the bet facton can be used as a dect estmate of the pobablty of showng. Ths estmate s compaed to the estmate obtaned by the Havlle fomula- That s why they call t a "dect" test. Howeve, they make an mpotant assumpton that ths bet facton s a hghly accuate estmate fo the pobablty of showng. Futhe, they use an eo ndex whch s the pecentage of eo fo the Havlle fomula fo the pupose of compason. Smple lnea egesson s then ftted to these two estmates. The esults ndcate that the Havlle fomula consstently undeestmates the tue pobabltes of showng fa low values and oveestmates the tue pobabltes undeestmates the of showng tue fo hgh pobabltes. values* The Oveall, amount of t the undeestmaton s found to vay consdeably but on aveage to be small. Fnally, dummy vaable egessons ae used to check whethe

36 dstance, feld sze, tack condton, class stakes and dates a f f e c t t h e e o n d e x. They f n d t h a t t h e undeestmaton tends t o b e smalle f o mddle t o long dstance aces fo lage f e l d s z e s and f o t h e w n t e months. Asch and Quandt (987) also u s e t h e Havlle f o m u l a f o analysng e x a c t a data (see s e c t o n 2.2. above f o detals). S t e n (988, 990 does n o t consde Heney's Nomal unnng t m e s model. Instead, a s Havlle's model s equvalent t o assumng ndependent Exponental unnng t m e s, h e suggests t o e x t e n d t h e Exponental dstbuton t o a Gamma dstbuton f o t h e unnng t m e s w t h a f x e d shape paamete () and d f f e e n t s c a l e paametes (¾.^) v a y n g accodng t o t h e mean unnng t m e s o f t h e hoses. Howeve, h s f o m u l a f o estmatng complcated pobabltes h a s no closed f o m, s d f f c u l t t o c a lc ula t e, and n o smple appoxmaton s gven. Based o n a v e y small numbe o f a c e s, h e empcally shows t h a t t h e f o m u l a w t h = 2 s b e t t e t h a n t h a t w t h = l w h c h educes t o Havlle's model. In f a c t, a s m l a d e a h a s aleady been dscussed n Heney (983) b u t Heney does n o t d e c t l y consde t h e h o s e - a c n g poblem and smlaly no s m p l f e d f o m u l a h a s b e e n gven. 24

37 2.4 Optmal bettng Hausch, Zemba and Rubnsten (98) fst show vesus subjectve pobabltes of wnnng, placng (smla to Havlle). "neffcences" ae They suggest that fo not and wn suffcently lage to the actual showng bets, make these postve pofts but etuns fo place and show bets may be possble. Based on the Havlle fomula, they fnd the fomulas of expected etuns fo place and show. The bettng model s : f the estmated expected etun s geate than a cetan value, fnd the optmum expected amount log of place and show bets by maxmzng the etun - the gowth ate of captal o the Kelly cteon (see, e.g. Beman (960)). Howeve, ths s so complcated that the bettos ae unable to fnd the soluton. Thus, they smplfy the pocedue by fttng egessons f o expected etun and optmal bets, s good espectvely. Fnally, they vefy the method by usng actual data and also show statstcally that the success s not andom. Hausch and Zemba (985) extend the 98 Ideas to complcated stuatons. As Zemba s one of the nventos of moe ths pocedue, the stategy s called D. Z's system. Followng the above eseaches, Bolton and Chapman (986} buld up anothe wageng system whch contans two components : a model of the hose ace pocess and a wageng stategy. A model of

38 the hose ace pocess pedcts the pobablty of each hose wnnng f o each ace. The model used s a Multnomal Logt model wth nput vaables weght, ncludng measues of a hose s -qualty, p o s t poston, jockey's c h a a c t e s t c s, e t c.. Usng the "pedcted" w n n n g pobabltes, t h e y e v a l u a t e d f f e e n t bettng stateges (e.g. stateges b y whch the bettos may ean pofts. suggest the to Rosne (975)) and t h e n suggest some bette Fnally, they l a t e e s e a c h c a n combne w a g e n g s t a t e g y smla t o o n e t h e y developed t o g e t h e w t h a t e c h n c a l appoach smla D.U s system. R e c e n t l y, Hausch and Zemba (990) e x t e n d e d t h e D.Z's system t o t h e c a s e o f c o s s - t a c k bettng- That s, t h e s t a t e g y can successfully obtan some pofts by lookng f o coss-tack n e f f c e n c y. Some empcal evdence s g v e n n t h e pape. 26

39 CHAPTER T H R EE MODELLING THE WINNING PROBABILITIES 3. Intoducton In hose-acng, wn bet factons have long been consdeed as a knd of subjectve pobabltes of wnnng snce they eflect the subjectve choces of all the bettos nvolved n the wn bet n a ace. Some people call them consensus pobabltes. whethe the wn bet factons ae good estmates of To study the tue objectve wnnng pobabltes, seveal eseaches n the U.S., the U.K., Austala and Hong Kong have done some empcal studes. Howeve, the statstcal methods wee udmentay. Among these studes, one of the best known s by Al (977). Based on hs method of analyss, he concluded that thee was stong evdence of a favoute-longshot bas,.e. bettos undebet favoutes and ovebet longshots. Howeve, hs method has some weaknesses. In ths chapte w e suggest anothe s t a t s t c a l technque t o a na ly se t h e w n b e t data. A smple l o g t model s poposed a f t e gong though a modellng pocess. Ths model, b y Cox's t e s t, s p e fe e d t o some pevous models. Empcal e s u l t s a e obtaned f o s e v e a l a c e t a c ks n t h e U.S., Hong Kong, Japan and Chna. 27

40 3.2 A l l ' s method of analysng wn bet data Fo each hose ace, Al (977) classfed the hoses by favoute poston (favoutsm). Let P be the bet facton of hose In ace, n so that P whee n j=l = total = 2 0 V and and u Y! P = j numbe of hoses n ace. Classfyng by favoutsm smply means odeng the bet factons as follows : p ⑴ that > P (2) > > p Is, hose (n ) (3.) V () s the most favoute hose and (n ) s the longshot. If the odds on hoses ae avalable but the bet factons ae unknowns (ths was tue n AlFs data set), Al suggested the followng : Let 0 = odds on hose n ace Recall that unde the pamutuel system, Ot s calculated as follows ( VSc ⑷ whee t = the tack take. 28 (3.2)

41 X = the w n bet amount on hose n ace, and total wn bet amount n ace E X and thus, P Fom (3.2), we have (3.3) V & Howeve, n the U.S., Is usually ounded downwads to cents and ths oundng effect 0 Is called Beakage. Due to ths beakage effect, (3.3) does not satsfy the unt sum constant : S P =. To satsfy ths constant, Al followng fomula to estmate the P suggested usng the : /( + 0 I I (3.4) V Sc E /( + 0 Aveage b e t f a c t o n f o t h f a v o u t e ⑴ CD (3,5) whee total numbe of aces n the data set. We also defne n hose n ace and u as the objectve wnnng pobablty fo (I ) as the objectve pobablty fo th favoute n ace. Futhe, we defne the followng ndcato vaable :

42 广 彳 f hose wns the ace 0 othewse Obvously, thee s only one Also, Y () s the ndcato vaable assocated wth P () The obseved wnnng f e q u e n c y f o t h e t h f a v o u t e s : Y = (n 去 ^ =l V (), (3 6) - Al expected the obseved fequency should be a bette estmato of the tue wnnng pobablty f m s suffcently lage because we do not know the accuacy of the bet factons. To test whethe the bet factons ae good estmates of the wnnng pobabltes, Al compaed the bet factons and the obseved wnnng fequences usng smple Z-statstcs : V whee s e (Y())=/Y⑴(l-Y⑴ / m (3.7) (3.8) As an llustaton of hs method, Al's data set s analysed by hs method as shown n table 3.L 30

43 Table 3. bet fac wnfeq? ( n , _ , , , (Note : t h e e s u l t above s d f f e e n t f o m t h a t In AH (977), Whee t h e e w e e equal odds b e t w e e n t h e w nnng h o s e a n d some othe h o s e s, h e a l w a y s chose t h e w nnng hose a s a moe f a v o u t e hose, thus basng t h e esults slghtly towads t h e ove be t t ng o f f a v o u t e s e s u l t. We u s e a n a l t e n a t v e method : w e a l t e n a t e b e t w e e n t h e f s t and t h e second h o s e a s t h e moe f a v o u t e w h e n e v e thee s a te.) Fom t h e above t a b l e, t h e f s t column s h o w s t h e f a v o u t e p o s t o n s,. e. f o m most f a v o u t e s t o longshots. T h e numbe o f a c e s n t h e second column a e d f f e e n t because t h e numbe o f h o s e s n d f f e e n t a c e s s n o t t h e same. In t h s d a t a s e t, t h e mnmum numbe o f h o s e s s 4. The the exstence of a s t a t s t c s n t h e l a s t column suggests f avoute-longshot b a s 3,. e. t h e bettos

44 undebet favoutes and ovebet longshots. Some weaknesses of Al's method ae suggested : 3.2. Bet factons n All's analyss ae teated as ndependent acoss hoses. The constant that bet factons must sum to one ntoduces a necessay but neglected negatve coelaton between bet f actons Al's method depends on the classfcaton (.e. the favoutsm). Howeve, 6th favoute n 6-hose-aces may be qute dffeent fom a 6th favoute n a 0-hose-aces and thus the aveage bet factons may not be vey meanngful When usng the Z-test, Al teated the aveage bet facton as a constant value athe than a andom vaable. We thnk that a andom vaable may be moe appopate hee snce P⑴ s essentally an estmato of the mean subjectve pobablty fo the th favoute hose n ace Moeove, nomal appoxmatons a e not vey accuate when t h e popotons a e v e y small o t h e numbe o f a c e s a e not lage. Hence, Al's Z - t e s t may not b e vald. (One smple w a y t o mpove Al's method s smply t o f n d t h e d f f e e n c e between Y u and P and then t e s t whethe t h e mean d f f e e n c e equals u z e o. Howeve, t h e esultng Z-value would not b e v e y d f f e e n t. ) Al used t h e Bnomal dstbuton f o t h e wnnng event of a 32

45 hose. To fully utlze the nfomaton, the Multnomal dstbuton should be bette because thee ae moe than two hoses. 3.3 Classes of Multnomal Logt Models We eque a model to f t the tue objectve pobablty on the subjectve pobablty of wnnng (.e. the bet facton) (See Atchson (986) o L (986) fo efeences). We stat by assumng that the wnnng pobablty follows the followng fom ln(7 () /tt (n) (omttng the subscpt fo ace ) : ) = a + p ln(p I () /P (n) ) =,2,...,n- (3.9) whee P, () 2 之 P (n) and n ( ) s assocated wth P ( ).e. P,, = bet f a c t o n f o f s t favou Ite and () P (n) = bet f a c t o n f o longshot, e t c. We call ths the A-class snce the logt tansfom, ln(p⑴/ P (n) ), s commonly used by Atchson (986). In ths class of model, the logt (n the multvaate case) of the wnnng pobablty s lnealy elated to the logt of the bet factons. Note howeve, that the esults ae dependent on the abtay choce of the dvso (hose n n (3.9) above) hose. 33 snce all measues ae elatve to that

46 Anothe smla class s exp( a + tt⑴ = I exp( a s In P⑴ + j3 S s (3.0) In Pf J (s) =l 2 " n (wth a n = 0) We c a l l (3.0) t h e L - c l a s s. I t s t h e one o f t h e common f o m s o f t h e Polytomous model (see, f o example, Hosme and Lemeshow (989)) o t h e Multnomal logt model (see fo example. Judge Gffths, H l l, Lutkepohl Sc Lee Both c l a s s e s (985)). a e m u l t va a t e e x t e n s o n s o f t h e bnay l o g t model, b u t t h e A-class depends o n w h c h f a v o u t e h o s e s chosen a s t h e dvso u n l e s s = 0 V. Thus t h e L - c l a s s h a s a n advantage o f s y m m e t y o f paametes s n c e n o dvso s equed. Both t h e A一class and L-class ae qute complcated and we beleve that a smple model should also f t the data qute well. Fo estmaton, we assume : Y 一 = (Y,...,Yn whee Y )T follows a MultnomaK 个) dstbuton s defned n (3.5) and = (tt,..., The lkelhood functon s - I, )T

47 and the log lkelhood can be wtten as = 乙 In tt⑴^ (3.2) =l whee denotes the wnnng pobablty assocated wth the wnnng hose n ace. Maxmum Lkelhood Estmaton can be easly appled to maxmze (3.2) wth espect to the paametes. Notce that when pobabltes equal 0 and bet, factons.e. Ths mples wnnng sk neutalty (See Al(977)). 3.4 Selecton of models Both t h e A-class and t h e L - c l a s s a e e x a m n e d usng some lage d a t a s e t s. Ou hypotheses nvolve t e s t n g w h e t h e and ae z e o s, constants o not. We u s e t h e 6, 7 and 8 - h o s e - a c e s o f Al's d a t a f o t h e pupose o f s e l e c t o n o f models because t h e y nclude q u t e a l o t o f aces. To k e e p t h e analyss smple, w e smply pck t h e a c e s w t h t h e same numbe o f hoses* The t o t a l numbe o f a c e s f o 6 7 and 8 - h o s e - a c e s a e 807, 3532 and 5450 e s p e c t v e l y. The e s u l t s f o A-class a e s h o w n n Fg. 3., 3. 2 and 3.3. And t h e e s u l t s f o L-class a e shown n Fg. 3. 4, 3. 5 and 3.6, The d e t a l e d estmated paametes f o d f f e e n t models a e epoted n 35

48 Appendx A. In these fgues, models ae aanged accodng to complexty. The top box s the smplest model and the lowest box s the most complcated model. Fo example, 0, nsde the top box n each fgue means a = 0 and p = V. Models wth constant a (^0) n the L-class ae gnoed snce ths mples settng one hose apat. The negatve numbes nsde the boxes ae the assocated log lkelhoods and thus lkelhood ato tests can be used. The numbes on the lnes ae the assocated p-values fo the tests that the next models ae bette than the pevous ones. Anothe set of p-values ae assocated wth the tests that the bottom model (quadatc whch wll be explaned below) s bette. Akake Infomaton Ctea (AIC) values ae gven n paentheses below the log lkelhoods. Apat fom fttng the above models, we also ty to epaametze the models as follows : B I = o c + = =a + and (3.3) n ode to educe the numbe of paametes and to captue the tend of ^and oy The detaled esults ae also epoted n Appendx A. Quadatc models ae also ted, but the addtonal paamete s qute sgnfcant n 8-hose-aces only. The followlng quadatc models ae ftted and the esults ae also shown n Fg- 3.-3,6. 36

49 A-class ln(7 (An)) =,2,...,n- L-class : exp[ a + C T = ⑴ ^ exp[ a s s In P ⑴ + 3 J + v (In P⑴ 2 In P s + y (In P (s) (s) 2 =l,2 n (wth a = 0) n Futhe, to see how much bette the estmaton wll be f a much moe complcated logt model s consdeed, the A-class model s extended to the followng one : InCt /tt () n- (n) ) = a + E p InCP / P j=l IJ (j) (n) =,2,...,n- (3.4) t h a t s, t h e wnnng pobablty o f a hose not only depends o n t h e assocat ed bet f a c t o n o f t h a t hose but also depends on t h e b e t f a c t o n s o f a l l t h e othe hoses. Model (3.4) can be eaanged a s follows : 37

50 exp[ a K U T () = ~ n- Ee x P [ + Z 3 J j= InCP j / P ) ] (j) (n) n- a, S + S 8 J= ln(p., / P, J + s J exp a + S ^ E exp[ a (j ) p j= ^ j + S p n) In P In P ( b y p u t t n g p =- S B ) In exp[ a ^ + E 0 j=l j In P J= j I (3-5} T e x pk [ a ^ whee a = a - a n s + I p j=' s j In P (j) and / 3 = / 3 j We may note that model (3.5) s actually t h e extended f o m o f L - c l a s s model n (3.0). Hence, model (3.4) s a n extenson of both A-class and L-class models. Though w e f t t h s complcated model on a v e y l a g e data s e t (5450 S-hose-aces), t h e addtonal paametes ae not sgnfcant at smple model 5% sgnfcance l e v e l w h e n compaed t o t h e and o f A-class model. The assocated esult Is also shown n Table 3.3 and 3.6. Wthn these classes of models, fom the popetes of the models and the esults obtaned, constant-p model ( L e. = 38 0 we pefe what we call the and = P V ). In t h e s e

51 ccumstances, both the A and L classes educe to the same constat-0 model. The easons we pefe ths ae : 3.4. Ths smple model s not affected by the choce of dvso (see (3.9)). Hence, the model s ndependent of the classfcaton Its estmated value dffeent ace-szes. (The s qute consstent n aces wth 入.426,.656 and.35 espectvely.) P's ae sgnfcantly dffeent fom one at 5又 sgnfcance level n all cases hee. Also, the educton of log lkelhood caused by constant-^ n the model fttng pocess s geat n geneal. In addton, usng the Akake Inf omaton Cteon (AIC) (.e. takng the numbe of estmated paametes nto account e.g. Sakamoto, Ishguo & Ktagawa (986)), the constant-p model attans the mnmum values among all the ftted models on both 6-hose-aces and 7-hose-aces n both A-class and L-class. Wth 8-hose-aces, models wth moe paametes acheve slghtly bette AIC values. The constant-p model s as follows : Infx /TT, = )3 l n ( P, ( ) (n) () /P/ ) w h c h c a n b e eaanged a s : 39 (n) = (3.6)

52 P(Y= J P fo =,2,. (3.7) Note that thee s no equement of any odeng conventon. Let n and P be the geometc means of the and espectvely, then l/n = = n j= L S P l/n z thus. Pa o, In In PNP It may be of nteest to notce that the left and ght sdes can be expessed n tems of log atos but the denomnatos ae geometc means. p can be thought of as measung the "Bas" (not exactly the statstcal bas) of the subjectve pobablty wth espect to the objectve pobablty. We would say that the subjectve pobabltes ae unbased f 3 / s not sgnfcantly dffeent fom one,.e. whch Is the sk neutal hypothess tested by many eseaches. If If favoutes and ovebet longshots (I.e. sk pef eence). By usng ths model fo analyss, the weaknesses n secton 3.2 dsappea 40