Probability m odels on horse-race outcomes

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1 Jour nl of Applied Sttistics, Vol. 25, No. 2, 1998, 221± 229 Probbility m odels on horse-rce outcomes M UKHTAR M. ALI, Deprtment of Economics, University of Kentucy, USA SUMMARY A number of models hve been exmined for modelling probbility bsed on rnings. M ost prominent mong these re the gmm nd nor ml probbility models. The ccurcy of these models in predicting the outcomes of horse rces is investigted in this pper. The prmeters of these models re estimted by the mximum lielihood method, using the informtion on win pool frctions. These models re used to estimte the probbilities tht rce entrnts nish second or third in rce. These probbilities re then compred with the corresponding objective probbilities estimted from ctul rce outcomes. The dt re obtined from over rces. it is found tht ll the models tend to overestimte the probbility of horse nishing second or third when the horse hs high probbility of such result, but underestimte the probbility of horse nishing second or third when this probbility is low. 1 Introduction In mny respects, the pri-mutuel horse-rce wgering mret is similr to the stoc mret. In both mrets, returns from investments re uncertin, there re mny prticipnts nd there is vriety of informtion concerning investments nd prticipnts. This hs generted considerble interest in studying the e ciency of the wgering mret (see, for exmple, Dowie, 1987; Ali, 1979; Figlewsi, 1979; H usch et l., 1981; Asch et l., 1982). The common method of ttcing this problem hs been to devise pro tble betting strtegy. If such strtegy exists, then the mret is ine cient. A prerequisite for developing pro tble betting strtegy is to hve ccurte prediction of the probbility of the outcomes of horse rce. Thus, probbility models which ssign ccurtely the probbility of the outcome of horse rce would be of utmost interest to cdemic reserchers who wnt to study the e ciency of the wgering mret. Hrville (1973) exmines one such probbility model. His nlysis of 335 Correspondence: M. M. Ali, Deprtment of Economics, University of Kentucy, Lexington, KY 40506, USA. Tel: /98/ $ Crfx Publishing Ltd

2 222 M. M. Ali thoroughbred rces suggests tht his model overestimtes the probbility of nishing second or third for horses tht hve high probbility of such result, nd underestimtes the probbility of nishing second or third for other horses. Stern (1990) exmines clss of models tht includes Hrville s model, nd pplies two models from this clss to nlyze 47 rces. Anlysis seems to corroborte the ndings of Hrville (1973). Unfortuntely, both these studies re limited in scope, in terms of the number of models nd the number of rces being nlyzed. Henery (1981) proposes n lterntive model. Bcon-Shone et l. (1992) propose logistic models bsed on probbility obtined from Hrville (1973), Henery (1981) nd number of Stern (1990) models. They t these logistic models to dt on rces held t rcetrcs in Hong K ong nd M edowlnds, NJ. U sing lielihood criterion, they found tht logistic models bsed on probbility obtined from the H enery (1981) model t the dt best. Their conclusion ws further con rmed by Lo nd Bcon-Shone (1994), who t logistic models bsed on probbility obtined from Henery (1981) nd H r ville (1973) probbility models. These studies suggest tht the Henery (1981) model is liely to provide ccurte estimtes of the rning probbility of horse-rce outcomes. Unfortuntely, these studies did not exmine the ccurcy of such estimtes. A mjor purpose of this pper is to investigte the ccurcy of number of commonly dvocted probbility models. The nlysis will be bsed on more thn rces. The models re described in Section 2. Also described in Section 2 is the mximum lielihood estimte (MLE) of the model prmeters. Detils of the dt nlysis nd ndings re reported in Section 3. Some concluding remrs re given in Section 4. 2 Probbility m odels nd their estim tes 2.1 Probbility models Assigning the probbility of the outcomes of horse-rces in which horses re competing is the sme s ssigning the probbility for the permuttions of the rst integers. The integers cn be interpreted s rns of objects. A number of probbility models for such rning ( permuttions) hve been proposed in the sttisticl nd psychologicl literture (see Critchlow et l., 1991). Among these models is clss of models which ssign to ech rning the probbility of the corresponding ordering of independent, not necessrily identiclly distributed rndom vribles. More speci clly, let X 1, X 2,..., X be independent rndom vribles with probbility distribution functions F(x; i )(i 5 1, 2,..., ), nd let p 5 (p 1, p 2,..., p ) represent permuttion of objects in which object p j hs rn j ( j5 1,..., ). Then, these models ssign the probbility to the permuttion p s Pr(p ) 5 Pr(X p 1 < X p 2 <... < X p ) The models re nown to be rning models. Two well-studied cses of the rning models re the model of Thurstone (1927), Dniels (1950) nd Mosteller (1951), lso nown s the norml rning model, where the rndom vribles re normlly distributed with men i(i5 1, 2,..., ) nd vrince 5 1; nd the Luce (1959) model, where the distribution of the rndom vribles is Gumbel. The Luce model is lso the rst-order model in the Plcett (1975) system of logistic models. H enery (1981) proposes the norml rning model for horse-rce outcomes. H enery (1983) nd Stern (1990) investigte rning model nown s the

3 Modelling horse-rce outcomes 223 gmm rning model, where the rndom vribles X i (i 5 1, 2,..., ) hve gmm distributions with scle prmeter i nd common shpe prmeter r. The probbility density of X i is given by f(x; i, r) 5 [ r i C (r)] x r2 1 exp (2 ix), x > 0 W ith shpe prmeter r 5 1, the rndom vribles hve exponentil distributions nd the model becomes the Luce model. Hrville (1973) pplies the Luce model nd Stern (1990) pplies the gmm rning models with shpe prmeter r 5 1, 2 to horse rcing. Bcon-Shone et l. (1992) nd Lo nd Bcon-Shone (1994) t logistic models bsed on the probbility obtined from both norml nd gmm rning models. In this pper, we exmine the norml rning model of Thurstone (1927), Dniels (1950) nd Mosteller (1951), nd the gmm rning model of Henery (1983) nd Stern (1990). The gmm rning model is clss of models where vriety of models re obtined by considering diœerent vlues of the shpe prm eter r. A number of these models will be exmined in this pper. 2.2 Prmeter estimtes It is simple mtter to obtin M LEs of the prmeters if we hve the complete set of dt. This is becuse, in the complete dt set, rndom smple of permuttions is observed nd the empiricl distribution p n (p ) of permuttion p is nown. Consequently, the log lielihood of the rning model cn be obtined s ln [L( p n (p ), )] 5 p np R n (p ) ln [ p(p )]+ C where 5 ( 1, 2,..., ) re the prmeters np n (p ) is the number of times tht the permuttion p is observed, nd p(p ) is the probbility of the permuttion p under the model (expressed s function of the unnown prmeters ). The lielihood estimte of mximizes the term ln L. Unfortuntely, we do not hve complete dt set. In fct, we observe only one outcome from rce nd the dt from diœerent rces cnnot be combined, becuse ech rce is diœerent, in the sense tht the model prmeters will vry from rce to rce. Fortuntely, however, not only is the outcome of rce observed but the empiricl probbility p n (i ) tht horse i wins cn lso be ccurtely estimted from the published win odds O i. The win odds re determined by the mount of money bet on ech horse 1 to win 2 the rce, the trc te nd brege 3. If W i is the m ount bet on horse i to win nd b is the te-out rte nd brege, then the totl win pool is W 5 R W i nd the odds O i re given by 1 + O i5 (1 2 b )W /W i (1) It hs been observed in severl studies (see, for exmple, Fbricnd, 1965; Weitzmn, 1965; Ali, 1977; Snyder, 1978) tht the empiricl probbility tht horse i wins cn be ccurtely estimted by ting it to be proportionl to (W i /W ) d, for some prmeter d. From eqution (1), the win pool frction W i /W is given by W i /W 5 [1 /(1 + O i )] /R [1 /(1 + O i )] (2) so tht it cn be obtined from the published win odds O i. Knowing the empiricl

4 U U ò U 224 M. M. Ali probbility p n (i ), it is esy to write the lielihood function nd, s shown by Stern (1990), the M LEs stisfy the system of equtions p(i ) 5 p n (i ) (3) where p(i ) is the model-bsed probbility tht horse i wins the rce: p(i ) 5 f(x; i) P j¹ i 5 1 [1 2 F(x; j)] dx (4) p(i ) is function of the prmeters. It my be noted tht p(i ) is unchnged if constnt is dded to ech i term in the cse of the norml rning model, nd if ech i term is multiplied by positive constnt in the cse of the gmm rning model. Therefore, without loss of generlity, we hve chosen i so tht R i5 0 for the norml rning model nd R i 5 1 for the gmm rning model. Hence, M LEs of i re obtined by solving eqution (3) subject to the condition tht R i5 0 for the norml rning model nd R i 5 1 for the gmm rning model. The solutions re obtined by minimizing S 5 R [ p(i )2 p n (i )] 2 (5) p(i ), s de ned in eqution (4), is obtined by numericl integrtion. The Guss± Newton method s modi ed by M rqurdt (1963) is used to minimize the function S numericlly. To strt oœ the itertive solution procedure, initil estimtes of i re required. These re obtined s follows. For the norml rning model, following Henery (1981), we hve i5 ( 2 1)u (z 0 )(z i2 z 0 ) 2 1 [(i2 3 /8) /( + 3 /4)] (6) U u R where z i5 1 [ p n (i )], z 0 1 (1 /), p n (i ) is the empiricl win probbility, nd (.) nd (.) re, respectively, the distribution nd density functions of the stndrd norml vrible. The initil estimtes of i re devitions, i2 i/, where the i terms re given in eqution (6). For the gmm rning model, the probbility p(i ) depends on the shpe prm eter r. To me this dependency explicit, let us write p(i, r) for p(i ). Some elementry computtions suggest tht p(i, r) is pproximtely proportionl to p(i, r5 1) g(r), where g(r) is positive nd monotoniclly decresing function of r. M oreover, it cn be shown tht p(i, r 1) i. Therefore, from the lielihood eqution (3), i g(r) is pproximtely proportionl to p n (i ) nd estimtes of i cn be ten to be proportionl to p n (i ) g(r). Thus, some preliminry estimtes of i cn be obtined once the function g(r) is nown. To determine g(r), we note tht, besides being positive nd decresing in r, it must be equl to 1 when r5 1. Tht g(r 1) 1 follows from the fct tht, when r 1, becuse p(i, r 1) i, the M LEs estimte of i, which solve eqution (3), re p n (i ). With some tril nd error, we found tht resonble estimtes of i cn be obtined by setting g(r) to 2 /(1 + r). This is wht is used in our subsequent nlysis. 3 Dt nlysis To chec the ccurcy of the norml nd gmm rning models in ssigning the probbility of horse-rce outcomes, dt on hrness horse rces were

5 Modelling horse-rce outcomes 225 TABLE 1. Dt description Averge bet Averge bet per Rcetrc No. of rcing Averge dily per rce person in rce nd yer dtes No. rces ttendnce (US$) (US$) Srtog Roosevelt b Yoners Note: All trcs re in the stte of New Yor. Includes ll possible betting opportunities. b October± December 1970 nd Mrch± October In the nlysis, the time period is treted s the yer collected. The dt re described in Tble 1. Previously, these dt hve been reported nd nlyzed by Ali (1977). Dt from rces tht involve `ded het t ny nishing position re not considered. For the se of comprbility, only rces with the sme number of betting interest re nlyzed. Speci clly, our nlysis is limited to rces with eight entrnts. In our dt, eight entrnts competed in out of the rces. Besides the `win bet, there re t lest two regulr betting opportunities in rce, nown s `plce nd `show bets. Betting on horse to plce is successful if the horse nishes rst or second, nd betting on horse to show is successful if the horse nishes rst, second or third. The probbilities tht horse i nishes second or third re of direct interest to those ming the plce nd show bets. To chec the model ccurcy, the model-bsed probbility of horse nishing second (or third) is compred with its corresponding objective probbility. The objective probbility of horse nishing second (third) is de ned to be the proportion of times tht the horse nishes second (third) when the rce is repeted n in nitely lrge number of times. Both model-bsed nd objective probbilities re diœerent for horses in rce, nd they lso diœer in diœerent rces. The model-bsed probbilities cn be obtined from the estimted model but, becuse we hve only one observtion, relible estimte of the objective probbilities cnnot be obtined. In our study, the horses re grouped nd their verge model-bsed probbility is compred with n unbised estimte of the corresponding verge objective probbility. The verge objective probbility is estimted by the reltive frequency of the horses in group nishing in speci ed position (second or third). If the estimte is u, then its stndrd error is estimted s the squre root of u (1 2 u ) /n, where n is the number of rces from which the estimte is obtined.

6 ò ò P 226 M. M. Ali Following Ali (1977), horses re grouped by `fvorites. The horse with the lowest odds O i to win rce is nown s the rst fvorite; the horse with the second lowest odds O i to win the rce is nown s the second fvorite, nd so on. On the bsis of informtion of win odds on the horses in rce, the lielihood estimtes of the prmeters i of ech model re obtined. Following the nlysis of this dt by Ali (1977) nd Lo (1992), the prmeter d is set to 1.16 for the rces t Srtog, nd to 1.13 for the rces t both Roosevelt nd Yoners. We then use the estimted model to estimte the probbility tht fvorite i nishes second ( p(i, 2)) or third ( p(i, 3)). These re model-bsed probbilities. These probbilities re obtined by numericl integrtion, utilizing the following formule: p(i, 3) 5 p(i, 2) 5 f(x; i) R f(x; i) R F(x; j) P j¹ i l¹ i,j F(x; ) F(x; ) j1 j2 j 1 j 2 ¹ i j 1< j 2 [1 2 F(x; l)] dx (7) l¹ i, j 1, j 2 [1 2 F(x; 1 )] dx (8) where f(x;.) nd F(x;.) re, respectively, the pproprite density nd distribution functions. Averge model-bsed probbilities re simple verges of p(i, 2) nd p(i, 3) over the number of rces being nlyzed. The results from comprisons of the probbilities of nishing second nd nishing third re given in Tbles 2 nd 3 respectively. For ech model, the results re reported in three rows. The probbilities estimted from the model re reported in the rst row, while the observed reltive frequencies (objective probbilities) re in the second row. The devitions of the model-bsed probbilities from the corresponding reltive frequencies re expressed s rtios to the respective stndrd errors nd re reported in the third row. In generl, irrespective of the rning model, the probbilities of nishing second nd third re overestimted for those horses which hve high probbilities of nishing second or third (or of winning the rce), nd re underestimted for those horses which hve low probbilities of nishing second or third (or of winning the rce). This grees with the ndings of Hrville (1973), who exmined the gmm rning model with shpe prmeter 1; it lso grees with the ndings of Stern (1990), who nlyzed 47 rces bsed on the gmm rning model with shpe prm eter r 5 1 nd 2. H owever, the stndrdized devitions of the model-bsed probbilities from the corresponding verge reltive frequencies (devitions divided by their stndrd errors) re lmost lwys sm ller when the probbilities re estimted from the norml rning model thn when the probbilities re estimted from ny of the gmm rning models. For exmple, for fvorite 1, in estimting the probbility of nishing second, the stndrdized devition is for the norml rning model, which is the sm llest stndrdized devition mong ll the models. The next smllest devition resulted for the gmm model with the shpe prmeter r This is expected, becuse the gmm model with shpe prm eter r5 20 is closer to the norml model thn is gmm model with shpe prmeter less thn 20. Thus, it seems tht the norml rning model best ts the dt. This is consistent with the ndings of Bcon-Shone et l. (1992) nd with those of Lo nd Bcon-Shone (1994). However, contrry to the conclusions drwn by Bcon-Shone et l. (1992) nd Lo nd Bcon-Shone (1994), this best- tting model signi cntly overestimtes the probbility of nishing second or third for

7 Modelling horse-rce outcomes 227 TABLE 2. Probbility of nishing second: model-bsed vs objective probbility Fvorite horse Model [n] Probbility N(0, 1) b M c [15402] O d (M O) /SE e G(0.5) M [14059] O (M O) /SE G(0.75) M [15400] O (M O) /SE G(1.0) M [15402] O (M O) /SE G(2.0) M [15400] O (M O) /SE G(5.0) M [15402] O (M O) /SE G(10.0) M [15402] O (M O) /SE G(20.0) M [15402] O (M O) /SE n is number of rces in the smple. b N(0, 1) is the norml rning model nd G(r) is the gmm rning model with shpe prmeter r. c M is the verge model-bsed probbility. d O is the verge objective probbility. e SE 5 [O(1 2 O)/n] 1 /2 is the stndrd error of O, where n is the number of rces in the smple. horses which hve high probbility of winning, nd overestimtes the probbility of nishing second or third for horses tht hve low probbility of winning. 4 Concluding rem rs We hve exmined the norml rning model nd number of gmm rning models for their ccurcy in estimting rning probbilities in horse rces. The nlysis is bsed on over rces. It is found tht the norml rning model best ts the dt, but ll models suœer from fvorite± longshot bis, i.e. the probbility of nishing second or third is overestimted for horses which hve high probbility of winning nd is underestimted for horses which hve low probbility of winning. The dt were further subdivided by yer nd rcetrc nd n nlysis of these dt led to the sme conclusion bout fvorite± longshot bis. This shows tht none of the rning models my be used to construct pro tble betting strtegy. Husch et l. (1981) reported positive pro ts, by pplying betting strtegy in

8 228 M. M. Ali TABLE 3. Probbility of nishing third: model-bsed vs objective probbility Fvorite horse Model [n] Probbility N(0, 1) b M c [15402] O d (M O)/SE e G(0.5) M [14059] O (M O)/SE G(0.75) M [15400] O (M O)/SE G(1.0) M [15402] O (M O)/SE G(2.0) M [15400] O (M O)/SE G(5.0) M [15402] O (M O)/SE G(10.0) M [15402] O (M O)/SE G(20.0) M [15402] O (M O)/SE n is the number of rces in the smple. b N(0, 1) is the norml rning model nd G(r) is the gmm rning model with shpe prmeter r. c M is the verge model-bsed probbility. d O is the verge objective probbility. e SE 5 [O(1 2 O)/n] 1 /2 is the stndrd error of O, where n is the number of rces. which the probbilities were estimted by the gmm rning model with shpe prm eter r 5 1 to 627 rces held during the 1973± 74 winter seson t Snt Anit Rcetrc in Arcdi, CA nd to 1065 rces held during the 1978 summer seson t Exhibition Pr, Vncouver, BC. The chievement of such positive pro ts my hve been s result of the peculirity of the smples of rces tht were nlyzed. Lo et l. (1994) pplied the betting system of Husch et l. (1981) nd its vrints to 705 rces held during 1984 t M edowlnds, NJ, s well s pplying it to 905 rces held during 1981± 82 in Hong Kong nd to 983 rces held during 1990 ± 91 in Jpn. In ech cse, the system of Husch et l. produced negtive pro ts. Notes 1. In some cses, severl horses re groupedð nown s n `entry or ` eld Ð for single betting interest, nd the group is ssigned single number. A bet on this number is successful if one of the horses in the group is successful. Thus, if it is `win bet, then the bet is successful if one of the horses in the group nishes rst. Without loss of generlity, this group is ten s single horse. 2. Rces with `ded het t ny nishing position re not considered.

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