Jour nal of A pplied Statistics, Vol. 25, N o. 2, 1998, 221± 229Probability models on horse-race outcomesM U KHTA R M . AL I, D epartm ent of Economics, University of K entucky, U SASUM M ARYA numb er of models have been examined for m odelling probability based onrankings. M ost prom inent am ong these are the gamma and norm al probability models.The accuracy of these m odels in predicting the outcomes of horse races is investigated inthis paper. The parameters of these models are estim ated by the maximum likelihoodmethod, using the inform ation on win pool fractions. These m odels are used to estimatethe probabilities that race entrants ® nish second or third in a ra ce. These probabilities arethen com pared with the corresponding objective probabilities estimated from actual raceoutcom es. The data are obtained from over 15 000 races. it is found that all the modelstend to overestimate the probability of a horse ® nishing second or third when the horse hasa high probability of such a result, but underestim ate the probability of a horse ® nishingsecond or third when this probability is low.1 Introd uctionIn many respects, the pari-mutuel horse-race wagering m ar k et is similar to thestock m arket. In both m arkets, retur ns from investments are uncertain, there arem any par ticipants and there is a variety of information concerning investments andparticipants. This has generated considerable interest in studying the e ciency ofthe wagering m arket (see, for exam ple, D owie, 1987; Ali, 1979; Figlewski, 1979;H auschet al., 1981; Aschet al., 1982). The com m on method of attacking thisproblem has been to devise a p ro ® table betting strategy. If such a strategy exists,then the m arket is ine cient. A prerequisite fo r developing a pro® table bettingstrategy is to have accurate prediction of the probability of the outcom es of a horserace. Thu s, pro bability m odels which assign accurately the probability of theoutcome of a horse race would be of utmost interest to academic researchers whowan t to study the e ciency of the wagering m arket.Harville (1973) examines one such probability m odel. H is analysis of 335Correspondence: M . M . Ali, Departm ent of Econom ics, University of Kentucky, Lexington, KY 40506 ,USA. Tel: 606 257 3626.0266-476 3/98 /020221-0 9 $7.0 0  199 8 C arfax Publishing Ltd222M. M . Alithorough bred races s u ggests that his model overestimates the probability of ® nish ingsecond or third for h orses that have a high probabilit y of such a result, andunderestim ates the probabili ty of ® nishing second or third fo r other horses. Ste r n(1990) examines a class of m odels that includes Harville’ s m odel, and applies twom odels from this class to analyze 47 races. A nalysis seems to corroborate the® ndings of Harville (1973). Unfortunately, both these studies are lim ited in scope,in term s of the number of models and the number of races being analyzed. H enery(1981) proposes an alter nativ e m odel. B acon-Shoneet al. (1992) propo se logisticm odels based on probability obtained from H arville (1973), Henery (1981) and anumber of Ster n (1990) models. T hey ® t these logistic m odels to data on racesheld at racetracks in Hong K ong and M eadowlands, N J. U sing a likelihoodcriterion, they found that log istic models based on probability obtai ned from theH enery (1981) m odel ® t the data best. T heir conclusion was further con® rm ed b yLo and Bacon-S hone (1994), who ® t logistic models based on probability obtainedfrom Hener y (1981) an d H ar ville (1973) probability models. These studies suggestthat the H enery (1981) m odel is likely to provide accurate estim ates of the rankingprobability of hor se-race outcom es.Unfortunately, these studies did not examine the accuracy of su ch estimates. Am ajor purpose of this paper is to investi gate the accuracy of a number of com m onlyad vo cated probab ility m odels. The analysis will be based on m ore than 15 000races. The m odels are described in Section 2. Also described in Section 2 is them aximum likelihood estim ate (MLE ) of the m odel param eters. D etails of the dataan alysis and ® ndings are reported in S ection 3. S ome concluding rem ark s are g ivenin Section 4.2 Probability m odels and their estimates2.1 Probability m odelsAssigning the probability of the outcom es of ho rse- races in whichkho rses arecom peting is the same as assig ning the pro bability for the permutations of the ® rstkintegers. T hekintegers can b e inter preted as ranks ofkobjects. A num ber ofprobability m odels for such ranking ( per mutations) have been proposed in thestatistical and psychological literature (see Critchlowet al., 1991). Am ong thesem odels is a cl ass of models which assig n to each ranking the probability of thecorresponding ordering of independent, not necessarily identically distrib utedrandom variables. More sp eci® cally, letX1,X2, . . . ,Xkbekindependent randomvariables w ith probab ility distribution functionsF(x; ai)(i51, 2, . . . ,k), and letp5(p1, p2, . . . , pk) represent a perm utation ofkobjects in which object pjhas rankj(j51, . . . ,k). Then, these m odels assign the probability to the perm utation p asPr(p )5Pr(Xp1<Xp2<. . .<Xpk)The models are known to be ranking models. Two well-studied cases of the rankingm odels are the m odel of Thurstone (1927), D an iels (1950) and Mosteller (1951),also known as the norm al ranking m odel, where the random variables are normallydistributed with m ean ai(i51, 2, . . . ,k) and variance51; and the L uce (1959)m odel, where the distrib ution of t he random variables is Gum bel. T he L uce modelis also the ® r st-order m o del in the Plackett (1975) system of log istic m odels.H enery (1981) proposes the norm al ranking m odel for horse-race outcom es.H enery (1983) and Stern (1990) investigate a ranking model known as theModelling horse-race outcom es223gam m a ranking m odel , where the random variablesXi(i51, 2, . . . ,k) have gamm adistributions with scale parameter aian d a com m on shape param eterr. T heprobability density ofXiis given byf(x; ai,r)5[ariC(r)]xr 2 1exp (2aix),x >0W ith shape param eterr51, the random variables have exponential distr ibutionsan d the m odel becomes the Luce m odel. Harville (1973) ap plies the Luce m odelan d Stern (1990) applies the gam m a ranking models w ith sh ape param eterr51, 2to horse racing. Bacon-Shoneet al. (1992) and Lo and B acon-S hone (1994) ® tlog istic models