Cluster Analysis Model for Selection of High Performing Greyhounds

Transcription

1 Internatonal Journal of Busness and Socal Scence Vol. 2 No. 21 [Specal Issue November 2011] Cluster Analyss Model for Selecton of Hgh Performng Greyhounds Anl Gulat 1 and Wllam Bosworth 2 Abstract Conventonal economc and fnance theory postulates that observed prces n a compettve market wll reflect an accurate assessment of expected outcomes. In legal sports wagerng the relatve value of a rsky asset (a wager) can be compared to ts relatve expected outcome to gan nsght nto the process of prce dscovery of the asset. At least 15 studes spannng 30 years and four contnents provde evdence that partcpants n par-mutuel bettng make consstent, explotable errors n ther bettng decsons. There are four explanatons for ths observed market neffcency, ncludng behavoral anomales, asymmetrc nformaton, measurement error, and nosy sgnalng nherent n the bddng process. Ths study tests a bettng rule based on the wn hstory of racng greyhounds usng data taken from sx greyhound race tracks. Applyng cluster analyss to racng greyhounds we show that readly avalable nformaton s not effcently exploted gvng rse to a statstcally and economcally sgnfcant proftable bettng rule. Our fndngs are consstent wth behavoral anomales and do not support the asymmetrc nformaton hypothess and the measurement error explanaton suggested n other studes. Key Words: par-mutuel bettng, gamblng, favorte longshot bas, greyhound racng. 1. Introducton and Revew of Lterature Par-mutuel bettng, the knd of wagerng that takes place at horse and greyhound racng venues n the U.S. and other natons, has provded a rch source of data wth whch to nvestgate the theory of market effcency. Accordng to ths theory, an effcent market s one n whch the prce of an asset s an unbased estmate of ts expected yeld. If the market for wagers n horse and greyhound racng s effcent, racng market partcpants wll respond to mbalances between payout ratos and reward probabltes. Eventually the market odds or payout ratos should reflect the relatve probabltes of wnnng. In general a partcpant should, on average, realze negatve returns equal to the porton of the pool retaned by the track operator. Whle many studes 3 have detected bettng patterns nconsstent wth effcent markets, others 4 have provded evdence that the methodologes lnkng relatve probabltes and market odds are based n favor of fndng aganst effcent markets n par-mutuel bettng. Ths study uses a new technque to overcome some of the measurement errors nvolved n prevous studes. We apply ths technque to greyhound racng and fnd statstcally and economcally sgnfcant unexploted bettng opportuntes that are nconsstent wth an effcent market. Par-mutuel bettng s a form of bettng n whch all smlar wagers are pooled, a certan porton s deducted from the pool by track operators for overhead, and the remander s proportonally dvded among the wnners. As bets are cast partcpants are able to vew the payout ratos on the Ttote board (an electronc dsplay) as the pools accumulate. The effcent markets hypothess holds that payout ratos for the racng entres that have a low (hgh) probablty of wnnng should be large (small). Any payout rato that s large relatve to the probablty of wnnng wll provde an opportunty for a bet wth an expected proftable outcome. A bet wth a potental payout that s small relatve to the probablty of wnnng has an expected loss. The former bettng opportunty wll attract more bets causng the pool to be dvded among more potental wnners thus reducng the payout rato. At the same tme the latter wll be avoded and the payout rato wll ncrease as the potental pool s dstrbuted over fewer wnnng bets. Grffth (1949) observed that the behavor of betters s consstent wth agents who can detect the dfferences n the probablty of dfferent outcomes. But a semnal study by Al (1977) uncovered an anomalous behavor. In hs study of horse race bettng he found that bettors tend to under bet horses wth a low payout rato (hgh probablty of wnnng) and over bet horses wth a hgh payout rato (low probablty of wnnng). 1 Correspondng Author, Western New England Unversty, 1215 Wlbraham Road, Sprngfeld MA , USA. 2 Western New England Unversty, 1215 Wlbraham Road, Sprngfeld MA , USA. 3 Al (1977), Snyder (1978), Thaler and Zemba (1988), and Asche and Quandt (1990) Zemba and Hausch (1986) Bolton and Chapman (1986), Garbrel and Marsdan (1990), Busche (1994), Gramm and Owens (2006), Wnter and Kukuk (2006), and Gulat and Shetty (2007) 4 Walls and Busche (2003) and Busche and Walls (2001). 15

2 The Specal Issue on Contemporary Research n Arts and Socal Scence 16 Centre for Promotng Ideas, USA Ths anomaly, known as the favorte long shot bas (FLB), has snce been detected n many other studes of horse racng ncludng Snyder (1978), Zemba and Hausch (1984, 1987), Thaler and Zemba (1988), and Asche and Quandt (1990), and Wnter and Kukuk (2006). Whle Gramm and Owens, (2006) detect FLB they also fnd evdence that ncreased access to bettng through smulcastng s reducng neffcency. The exstence of the FLB s not wthout controversy. Snyder (1978), Zemba and Hausch (1984, 1987) show that the wn market s weakly effcent n the sense that there s no proftable bettng strategy. Gramm et. al. (2012) argue that FLB s not economcally sgnfcant. Busche and Walls (2003) and Walls and Busche (2003) offer an alternatve explanaton of the observed FLB. Par-mutuel bettng ncludes not only a track take-out but also a protocol of roundng payout ratos down to the nearest amount dvsble by 20 cents n the case of horse racng and to the nearest 10 cents n the case of greyhound racng. That s, f a correctly calculated payout s $16.59 on a two dollar bet, the actual amount pad out s only $ The track retans the 19 cents n addton to ts percentage take-out of the total pool. Busche and Walls (2003) pont out that ths fee, called breakage s an average 10 cents per bet and s unformly dstrbuted across all payments. The favortes have low payout ratos so that the breakage represents a hgher percentage of the wnnngs than when a long shot wns wth a relatvely hgh payout rato. Walls and Busche (2003) usng data from horse tracks n Hong Kong and Japan and Busche and Walls (2003) usng Al s (1977) data arranged the reported racng results n order of the sze of the realzed breakage amounts. They found that FLB exsted only n the hgh breakage races. Gulat and Shetty (2007) appled Al s (1977) methodology to greyhound racng. Not only dd they fnd evdence that rejects FLB, they found that hgh probablty greyhounds were over bet and low probablty greyhounds were under bet, that s, the opposte of FLB. They conjectured that the dscrepancy reflects the manner n whch racng felds are selected n greyhound racng. In greyhound racng entrants are entered nto a partcular race based on ther recent success/falure hstory. Wnnng entrants are moved up to a hgher/more compettve grade and vce versa. Wnners of the most compettve races are bound by the lack of more compettve races to enter. Entres n a race of any grade, other than the hghest, may be one of three types: those just moved up by wnnng a race n the next lower grade, those just moved down from a hgher grade due to poor showng and those whch mantanng the current grade based on above average performance. Ths composton of the feld creates an lluson that favorte greyhounds are more lkely to wn n less compettve races and these are the greyhounds that are over bet. The explanatons for why FLB s observed n par-mutuel bettng fall nto two categores, behavoral explanatons and game theoretc explanatons. Behavoral explanatons nclude Grffth (1949) - mscalculaton of the odds; Rosett (1965), Wetzman (1965) Quand (1986) and Al (1977) rsk seekng behavor; Kahneman and Tversky (1984) and Thaler (1985) mental accountng or compartmentalzaton of wealth; and Canfeld, Fauman, and Zemba (1987) the value of braggng rghts from successful long shot bets. The game theoretc assumes that there are both prvately nformed and nosy bettors. Embedded n the subjectve probabltes dstrbuton of the nosy bettors are the objectve probabltes known to the nformed bettors. Therefore the posted odds nfluence the nosy bettors decsons. That s, nosy bettors observe posted odds and try to extract nformaton from them. In Potters and Wt (1996), Feeney and Kng (2001), and Koessler and Zegelmeyer (2003), nformed bettors are able to nfluence the fnal odds after the nosy bettors have cast ther bets. Ottavan and Sorensen (2003) show that the strength of belef of nformed bettors s hgher for hgh probablty entrants. In the moments leadng up to the race, nosy bettors update ther subjectve probabltes by observng posted odds. Therefore nformed bettors wat untl the last moment to bet because an early bet can provde a sgnal to nosy bettors that would dlute ther potental wnnngs as bets already placed cannot be wthdrawn. Koessler et. al. (2008) and Axelrod et. al. (2009) model smlar bettng externaltes. These models predct the behavor observed by Asch, Malkel, and Quandt (1982) and Gandar, Zuber, and Johnson (2001) n whch the fnal odds of the ultmate wnners declne over the bettng perod. In these studes the problem of unequal dstrbuton of nformaton s unresolved due to an ncomplete tâtonnement process resultng n FLB. All of these game theoretc models assume that there s asymmetrc nformaton. If, n fact, there s publcly avalable nformaton that could lead to a proftable bettng rule then the game theoretc could be rejected n favor of alternatve explanatons ncludng behavoral anomales. Brately (1973) attempted to use wn hstory n the context of horse racng to form a bettng rule but faled to do so. Bolton and Chapman (1986) use a multnomal Logt model to derve a successful horse racetrack bettng rule usng a lst of explanatory varables that ncludes wn percentage, average speed n prevous races, weght, post poston, whether the dstance of the race s wthn the horse s experence, the wn percent of the jockey, and the number of wns of the jockey.

3 Internatonal Journal of Busness and Socal Scence Vol. 2 No. 21 [Specal Issue November 2011] Goodwn (1996) argued that models based on an assumpton of any partcular dstrbuton of expectatonal errors (e.g. normal, logstc, etc.) can lead to based estmators of underlyng parameters. Further, heteroscedastcty n greyhound racng due to the groupngs of smlar dogs (e.g. successful dogs, less successful dogs) n dfferent races can lead to nconsstent parameter estmates. He bult a model that uses varous hstorcal varables, handcappng and posted odds as predctors. Hs model sgnfcantly outperforms the predctons of professonal handcappers. In contrast to both these studes, we drectly test a smpler bettng strategy based only on wn hstory. We avod parameterzaton by usng cluster analyss whch mnmzes the dstance functons among greyhounds based on ther wn hstory. All of the emprcal studes cted so far (except Grffth, 1949) have grouped the data and calculated the statstcs to test the hypothess of market effcency n the manner ntroduced by Al (1977). In ths method the racng entrants are ranked based on ther bettng odds. That s, the entrant wth the lowest (hghest) payout rato s ranked frst (last). The gross payout rato for the th entry s a functon of the entry s relatve share of the bettng pool. P (1 t) where, G 1 X / X, P = The gross return, t = The percentage take-out retaned by the track operator, X = The amount of money bet on the th entry, G = The number of entres racng n the race, X = The total amount n the bettng pool across all entres. The probablty of wnnng mpled by the pattern of wagers (IP) s, IP X G / X. (2) 1 To adjust for the breakage the mpled probablty s calculated as, G RD IP X /( X ) (3) 2 1 Where, RD s the maxmum possble breakage. Wall and Busche (2003) and Busche and Wall (2003) departed from ths procedure by omttng the RD/2 term. Instead, they further grouped observatons ranked by the amount of breakage actually realzed. Thus, each entry was grouped n two ways, once by IP and once by the breakage that resulted n the fnal payout. When they dd so they found statstcally sgnfcant evdence that gnorng breakage costs bases the results n favor of FLB. The present study departs from the method of Al (1977) by usng cluster analyss n leu of the mpled probablty calculaton of Equaton 3. In greyhound racng the wn hstory of every entry s avalable to all race track partcpants. Therefore, t s possble to calculate the actual probablty of ts wn predcated on ts hstory. If, usng hstorcal data nstead of the posted payout ratos, any bas can stll be detected then the effcent markets hypothess s rejected by the data. Secton II reports on the data collecton and the methodology, Secton III reports the results, and Secton IV concludes. 2. Data and Methodology We chose sx greyhound racetracks for ths study wth the purpose of creatng a geographcally dverse sample. Collectvely, the observaton perods on these sx tracks represent about 230,000 races. The track names and the observaton perods are gven below. Insert Table 1 Here Data Selecton To gve our study a clear focus we narrowed our study by usng nformaton that s publcly avalable to bettors. On each track we only consdered races that started wth a full box.e. eght greyhounds. 17 (1)

4 The Specal Issue on Contemporary Research n Arts and Socal Scence Centre for Promotng Ideas, USA Further, all greyhounds that had not competed n each of the top three grades n ther racng careers were removed. Havng defned our feld of study we calculated the wn rato of each of the greyhounds. The result s a table of every greyhound that competed n the top three grades and ther correspondng wn rato n each of the top three grades. 2.1 Data Analyss Usng hstorcal performance as the defnng characterstc we clustered the greyhounds on each track nto two groups: Hgh wn frequency and Low wn frequency. The clusterng was performed usng the k-means cluster analyss n SPSS. All groups converged wthn 20 teratons.table II shows the number of greyhounds classfed as Hgh wn frequency, and Low wn frequency, for each of the sx tracks. 18 Insert Table 2 Here For the entre feld, the number of Low wn frequency greyhounds outnumbered the Hgh wn frequency counts by about a multple of seven. The range was from 5.97 tmes (Gulf) to 7.04 tmes (Brmngham).The k- means analyss tool n SPSS was used to classfy the greyhounds nto the two clusters. The fnal centers of the Hgh wn frequency cluster are shown n Table III below. The wn percentage s decomposed by grade (hghest, second and thrd). Insert Table 3 Here For the hghest grade, the wn proporton of the Hgh wn frequency cluster s low relatve to the other two grades. Greyhounds are moved up a grade wth superor performance and are downgraded to the next lower grade due to sub-par performance. Superor performers n the top grade cannot move any hgher. Better performers reman n the hghest grade and sub-par performers are downgraded. Therefore the feld n a top grade race s composed of greyhounds mantanng that grade or movng up by wnnng the next lower grade race. Therefore, the feld n the top grade s more compettve (evenly matched) whch results n lower wn percentage. The fnal centers of the Low wn frequency greyhounds cluster are shown n Table IV, below. The wn percentage s decomposed by grade, as before. Insert Table 4 Here As noted n the case of the Hgh wn frequency greyhounds, the Low wn frequency greyhounds also won the least n the hghest grade. 2.2 Methodology Our hypothess s that the greyhound racng market s domnated by ratonal racng partcpants. The null hypothess s the weak form of the effcent market hypothess: no bets should have postve expected values. We assume that racng partcpants have access to wn hstores of the greyhounds and are able to form expectatons based on those hstores. We do not attempt to model the nformaton avalable from handcappers, odds posted by the track operators, or evolvng posted odds. We assume that bettng partcpants study the hstory of the greyhounds n a race and determne the probablty of ther wnnng based on ther past performance, compare those probabltes to the posted odds before makng bettng decsons.the clusters therefore classfy the greyhounds n two groups,.e. the group wth hgh wnnng potental and the other wth low probablty of wnnng the race. If bettors are able to assess these probabltes accurately and bet accordngly, there should be no opportuntes for excess returns. Actually, the returns should be negatve and equal the track take-out. To test our hypothess, we placed two dollar hypothetcal bets on each greyhound n the two clusters. We segregated the returns by cluster for Favortes and Longshots and the results are presented n the next secton. 3. Results Table V shows results of a hypothetcal $2 bet to wn wagered on the greyhound n the Hgh cluster when the entry was the favorte to wn. Smlarly a hypothetcal $2 bet to wn was wagered on the favortes n the Low cluster, for the top three grades studed. These returns are calculated net of the track take-out. Insert Table 5 Here The returns n Hgh cluster are postve for each of the sx tracks, thus representng proft makng opportuntes. Conversely, the returns generated n Low cluster are negatve across the board, wth one excepton of Derby Lane. Explanaton of ths exceptonal behavor would requre deeper analyss whch s beyond the scope of the present study. For now, we attrbute ths excepton to shorter observaton perod.

5 Internatonal Journal of Busness and Socal Scence Vol. 2 No. 21 [Specal Issue November 2011] Table VI shows results of smlar hypothetcal $2 bet to wn wagered on the greyhound n the Hgh cluster when t was the Longshot. Smlarly a hypothetcal $2 bet to wn was wagered on the Longshot n the Low cluster, for the top three grades studed. These returns are calculated net of the track takeout. Insert Table 6 Here These results generally follow the pattern that returns n the Hgh cluster are far superor than the Low cluster. Also, we must underscore the postve returns experenced by the Low cluster greyhounds. We note that these results sgnal that the FLV bas s present n our data set. 3.1 Dscusson of Results Our results show that bettors are unable to dfferentate between the Hgh and Low potental (cluster) entres. Sgnfcant and postve returns n the Hgh cluster result for these entres, collectvely, attractng smaller percentage of the Wn pool as compared to the true odds of wnnng. Therefore, we conclude that these entres are sgnfcantly under-bet. Conversely, the entres n Low cluster are ether over-bet, or at the very least represent far bets. Our method of analyss dscrmnates between entres wth a potental to consstently wn races and between a second group wth more unpredctable outcomes. Second, we show that market partcpants are unable to dscrmnate between the two groups. 4. Conclusons In ths study we tested a smple model n whch racng partcpants used hstorcal data to bet on greyhound races. Our hypothess s that racng partcpants effcently explot publcly avalable nformaton. Our test conssted of explotng the wn hstory of the greyhounds n each race as the sole predctor of the wn probablty. The results of our study reject the hypothess of an effcent market n par-mutuel greyhound racng. Devatons from ratonal bettng payouts were both statstcally and economcally sgnfcant for the sx race tracks observed n ths study. The results of ths study do not support the asymmetrcal explanaton of systematc bas n par-mutuel bettng of Potters and Wt (1996), Feeney and Kng (2001), Koessler and Zegelmeyer (2003), Ottavan and Sorensen (2003), Koessler et. al. (2008), and Axelrod et. al. (2009). Moreover, ths study overcomes the measurement problems of Walls and Busche (2003) and Busche and Walls (2003). We therefore conjecture that we are observng a behavoral phenomenon nconsstent wth effcent markets. References Al, Mukhtar M. (1977). Probablty and Utlty Estmates for Racetrack Bettors. Journal of Poltcal Economy 85, Asch, Peter, Burton G, Malkel and Rchard.E. Quant. (1982). Rsk Love. Journal of Economc Educaton 10, Asch, Peter and Rchard.E. Quant. (1990). Rsk Love. Journal of Economc Educaton 21, Axelrod, Bors S., Kulck, Ben J., Plott, Charles R., and Roust, Kevn A. (2009). The desgn of mproved parmutueltype nformaton aggregaton mechansms: Inaccuraces and the long-shot bas as dsequlbrum phenomena. Journal of Economc Behavor & Organzaton, 69, Bolton, Ruth N., and Randall G. Chapman (1986). Searchng for Postve Returns at the Trak: A Multnomal Logt Model for Handcappng Horse Races. Management Scence 37, Bratley, P. (1973). A Day at the Races. Infor 11 (2) Busche, Kelly. and C.D. Hall (1988). An Excepton to the Rsk Preference Anomaly. Journal of Busness 61, Busche, Kelly. (1994) Effcent Markets Results n an Asan Settng. In D. Hausch, V. Lo, and W.T. Zemba eds. Effcency of Racetrack Bettng Markets, Academc Press, New York, , Busche, Kelly and Walls, W. Davd (2000). Decson Costs and Bettng Market Effcency. Ratonalty and Socety 12, Busche, Kelly and Walls, W. Davd (2001) Breakage and Bettng Market Effcency: Evdence from the Horse Track. Appled Economcs Letters 8, Canfeld, Bran R. Bruce C. Fauman, and Wllam T. Zemba (1987). Effcent Market Adjustment of Odds Prces to Reflect Track Bases. Management Scence 33, Fama, Eugene F. (1991). Effcent Captal Markets: II. Journal of Fnance 46, Feeney, Rob and Stephen P. Kng (2001). Sequental Parmutuel Games. Economc Letters 72, Gabrel, Paul E. and James Marsden (1990). An Examnaton of Market Effcency n Brtsh Racetrack Bettng. Journal of Poltcal Economy 98,

6 The Specal Issue on Contemporary Research n Arts and Socal Scence Centre for Promotng Ideas, USA Gandar, John M., Rchard A Zubar, and R. Stafford Johnson (2001). Searchng for the Favorte-Longshot Bas Down Under: An Examnaton of the New Zealand Par-Mutuel Bettng Market. Appled Economcs 33, Goodwn, Barry K. (1996). Semparametrc (Dstrbuton Free) Testng of the Expectatons Hypothess n a Parmutuel Gamblng Market. Journal of Busness & Economc Statstcs 14, Gramm, Marshall and Douglass H. Owens (2006). Effcency n ParMutuel Bettng Markets across Wagerng Pools n the Smulcast Era. Southern Economc Journal 72: Gramm, Marshall, McKnney, C. Ncholas, Owens, Douglas (2012). Effcency and arbtrage across parmutuel wagerng pools. Appled Economcs 44, Grffth, R.M. (1949). Odds Adjustment by Amercan Horse Race Bettors. Amercan Journal of Psychology 44, Gulat, Anl and Shekar Shetty (2007) Emprcal Evdence of Error n Prcng of Favortes and Longshots n Greyhound Racng. Journal of Economcs and Fnance 31, Hausche, Donald B., Wllam.T. Zemba, and Mark Rubnsten (1981). Effcency n the Marker for Race Track a Bettng. Management Scence 27, Hurley, Wllam and Lawrence McDonough (1995). Note on the Hayek Hypothess and the Favorte-Longshot Bas n Parmutuel Bettng. Amercan Economc Revew 85, Kahneman, Danel, and Amos Tversky (1984). Choces Values and Frames. Amercan Psychologst 39, Koessler, Frédérc and Anthony Zegelmeyer (2002) Parmutuel Bettng under Asymmetrc Informaton THEMAS mmeo Cergy-Pontose. Koessler, Frédérc, Noussar, Charles, and Zegelmeyer, Anthony (2008). Parmutuel bettng under asymmetrc nformaton. Journal of Mathematcal Economcs 44, Ottavan, Marco and Peter Norman Sorensen (2003). Late Informed Bettng and the Favorte-Longshot Bas. Workng paper, Insttute of Economcs, Unversty of Copenhagen. Ottavan, Marco and Peter Norman Sorensen (2010). Nose, Informaton, and the Favorte-Longshot Bas n Parmutuel Predcton. Amercan Economc Journal: Mcroeconomcs 2, Potters, Jan and Jorgen Wt (1996). Bets and Bds: Favorte Longshot Bas and the Wnner s Curse. CentER Dscusson Paper Quandt, Rcjard E. (1986). Bettng and Equlbrum Quarterly Journal of Economcs 101, Rodrguez, Alvaro (2011). Computng the Probablty of Wnnng a Competton wth an Applcaton to Horse Races. Journal of Quanttatve Analyss n Sports 7, 9p. Rosett, Rchard N. (1965) Gamblng and Ratonalty. Journal of Poltcal Economy 73, Shn, Hyun Song (1991) Prces of State Contngent Clams wth Insders and the Favorte Longshot Bas. Economc Journal 102: Snyder, Wayne.W. (1978) Horse Racng: Testng the Market Effcency Model. The Journal of Fnance 4: Thaler, Rchard H. (1985). Mental Accountng and Consumer Choce. Marketng Scence, 4, Thaler, Rchard. and Wllam T. Zemba (1988) Anomales: Parmutuel Bettng Markets: Racetracks and Lotteres. Journal of Economc Perspectves 2: Tercell, Dek (1998). Bases n Assessments of Probabltes: New Evdence from Greyhound Races. Journal of Rsk and Uncertanty 17: Walls, W.Davd and Kelly. Busche (2003). Broken Odds and the Favorte-Longshot Bas n Par-Mutuel Bettng: A Drect Test. Appled Economc Letters 10, Wetzman, Martn (1965). Utlty Analyss and Group Behavor: An Emprcal Analyss. Journal of Poltcal Economy. 73: Wnter, Stephan, and Martn Kukuk (2006). Rsk Love and the Favorte Longshot Bas: Evdence from German Harness Horse Racng. Southern Busness Revew 58, Zemba, Wllam T. and Donald B Hausche (1987) Dr. Z s Beat the Racetrack. Wllam Morrow, New York. Table I. Track Names and Observaton Perod Track Name Track Code Racng results from 1 Brmngham Race Course Brmngham Apr Aug Bluffs Run Greyhound park Bluffs Jan Aug Derby Lane Derby Mar Aug Gulf Greyhound Park Gulf Jan Aug Jacksonvlle Greyhound Racng Jacksonvlle May Aug Palm Beach Kennel Club Palm Jun Aug

7 Internatonal Journal of Busness and Socal Scence Vol. 2 No. 21 [Specal Issue November 2011] Table II. Cluster Analyss Results, Number of Greyhounds Classfed n the Two Clusters Track Code Hgh Low 1 Brmngham Bluffs Derby Gulf Jacksonvlle Palm Table III. Fnal Centers of the "Hgh" Cluster (Percent of Races Won n Top Three Grades) Track Code Hghest Second Thrd 1 Brmngham 14.64% 46.41% 73.31% 2 Bluffs 14.41% 34.10% 69.69% 3 Derby 15.01% 31.96% 70.61% 4 Gulf 17.46% 48.08% 84.02% 5 Jacksonvlle 13.21% 29.70% 69.83% 6 Palm 15.39% 37.11% 79.76% Table IV. Fnal Centers of the "Low" Cluster (Percent of Races Won n Top Three Grades) Track Code Hghest Second Thrd 1 Brmngham 8.29% 12.59% 17.46% 2 Bluffs 5.05% 13.61% 15.48% 3 Derby 5.36% 15.42% 18.72% 4 Gulf 6.09% 16.00% 20.23% 5 Jacksonvlle 5.00% 15.23% 17.84% 6 Palm 5.25% 16.08% 19.87% Table V. Hypothetcal Bettng Results for "Hgh" and "Low" Clusters - Rates of Return for $2 Bet to Wn on Favortes. Track Code Hgh Low 1 Brmngham 23.72% % 2 Bluffs 7.92% % 3 Derby 26.15% 23.92% 4 Gulf 8.84% % 5 Jacksonvlle 9.25% % 6 Palm 1.61% % Table VI. Hypothetcal Bettng Results for "Hgh" and "Low" Clusters - Rates of Return for $2 Bet to Wn on Longshots. Track Code Hgh Low 1 Brmngham % 31.91% 2 Bluffs 91.28% 23.33% 3 Derby 28.85% % 4 Gulf 49.79% 10.90% 5 Jacksonvlle 42.20% 2.34% 6 Palm 84.64% -1.65% 21