SP Betting as a Self-Enforcing Implicit Cartel

Transcription

1 SP Bettg as a Self-Eforcg Implct Cartel by Ad Schytzer ad Avcha Sr Departmet of Ecoomcs Bar-Ila Uversty Ramat Ga Israel e-mal: schyta@mal.bu.ac.l srav@mal.bu.ac.l Abstract A large share of the UK off-course horse racg bettg market volves wg payouts determed at Startg Prces (SP). Ths mples that gamblers ca bet wth off-course bookes o ay horse before a race at the fal pre-race odds as set by o-course bookes for that horse. Gve the olgopolstc structure of the off-course gamblg market the UK, a market that s domated by a small umber of large bookmakg frms, we study the pheomeo of SP as a type of self-eforcg mplct colluso. We show that gve the ucertaty about a race outcome, ad ther ablty to fluece the prces set by o-course bookes, agreeg to lay bets at SP s superor for off-course bookes as compared wth offerg fxed odds. We thus exted the results of Rotemberg ad Saloer (990) to markets wth ucertaty about both demad ad outcomes, We test our model by studyg the predcted effects of SP bettg o the behavor of o-course bookes. Usg data draw from both the UK ad Australa o-course bettg markets, we show that the dffereces betwee these markets are cosstet wth the predcted effects of SP bettg the UK off-course market ad ts absece from the Australa market. Thursday, Jue 4, 2007

2 A. Itroducto It s ofte argued that horse bettg markets share may commo features wth facal markets (Thaler ad Zemba, 988, Sh, 992, 993, Vaugha Wllams ad Pato, 997, amog others). I both markets, well formed partcpats trade rsky assets, ad termedares play a mportat role settg prces (Sh, 992). At the same tme, ulke more complcated facal markets, bettg markets are bouded tme ad yeld a sgle ad uequvocal outcome. Thus, bettg markets "provde a clear vew of prcg ssues whch are more complcated elsewhere" (Sauer, 998, p. 202). I ths paper we study the UK market for SP bettg o horse races. I the UK, bettors may place bets wth bookmakers both at the track (o-course bookmakers) ad bookmakers' offces aroud the coutry (off-course bookmakers). Most of the offcourse bettg s carred out at Startg Prces (SP), 2 whch are the odds offered by o-course bookmakers at the ed of the bettg perod. Thus bettors who bet at SP are ot oly gamblg o the outcome of the race, but also o the returs. 3 SP bettg also seems to be rsky for off-course bookmakers: Whereas o-course bookmakers hedge agast the cotgecy of havg a large share of the bets placed o hgh odds horses by cotually adjustg odds, off-course bookmakers do ot have drect cotrol over SP. Offerg bets at SP thus seems to be cosstet wth commo models that vew the bookmakers' target fucto as ether guarateeg proft at mmum rsk or maxmzg profts by outperformg bettors (Crafts, 985, Vaugha Wllams ad Pato, 997, Lee ad Smth, 2002, Levtt, 2003, Strumpf, 2003) To expla why SP bettg s evertheless the preferred form of bettg the UK, we develop a model that focuses o the cocetrated structure of the UK offcourse market. I our model there are two types of bookmaker: Large off-course bookmakers wth the ablty to fluece market outcomes ad small o-course bookmakers who operate compettve markets. Our results mply that equlbrum, the off-course bookmakers may choose to collaborate mplctly ad sell bets at SP. Owg to the tact colluso, SP bettg allows the off-course bookmakers The authors wsh to thak Arthur Fshma, Ha Guowe ad Dael Levy for ther helpful commets. 2 Dowe (976), for example, estmates that 95% of the off-course bettg are made at SP. 3 Iterestgly, J.K. Rowlg satrzes SP bettg "Harry Potter ad the Goblet of Fre." There, a bookmaker offers bets wthout amg the odds. ad oly promses to gve "excellet odds" later. The egatve atttude towards SP bettg cotues whe the bookmaker cheats ad pays wers wth fake moey. 2

3 to mapulate the o-course prces ad thus creases ther expected profts. 4 Our aalyss thus exteds the results of Rotemberg ad Saloer (990) who show that sellers who face demad ucertaty have a cetve to collude. We show that whe both sellers ad clets face ucertaty about outcomes, the sellers have a eve stroger cetve to collude. We test our emprcal hypotheses by comparg the UK bookmakers market wth the Australa market. These two markets share may commo features, wth the excepto that SP bettg s llegal Australa. We show that the data support our theoretcal predctos suggestg that the UK horse racg market s less effcet tha the Australa market predctable ways. Our fdgs are stregtheed by recet fdgs o effceces the UK horse racg market (see for example: Gabrel ad Marsde, 990, Vaugha Wllams ad Pato, 997, Bruce ad Johso, 2005, Johso ad Sug, 2006). Our results may also cotrbute to the study of tact colluso cocetrated markets (Rotemberg ad Saloer, 990, Slade, 992, Boreste ad Shepard, 996). B. Market Structure Before developg our model, we brefly descrbe the structure of the UK ad Australa horse bettg markets. I the UK, bettors may place bets wth both the par-mutuel or wth o- ad off- course bookmakers. The par-mutuel market ormally opes for bets aroud 24 hours before races beg, ad the odds are settled accordg to the fal sums placed o each horse. 5 The off-course market s cotrolled by a small umber of bookmakg frms, who operate a large umber of offces throughout the coutry. 6 These outlets ormally ope for bettg (lke the par-mutuel market) aroud 24 hours before each 4 Prevous authors have already oted that off-course bookmakers may tervee the o-course markets. Dowe (976, p. 40), for example, otes that: "actual SP s thus determed by both off- ad o- course actvty." 5 For more o par-mutuel markets see: Thaler ad Zemba (988), Sauer (998) ad Gadar et al. (200), amog others. 6 Accordg to the teret ste Wkpeda, there are four ma off-course bookmakg frms the UK. I 2004, those four frms operated over 8,000 bettg offces (e.wkpeda.org/wk/bookmaker), ad had a cosderable turover ad market power; Wllam Hll, for example, had a total turover of over 8 bllo 2004 ( 3

4 race. 7 They offer bets maly at SP, whch are the fal odds offered by o-course bookmakers ad are thus ukow at the tme the bet s made. Ulke the cocetrated off-course bettg market, the o-course market s ormally far more compettve, wth 0 50 o-course bookmakers at each track. The o-course bookmakers ope for bettg about twety mutes to half a hour before each race, ad they offer bettg at fxed odds. The frst set of odds they post at the tme they ope for bets are kow as Opeg Prces (OP). As the market progresses, they ofte chage odds order to maage ther exposure. 8 I partcular, bookmakers ofte shorte the odds o horses that are pluged. A pluge occurs whe a large bet o a horse s made smultaeously wth may bookmakers; past research shows that pluges usually dcate the presece of bettors wth superor formato (sders), such as horse owers ad traers (Sh 99, 992, 994 ad Schytzer ad Shloy 995, 2003, 2005). The last set of prces that the bookmakers post before the market closes are kow as the Startg Prces (SP). As oted above, these are also the set of prces that determe the payouts for most of the bets lad by the off-course bookmakers. The Australa o-course market s structured the same way as the UK market. The dffereces occur off-course, where SP bettg s llegal ad bets are made ether at fxed odds offered o selected races, more or less as the UK, or wth varous par-mutuels. Fally, there are o-le bookmakers who offer bettg at parmutuel odds. 9 7 The off-course bookmakers also offer bettg at fxed odds. However for most races, they ope for bets at fxed odds oly aroud half a hour before the races start. The oly exceptos are very mportat races; such cases fxed odds are sometmes offered eve moths before the race. These exceptos otwthstadg, the vast majorty of bettors who bet wth the off-course bookmakers choose to do so at SP. 8 I over 42,000 observatos that we have our UK data set, fewer tha 8.5% of the ruers mataed a costat prce from the tme the market opeed ad utl t closed. 9 Eve 2007, whe there are legal o-le bookmakers Australa, SP bettg remas llegal ad, for most races, the bookmakers offer the odds whch are determed by the varous par-mutuels operatg the dfferet states. These bookmakers compete by permttg bettors to choose accordg to whch par-mutuel they wsh to be pad f successful or by offerg to pay the maxmum dvded amog the par-mutuels. Thus, to the extet that Australa off-course bookes themselves bet aywhere t wll be wth the par-mutuel. It s terestg to ote that UK off-course bookmakers do ot compete by offerg dscouts from SP, although loss rebates are occasoally employed. 4

5 C. The model, Assumptos Our model combes elemets from Sh (992, 993), Rotemberg ad Saloer (990) ad Schytzer ad Shloy (2005). It descrbes a market for bets o a horse race wth N horses, labeled by the set {,2 }. Each horse, {,2... ) has (ukow) probablty p of wg, such that p = ad p > 0. Sce our focus s o the behavor of off-course bookmakers, we defe the settg as follows. Frst, there are two off-course bookmakers, detfed as booke ad booke 2. Each of the bookes offers a set of N tckets that are labeled,2. Each booke N = > chooses the prce of each tcket, such that the prce of tcket {,2...,} 0 sold by booke j {,2} s π where 0 π <. The th tcket costs ad pays j < j the π j evet that horse ws the race ad zero otherwse. We show below that equlbrum, both bookes set SP prces rather tha offer tckets at fxed odds. We assume that the goal of the off-course bookes s to maxmze ther expected profts ad that they are rsk averse the followg sese: Assumpto (Rsk Averso): The bookes always mata a balaced book, such that for every tcket {,2... } the expected cotget payout s less tha or equal to the total sum accumulated by the booke. Deotg by T j the total sum of moey wagered by all bettors o horse wth booke j, ths mples that: {,2... } j {,2} ( p ) E Tj π T j =, 2 where E deotes expectatos. We assume the demad for each of the tckets s gve by: 0 Sh (992) vestgates the case where a horse has a probablty of of wg. However, such cases bookmakers ormally refuse to take bets. As our terest s essetally emprcal, we do ot cosder such cases. Normally, bookmakers post prces terms of odds rather tha prces. The relatoshp betwee the π, t odds ad the prces s gve by: O, t = where O, t s the odds offered o horse race t ad π, t π,t s ts prce. 2 Ths assumpto s ofte made the lterature. For example, Crafts (985, p. 295) assumes that bookmakers goal s to have a "perfect book," meag that "the bookmaker make[s] moey whchever horse ws." 5

6 T j j β = j + ak 0 [ π j Eoff ( p )] β [ π E ( p )] j off f π j f = π π j otherwse k < π k j Where j s the demad for horse at booke j f booke j's prce s equal to the expected wg probablty, β s the margal chage the demad as a result k () of a chage the prce ad E off deotes the expectato formed by off-course bettors based o all publc formato. We assume that all the off-course bettors have the same set of formato ad smlar expectatos about the prospects of ruers. Ths leads us to follow Rotemberg ad Saloer (990) ad assume that f the bookes offer equal prces they expect equal demad. That s, we assume that j ~ N (, σ ) ad that, the case of equal prces, demad vares betwee them oly by a radom compoet wth equal devatos. We assume further that the off-course bookes have the same set of formato as the bettors. Therefore, they also have the same expectatos as off-course bettors. 3 Sce expected demad depeds o posted prces, the two bookes face Bertrad-type competto (Carlto ad Perloff, 994), where bettors buy tckets from the cheaper booke a amout proportoal to the dfferece betwee the prce ad the expected probablty of each ruer. Note that the case where prces equal expected probablty, the expected demad for each horse s. Therefore, uder the (strog) assumpto that we make about j, t s possble to deduce the subjectve probablty that off-course bettors assg to each horse by calculatg: E off ( p ) = E off j = j = =. Aother set of players that exst the model s the o-course bookmakers ad bettors. We assume that o-course bookmakers maxmze ther expected profts ad that the o-course bettg market s effcet, so that expected profts are zero. Ths mples that the o-course bookmakers behave a way that s smlar to that modeled 3 Ths assumpto ca be defeded o the grouds that off-course bookmakers ope for bets a relatvely log tme before races beg. It s well documeted that prces o-course markets ofte chage utl the fal momets before they close. Ths mples that ulke, for example, the US markets for bets o pot spreads (Levtt, 2003), bookmakers do ot possess all the relevat formato utl the ed of the bettg perod. 6

7 by Schytzer ad Shloy (2005) ad Sh (992). We further assume that bettors the o-course market are better formed tha off-course bettors ad that they bet proporto to ther expectatos. 4 Deotg by E o ( p ) the expectato formed by o-course bettors, we arrve at assumpto 2: Assumpto 2 (O-Course \ Off-Course Expectatos): For every ruer < o ( off >, E p ) = E ( p ) ad E p ) has greater predctve power tha E off p ). o ( Assumpto 2 s justfed o both theoretc ad emprcal grouds: Theoretcally, f oe assumes that the ma goal of the bettors s to proft, the comg to the racetrack volves hgher costs tha bettg off-course. Thus, bettors travel to the track oly f they could ga addtoal formato there, or f they were determed to explot sde formato at fxed odds a compettve settg, or both. From a emprcal stad pot, o-course bettors have a better opportuty to spect the codto of the ruers, ad also to observe the actos of sders. The fdgs the lterature also support the vew that o-course bettors out perform offcourse bettors (Fglewsk, 979; Schytzer ad Shloy, 995; Gadar et al., 200). Our remag two assumptos cocer the relatoshp betwee the offcourse ad o-course markets: Assumpto 3 (Off-Course Bookes Iterveto): Off course bookes ca chage the prces the o-course market by plugg ruers. As descrbed above, o course bookmakers respod to pluges by creasg the prce of the pluged horse order to avod loses the case that the pluges horses w. Off-course bookmakers ca take advatage of ths ad because the off course market s much larger tha the ocourse market. Off-course bookmakers ca thus pluge horses o-course wthout offsettg ther books sgfcatly. Moreover, the case that they receve large bets o horses wth a low o-course prce, off-course bookes ca hedge some of ther rsk by bettg o those horses, ad thus dog so s almost always proftable. The last assumpto cocers the ablty of off-course bookmakers to observe the SP prces before the close of bettg. I order to smplfy the model we make the followg assumpto: ( 4 Ths s smlar to the assumpto made by Schytzer ad Shloy (2005) where they assume that there s a herarchy of formato. At the top of the herarchy there are bettors wth sde formato ad bet o-course, ad at the bottom there s the geeral gamblg publc. 7

8 Assumpto 4 (Observg SP): The off-course bookmakers ca observe the equlbrum prces the o-course market just before the close of bettg ad thereby mply startg prces. Oce they compare these ear fal equlbrum prces wth ther desred SP, they ca fluece prces by plugg horses before the close of bettg. 5 We therefore model the sequece of evets as follows: Frst, the off-course market s opeed ad the off-course bookes set ther prces for each tcket. 6 I the secod stage, off-course bettors place ther bets by buyg tckets from the offcourse bookes at the prce offered, accordg to the demad fucto gve by () ad ther expectatos about outcomes. I the thrd stage, equlbrum prces are reached the o-course market ad are observed by the off-course bookes. I the fal stage the off-course bookes may pluge horses order to fluece SP. 2. Soluto Each booke sells tckets,2 at a prce of π, ad pays out ut of tcket sold the evet that horse ws, a evet that occurs wth for each π probablty p. Hs profts o each horse are therefore gve by () tmes the prce of a tcket mus the sum he pays the evet that the horse ws. Hs total profts are therefore: R j = p { [ ]} j β π Eoff ( p ) π = Sce the bookes have o formatoal advatage over the bettors, t follows that f they try to compete by prces, the oly possble equlbrum s a Bertrad equlbrum wth zero expected profts. The prce of each horse such case s gve, (2) 5 Assumg that the off-course bookmakers do ot actually kow the equlbrum levels but employ stochastc lear programmg to determe the actos accordg to expected equlbrum prces close to the ed of bettg would ot chage the results, so log as the ose ther predctos was ot too great. 6 We assume that off-course bookmakers offer oly oe type of asset, although realty they may offer a mxture of assets. For example, off-course bookmakers ormally ope for bets at fxed odds about twety mutes from the start of each race. However, at that tme most bettors bet at fxed odds ad ot at SP, ad addto the SP market s may tmes larger tha the off-course fxed-odds market. 8

9 uder our assumptos about the dstrbuto of j, by: j = {,2} ad expected demad s j π = Eoff ( p ) = Eoff =, j = = E ) =. ( j However, ths model, ths s ot a uque equlbrum. Assume (wthout loss of geeralty) that booke declares that he would pay bettors accordg to SP as set the o-course market, ad that booke 2 follows. Note that we defe the SP of horse a smlar fasho to our defto of π ;.e. the startg prce of horse s the prce of a cotget clam to the evet that horse ws the race. We show that ths s a stable equlbrum. To see that ths s deed a equlbrum outcome, frst assume that for every horse the race, off-course bettors take the expected SP of that horse to equal ther expected probablty. If bettors take SP to equal expected probabltes, expected demad for each booke aga equals profts for each booke are gve by: R = = j j j E j = E = = π = π = E( π ) = E( ) = ad expected 2 E( π ) p π P (3) where E deotes the expectatos formed by the bookes. Assumpto 4 states that each off-course booke kows the equlbrum prce set the o-course market before t closes. We deote the equlbrum prce of horse that s observed before the close of the market by e SP j. Whe the off-course booke observes e SP he has two possble cases to cosder: SP e = Frst, SP 0. (4) = SP e e = 2 I ths case, the o-course bettors vew the horse as at least as hot a favorte as the offcourse bettors. I cosequece, the off-course bookes make postve expected profts from adoptg SP as the quoted prce. 9

10 I the other case: SP e < = e SP SP e = = 2 < 0. (5) Here, the off-course bookes have egatve expected gas from the horse. Moreover, ths case, ther books are ot balaced ad ths breaches Assumpto. However, each booke kows that he ca fluece SP by plugg. Sce the other booke faces, expectato, the same demad for that horse, each booke predcts that f (5) holds, both of them would pluge the horse to the pot where ts prce equals. 7 If, respose, the o-course bookmakers decrease the prce of ay = other horse j so that ts prce falls below = j, the the off-course bookmakers respod the same way wth regard to that horse as well, so that evetually, were there suffcet tme remag before the start of the race for complete adjustmet to equlbrum, the prce of all ruers would be ether greater tha or equal to the share of bets placed o them the off-course market. Now, sce o-course bookes presumably oly chage prces whe there s suffcet tme remag for further bets, t s reasoable to assume that o horse s prce wll fall suffcetly respose to a off-course bookmaker s pluge, o a dfferet horse, to ecesstate a further pluge for whch there s suffcet tme. As a cosequece, uder Assumptos () (4), sellg SP bets allows the bookes to make o-egatve expected profts o each horse, whereas he would make zero expected profts f he offers bets at fxed odds. The tuto that drves ths result s smple: By deferrg the decso to the o-course market, the off-course bookes ca make expected profts o horses that off-course bettors favor more tha o-course bettors. As log as they collude tactly to offer SP, they ca also use ther market power to hedge agast expected losses o horses that o-course bettors favor more tha off-course bettors. 7 Note that plugg the horse helps the bookes to hedge ther ow rsks, so plugg does ot create a publc good dlemma. 0

11 To prove that SP prcg s deed a equlbrum, t remas to be show that the bookes caot ga from devatg, ad that off-course bettors deed take SP to equal expected prces. Frst we show that the bookes caot ga from devatg. Assume that booke aouces hs prces to be SP, ad booke 2 cosders devatg. Ths ca oly be proftable to booke 2 f he sets hs prce to be below the expected SP prce. Assume that booke 2 aouces that for tcket, the prce s set below ts expected prce, so that t equals: ε, ε > 0. I ths case the demad for horse would equal = j + k b ε E off ( p ) > j. = At the same tme, f he sets the prce below the expected probablty, t follows that: E off p < 0. π Substtutg ths result the booke's proft fucto (equato (2)), t follows that the expected profts of booke 2 are egatve ad hs book s ot balaced. Ths breaches Assumpto, ad therefore caot hold equlbrum. Ths s true for the prce of ay horse, ad therefore booke 2 caot ga from devatg. 8 Cosderg the bettors, beg ratoal, they kow the cetves of the bookes. They therefore kow that, equlbrum: SP = E o ( p ) = E off = ( E o ( p )) = = f E o ( p ) E off otherwse ( p ) = Sce off-course bettors use all the formato avalable to them, the best expectatos they ca form s to expect the o-course bettors to have the same = 8 Note that eve f booke 2 s sestve to breachg Assumpto, booke would fd that he ears o moey o horse ad that he ca therefore mprove hs stuato by sellg tcket at a lower prce: ad therefore eve f booke 2 agrees to crease hs persoal rsk, the stuato s ot a equlbrum.

12 expectatos. It thus follows that the off-course bettors expect SP prces to equal ther expectato. Thus, although they kow the cetves of the off-course bookmakers, bettg at SP prces s as worthwhle for them as bettg accordg to ther ow expectatos. Bettg wth the par-mutuel also volves bettg at odds whch are determed oly at the tme the race begs, ad thus bettg wth the par-mutuel does ot a pror mprove ther expected returs. Off-course bettors could therefore crease ther profts oly by collectg more data ad bettg at fxed odds ether ocourse or wth off-course bookes shortly before race tme; however t seems that for most bettors the costs assocated wth dog so exceed the gas. Although SP bettg s ot a uque Nash equlbrum the model, t s evdetly preferred from the stadpot of the off-course bookmakers. I addto, the hstorcal codtos have also favored the emergece of SP bettg as the commo bettg prce. 9 Oce t became accepted, the ts propertes as a equlbrum soluto expla ts persstece ad robustess over tme. C. Emprcal hypotheses, data ad tests. Emprcal Hypotheses The theoretcal model developed secto B predcts that off-course bookes wll tervee o-course markets whe horses whch are heavly backed by offcourse bettors receve a lower prce the o-course market. Our model mples that the off-course bookmakers are most lkely to tervee the o-course market close to the tme that t closes, by plugg those horses that are, terms of ther proft cosderatos, uder-prced ad, thereby, rase ther prces. Sce off-course bookmakers who offer SP exst the UK but ot Australa, ad sce the two markets are smlar most of ther other features, ths leads to the followg testable hypotheses: Hypothess : OPs the UK wll be hgher tha elsewhere, but SPs the UK should ot be sgfcatly dfferet tha other markets wth a smlar level of competto. Ths motvato for ths hypothess comes from Sh (99, 992 ad 993) ad Schytzer ad Shloy (2005) who show that o-course bookmakers crease opeg prces (OPs) respose to a crease the rate of pluges. As the latter 9 See for example: 2

13 paper shows, by rasg the opeg prces the bookmakers gve themselves a greater marg for adjustg prces respose to pluges. It s also show that startg prces are mostly determed by competto cosderatos ad that ther level s therefore depedet of the threat of pluges. Hypothess 2 cosders the effect of terveto by off-course bookmakers o the favorte-logshot bas. Sh (99, 992, 994), Vaugha-Wllams ad Pato (997), Schytzer ad Shloy (2003, 2005) ad Bruce ad Johso (2005) show that bookmakers ted to crease ot oly ther OP respose to the threat of pluges, but that the threat of pluges also makes them accetuate the favorte-logshot bas. We therefore predct that the UK market wll exhbt a greater extet of favortelogshot bas ts opeg prces tha the Australa markets. Moreover, wthout the terveto of off-course bookmakers, the actos of bettors wth superor sde formato should reduce the favorte-logshot bas by the tme that the bettg perod closes (Crafts, 985, Hurley ad McDoough, 995, Sauer, 998, Gadar et al., 200). However, f there s a group of bettors (such as off-course bookmakers) whch pluges horses accordace wth the bettg patters of uformed bettors, the they wll couter the effect of the formed traders. We therefore predct that the UK market should preset a hgher level of favorte-logshot bas tha other markets. Ths hypothess s supported by the recet fdgs of Johso ad Sug (2006) who report that the favorte-logshot bas the UK s suffcetly severe to permt the exstece of proftable wagerg rules eve at SP. I addto, the model predcts that off-course bookmakers have a cetve to tervee the o-course market whe the prces of ruers that are favored by off-course bettors drop that market. I such cases, the off-course bookmakers have a cetve to pluge those horses ad force the o-course bookmakers to rase the relevat prces. 20 If ths hypothess s true, the we should fd that horses whose prce fell ad the creased the late stages of the bettg, should be over-prced relatve to ther true wg probabltes. Ad there wll be other horses whose prces fell cosequece, to the pot where they are ow uder-prced. If, o the other had, the markets are effcet ad the chages prces arse cosequece of ew relevat formato, the we should fd that the startg prces of all ruers 20 Dowe (976, p. 40), metos that: "staces of mafestly mperfect equlbrum are the focus of complats by both smaller off-course bookmakers ad puters." 3

14 should be the best predctors of performace. We are therefore led to the followg hypothess: Hypothess 3: Ruers whose prces fell ad the rose the UK market should have a startg prce that s too hgh relatve to ther actual performace. I cotrast, horses whose prces rose ad the fell should have startg prces low relatve to actual performaces. We do ot expect to fd a smlar patter markets wthout offcourse SP bettg. 2. Data Our testg procedure volves comparg the UK o-course bookmakers market wth smlar data collected from the Australa o-course bettg market. The dfferece betwee these markets s that SP bettg s llegal Australa ad offcourse bettg s domated by the par-mutuel. 2 I most other respects, the two markets are smlar, wth off-course bookmakers both coutres offerg the same types of bets. I addto, the two o-course markets share may commo features: I both markets there are about the same umber of bookmakers who compete amog themselves ad wth the par-mutuel. Pror research reveals that both markets there s smlar actvty o the part of sders (Schytzer ad Shloy, 995, Vaugha- Wllams ad Pato, 997, Gadar et al., 200, Bruce ad Johso, 2005). Further, ulke, for example, the Hog Kog market, both the Australa ad UK bettg markets reveal the same patters of bases, cludg the favorte-logshot bas (Busche ad Hall, 988, Sauer, 998). The database we use for the UK cotas formato o all flat races wth sx ruers or more held the UK over the etre 995 seaso, whch was provded by Tmeform. Ths data set cludes formato o 39,098 ruers that took part 3,562 races. We compare these data wth two dfferet data sets o the bookmakers market Australa that derve from the CD verso of the "Australasa Racg Ecyclopeda '98." The frst data set cotas formato o all the flat races wth sx starters or more ru at Metropolta race tracks Australa the 997/8 seaso. It cludes data o 43,056 ruers that took part 3,654 races. Sce Metropolta races ted to attract large audeces, offer large przes ad attract cosderable meda 2 Cocers have bee expressed the UK as to whether SP bettg s really far. For example: Crafts (985), p. 303 quotes a home offce report that expresses cocer over " whether puters should be ureasoably pealzed for ther gorace." 4

15 atteto, ths data set may be expected to exhbt the smallest degree of bas ad also to offer relatvely fewer opportutes for sders (Vaugha Wllams ad Pato, 997 ad Bruce ad Johso, 2005). Ths mples that for some of the tests we perform, comparg ths data set wth bets o races that took place at all the UK racetracks would be expected to yeld coservatve results. Thus, addto, we use aother data set that cludes observatos o all the flat races wth sx ruers or more ru the state of Vctora the 997/998 seaso. Ths data set s more smlar to that of the UK terms of prze moey ad the combato of large ad small tracks, although some Vctora tracks are certaly smaller ad more remote tha ay of those foud the UK. Ths s evdet from the far greater varace of prze moey relatve to the average the Vctora market as show Table. It s possbly a more atural bechmark for comparsos betwee the UK ad Australa markets tha the Metropolta market. Thus, f the exstece of SP bettg the UK s of o partcular cosequece ad the ma force that drves prces s the belefs of bettors ad the exstece of sders (Sh, 99, 992), the we would expect the UK results to fall somewhere betwee those provded by the two Australa markets. All data sets clude formato o opeg ad startg odds offered o each horse ad o ts actual place the race. For some ruers, the data sets also clude formato o ther prce at some md-pot the bettg. Ths formato s apparetly made avalable, geeral, for horses whose prce dd ot chage mootocally durg the bettg perod. That s, ths formato s gve for ruers whose prces decreased the frst stages of the bettg perod ad the creased or vce versa. 22 Table provdes summarzed data for the three markets. *** Table about here *** 2. Results: Testg Hypothess : The presece of bettors who are wllg to place large sums of moey o horses (pluges) s a rsk to bookmakers who must hedge themselves agast ths cotgecy. Uder our hypothess that off-course 22 I the full data set that cludes formato o the UK, Vctora ad Metropolta meetgs throughout Australa, there were oly 64 cases where the mddle prce was gve for horses whose mddle prce was betwee ther OP ad SP. 5

16 bookmakers the UK have a cetve to pluge horses whose prce drops ocourse durg the bettg perod, t follows that UK o-course bookmakers face a greater rsk of pluges tha do ther Australa couterparts, where off-course bookmakers are ot a source of cocer. We therefore predct that the average OP should be hgher the UK. To test ths hypothess we frst defe the Opeg Prce of horse that partcpates race j, p j, as the prce of a cotget clam to the evet that horse ws. Defg Oj as the opeg odds for ruer race j, the the opeg prce of ruer s the oe that solves: O j p j =. p j We summarze the data o OP the three data sets ad compare them by usg both a t-test ad the Ma-Whtey test. Table 2 summarzes the results: We fd that the average opeg prces of ruers the two Australa samples are sgfcatly lower tha the UK at the % level. *** Table 2 about here *** To check further that the hgher opeg prces the UK are a respose to pluges ad ot the result of less compettve markets, we compare the average SP prces all the markets. Uder the commo assumpto the lterature that o-course markets are compettve (Sh, 993, Vaugha Wllams ad Pato, 997, Schytzer ad Shloy, 2004), the startg prces should be set at a level that gves o-course bookmakers zero profts. The average level of SP should therefore be depedet of the rate of pluges. Table 3 summarzes our fdgs. The results dcate that the startg prces the UK are, o average, hgher tha the Australa Metropolta races but lower tha races held Vctora. The lower average startg prces the Australa Metropolta races may be explaed by the fact that the Australa Metropolta races are larger tha most of the races the UK ad attract hgher przes, hgher meda atteto ad a large umber of bookmakers. However, the startg prces the UK markets are lower o average tha the startg prces Vctora. Ths mples that the UK o-course bookmakers market s at least as compettve as the Vctora market. Ths may be explaed by the presece, the Vctora data set, of remote race tracks whch attract far fewer bookmakers ad whose markets are thus ot very compettve. Nevertheless, despte the more tesve 6

17 competto, the Opeg Prces the UK are, as we showed Table 2, sgfcatly hgher, whch further stregthes our asserto that they are set order to allow UK bookmakers greater maeuverablty respose to pluges tha that perceved as ecessary by Australa o-course bookmakers. *** Table 3 about here *** Testg Hypothess 2: Accordg to ths hypothess, the favorte-logshot bas wll be more proouced the UK market tha other markets. To test ths hypothess we use a smlar methodology to that employed by Gadar et al (200). 23 Ths methodology relates the returs from each ruer to the odds offered o t by the bookmakers. We defe the opeg et returs ( OR j ) from ruer race j as the et returs from placg a bet o t at the opeg prce race j. That s, the opeg returs to ruer race j equal ts opeg odds the case that t ws the race ad - otherwse. Smlarly we defe the startg et returs ( ) as the returs for placg a bet o a ruer at ts startg prce. We test for the exstece of a favorte-logshot bas by rug regressos where the explaed varables are the et opeg ad startg returs ad the explaatory varables are the relevat odds; f there s o favorte-logshot bas there should be o correlato betwee the odds set by the bookmakers ad the returs. If, o the other had, there s a favortelogshot bas the data, the there should be a egatve correlato betwee the odds ad the returs, dcatg that low probablty (hgh odds) ruers are over-prced. Sce we are terested comparg the favorte-logshot bas betwee the UK ad both the Australa Metropolta ad the Vctora data sets, we created two data sets: The frst combes the observatos from the UK wth observatos from the Australa Metropolta races ad the secod combes all the observatos from the UK ad Vctora. We defed two dummy varables, Metropolta ad Vctora for dfferetatg betwee observatos from the UK ad observatos from Metropolta races ad from races Vctora. For each data set we the ra two regressos: oe checkg for the exstece of a favorte-logshot bas opeg prces ad oe testg for the exstece of a favorte-logshot bas startg prces. To check for the dfferece betwee the UK ad the Metropolta ad Vctora races, SR j 23 We also performed the test suggested by Bruce ad Johso (2005) ad obtaed smlar results. 7

18 respectvely, we added dummy ad teracto varables to cotrol for possble shfts the tercept ad the slope. Ths gave us four equatos of the followg structure: R j = 2 + β Odds + δ Dummy + δ Dummy Odds + ε where R s ether OR or SR, Odds s the relevat odds type (opeg odds or startg odds) ad Dummy deotes ether the Metropolta dummy or the Vctora dummy, as approprate. Sce the et returs are cesored at - we use Tobt regressos ad we clustered observatos by races because the returs o horses each race are ot depedet. 24 The results for the opeg et returs are reported Table 4 ad for the startg et returs Table 5. The results support our hypothess. There s a sgfcat favorte-logshot bas the opeg prces all three data sets as dcated by the egatve coeffcet of the opeg odds all the regressos. However, the favorte-logshot bas s sgfcatly more proouced the UK, as predcted by our hypothess ad as dcated by the postve sg of the teracto coeffcet. I addto, the favorte-logshot bas s smaller all data sets at SP prces, but we fd that the smallest decrease occurs the UK. *** Table 4 about here *** *** Table 5 about here *** A alteratve explaato for the stroger favorte-logshot bas SP the UK s that sders bet less the UK, ad that the bas OP s the result of some pheomeo other tha a respose by bookmakers to the threat of pluges. To cotrol for such a possblty, we employ Sh's (993) procedure to estmate the share of sde moey the dfferet markets. Sh (993) developed a model that relates the sum of startg prces a race ad the share of bets places by sders wth superor formato. The model s based o the assumpto that sders have perfect foresght ad therefore always bet correctly. Sh (993) shows that bookmakers respod to the exstece of such superorly formed bettors a way that makes the total sum of 24 We also ra the regressos uder more geeral hetroscedastcty assumptos. The results remaed vrtually detcal. 8

19 startg prces deped o the umber of horses a race. Vaugha-Wllams ad Pato (997) also employ ths procedure ad report that t s a relable dcator of sders' actvty. We therefore employ ths procedure for the three markets wth whch we are cocered ad report the estmated share of sder tradg each of the markets our data sets. The results are reported Table 6. Our estmate of sder share the UK s of the same magtude as those of Vaugha-Wllams ad Pato (997) ad somewhat greater tha those reported by Sh (993). However our data set cludes may more races tha Sh's, whch mght expla most of the dfferece (Vaugha ad Pato, 997). Table 6 also dcates that the share of sder tradg the UK s somewhat larger tha the Australa Metropolta meetgs, whch s to be expected, gve the smaller returs for sder formato races wth relatve hgh publc ad meda atteto (Vaugha ad Pato, 997, Bruce ad Johso, 2005). Further, the share of smart moey the UK s a lttle less tha Vctora as s to be expected from the greater opportutes for sders offered rural Vctora. However, the dffereces betwee the three markets are small. Thus, t s ulkely that t s the actvty of sde traders that duces the dfferet favortelogshot bases the three markets. Ths stregthes our hypothess that the bas s duced by a force that exsts the UK but ot Australa. *** Table 6 about here *** The tests o Hypotheses ad 2 provde evdece for the exstece of a elemet that affects the UK market a way that does ot exst the Australa markets. We therefore tur to Hypothess 3, whch tests the possble effects o the prces of ruers that are see by o-course bettors (cludg the formed bettors) as havg a low probablty of wg. Our theoretcal model suggests that such cases, the off-course bookmakers may have a cetve to tervee the market ad push the prces of those ruers by plugg them. As a cosequece, the startg prce of such ruers should over-estmate ther ablty. I markets where there s o such terveto, the prces may vary radomly durg the bettg to reflect chages formato. I such markets, the startg prce should be the best predctor of outcomes, ad the patter of chages the prces should ot cota ay extra formato. More specfcally, we use the mddle prces that we have the data set 9

20 to defe two dummy varables: up_dow, whch equals f the mddle prce of a ruer s hgher tha both ts opeg ad startg prce ad dow_up, whch equals f the mddle prce of a ruer s lower tha both ts opeg ad startg prce. We the multply these dummes by the absolute value of the dfferece betwee the mddle ad startg odds, so that we have a measure of the relatoshp betwee the chage the odds ad the wg probablty of the ruer. We use these varables to estmate the followg regresso: SR j = + β Sodds + δ up _ dow abs( Sodds Modds ) + j j j j + δ 2 dow _ upj abs( Soddsj Moddsj ) + ε where Sodds are the startg odds ad Modds are the mddle odds, Ths s a smlar regresso to the oe we used to test hypothess 2; however, our ma terest here s the coeffcets, δ ad δ 2 : a egatve ad sgfcat coeffcet would allow us to reject the ull hypothess that ruers whose prces fell ad the rose durg the ocourse bettg have the same wg probabltes as other horses ther prce group. We estmate the regressos usg the Tobt estmato procedure, ad we assume that observatos are clustered by races. The results are reported Table 7, ad they are as predcted by our hypothess. Thus, Vctora ad the Australa Metropolta races, the patter of chages the prces durg the bettg perod s ot sgfcat as a predctor of the returs whe startg prces are cluded the regresso. I the UK, however, horses whose prces fell ad the rose are sgfcatly over-prced. The greater s the dfferece betwee ther mddle ad startg prces, the less lkely are those horses to w: ths s cosstet wth the assumpto that the chages the prces betwee mddle ad startg prces are the result of ew formato. It does, however, support our hypothess that off-course bookmakers try to crease the prces of ruers that are favored by off-course bettors but ot by the bettors o-course. Furthermore, horses whose prces tally rose the UK - dcatg the presece of postve sde formato, but the fell - suggestg that other horses had bee pluged by off-course bookmakers, forcg the o-course bookmakers to rase the odds of o-pluged horses a attempt to balace ther 25 j 25 abs( Sodds Modds) stads for the absolute value of the dfferece betwee the Startg ad Mddle odds. 20

21 books, were uder-prced (albet at a 0% level of sgfcace) ad thus had a creased probablty of wg. *** Table 7 about here *** D. Cocluso I ths paper we study the optmalty of SP bettg for off-course bookmakers the UK. We do so by presetg a model that combes elemets from Rotemberg ad Saloer (990) together wth Sh's (99, 992) ad Schytzer ad Shloy's (2004) models of horse bettg bookmakg markets. Rotemberg ad Saloer (990) show that whe sellers a repeated game face demad ucertaty, they may tactly collude. We thus exted ther cotrbuto by showg that ths also apples to a evromet where both sellers ad clets face ucertaty about outcomes. More specfcally, we aalyze the UK off-course bookmakg market, where most bettg s carred out at Startg Prces. We show that despte the fact that sellg bets at SP seems to volate the commo assumpto that bookmakers proft by offerg fxed odds ad matag a balaced book,.e. by adjustg odds to reflect the share of bets placed o each horse, t may be recocled wth proft maxmzato a cocetrated market, because t eables bookmakers to form a mplct cartel. Ideed, sellg bets at SP s especally sutable for coordatg mplct colluso; as Rotemberg ad Saloer (990) ote, order to mata tact colluso sellers have to set easy to follow prces ad mata them over log perods. Startg prces clearly comply wth ths rule because, by defto, they are a fxed referece prce whch caot be chaged drectly by the actos of a sgle off-course bookmaker. Our model mples that ths tact colluso creases the profts of the offcourse bookmakers by allowg them to fluece the prces the o-course market ther favor. We test some emprcal mplcatos of the model by comparg the UK market wth smlar o-course bookmakg markets Australa, the dfferece beg that Australa SP bettg s llegal. We fd evdece that the UK market s dfferet from the Australa markets ways that are cosstet wth the mplcatos of our model. Especally, we fd evdece that UK markets are less effcet ways that may be explaed by the exstece of a large bettor (or a few large bettors) that make prces the market overstate the wg probablty of ruers that o-course 2

22 bettors back less tha merted by the fal, pre-race prces. Ths evdece strogly supports our hypothess that the UK market s dfferet from other markets owg to the terveto of large off-course bookmakers who offer SP bettg. Ths also leads the UK bookmakg market to be less effcet tha other markets, the sese that prces covey less accurate formato tha Australa. Thus, the terveto of the off-course bookmakers may be agast the terest of the bettors, gve that bettg s a zero-sum game. Our fdgs are cosstet wth other recet fdgs the lterature o the UK bookmakg market that show that the bases that market are larger tha other markets ad mght be large eough to volate, practcal terms, weak-form effcecy. For example, Gabrel ad Marsde (990) report that bettg wth the Tote (par-mutuel) gves bettors cosstetly hgher reveues tha bettg wth bookmakers at SP. Johso ad Sug (2006) also report that the UK market s less effcet tha other bettg markets, ad that the bases SP may be large eough to allow postve expected profts. These fdg are cosstet wth the assumpto that SP represet equlbrum prces (Dowe, 976), but our fdgs suggest a possble explaato for ths seemg aomaly; amely, the exstece of large bettors who bet proporto to bets placed by the uformed publc. A mportat caveat s that our fdgs are for the 995 UK seaso. It s possble that chages the market structure that have occurred sce of the troducto of o-le bettg ad the fact that SP bettg s curretly losg some of ts former popularty may have lesseed the motvato for off-course bookmakers tactly to collude See for example: 22

23 E. Refereces: Boreste Sever ad Adrea Shepard (996), "Dyamc Prcg Retal Gasole Markets," 27,3: Bruce, Alstar C. ad Johe E.V. Johso (2005), "Market Ecology ad Decso Behavor State-Cotget Clams Markets," Joural of Ecoomc Behavor ad Orgazato 56: Busche, Kelly ad Chrstopher Hall (988), "A Excepto to the Rsk Preferece Aomaly," Joural of Busess 6: Carlto, D.W. ad J. M. Perloff (994), Moder Idustral Orgazato, 3 rd edto, (New York, NY: Harper-Colls-College-Publshers). Crafts N.F.R (985), "Some Evdece of Isder Kowledge Horse Race Bettg Brta," Ecoomca 52,207: Dowe, Jack (976), "O the Effcecy ad Equty of Bettg Markets," Ecoomca 43,70: Fglewsk, Stephe (979), "Subjectve Iformato ad Market Effcecy a Bettg Market," Joural of Poltcal Ecoomy 87,: Gabrel, Paul E. ad Marsde James R. (990), "A Examato of Market Effcecy Brtsh Racetrack Bettg," Joural of Poltcal Ecoomy 98,4: Gadar, Joh, M., Rchard A. Zuber ad R. Stafford Johso (200), "Searchg for the Favourte-Logshot Bas Dow Uder: A Examato of the New-Zealad Par- Mutuel Bettg Market," Appled Ecoomcs 33: Hurley, Wllams ad Lawrece McDoough (995), "A Note o the Hayek Hypothess ad the Favorte Log Shot Bas Par-mutuel Bettg," Amerca Ecoomc Revew 85,4: Johso, Johe E.V. ad Sug (2006), "Revealg Weak Form Ieffcecy a Market for State Cotget Clams," Workg Paper. Lee, Marcus ad Gary Smth (2002), "Regresso to the Mea ad Football Wagers," Joural of Behavoral Decso Makg 5: Levtt, Steve D. (2004), "Why are Gamblg Markets Orgazed so Dfferetly from Facal Markets?" Ecoomc Joural 4:

24 Rotemberg, ad Saloer (990), "Collusve Prce Leadershp," joural of Idustral Ecoomcs 39,: 93-. Sauer, Raymod D. (998), "The Ecoomcs of Wagerg Markets," Joural of Ecoomc Lterature 36,4: Schytzer, Ad ad Yuval Shloy (995),"Isde Iformato a Bettg Market," Ecoomc Joural 05: Schytzer, Ad ad Yuval Shloy (2003), "Is the Presece of Isde Traders Necessary to Gve Rse to a Favorte-Logshot Bas?" L. Vaugha Wllams (ed.) The Ecoomcs of Gamblg, Routledge, 4-7. Schytzer, Ad ad Yuval Shloy (2005), "Isder-Tradg ad Bas a Market for State-Cotget Clams," L. Vaugh Wllams (ed.) Iformato Effcecy Facal ad Bettg Markets, Cambrdge Uversty Press, Sh, Hyu Sog (993), "Measurg the Icdece of Isder Tradg a Market for State-Cotget Clams," Ecoomc Joural 03,420: Sh, Hyu Sog (992), "Prces of State Cotget Clams wth Isder Traders ad Favorte-Logshot Bas," Ecoomc Joural 02,4: Sh, Hyu Sog (99), "Optmal Bettg Odds agast Isder Traders," Ecoomc Joural 0,408: Slade, Margaret E. (992), "Vacouver's Gasole Prce Wars: A Emprcal Exercse Ucoverg Supergame Strateges," Revew of Ecoomc Studes 59,2: Strumpf, Kolema S. (2003), "Illegal Sports Bookmakers," Workg Paper. Thaler, Rchard H. ad Wllams T. Zemba (988), "Aomales: Par-mutuel Bettg Markets: Racetrack ad Lotteres," Joural of Ecoomc Perspectve 2,2:6-74. Vaugha Wllams, Leghto ad Davd Pato (997), "Why s there a favorte- Logshot Bas Brtsh Racetrack Bettg Markets?" Ecoomc Joural 07,440:

25 Number of horses Table : Summary Statstcs Number of races UK 39,098 3,562 Metropolta 43,056 3,654 Vctora 44,36 3,98 Average Prze Moey 6,255 (4,735) 4,409 (37,33) 9400 (35,625) Meda Number of Ruers Table : Number of horses s the total umber of starters. The average prze moey s gve UK pouds (For observatos o races ru Australa we coverted the prze moey to UK pouds at the the rulg exchage rate betwee the Australa dollar ad the UK poud, whch averaged 0.47 /Aus$. Stadard devatos are parethess

26 Table 2: Average Opeg Prces Metropolta Vctora UK AVERAGE OP S.D Whtey-Ma 5.52*** 6.854*** t-test 8.73*** 3.9*** Table 2: Comparg Opeg Prces (OP) the UK ad Australa. The colum marked as Metropolta gves the results for ruers all the Metropolta meetgs Australa the 997/8 seaso. The Vctora colum gves the results for ruers partcpatg all races held Vctora durg the 997/998 seaso. The UK colum s for all the ruers the UK durg the 995 seaso. The le marked as Whtey-Ma gves the value of the Whtey Ma test for comparg the relevat data set wth the data set o the UK races. The t-test le gves the value of the t-test for equal meas the UK ad the relevat Australa data set. ***- sgfcat at %. 26

27 Table 3: Average Startg Prces Metropolta Vctora UK AVERAGE SP S.D Whtey-Ma 5.52*** -3.85*** t-test.38*** -0.2*** Table 3: Comparg Startg Prces (SP) the UK ad Australa. The colum marked as Metropolta gves the results for ruers all the Metropolta meetgs Australa the 997/8 seaso. The Vctora colum gves the results for ruers partcpatg races held Vctora the 998 seaso. The UK colum s for all the ruers the UK the 995 seaso. The le marked as Whtey-Ma gves the value of the Whtey Ma test for comparg the relevat data set wth the UK. The t-test le gves the value of the t- test for equal meas the UK ad the relevat Australa data set. ***- sgfcat at %. 27

28 Table 4: Favorte-Logshot bas at OP Vctora Opeg Odds *** (0.023) Metropolta -0.89*** (0.034) Opeg Odds Vctora dummy 0.222*** (0.023) Vctora dummy -.65*** (0.238) Opeg Odds Metropolta dummy 0.325*** (0.069) Metropolta dummy -2.4*** Costat -.5*** (0.506) *** (0.26) (0.395) 2 χ *** *** Table 4: The results of Tobt models estmatg the favorte-logshot bas the three markets. Depedet varable: the returs to each horse o each race calculated accordg to ts Opeg Prces (OP). The Vctora colum gves the results of a regresso whch cludes observatos from the UK ad Vctora. The Metropolta colum gves the results of a regresso o all observatos from the UK ad the Australa Metropolta races. Opeg Odds Vctora (Metropolta) s a teracto varable that measures the margal effect of a chage the odds Vctora (Metropolta) as compared wth the effect the UK. Stadard devatos are parethess. ***- sgfcat at %. 28