SP Bettng as a Self-Enforcng Implct Cartel by Ad Schnytzer and Avcha Snr Department of Economcs Bar-Ilan Unversty Ramat Gan Israel 52800 e-mal: schnyta@mal.bu.ac.l snrav@mal.bu.ac.l Abstract A large share of the UK off-course horse racng bettng market nvolves wnnng payouts determned at Startng Prces (SP). Ths mples that gamblers can bet wth off-course bookes on any horse before a race at the fnal pre-race odds as set by on-course bookes for that horse. Gven the olgopolstc structure of the off-course gamblng market n the UK, a market that s domnated by a small number of large bookmakng frms, we study the phenomenon of SP as a type of self-enforcng mplct colluson. We show that gven the uncertanty about a race outcome, and ther ablty to nfluence the prces set by on-course bookes, agreeng to lay bets at SP s superor for off-course bookes as compared wth offerng fxed odds. We thus extend the results of Rotemberg and Saloner (1990) to markets wth uncertanty about both demand and outcomes, We test our model by studyng the predcted effects of SP bettng on the behavor of on-course bookes. Usng data drawn from both the UK and Australan on-course bettng markets, we show that the dfferences between these markets are consstent wth the predcted effects of SP bettng n the UK off-course market and ts absence from the Australan market. Saturday, November 10, 2007 1
A. Introducton 1 It s often argued that horse bettng markets share many common features wth fnancal markets (Thaler and Zemba, 1988, Shn, 1992, 1993, Vaughan Wllams and Paton, 1997, among others). In both markets, well nformed partcpants trade rsky assets, and ntermedares play an mportant role n settng prces (Shn, 1992). At the same tme, unlke more complcated fnancal markets, bettng markets are bounded n tme and yeld a sngle and unequvocal outcome. Thus, bettng markets "provde a clear vew of prcng ssues whch are more complcated elsewhere" (Sauer, 1998, p. 2021). In ths paper we study the UK market for SP bettng on horse races. In the UK, bettors may place bets wth bookmakers both at the track (on-course bookmakers) and n bookmakers' offces around the country (off-course bookmakers). Most of the offcourse bettng s carred out at Startng Prces (SP), 2 whch are the odds offered by on-course bookmakers at the end of the bettng perod. Thus bettors who bet at SP are not only gamblng on the outcome of the race, but also on the returns. 3 SP bettng also seems to be rsky for off-course bookmakers: Whereas on-course bookmakers hedge aganst the contngency of havng a large share of the bets placed on hgh odds horses by contnually adjustng odds, off-course bookmakers do not have drect control over SP. Offerng bets at SP thus seems to be nconsstent wth common models that vew the bookmakers' target functon as ether guaranteeng proft at mnmum rsk or maxmzng profts by outperformng bettors (Crafts, 1985, Vaughan Wllams and Paton, 1997, Lee and Smth, 2002, Levtt, 2003, Strumpf, 2003) To explan why SP bettng s nevertheless the preferred form of bettng n the UK, we develop a model that focuses on the concentrated structure of the UK offcourse market. In our model there are two types of bookmakers: Large off-course bookmakers wth the ablty to nfluence market outcomes and small on-course bookmakers who operate n compettve markets. Our results mply that n equlbrum, the off-course bookmakers may choose to collaborate mplctly and sell bets at SP. Owng to the tact colluson, SP bettng allows the off-course bookmakers 1 The authors wsh to thank Arthur Fshman, Han Guowen and Danel Levy for ther helpful comments. 2 Dowe (1976), for example, estmates that 95% of the off-course bettng are made at SP. 3 Interestngly, J.K. Rowlng satrzes SP bettng n "Harry Potter and the Goblet of Fre." There, a bookmaker offers bets wthout namng the odds. and only promses to gve "excellent odds" later. The negatve atttude towards SP bettng contnues when the bookmaker cheats and pays wnners wth fake money. 2
to manpulate the on-course prces and thus ncreases ther expected profts. 4 Our analyss thus extends the results of Rotemberg and Saloner (1990) who show that sellers who face demand uncertanty have an ncentve to collude n stuatons where both the clents and the sellers who face uncertanty. We test our emprcal hypotheses by comparng the UK bookmakers market wth the Australan market. These two markets share many common features, wth the excepton that SP bettng s llegal n Australa. We show that the data support our theoretcal predctons n suggestng that the UK horse racng market s less effcent than the Australan market n predctable ways. Our fndngs are strengthened by recent fndngs on neffcences n the UK horse racng market (see for example: Gabrel and Marsden, 1990, Vaughan Wllams and Paton, 1997, Bruce and Johnson, 2005, Johnson and Sung, 2006). Our results may also contrbute to the study of tact colluson n concentrated markets (Rotemberg and Saloner, 1990, Slade, 1992, Borensten and Shepard, 1996). B. Market Structure Before developng our model, we brefly descrbe the structure of the UK and Australan horse bettng markets. In the UK, bettors may place bets wth both the par-mutuel or wth on- and off- course bookmakers. The par-mutuel market normally opens for bets around 24 hours before races begn, and the odds are settled accordng to the fnal sums placed on each horse. 5 The off-course market s controlled by a small number of bookmakng frms, who operate a large number of offces throughout the country. 6 These outlets normally open for bettng (lke the par-mutuel market) around 24 hours before each race. 7 They offer bets manly at SP, whch are the fnal odds offered by on-course bookmakers and are thus unknown at the tme the bet s made. 4 Prevous authors have already noted that off-course bookmakers may ntervene n the on-course markets. Dowe (1976, p. 140), for example, notes that: "actual SP s thus determned by both off- and on- course actvty." 5 For more on par-mutuel markets see: Thaler and Zemba (1988), Sauer (1998) and Gandar et al. (2001), among others. 6 Accordng to the nternet ste Wkpeda, there are four man off-course bookmakng frms n the UK. In 2004, those four frms operated over 8,000 bettng offces (en.wkpeda.org/wk/bookmaker), and had a consderable turnover and market power; Wllam Hll, for example, had a total turnover of over 8 bllon n 2004 (http://mranda.hemscott.com/r/wmh/pdf/p_and_l2004.pdf). 7 The off-course bookmakers also offer bettng at fxed odds. However for most races, they open for bets at fxed odds only around half an hour before the races start. The only exceptons are very mportant races; n such cases fxed odds are sometmes offered even months before the race. These 3
Unlke the concentrated off-course bettng market, the on-course market s normally far more compettve, wth 10 50 on-course bookmakers at each track. The on-course bookmakers open for bettng about twenty mnutes to half an hour before each race, and they offer bettng at fxed odds. The frst set of odds they post at the tme they open for bets are known as Openng Prces (OP). As the market progresses, they often change odds n order to manage ther exposure. 8 In partcular, bookmakers often shorten the odds on horses that are plunged. A plunge occurs when a large bet on a horse s made smultaneously wth many bookmakers; past research shows that plunges usually ndcate the presence of bettors wth superor nformaton (nsders), such as horse owners and traners (Shn 1991, 1992, 1994 and Schnytzer and Shlony 1995, 2003, 2005). The last set of prces that the bookmakers post before the market closes are known as the Startng Prces (SP). As noted above, these are also the set of prces that determne the payouts for most of the bets lad by the off-course bookmakers. The Australan on-course market s structured n the same way as the UK market. The dfferences occur off-course, where SP bettng s llegal and bets are made ether at fxed odds offered on selected races, more or less as n the UK, or wth varous par-mutuels. Fnally, there are on-lne bookmakers who offer bettng at parmutuel odds. 9 C. The model 1, Assumptons Our model combnes elements from Shn (1992, 1993), Rotemberg and Saloner (1990) and Schnytzer and Shlony (2005). It descrbes a market for bets on a exceptons notwthstandng, the vast majorty of bettors who bet wth the off-course bookmakers choose to do so at SP. 8 In over 42,000 observatons that we have n our UK data set, fewer than 8.5% of the runners mantaned a constant prce from the tme the market opened and untl t closed. 9 Even n 2007, when there are legal on-lne bookmakers n Australa, SP bettng remans llegal and, for most races, the bookmakers offer the odds whch are determned by the varous par-mutuels operatng n the dfferent states. These bookmakers compete by permttng bettors to choose accordng to whch par-mutuel they wsh to be pad f successful or by offerng to pay the maxmum dvdend among the par-mutuels. Thus, to the extent that Australan off-course bookes themselves bet anywhere t wll be wth the par-mutuel. It s nterestng to note that UK off-course bookmakers do not compete by offerng dscounts from SP, although loss rebates are occasonally employed. 4
horse race wth N horses, labeled by the set {1,2 n }. Each horse, { 1,2... n) has (unknown) probablty p of wnnng, such that p = 1 and 1 p > 0. Snce our focus s on the behavor of off-course bookmakers, we defne the settng as follows. Frst, there are two off-course bookmakers, dentfed as booke 1 and booke 2. Each of the bookes offers a set of N tckets that are labeled 1,2 n. Each booke N = 1 > chooses the prce of each tcket, such that the prce of tcket { 1,2...,n} 10 sold by booke j {1,2} s π where 0 π < 1. The th tcket costs 1 and pays j < j 1 n the π j event that horse wns the race and zero otherwse. 11 We show below that n equlbrum, both bookes set SP prces rather than offer tckets at fxed odds. We assume that the goal of the off-course bookes s to maxmze ther expected profts and that they are rsk averse n the followng sense: Assumpton 1 (Rsk Averson): The bookes always mantan a balanced book, such that for every tcket { 1,2... n} the expected contngent payout s less than or equal to the total sum accumulated by the booke. Denotng by T j the total sum of money wagered by all bettors on horse wth booke j, ths mples that: {1,2... n} j {1,2} T j ( p ) E Tj π n T j = 1, 12 where E denotes expectatons. We assume the demand for each of the tckets s gven by: α j β = α 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 α j s the demand for horse at booke j f booke j's prce s equal to the expected wnnng probablty, β s the margnal change n the demand as a result k (1) 10 Shn (1992) nvestgates the case where a horse has a probablty of 1 of wnnng. However, n such cases bookmakers normally refuse to take bets. As our nterest s essentally emprcal, we do not consder such cases. 11 Normally, bookmakers post prces n terms of odds rather than prces. The relatonshp between the 1 π, t odds and the prces s gven by: O, t = where O, t s the odds offered on horse n race t and π, t π,t s ts prce. 12 Ths assumpton s often made n the lterature. For example, Crafts (1985, p. 295) assumes that bookmakers goal s to have a "perfect book," meanng that "the bookmaker make[s] money whchever horse wns." 5
of a change n the prce and E off denotes the expectaton formed by off-course bettors based on all publc nformaton. We assume that all the off-course bettors have the same set of nformaton and smlar expectatons about the prospects of runners. Ths leads us to follow Rotemberg and Saloner (1990) and assume that f the bookes offer equal prces they expect equal demand. That s, we assume that αj ~ N ( α, σ ) and that, n the case of equal prces, demand vares between them only by a random component wth equal devatons. We assume further that the off-course bookes have the same set of nformaton as the bettors. Therefore, they also have the same expectatons as off-course bettors. 13 Snce expected demand depends on posted prces, the two bookes face Bertrand-type competton (Carlton and Perloff, 1994), where bettors buy tckets from the cheaper booke n an amount proportonal to the dfference between the prce and the expected probablty of each runner. Note that n the case where prces equal expected probablty, the expected demand for each horse s α. Therefore, under the (strong) assumpton that we make about α j, t s possble to deduce the subjectve probablty that off-course bettors assgn to each horse by calculatng: E off ( p ) = E off αj = n αj = 1 α n = 1 α. Another set of players that exst n the model s the on-course bookmakers and bettors. We assume that on-course bookmakers maxmze ther expected profts and that the on-course bettng market s effcent, so that expected profts are zero. Ths mples that the on-course bookmakers behave n a way that s smlar to that modeled by Schnytzer and Shlony (2005) and Shn (1992). Especally, ths mples that oncourse bookmakers know that there may be nsders, and protect ther profts by ncreasng the prces of low-probablty runners when the probablty of plunges ncreases. In the emprcal secton we test some mplcatons of ths predcton. 13 Ths assumpton can be defended on the grounds that off-course bookmakers open for bets a relatvely long tme before races begn. It s well documented that prces n on-course markets often change untl the fnal moments before they close. Ths mples that unlke, for example, the US markets for bets on pont spreads (Levtt, 2003), bookmakers do not possess all the relevant nformaton untl the end of the bettng perod. 6
We further assume that bettors n the on-course market are better nformed than off-course bettors and that they bet n proporton to ther expectatons. 14 Denotng by assumpton 2: E on ( p ) the expectaton formed by on-course bettors, we arrve at Assumpton 2 (On-Course \ Off-Course Expectatons): For every runner < on ( off >, E p ) = E ( p ) and E p ) has greater predctve power than E off p ). on ( Assumpton 2 s justfed on both theoretc and emprcal grounds: Theoretcally, f one assumes that the man goal of the bettors s to proft, then comng to the racetrack nvolves hgher costs than bettng off-course. Thus, bettors travel to the track only f they could gan addtonal nformaton there, or f they were determned to explot nsde nformaton at fxed odds n a compettve settng, or both. From an emprcal stand pont, on-course bettors have a better opportunty to nspect the condton of the runners, and also to observe the actons of nsders. The fndngs n the lterature also support the vew that on-course bettors out perform offcourse bettors (Fglewsk, 1979; Schnytzer and Shlony, 1995; Gandar et al., 2001). Our remanng two assumptons concern the relatonshp between the offcourse and on-course markets: Assumpton 3 (Off-Course Bookes Interventon): Off course bookes can change the prces n the on-course market by plungng runners. As descrbed above, on course bookmakers respond to plunges by ncreasng the prce of the plunged horse n order to avod loses n the case that the plunges horses wn. Off-course bookmakers can take advantage of ths and because the off course market s much larger than the oncourse market. Off-course bookmakers can thus plunge horses on-course wthout offsettng ther books sgnfcantly. Moreover, n the case that they receve large bets on horses wth a low on-course prce, off-course bookes can hedge some of ther rsk by bettng on those horses, and thus dong so s almost always proftable. The last assumpton concerns the ablty of off-course bookmakers to observe the SP prces before the close of bettng. In order to smplfy the model we make the followng assumpton: ( 14 Ths s smlar to the assumpton made by Schnytzer and Shlony (2005) where they assume that there s a herarchy of nformaton. At the top of the herarchy there are bettors wth nsde nformaton and bet on-course, and at the bottom there s the general gamblng publc. 7
Assumpton 4 (Observng SP): The off-course bookmakers can observe the equlbrum prces n the on-course market just before the close of bettng and thereby mply startng prces. Once they compare these near fnal equlbrum prces wth ther desred SP, they can nfluence prces by plungng horses before the close of bettng. 15 We therefore model the sequence of events n as follows: Frst, the off-course market s opened and the off-course bookes set ther prces for each tcket. 16 In the second stage, off-course bettors place ther bets by buyng tckets from the offcourse bookes at the prce offered, accordng to the demand functon gven by (1) and ther expectatons about outcomes. In the thrd stage, equlbrum prces are reached n the on-course market and are observed by the off-course bookes. In the fnal stage the off-course bookes may plunge horses n order to nfluence SP. 2. Soluton Each booke sells tckets 1,2 n at a prce of π, and pays out unt of tcket sold n the event that horse wns, an event that occurs wth 1 for each π probablty p. Hs profts on each horse are therefore gven by (1) tmes the prce of a tcket mnus the sum he pays n the event that the horse wns. Hs total profts are therefore: R j n = p { [ ]} αj β π Eoff ( p ) 1 π = 1 Snce the bookes have no nformatonal advantage over the bettors, t follows that f they try to compete by prces, the only possble equlbrum s a Bertrand equlbrum wth zero expected profts. The prce of each horse n such case s gven, (2) 15 Assumng that the off-course bookmakers do not actually know the equlbrum levels but employ stochastc lnear programmng to determne the actons accordng to expected equlbrum prces close to the end of bettng would not change the results, so long as the nose n ther predctons was not too great. 16 We assume that off-course bookmakers offer only one type of asset, although n realty they may offer a mxture of assets. For example, off-course bookmakers normally open for bets at fxed odds about twenty mnutes from the start of each race. However, at that tme most bettors bet at fxed odds and not at SP, and n addton the SP market s many tmes larger than the off-course fxed-odds market. 8
under our assumptons about the dstrbuton of α j, by: j = {1,2} and expected demand s αj α π = Eoff ( p ) = Eoff =, n n αj α = 1 = 1 E α ) = α. ( j However, n ths model, ths s not a unque equlbrum. Assume (wthout loss of generalty) that booke 1 declares that he would pay bettors accordng to SP as set n the on-course market, and that booke 2 follows. Note that we defne the SP of horse n a smlar fashon to our defnton of π ;.e. the startng prce of horse s the prce of a contngent clam to 1 n the event that horse wns the race. We show that ths s a stable equlbrum. To see that ths s ndeed an equlbrum outcome, frst assume that for every horse n 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 demand for each booke agan equals profts for each booke are gven by: R n n n = = j j 1 j E j 1 = E = n = 1 π = 1 π = 1 E( π ) = 1 E( α ) = α and expected 2 α E( π ) α α p α π α P α (3) α where E denotes the expectatons formed by the bookes. Assumpton 4 states that each off-course booke knows the equlbrum prce set n the on-course market before t closes. We denote the equlbrum prce of horse that s observed before the close of the market by e SP j n. When the off-course booke observes e SP he has two possble cases to consder: α SP α α α e = 1 Frst, SP 0. (4) n = 1 α SP e e n n = 1 α 2 In ths case, the on-course bettors vew the horse as at least as hot a favorte as the offcourse bettors. In consequence, the off-course bookes make postve expected profts from adoptng SP as the quoted prce. 9
In the other case: SP e < α n = 1 α e α SP SP e n = 1 n = 1 α α α 2 < 0. (5) Here, the off-course bookes have negatve expected gans from the horse. Moreover, n ths case, ther books are not balanced and ths breaches Assumpton 1. However, each booke knows that he can nfluence SP by plungng. Snce the other booke faces, n expectaton, the same demand for that horse, each booke predcts that f (5) holds, both of them would plunge the horse to the pont where ts prce α equals n. 17 If, n response, the on-course bookmakers decrease the prce of any α = 1 other horse j so that ts prce falls below α n = 1 j α, then the off-course bookmakers respond n the same way wth regard to that horse as well, so that eventually, were there suffcent tme remanng before the start of the race for complete adjustment to equlbrum, the prce of all runners would be ether greater than or equal to the share of bets placed on them n the off-course market. Now, snce on-course bookes presumably only change prces when there s suffcent tme remanng for further bets, t s reasonable to assume that no horse s prce wll fall suffcently n response to an off-course bookmaker s plunge, on a dfferent horse, to necesstate a further plunge for whch there s nsuffcent tme. As a consequence, under Assumptons (1) (4), sellng SP bets allows the bookes to make non-negatve expected profts on each horse, whereas he would make zero expected profts f he offers bets at fxed odds. The ntuton that drves ths result s smple: By deferrng the decson to the on-course market, the off-course bookes can make expected profts on horses that off-course bettors favor more than on-course bettors. As long as they collude tactly to offer SP, they can also use ther market power to hedge aganst expected losses on horses that on-course bettors favor more than off-course bettors. 17 Note that plungng the horse helps the bookes to hedge ther own rsks, so plungng does not create a publc good dlemma. 10
To prove that SP prcng s ndeed an equlbrum, t remans to be shown that the bookes cannot gan from devatng, and that off-course bettors ndeed take SP to equal expected prces. Frst we show that the bookes cannot gan from devatng. Assume that booke 1 announces hs prces to be SP, and booke 2 consders devatng. Ths can only be proftable to booke 2 f he sets hs prce to be below the expected SP prce. Assume that booke 2 announces that for tcket, the prce s set below ts expected prce, so that t equals: α n ε, ε > 0. In ths case the demand for horse would α equal = 1 α α j + αk b ε E n off ( p ) > α j. α = 1 At the same tme, f he sets the prce below the expected probablty, t follows that: E off p 1 < 0. π Substtutng ths result n the booke's proft functon (equaton (2)), t follows that the expected profts of booke 2 are negatve and hs book s not balanced. Ths breaches Assumpton 1, and therefore cannot hold n equlbrum. Ths s true for the prce of any horse, and therefore booke 2 cannot gan from devatng. 18 Consderng the bettors, beng ratonal, they know the ncentves of the bookes. They therefore know that, n equlbrum: SP = E on ( p ) = E off α n = 1 α ( E on ( p )) = α n = 1 α f E on ( p ) E off otherwse ( p ) = Snce off-course bettors use all the nformaton avalable to them, the best expectatons they can form s to expect the on-course bettors to have the same α n = 1 α 18 Note that even f booke 2 s nsenstve to breachng Assumpton 1, booke 1 would fnd that he earns no money on horse and that he can therefore mprove hs stuaton by sellng tcket at a lower prce: and therefore even f booke 2 agrees to ncrease hs personal rsk, the stuaton s not an equlbrum. 11
expectatons. It thus follows that the off-course bettors expect SP prces to equal ther expectaton. Thus, although they know the ncentves of the off-course bookmakers, bettng at SP prces s as worthwhle for them as bettng accordng to ther own expectatons. Bettng wth the par-mutuel also nvolves bettng at odds whch are determned only at the tme the race begns, and thus bettng wth the par-mutuel does not a pror mprove ther expected returns. Off-course bettors could therefore ncrease ther profts only by collectng more data and bettng at fxed odds ether oncourse or wth off-course bookes shortly before race tme; however t seems that for most bettors the costs assocated wth dong so exceed the gans. Although SP bettng s not a unque Nash equlbrum n the model, t s evdently preferred from the standpont of the off-course bookmakers. In addton, the hstorcal condtons have also favored the emergence of SP bettng as the common bettng prce. 19 Once t became accepted, then ts propertes as an equlbrum soluton explan ts persstence and robustness over tme. C. Emprcal hypotheses, data and tests 1. Emprcal Hypotheses The theoretcal model developed n secton B predcts that off-course bookes wll ntervene n on-course markets when horses whch are heavly backed by offcourse bettors receve a lower prce n the on-course market. Our model mples that the off-course bookmakers are most lkely to ntervene n the on-course market close to the tme that t closes, by plungng those horses that are, n terms of ther proft consderatons, under-prced and, thereby, rase ther prces. Snce off-course bookmakers who offer SP exst n the UK but not n Australa, and snce the two markets are smlar n most of ther other features, ths leads to the followng testable hypotheses: Hypothess 1: OPs n the UK wll be hgher than elsewhere, but SPs n the UK should not be sgnfcantly dfferent than n other markets wth a smlar level of competton. Ths motvaton for ths hypothess comes from Shn (1991, 1992 and 1993) and Schnytzer and Shlony (2005) who show that on-course bookmakers ncrease openng prces (OPs) n response to an ncrease n the rate of plunges. As the latter 19 See for example:http://excte.gamblngtmes.com/wrters/jcoates/jcoates_wnter2001.html 12
paper shows, by rasng the openng prces the bookmakers gve themselves a greater margn for adjustng prces n response to plunges. It s also shown that startng prces are mostly determned by competton consderatons and that ther level s therefore ndependent of the threat of plunges. Our model predcts that n the UK, plunges are performed by both nformed bettors and off-course bookmakers whereas n Australa plunges are only made by nformed bettors. Thus we predct that there are more plunges n the UK and ths should drve UK on course bookmakers to offer hgher OPs. Hypothess 2 consders the effect of nterventon by off-course bookmakers on the favorte-longshot bas. Shn (1991, 1992, 1994), Vaughan-Wllams and Paton (1997), Schnytzer and Shlony (2003, 2005) and Bruce and Johnson (2005) show that bookmakers tend to ncrease not only ther OP n response to the threat of plunges, but that the threat of plunges also makes them accentuate the favorte-longshot bas. We therefore predct that because our model predcts that there should be more plunges n the UK, the UK should also exhbt a greater extent of favorte-longshot bas n ts openng prces than the Australan markets. Moreover, wthout the nterventon of off-course bookmakers, the actons of bettors wth superor nsde nformaton should reduce the favorte-longshot bas by the tme that the bettng perod closes (Crafts, 1985, Hurley and McDonough, 1995, Sauer, 1998, Gandar et al., 2001). However, f there s a group of bettors (such as off-course bookmakers) whch plunges horses n accordance wth the bettng patterns of unnformed bettors, then they wll counter the effect of the nformed traders. We therefore predct that the UK market should present a hgher level of favorte-longshot bas than other markets. Ths hypothess s supported by the recent fndngs of Johnson and Sung (2006) who report that the favorte-longshot bas n the UK s suffcently severe to permt the exstence of proftable wagerng rules even at SP. In addton, the model predcts that off-course bookmakers have an ncentve to ntervene n the on-course market when the prces of runners that are favored by off-course bettors drop n that market. In such cases, the off-course bookmakers have an ncentve to plunge those horses and force the on-course bookmakers to rase the relevant prces. 20 If ths hypothess s true, then we should fnd that horses whose prce fell and then ncreased n the late stages of the bettng, should be over-prced 20 Dowe (1976, p. 140), mentons that: "nstances of manfestly mperfect equlbrum are the focus of complants by both smaller off-course bookmakers and punters." 13
relatve to ther true wnnng probabltes. And there wll be other horses whose prces fell n consequence, to the pont where they are now under-prced. If, on the other hand, the markets are effcent and the changes n prces arse n consequence of new relevant nformaton, then we should fnd that the startng prces of all runners should be the best predctors of performance. We are therefore led to the followng hypothess: Hypothess 3: Runners whose prces fell and then rose n the UK market should have a startng prce that s too hgh relatve to ther actual performance. In contrast, horses whose prces rose and then fell should have startng prces low relatve to actual performances. We do not expect to fnd a smlar pattern n markets wthout offcourse SP bettng. 2. Data Our testng procedure nvolves comparng the UK on-course bookmakers market wth smlar data collected from the Australan on-course bettng market. The dfference between these markets s that SP bettng s llegal n Australa and offcourse bettng s domnated by the par-mutuel. 21 In most other respects, the two markets are smlar, wth off-course bookmakers n both countres offerng the same types of bets. In addton, the two on-course markets share many common features: In both markets there are about the same number of bookmakers who compete among themselves and wth the par-mutuel. Pror research reveals that n both markets there s smlar actvty on the part of nsders (Schnytzer and Shlony, 1995, Vaughan- Wllams and Paton, 1997, Gandar et al., 2001, Bruce and Johnson, 2005). Further, unlke, for example, the Hong Kong market, both the Australan and UK bettng markets reveal the same patterns of bases, ncludng the favorte-longshot bas (Busche and Hall, 1988, Sauer, 1998). The database we use for the UK contans nformaton on all flat races wth sx runners or more held n the UK over the entre 1995 season, whch was provded by Tmeform. Ths data set ncludes nformaton on 39,098 runners that took part n 3,562 races. We compare these data wth two dfferent data sets on the bookmakers market n Australa that derve from the CD verson of the "Australasan Racng 21 Concerns have been expressed n the UK as to whether SP bettng s really far. For example: Crafts (1985), p. 303 quotes a home offce report that expresses concern over " whether punters should be unreasonably penalzed for ther gnorance." 14
Encyclopeda '98." The frst data set contans nformaton on all the flat races wth sx starters or more run at Metropoltan race tracks n Australa n the 1997/8 season. It ncludes data on 43,056 runners that took part n 3,654 races. Snce Metropoltan races tend to attract large audences, offer large przes and attract consderable meda attenton, ths data set may be expected to exhbt the smallest degree of bas and also to offer relatvely fewer opportuntes for nsders (Vaughan Wllams and Paton, 1997 and Bruce and Johnson, 2005). Ths mples that for some of the tests we perform, comparng ths data set wth bets on races that took place at all the UK racetracks would be expected to yeld conservatve results. Thus, n addton, we use another data set that ncludes observatons on all the flat races wth sx runners or more run n the state of Vctora n the 1997/1998 season. Ths data set s more smlar to that of the UK n terms of prze money and n the combnaton of large and small tracks, although some Vctoran tracks are certanly smaller and more remote than any of those found n the UK. Ths s evdent from the far greater varance of prze money relatve to the average n the Vctoran market as shown n Table 1. It s possbly a more natural benchmark for comparsons between the UK and Australan markets than the Metropoltan market. Thus, f the exstence of SP bettng n the UK s of no partcular consequence and the man force that drves prces s the belefs of bettors and the exstence of nsders (Shn, 1991, 1992), then we would expect the UK results to fall somewhere between those provded by the two Australan markets. All data sets nclude nformaton on openng and startng odds offered on each horse and on ts actual place n the race. For some runners, the data sets also nclude nformaton on ther prce at some md-pont n the bettng. Ths nformaton s apparently made avalable, n general, for horses whose prce dd not change monotoncally durng the bettng perod. That s, ths nformaton s gven for runners whose prces decreased n the frst stages of the bettng perod and then ncreased or vce versa. 22 Table 1 provdes summarzed data for the three markets. *** Table 1 about here *** 22 In the full data set that ncludes nformaton on the UK, Vctoran and Metropoltan meetngs throughout Australa, there were only 164 cases where the mddle prce was gven for horses whose mddle prce was between ther OP and SP. 15
2. Results: Testng Hypothess 1: The presence of bettors who are wllng to place large sums of money on horses (plunges) s a rsk to bookmakers who must hedge themselves aganst ths contngency. Under our hypothess that off-course bookmakers n the UK have an ncentve to plunge horses whose prce drops oncourse durng the bettng perod, t follows that UK on-course bookmakers face a greater rsk of plunges than do ther Australan counterparts, where off-course bookmakers are not a source of concern. We therefore predct that the average OP should be hgher n the UK. To test ths hypothess we frst defne the Openng Prce of horse that partcpates n race j, p j, as the prce of a contngent clam to 1 n the event that horse wns. Defnng Oj as the openng odds for runner n race j, then the openng prce of runner s the one that solves: O j p j = 1. p j We summarze the data on OP n the three data sets and compare them by usng both a t-test and the Mann-Whtney test. Table 2 summarzes the results: We fnd that the average openng prces of runners n the two Australan samples are sgnfcantly lower than n the UK at the 1% level. *** Table 2 about here *** To check further that the hgher openng prces n the UK are a response to plunges and not the result of less compettve markets, we compare the average SP prces n all the markets. Under the common assumpton n the lterature that on-course markets are compettve (Shn, 1993, Vaughan Wllams and Paton, 1997, Schnytzer and Shlony, 2004), the startng prces should be set at a level that gves on-course bookmakers zero profts. The average level of SP should therefore be ndependent of the rate of plunges. Table 3 summarzes our fndngs. The results ndcate that the startng prces n the UK are, on average, hgher than n the Australan Metropoltan races but lower than n races held n Vctora. The lower average startng prces n the Australan Metropoltan races may be explaned by the fact that the Australan Metropoltan races are larger than most of the races n the UK and attract hgher przes, hgher meda attenton and a large number of bookmakers. However, the 16
startng prces n the UK markets are lower on average than the startng prces n Vctora. Ths mples that the UK on-course bookmakers market s at least as compettve as the Vctora market. Ths may be explaned by the presence, n the Vctoran data set, of remote race tracks whch attract far fewer bookmakers and whose markets are thus not very compettve. Nevertheless, despte the more ntensve competton, the Openng Prces n the UK are, as we showed n Table 2, sgnfcantly hgher, whch further strengthens our asserton that they are set n order to allow UK bookmakers greater maneuverablty n response to plunges than that perceved as necessary by Australan on-course bookmakers. *** Table 3 about here *** Testng Hypothess 2: Accordng to ths hypothess, the favorte-longshot bas wll be more pronounced n the UK market than n other markets. To test ths hypothess we use a smlar methodology to that employed by Gandar et al (2001). 23 Ths methodology relates the returns from each runner to the odds offered on t by the bookmakers. We defne the openng net returns ( OR j ) from runner n race j as the net returns from placng a 1 bet on t at the openng prce n race j. That s, the openng returns to runner n race j equal ts openng odds n the case that t wns the race and -1 otherwse. Smlarly we defne the startng net returns ( ) as the returns for placng a 1 bet on a runner at ts startng prce. We test for the exstence of a favorte-longshot bas by runnng regressons where the explaned varables are the net openng and startng returns and the explanatory varables are the relevant odds; f there s no favorte-longshot bas there should be no correlaton between the odds set by the bookmakers and the returns. If, on the other hand, there s a favortelongshot bas n the data, then there should be a negatve correlaton between the odds and the returns, ndcatng that low probablty (hgh odds) runners are over-prced. Snce we are nterested n comparng the favorte-longshot bas between the UK and both the Australan Metropoltan and the Vctoran data sets, we created two data sets: The frst combnes the observatons from the UK wth observatons from the Australan Metropoltan races and the second combnes all the observatons from the UK and Vctora. We defned two dummy varables, Metropoltan and Vctora for SR j 23 We also performed the test suggested by Bruce and Johnson (2005) and obtaned smlar results. 17
dfferentatng between observatons from the UK and observatons from Metropoltan races and from races n Vctora. For each data set we then ran two regressons: one checkng for the exstence of a favorte-longshot bas n openng prces and one testng for the exstence of a favorte-longshot bas n startng prces. To check for the dfference between the UK and the Metropoltan and Vctoran races, respectvely, we added dummy and nteracton varables to control for possble shfts n the ntercept and n the slope. Ths gave us four equatons of the followng structure: R j = 1 2 α + β Odds + δ Dummy + δ Dummy Odds + ε where R s ether OR or SR, Odds s the relevant odds type (openng odds or startng odds) and Dummy denotes ether the Metropoltan dummy or the Vctora dummy, as approprate. Snce the net returns are censored at -1 we use Tobt regressons and we clustered observatons by races because the returns on horses n each race are not ndependent. 24 The results for the openng net returns are reported n Table 4 and for the startng net returns n Table 5. The results support our hypothess. There s a sgnfcant favorte-longshot bas n the openng prces n all three data sets as ndcated by the negatve coeffcent of the openng odds n all the regressons. However, the favorte-longshot bas whch s captured by the slope of the regresson s sgnfcantly more pronounced n the UK, as predcted by our hypothess and as ndcated by the postve sgn of the nteracton coeffcent. In addton, the favortelongshot bas s smaller n all data sets at SP prces, but we fnd that the smallest decrease occurs n the UK. *** Table 4 about here *** *** Table 5 about here *** An alternatve explanaton for the stronger favorte-longshot bas n SP n the UK s that nsders bet less n the UK, and that the bas n OP s the result of some phenomenon other than a response by bookmakers to the threat of plunges. To control 24 We also ran the regressons under more general hetroscedastcty assumptons. The results remaned vrtually dentcal. 18
for such a possblty, we employ Shn's (1993) procedure to estmate the share of nsde money n the dfferent markets. Shn (1993) developed a model that relates the sum of startng prces n a race and the share of bets places by nsders wth superor nformaton. The model s based on the assumpton that nsders have perfect foresght and therefore always bet correctly. Shn (1993) shows that bookmakers respond to the exstence of such superorly nformed bettors n a way that makes the total sum of startng prces depend on the number of horses n a race. Vaughan-Wllams and Paton (1997) also employ ths procedure and report that t s a relable ndcator of nsders' actvty. We therefore employ ths procedure for the three markets wth whch we are concerned and report the estmated share of nsder tradng n each of the markets n our data sets. The results are reported n Table 6. Our estmate of nsder share n the UK s of the same magntude as those of Vaughan-Wllams and Paton (1997) and somewhat greater than those reported by Shn (1993). However our data set ncludes many more races than Shn's, whch mght explan most of the dfference (Vaughan and Paton, 1997). Table 6 also ndcates that the share of nsder tradng n the UK s somewhat larger than n the Australan Metropoltan meetngs, whch s to be expected, gven the smaller returns for nsder nformaton n races wth relatve hgh publc and meda attenton (Vaughan and Paton, 1997, Bruce and Johnson, 2005). Further, the share of smart money n the UK s a lttle less than n Vctora as s to be expected from the greater opportuntes for nsders offered n rural Vctora. However, the dfferences between the three markets are small. Thus, t s unlkely that t s the actvty of nsde traders that nduces the dfferent favortelongshot bases n the three markets. Ths strengthens our hypothess that the bas s nduced by a force that exsts n the UK but not n Australa. *** Table 6 about here *** The tests on Hypotheses 1 and 2 provde evdence for the exstence of an element that affects the UK market n a way that does not exst n the Australan markets. We therefore turn to Hypothess 3, whch tests the possble effects on the prces of runners that are seen by on-course bettors (ncludng the nformed bettors) as havng a low probablty of wnnng. Our theoretcal model suggests that n such cases, the off-course bookmakers may have an ncentve to ntervene n the market 19
and push the prces of those runners by plungng them. As a consequence, the startng prce of such runners should over-estmate ther ablty. In markets where there s no such nterventon, then prces may vary randomly durng the bettng to reflect changes n nformaton. In such markets, the startng prce should be the best predctor of outcomes, and the pattern of changes n the prces should not contan any extra nformaton. More specfcally, we use the mddle prces that we have n the data set to defne two dummy varables: up_down, whch equals 1 f the mddle prce of a runner s hgher than both ts openng and startng prce and down_up, whch equals 1 f the mddle prce of a runner s lower than both ts openng and startng prce. We then multply these dummes by the absolute value of the dfference between the mddle and startng odds, so that we have a measure of the relatonshp between the change n the odds and the wnnng probablty of the runner. We use these varables to estmate the followng regresson: 25 SR = α + β Sodds + δ up _ down abs( Sodds Modds ) + j j 1 j j j + δ 2 down _ upj abs( Soddsj Moddsj ) + ε where Sodds are the startng odds and Modds are the mddle odds, Ths s a smlar regresson to the one we used to test hypothess 2; however, our man nterest here s n the coeffcents, δ 1 and δ 2 : a negatve and sgnfcant coeffcent would allow us to reject the null hypothess that runners whose prces fell and then rose durng the oncourse bettng have the same wnnng probabltes as other horses n ther prce group. We estmate the regressons usng the Tobt estmaton procedure, and we assume that observatons are clustered by races. The results are reported n Table 7, and they are as predcted by our hypothess. Thus, n Vctora and n the Australan Metropoltan races, the pattern of changes n the prces durng the bettng perod s not sgnfcant as a predctor of the returns when startng prces are ncluded n the regresson. In the UK, however, horses whose prces fell and then rose are sgnfcantly over-prced. The greater s the dfference between ther mddle and startng prces, the less lkely are those horses to wn: ths s nconsstent wth the assumpton that the changes n the prces between mddle and startng prces are the 25 Followng a comment by one of the referees, we also estmated those regressons by gvng greater weght to observatons wth hgh odds by usng the logs of the prces (as n Law and Peel, 2002). Ths procedure gave smlar results, and we thus report the results of the smpler to nterpret regressons. 26 abs( Sodds Modds ) stands for the absolute value of the dfference between the Startng and Mddle odds. 26 j 20
result of new nformaton. It does, however, support our hypothess that off-course bookmakers try to ncrease the prces of runners that are favored by off-course bettors but not by the bettors on-course. Furthermore, horses whose prces ntally rose n the UK - ndcatng the presence of postve nsde nformaton, but then fell - suggestng that other horses had been plunged by off-course bookmakers, forcng the on-course bookmakers to rase the odds of non-plunged horses n an attempt to balance ther books, were under-prced (albet at a 10% level of sgnfcance) and thus had an ncreased probablty of wnnng. *** Table 7 about here *** D. Concluson In ths paper we study the optmalty of SP bettng for off-course bookmakers n the UK. We do so by presentng a model that combnes elements from Rotemberg and Saloner (1990) together wth Shn's (1991, 1992) and Schnytzer and Shlony's (2004) models of horse bettng bookmakng markets. Rotemberg and Saloner (1990) show that when sellers n a repeated game face demand uncertanty, they may tactly collude. We thus extend ther contrbuton by showng that ths also apples to an envronment where both sellers and clents face uncertanty about outcomes. More specfcally, we analyze the UK off-course bookmakng market, where most bettng s carred out at Startng Prces. We show that despte the fact that sellng bets at SP seems to volate the common assumpton that bookmakers proft by offerng fxed odds and mantanng a balanced book,.e. by adjustng odds to reflect the share of bets placed on each horse, t may be reconcled wth proft maxmzaton n a concentrated market, because t enables bookmakers to form an mplct cartel. Indeed, sellng bets at SP s especally sutable for coordnatng mplct colluson; as Rotemberg and Saloner (1990) note, n order to mantan tact colluson sellers have to set easy to follow prces and mantan them over long perods. Startng prces clearly comply wth ths rule because, by defnton, they are a fxed reference prce whch cannot be changed drectly by the actons of a sngle off-course bookmaker. Our model mples that ths tact colluson ncreases the profts of the offcourse bookmakers by allowng them to nfluence the prces n the on-course market n ther favor. We test some emprcal mplcatons of the model by comparng the UK 21
market wth smlar on-course bookmakng markets n Australa, the dfference beng that n Australa SP bettng s llegal. We fnd evdence that the UK market s dfferent from the Australan markets n ways that are consstent wth the mplcatons of our model. Especally, we fnd evdence that UK markets are less effcent n ways that may be explaned by the exstence of a large bettor (or a few large bettors) that make prces n the market overstate the wnnng probablty of runners that on-course bettors back less than merted by the fnal, pre-race prces. Ths evdence strongly supports our hypothess that the UK market s dfferent from other markets owng to the nterventon of large off-course bookmakers who offer SP bettng. Ths also leads the UK bookmakng market to be less effcent than other markets, n the sense that prces convey less accurate nformaton than n Australa. Thus, the nterventon of the off-course bookmakers may be aganst the nterest of the bettors, gven that bettng s a zero-sum game. Our fndngs are consstent wth other recent fndngs n the lterature on the UK bookmakng market that show that the bases n that market are larger than n other markets and mght be large enough to volate, n practcal terms, weak-form effcency. For example, Gabrel and Marsden (1990) report that bettng wth the Tote (par-mutuel) gves bettors consstently hgher revenues than bettng wth bookmakers at SP. Johnson and Sung (2006) also report that the UK market s less effcent than other bettng markets, and that the bases n SP may be large enough to allow postve expected profts. These fndng are nconsstent wth the assumpton that SP represent equlbrum prces (Dowe, 1976), but our fndngs suggest a possble explanaton for ths seemng anomaly; namely, the exstence of large bettors who bet n proporton to bets placed by the unnformed publc. An mportant caveat s that our fndngs are for the 1995 UK season. It s possble that changes n the market structure that have occurred snce of the ntroducton of on-lne bettng and the fact that SP bettng s currently losng some of ts former popularty may have lessened the motvaton for off-course bookmakers tactly to collude. 27 27 See for example: http://www.expertform.com/bettng.htm. 22
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