A statistical evelopment of fixe os betting ules in socce Ian Milline 1, Paul White 1 an Don J. Webbe 2 1 Depatment of Mathematics an Statistics, Univesity of the West of Englan, Bistol, UK 2 Depatment of Economics, Univesity of the West of Englan, Bistol, UK Abstact Two simple but seemingly pofitable betting ules fo betting on the away win in association football ae evelope. One ule is consistent with avoiing those games in which thee is a clea favouite. The secon ule is base iectly on moelling bookmake os an assessing the esiuals une the fitte moel. Contay to pevious eseach the betting ule using the esiuals suggests avoiing betting on those games whee thee ae lage iscepancies between bookmake os an peicte-moel os. Keywos: Fixe os betting ules; away win; bookmakes pobabilities JEL Classification: D81 Coepsoning autho: Don J Webbe, Depatment of Economics, Univesity of the West of Englan, Bistol, UK. E-mail: Don.Webbe@uwe.ac.uk. Tel.: (+44/0) 117 328 2741. Fax: (+44/0) 117 328 2289. 1
Betting on the outcome of UK association football matches (socce matches) is big business with in excess of 3% of the UK ault population egulaly placing bets on the football fixe os maket (Depatment fo Cultue, Meia an Spot, 2007). It has also been the subject of ecent acaemic eseach (see, fo example, Achontakis an Osbone, 2007). In the football fixe os maket bookmakes offe betting os fo each of the thee mutually exclusive an exhaustive outcomes which ae fo the home team to win o the away team to win o fo the game to en in a aw. In the UK, it is customay pactice to quote os in the fom a to b fo the home win, c to fo the away win, an e to f fo the aw, whee a, b, c,, e an f ae integes. Thus, fo instance, if a betto wages b pouns on a home win an the outcome is a home win then the total etun woul be a + b pouns (pofit a pouns) but othewise woul esult in a loss to the betto of b pouns. The pecise poceues with which bookmakes eive os ae best viewe as a commecially guae secet although it is wiely known that the os ae essentially jugement foecasts by panels of expets employe by bookmakes (Shape, 1997). The os offee ae usually mae available appoximately one-week pio to the game taking place. Although bookmakes always eseve the ight to alte the os on offe they selom o so iespective of betting volumes an iespective of new infomation that may come to light uing the couse of the week (Foest et al., 2005). Typically os woul only alte in the ae event of incoect os being poste though a typogaphical eo. It is in these senses that the os ae consiee fixe. Bookmakes os may be convete eaily into ecimal os, wheeby the ecimalise os fo the home win ( h ), away win ( a ) an aw ( ) ae given by h = a ( a + b) ; c = a ( c + ; ) e = ( e + espectively. The total of these ecimalise os, f ), will always excee unity an ( 1) 100, known as the ove-oun, eflects an 2
anticipate inbuilt pofit fo the bookmake to offset cost of unning the maket assuming liabilities ae evenly spea ove the thee outcomes. In the UK fixe os football maket the ove-oun is cuently about 10% pe game. The values h B, h = ; a B, a = an B, = may be thought of as estimate bookmake pobabilities of match outcomes. A small but pecious point notewothy of futhe explanation is the teminology estimate bookmake pobability. The estimate bookmakes pobabilities ae a simple multiplicative escaling of the ecimal os une the assumption that the ove-oun is spea popotionally amongst the ecimalise os (which may not necessaily be the case). In aition iffeent bookmakes ae at libety to offe iffeent os giving ise to iffeent estimate pobabilities. Bookmakes os ae set with commecial objectives in min an as such the eive pobabilities may not eflect thei best estimates of the pobabilities that they might othewise eive. Figlewski (1979) an Knight (1965) make the impotant point that in games such as Roulette the pobability of winning ae known in avance an is theefoe a game of isk but with no uncetainty. In the case of betting on a football match outcome the betto is pesente with both isk an uncetainty since although each team can be thought of as having a cetain chance of winning the tue chance will not be known. Thee ae lage iniviual iffeences amongst people elating to gambling. Some will not gamble. Some will be isk-positive being attacte by lage os with the attenant bagging ights shoul a long shot pay iviens (Woolan, 1994). Avey an Chevalie (1999) foun evience of sentimental betting such as betting on well-known teams o on teams that have been covee in the meia ecently. Othes may be attacte to betting on socalle cetainties iespective of the os on offe. Othes may bet on the football team that they suppot to win out of loyalty, o bet on thei team to not win so as to have some financial 3
comfot shoul thei team not win. The iational behaviou of gambles cannot be efine on a simple continuum. In contast a pevalent theoy amongst statisticians an econometicians is the iea of eveloping betting stategies on value bets by systematically ientifying football games in which thee has been a peceive incoect setting of bookmakes os though an inefficient use of infomation available i.e. the maket has faile to captue the infomation use by a supeio analyst Pankoff (1968, p.204). Ou moelling appoach patly evolves aoun the peiction of bookmake estimate pobabilities. A numbe of authos have opeationalise the iea of a value bet by builing statistical moels to peict the pobabilities of match outcomes (say p h, p a, p ) an to compae these estimate moel pobabilities with estimate bookmakes pobabilities by consieing the atios p h h = ; B, h p a a= an B, a p =. Betting stategies then B, take the fom, fo example, to only bet on a home win outcome if h > * h whee * h is some etemine constant chosen to balance isk an pofitability. These stategies have been successfully implemente by Dixon an Coles (1997) an by Rue an Salvesen (2000) amongst othes. In passing we comment that thee ae othe ways of quantifying the egee of mismatch. The statistical moels fo peicting match outcomes eithe moel the match outcome iectly using iscete choice moels such as binay o oinal logistic egession moels o though moels peicting pobabilities of match scoes an then aggegating to peict match outcomes. The Poisson istibution o negative Poisson istibution ae invaiably use in this latte appoach. In an ealy attempt to moel match scoes Reep et al. (1971) conclue that it was ifficult to peict match outcomes confiently though this oute an 4
that chance ominates the game. Since then othes have successfully use moe computationally intensive complicate moels (e.g. Dixon an Coles, 1997). It is ou view that woking iectly with goals scoe an concee may not be the coect stance. In league football a home win with a fou-nil scoe line will be awae the same numbe of points as a two-nil home win. Duing a match a team winning by two clea goals may change tactics o pesonnel an be content with that magin of victoy athe than aiming to beat the opposition by the geatest possible scoe line. In othe cases a vey stong club may fiel a compaatively weakene team against some opposition with a view to esting sta playes fo futue games at the expense of a compaatively low winning magin. The pimay aim fo teams involve in a football match is the match outcome an (except in vey esticte cicumstances) the extent of goal iffeence is seconay. The pimay aim of betting on the fixe os maket evolves aoun the match outcome an financial etuns ae not linke to the magin of victoy. In a compaative stuy Goa (2005) foun vey little iffeence in peictive ability between the iscete choice appoach an moels iectly moelling goals scoe an concee. Thee is an agument that moelling exact scoes may be moe susceptible to the effect of outlies than a moel base on match outcomes. Fo all of these easons we have opte to wok iectly with iscete choice moels an to moel the match outcome iectly. Betting os on the aw outcome invaiably fall ove a naow ange elative to the ange of os on offe fo the home win o the away win. Pope an Peel (1989) suggest that this eflects a lack of ability of expets to foecast aws an Achontakis an Osbone (2007) ague that this coul simply eflect a geneal inability to peict aw outcomes with any egee of eliability. Taken at face value this suggests that thee may be inefficiencies in the aw os maket. Howeve the peiction of the aw outcome is notoiously ifficult. Pio to the evelopment of fixe os a common fom of gambling amongst UK football 5
fans was the teble chance football pools wheeby bettos woul ty to ientify aws fom games to be playe; monies stake woul go into a pool an iviens pai on elative pefomance. Although highly popula this fom of gambling was viewe by most as essentially a lottey (Foest, 1999). Accoingly, although inefficiencies may exist in the aw os, it oes not necessaily follow that they may be systematically exploitable an fo these easons we will not consie the evelopment of a betting stategy fo the aw os in this note. In league football thee is a well establishe home avantage effect (Clake an Noman, 1995), an Dixon an Coles (1997) epot that appoximately 46 pecent of games in the English football leagues esult in a home win. Fo these easons it natually follows that the os offee fo a home win ae typically lowe than those on offe fo an away win. Consequently a betting stategy base on the away win outcome, although occuing less often than one base on the home win outcome, may have the potential fo geate pofits than a betting stategy fo the home win. Fo these easons ou moelling stategy will focus on the away win only. Most of the publishe betting stategies in the statistical liteatue that have a positive expecte etun ae base on moels with estimate team specific paametes that ae continually upate. This high imensionally avesely impacts on the evelopment of a pactical betting ule. Fo these easons we consie a low imensional pasimonious moel specification to unepin the betting ule. Stategies base aoun betting on long shots o uneogs have been epote to be toublesome. Fo instance Thale an Ziemba (1988) epot on the favouite - long shot bias in acetack betting wheeby favouites ten to be une-backe an long shots ten to be ove-backe. Bi an McCae (1987) epote the stategy of betting on favouites o on long shots in hose acing woul not yiel a positive etun. Likewise Woolan an 6
Woolan (1994) epot that the favouite long shot bias is evese in baseball betting an that no simple favouite o long shot betting stategy woul pouce a positive etun. It is ou contention that betting ules base on the fom bet on the home win if h > * h may be paticulaly susceptible to the inclusion of too many matches whee thee is a clea favouite. Ou appoach is base aoun a moification of this fom of betting ule so as to allow the possibility of avoiing placing bets on matches involving clealy ientifie long-shot / favouite pais. Section 1 gives an oveview of the methoological appoach utilise to evelop betting ules fo the away win. A bief account of the ata use an the ationale fo the vaiables use in the moel is given in Section 2. Deive statistical moels an betting ules ae given in Section 3 an the utility of the moels is iscusse in Section 4. 1. Moelling Appoach Ou moelling appoach is base aoun sample ata ( i = 1,..., I ) fo eiving a betting ule an a secon sample ( j = 1,..., J ) fo out of sample assessment of the efficacy of the ules eive. Ou fist moel is to iectly estimate the pobability of an away win using a iscete choice moel. Let p a, i enote a moel base estimate pobability that a match inexe by i will esult in an away win, an let B a, i, enote the coesponing estimate pobability eive fom the bookmake s fixe os. Let a, i = p a i B, a, i, /. We will consie a betting ule of the fom bet on the away win fo match i if, an only if, l a, i u whee l an u ae constants chosen on citeia such as maximum pofit pe game o maximum pofit. This fom 7
of ule is a moe geneal vesion of the stuctue use by Dixon an Coles (1997) an Rue an Salvesen (2000) which use a one-sie limit utilising l only. One appoach to etemine optimum values fo l an u is to consie all possible values fo l an u ( l u ) an to apply these to sample values an choose the estimate paametes to be those values that maximise within sample pofit pe game o maximise within sample pofit. This appoach howeve may pouce many small seemingly goo pofitable intevals but which not may be eplicate on unseen ata (paticulaly so if pofit pe game is consiee). Fo this eason we consie etemining l an u sepaately. The optimum value fo l is chosen so that the betting ule bet on the away win fo game i if l esults in maximum within sample pofit. Similaly, u is chosen so that the betting ule bet on the away win if a, i esults in maximum pofit (an not maximum pofit u pe game). This appoach is intene to pouce a betting ule with geate elative obustness which oes not ovely capitalise on chance iiosyncatic sample chaacteistics. In aition we consie the evelopment of a betting ule of the fom bet on the a, i away win fo match i if an only if pl p a, i p. The ationale behin this ule is to u etemine whethe pofitable ules can be evelope wheeby the away team is a stong favouite (in which case the ule woul efault to betting on the away win if p a, i p l ), o in opposing a stong home team (in which case the ule woul efault to betting on the away win if p a, i i p u ) o whethe it is bette to focus betting on the away win when thee is no seemingly clea favouite. Ou secon appoach is to use oinay least squaes egession with bookmakes os as the epenent vaiable. Resiuals ( e i ) une the moel may be use to assess the elative extent of isageement between the bookmakes os fo the away win an the 8
peicte bookmakes os une the moel. Following the easoning given above, we consie a betting ule of the fom bet on the away win fo match i, if an only if, el ei e u. 2. Sample ata Data was ecoe on the 194 league football games that took places between the 2 n Octobe 2007 an 22 n Octobe 2007 fom the games playe in the top fou English football leagues an the top fou Scottish football leagues. The outcome of each game was ecoe (home win, aw, away win) along with fixe os fo each outcome offee by Labokes plc, the UK s lagest bookmake. Fixe os ae set with commecial an financial gains in min an may not necessaily eflect the best assessment of match outcomes since they may be set with anticipate betting volumes in min o inee set to influence betting volumes. Fo these easons we consie as peicto vaiables the home an away team pefomance atings publishe weekly by the Racing an Football Outlook (RFO) which is a weekly newspape publishe by Tinity Mio plc, eicate to betting on hoseacing an association football. The RFO inex is an inex base on the esults of the past 60,000 games an povies a fom ating on a scale of 0 to 1000 fo each team in the English an Scottish football leagues. Inceasing atings ae intene to eflect inceasing ability of a team an the iffeence in RFO atings between two teams is intene to eflect the extent of the egee of mismatch between the two chosen teams. The RFO pouces a sepaate inex fo home an away pefomance to account fo the home avantage effect an the extent of club specific home avantage effect (the home effect cannot be consiee to be of the same influence fo all 9
teams). We theefoe consie the RFO home ating fo the home team an the RFO away ating fo the away team as peicto vaiables fo match outcomes. Ou secon appoach is to use a goo peicto of betting os which utilises infomation that might not be use by bookmakes in eiving os. Fo this eason, fo each team in each game, we consie the aveage popotion of time that the team was winning, iespective of goal magin, in thei pevious thee league games as a peicto vaiable. This choice of peicto is patly infome by the eay availability of the ata an patly infome by the iea that the magin of victoy is not of pimay impotance but that the pecentage of time winning in pevious games will still povie an inication of elative ominance in ecent games against teams fom the same league. We theefoe consie aveage measue of time winning in pevious thee games an RFO atings as peicto vaiables of estimate bookmake pobabilities. A secon ata set compising all of those matches hel in the English an Scottish ivisions (63 games) between 15 th Januay 2008 an 21 st Januay 2008 was use to assess inepenently the out of sample usefulness of the eive betting ules. 3. Deive betting ules Table 1 summaises the iscete choice complementay log-log moel fo peicting the pobability of an away win. Oveall the moel is statistically significant (Log-likelihoo chisquae = 17.50, f = 2, p < 0.001), the iniviual peictos ae statistically significant (p < 0.001) an the iection of effects fo the RFO atings fo the home team at home (RFO HH) an the RFO atings fo the away team playing away (RFO AA) make goo conceptual sense. The moel aequately captues the stuctue in the ata (pecentage concoant pais between moel peictions an outcomes is 66.6%) an gooness-of-fit tests using Peason s esiuals 10
(p = 0.246) an eviance esiuals (p = 0.106) o not cast oubt on the appopiateness of the moel specification. Inspection of elta beta an elta eviance gaphics inicate that moel oes not suffe fom the pesence of ovely influential obsevations. Pio to fitting this moel we i consie a simple logistic specification howeve application of Bown s test inicate that a moel with a non-symmetic link function woul be moe appopiate. {Table 1 about hee} Figue 1 is a plot of within sample pofit against possible choices fo l fo the betting ule bet on the away win in match i if an only if a, i l with a, i estimate fo match i in the ata set using the complementay log-log egession equation an with a one poun bet wagee each time the ule is fie. In this way the optimal value fo l was foun to be 1.596. In a simila way the value fo the uppe boun * l = * u was etemine to be 7.597, which is the lagest obseve atio in the ata set. Fo the within sample ata the ule bet on the away win in match i if, an only if, 1.596 a, i 7.597 effectively efaults to bet on the away win if a, i 1.596 an fie 29 times yieling an absolute pofit of 19.43 giving a 67% pofit on monies stake. When applie to the test ata, the ule fie on 13 occasions giving an essentially beak-even etun of 0.80. {Figue 1 about hee} Applying the same poceue but using the peicte pobabilities fo an away win fom the complementay log-log moel gives the betting ule bet on match i to be an away win if 0.4470 p a, i 0.7146. This ule fie on 22 occasions an with 1 stake on each 11
game an oveall pofit of 16.34 was obtaine (i.e. a 74% etun). When applie to the test ata the ule fie on five occasions giving an oveall pecentage pofit of 62.8%. Table 2 summaises the fitte oinay least squaes moel with the estimate bookmake os fo the away win as the epenent vaiable. The oveall moel is 2 statistically significant ( R = 41.9%, F(4, 195) = 34.04, MSE = 1.186, p < 0.001), each peicto povies a unique statistically significant contibution to the moel an the iection of the effects in the moel make goo conceptual sense. The moel oes not suffe with poblems associate with multicolineaity (all vaiance inflation factos ae less than 4). A visual examination of the esiuals une the moel suggests that the assumption of inepenence of eos has not been gossly violate although thee is some evience of a small epatue fom nomality (Kolmogoov-Sminov test statistics fo nomality has a p- value of 0.01). Aopting the same poceue as ealie, but using the esiuals, gives a betting ule of the fom bet on the away win in match i if an only if 0.1489 ei 0.2042 whee ei is the esiual fo match i. Application of this ule to the sample ata gives ise to placing 28 bets yieling an oveall pofit of 9.75 (34.8% pofit). Applying the ule to the out of sample ata gives a pecentage pofit of 19.8%. {Table 2 about hee} 4. Discussion an conclusions The peceing analyses inicate that a pofitable betting stategy base on gambling on the away win may be possible. The esults of the iscete choice moel inicate that pofitability may be obtaine by avoiing those matches whee thee is a lage estimate pobability of an away win o a small estimate pobability of an away win. This fining is consistent with 12
pevious eseach cautioning against a betting stategy base on a long-shot o on a clea favouite. Instea the eive ule suggests that it may be pofitable to wage on the away win outcome on those seemingly ifficult to call matches. This may be a easonable fining if the extent of the home effect avantage has been incoectly estimate by the bookmake. The esults fom the value bet appoach which consies the atio of moel estimate pobability of the away win to the eive bookmake pobability of an away win as a betting tigge seem to be less spectacula. Distinct fom othe appoaches we consiee the iect moelling of bookmake os using OLS egession. This analysis suppote ou pio easone hypothesis that aveage time winning in pevious games is associate with the os on offe. Distinct fom othe appoaches we consiee the esiuals une the egession as quantifying the extent of mismatch between the bookmake os fo the away win an the moel peicte os. The eive betting ule fom this appoach suggests avoiing betting on the away win when thee is a lage iscepancy between peicte values an bookmake values an this fining is quite contay to the usual stance of betting on so calle value matches. The esults pesente elate to league football only an ue to the small sample size shoul be teate with caution. Howeve we have only fitte pio easone moels an have not unetaken a ata eging execise which othewise may have lea to too many false finings. In eiving an assessing the betting ules we have simply place a one-unit stake pe game. In pactice it might be favouable to vay the stake in some optimal way (e.g. betting stakes in popotion to peceive isk) an on this basis the pecentage etuns quote might be optimistically consiee as an unestatement. Likewise in pactice a betto will be in a position to shop aoun the iffeent bookmakes fo best pices fo the away win an oing so woul give a non-tivial positive impact on the pecentage etuns offee. We chose to consie aveage winning times in the past thee games a peicto vaiable although 13
thee may be futhe meit in extening this peicto vaiable ove a iffeent numbe of pevious games. A simila stategy coul be consiee fo betting on the home win, howeve if betting ules fo both the home win an away win ae to be consiee then some aitional thought woul have to be given to the possibility o pevention, of both ules fiing on the same game. 14
Refeences Achontakis, F. & Osbone, E. (2007). Playing it safe? A Fibonacci stategy fo socce betting. Jounal of Spots Economics, 8(3), 295-308. Avey, C. & Chevalie, J. (1999). Ientifying investo sentiment fom pice paths: The case of football betting. Jounal of Business, 72(4), 493-521. Bi, R. & McCae, M. (1987). Tests of the efficiency of acetack betting using bookmake os. Management Science, 33, 1552-1562. Clake, S. R & Noman, J. M. (1995). Home goun avantage of iniviual clubs in English socce. Statistician, 44, 509-521. Depatment of Cultue, Meia & Spot (2007). Taking Pat: The National Suvey of Cultue, Leisue an Spot, Chapte 9, Gambling available fom http://www.cultue.gov.uk/images/eseach/tpmay2007_9_gambling.pf Dixon, M. J. & Coles, S. G. (1997). Moelling association football scoes an inefficiencies in the football betting maket, Applie Statistics, 46(2), 265-280. Figlewski, S. (1979). Subjective infomation an maket efficiency in a betting maket. Jounal of Political Economy, 87(1), 75-88. Foest, D. (1999). The past an the futue of Bitish football pools. Jounal of Gambling Stuies, 15(2), 161-176. Foest, D., Goa, J. & Simmons, R. (2005). Os-settes as foecastes: The case of English football. Intenational Jounal of Foecasting, 21, 551-564 Goa, J. A. (2005). Regession moels fo foecasting goals an match esults in association football. Intenational Jounal of Foecasting, 21, 331-340. Knight, F. H. (1965). Risk, Uncetainty an Pofit. New Yok: Hape Tochbooks. 15
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Table 1 Complementay log-log moel fo the pobability of the away win Vaiable Coefficient SE(B) Z p (B) Constant -1.1897 1.7051-0.70 0.485 RFO HH -0.0157 0.0038-4.12 <0.001 RFO AA 0.0165 0.0042 3.90 <0.001 17
Table 2 OLS egession moel with bookmake os of away win as the epenent vaiable Vaiable Coefficient SE(B) t P B Constant 1.057 1.0060 1.05 0.295 RFO HH 0.0202 0.0022 9.32 <0.001 RFO AA -0.0194 0.0025-7.79 <0.001 Time 1+ Home Team 0.0164 0.0047 3.50 0.001 Time 1+ Away Team -0.0129 0.0048-2.66 0.008 18
20 1.595 10 Pofit fo unit stakes 0-10 -20-30 0-40 -50 0 1 2 3 4 5 6 7 8 > * Figue 1: Pofit fom ule bet on away win if > * 19