Globalisation and Efficiency in the. Fixed-odds Soccer Betting Market. David Forrest. and. Robert Simmons

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1 Globalisation and Efficiency in the Fixed-odds Soccer Betting Market by David Forrest and Robert Simmons Centre for the Study of Gambling and Commercial Gaming University of Salford December, 2001 Address for correspondence: Dr David Forrest Centre for the Study of Gambling and Commercial Gaming University of Salford Salford M5 4WT telephone: telefax:

2 ABSTRACT The paper analyses patterns of odds in the British soccer betting market over four seasons from It uncovers limited evidence for home-away and favourite-longshot bias identified in previous research. It also considers the hitherto unexplored issue of the extent to which odds set by bookmakers reflect the different numbers of supporters that different football clubs have. However, by the end of the period, the growth, from 1999, of a significant offshore sector with tax-free betting opportunities and greater competition had moved the market strongly towards efficiency. JEL Classification: L83 1

3 1. Introduction A voluminous literature, comprehensively surveyed by Sauer (1998) and Vaughan Williams (1999), concerns itself with efficiency in betting markets. The core question addressed is whether and to what extent the terms of bets on horse races or sports events capture accurately the objective probabilities of the possible outcomes. Given the size of wagering turnover in many jurisdictions, the question is interesting in itself; but most authors give it further and wider significance by portraying these specialised financial markets (where bettors are purchasing state contingent claims on winnings ) as offering opportunities for learning general lessons about the ability of economic agents to process information and about how successfully the information is transmitted into market prices. The largest number of studies relates to horse racing in the United States. 1 There, betting is organised on a pari mutuel (or pool) basis such that the relative odds of the competitors in a race depend only on the relative amounts wagered on each horse. Relative prices are therefore entirely demand-determined. If observed prices are inefficient in the sense of failing to reflect objective probabilities, this can be attributable only to bettor behaviour: either bettors suffer systematic misperceptions (uncorrected by arbitrageurs) or the explanation lies in their preferences (e.g. the well-known favourite-longshot bias may be generated by utility functions characterisable as risk-loving). A second way in which betting may be organised is in a bookmaker market. This is the dominant form of gambling on horse races in Britain, Ireland and many Commonwealth countries and is also favoured for wagering on team sports in the United States. In these 2

4 markets, a bookmaker announces the terms on which bets may be made (these may be summarised either by a quotation of the odds or, in the cases of American football and basketball, by points spreads). Clients may strike a bet at whatever terms are offered at the time but these terms will be adjusted in light of any subsequent bets as bookmakers receive guidance from the weight of money wagered on the various outcomes. The prices observed at the end of the betting period will therefore reflect interaction between the demand and supply sides of the market. Any inefficiency observed in the closing odds (or points spread) could be the result of bettor behaviour (as in pari mutuel markets) but authors such as Shin (1993) have drawn attention to the potential for sources of inefficiency alternatively to lie on the supply side (where bookmakers may, e.g. be responding to concern over insider trading). There is however, a third and unusual form of betting market, which is the focus of this paper. In fixed-odds betting on British football (soccer), bookmakers offer odds at the beginning of the trading period (which is of several days duration) and these are then available up to the time of the game, regardless of bettors response to the terms offered. Prices are purely supply-determined (though, of course, the bookmakers may take into account predicted client response to different odds). We examine this market to assess the degree of efficiency exhibited by the odds. In a sense, this is a particularly challenging market on which to test an efficiency hypothesis: where the odds are allowed to move, as in pari mutuel or in conventional bookmaker markets, price is influenced by diverse sources of information driving the trading decisions of bettors; these extra sources of information are unused in fixed-odds betting where bookmakers must rely on their own expertise (and perhaps their experience of patterns of demand). 3

5 Section 2 outlines the main features of the football betting market in the United Kingdom, Section 3 considers the small but growing body of previous literature and Section 4 describes our data. In Sections 5 and 6, we propose and implement tests for specific biases in odds and include consideration of a previously ignored issue, whether odds reflect levels of interest in and support for particular football clubs and whether this sentiment in the market can be exploited for wagering gain. Section 7 offers interpretation and concludes. 2. Betting on UK soccer matches All large British bookmakers offer odds on the possible outcomes (home team win, away team win, draw) of all games played under the auspices of the English (as well as the Scottish) professional leagues. 2 The bulk of matches are played on Saturday afternoons between August and May and a typical Saturday may feature sixty such games in England and Scotland. 3 The odds setters of an individual bookmaking firm meet at the beginning of the week (Sharpe, 1997) and the odds they decide are printed on to entry forms (known as coupons) which are then available all week. Bets may be made right up to the start of the relevant match(es) by visiting the bookmaker s shop or, in most cases, by telephone or internet, using a credit card. Such betting has been increasingly popular. Mintel Intelligence Report (2001) commented that football betting is the fastest-growing form of gambling in the UK. Global Betting and Gaming Consultants (2001) cite an opinion poll in 1998 that indicated nearly four million adults betting weekly on sports and claimed annual UK turnover of sports betting to be about 2bn. The huge majority of this turnover will relate to football though the figure includes not only betting on results but also wagering on other aspects of the game such as the identity of the first scorer in a match. 4

6 Crafts (1985), in a paper on insider trading in horse betting markets, remarked that the willingness of bookmakers to commit themselves to fixed-odds in respect of soccer games is testimony to the low relevance of inside information in the sport. We would not necessarily share this interpretation. Bookmaker betting grew during the 1980s to compete with an alternative form of betting on football, the pools, at that time a mass-participation long-odds (and pari mutuel) gambling medium where the idea was to select a group of games likely to end in draws. It was natural that bookmakers would challenge the pre-existing competition by aping their entry forms. 4 Printing bookmaker coupons similar to those of the pools necessitated fixing the odds in advance of the betting period. However, giving up the right to shift the odds in response to demand was only feasible and prudent because of alternative rules available to bookmakers to protect themselves against the exploitation of inside (or just new) information. The rules to which we refer constitute an important feature of the market, only occasionally and incidentally mentioned in previous academic studies of football betting: it is not normally possible to bet on the outcome of a single match. Only combination bets have traditionally been allowed. 5 This rule had its origin with the football authorities themselves which licensed use of the fixture lists and imposed a prohibition on single game bets in order to protect the integrity of the sport against temptations of match fixing. However, bookmakers have imposed more restrictive rules than the Football Association requires (and, moreover, differ from each other in the severity of their restrictions on types of bet) and their rules must therefore be understood as a part of business strategy rather than something imposed exogenously. 5

7 With some exceptions (e.g. televised matches), bookmakers have not normally permitted anything less than a treble bet. Such a rule is protection against those with inside or new knowledge. If one knew, as if with certainty, that team A would defeat team B, one could only be sure of using the knowledge to make a winning bet by choosing two other matches and making a series of treble bets in each of which A was wagered to win but each time with a different pair of forecasts from the other two games. The expense would normally make it impossible to take a position where a profit was certain and, even taking the risk of covering only some of the combinations of outcomes of the other two games, expected profit would be eroded or eliminated altogether. An alternative to betting on three or more matches has, however, emerged in recent years. Bookmakers permit double bets on a single game where the two halves of the bet relate to (i) the half-time score in the game (home, away, draw) and (ii) the full-time score in the game (again home, away or draw). Each bookmaker will offer the chance of such a double on any match on the coupon. The odds are not fixed match-by-match but are linked to the odds quoted for the chosen team on the main entry form. As a typical example, suppose the odds against home team A defeating away team B are 6/4. Corresponding to these odds, a bookmaker offers bets including the following: A leads at half-time, A leads at full-time 7/2. Draw at half-time, A leads at full-time 9/2. B leads at half-time, A leads at full-time 22/1. 6

8 Were a single bet permitted on A at 6/4, one would need to spend 0.40 to receive 1 in the event of an A victory. By employing the device of betting on the three combinations which can end in a win for A, one can ensure a pay-back of 1 for a spend of This represents a shading of the odds by 17.7% from 1.5 to 1 to 1.24 to 1. We interpret this 17.7% as a sort of insurance premium to protect bookmakers against insider dealing: if a client wants to bet on a single match he can do so but, in case his wishes are inspired by the possession of fresh news or inside information, he must accept odds approximately 18% less favourable than the published odds. The significance of the insurance premium is that it reduces the likelihood that any bias identified in the published odds can be exploited via the formulation of appropriate trading rules. For such betting to have been profitable in the past, the bias would have had to be sufficiently pronounced that bets could overcome not only the 9% tax and the (approximately) 10.5% bookmaker take 7 but also the implicit charge of 18% associated with being allowed to wager on a single game. The total take-out rate accounting for all three deductions, (1-(0.91*0.823)) , is 35.6%, which is not much less than the figure of 50% for the UK National Lottery. Transactions costs in soccer betting have, however, been falling. From May, 1999, a number of large British bookmakers set up offshore betting firms in jurisdictions such as Gibraltar and Guernsey to permit wagers to be placed via telephone or internet without any liability to betting duty. Moreover, competition amongst offshore operations eroded the severity of restrictions on single bets so that, e.g., it is now possible to make a single bet on any Premier 7

9 League fixture. In response to the move of bookmaking offshore, betting duty was removed in the United Kingdom in October, This fall in transactions costs would be expected to diminish the extent to which bookmakers could deliberately bias their odds without attracting the attention of professional bettors seeking opportunities to make positive expected profits. 3. Previous studies of UK soccer betting In view of the novelty of the market, there have been surprisingly few academic studies until very recently. Pope and Peel (1989) employed data from the 1981/2 football season and regressed the outcome of the match (defined by a binary variable equal to one for a named result) on the bookmaker odds quoted for that result. In the case of home wins and away wins, the coefficient on the bookmaker odds was never significantly different from one (at the 5% level) indicating that there was no statistically significant bias in odds-setting. Pope and Peel next estimated regressions in which measures of newspapers tipster forecasts were included as additional explanatory variables. Though tips had only low informational content, there was some indication that bettors could use them to reduce their expected losses and it was also possible for bettors to benefit by studying the odds set by different bookmakers, the pooling of information unsurprisingly assisting in the identification of games where a fairer bet might be made at a selected bookmaker. However, a 10% betting tax was noted to be an impediment to profitable trading rules, so that that market appeared to be weak-efficient in the Thaler-Ziemba (1988) sense of there being no opportunity to find bets with positive expected value. Our comment that one should take account of the worsening of odds implicit in restrictions against single bets would reinforce their conclusion. One should also note 8

10 that the data analysed by Pope and Peel are now nearly 20 years old and pertain to a period when betting volumes were negligible compared with their present importance (Jackson, 1994). Bookmakers now have much more incentive to pay for the specialised talent necessary to ensure accuracy in odds setting. Conceivably therefore, any tendency to inefficiency may have been eliminated since Pope and Peel s pioneering contributions. Dixon and Coles (1997) adopt a model proposed by Maher (1982) to forecast the results of football games played over the period. The score of each team is modelled as an independent Poisson distribution with mean determined by the past goal-scoring record of that team and the past goal-conceding record of the opposing team. 8 Probabilities of home, away and draw outcomes are extracted from the estimated model and a simulation performed for a filterbetting strategy whereby a bet is placed on any outcome for which the probability indicated by the model exceeds (by a nominated margin) the probability implicit in the bookmaker s odds. With the required ratio of model probability to bookmaker probability set at 1.2, the expected return to such betting is reported as borderline significantly different from 0.11, the expected return to a random betting strategy. This constitutes evidence against strong efficiency in odds to the extent that some bets indeed appear to yield higher expected returns than other bets. Whether there is sufficient evidence of a potentially profitable betting strategy is more open to question. From the authors figure 4, allowing for the worse odds associated with betting on a single match would require the filter ratio to be raised to about 1.3 for the expected return to become positive. Figures are not presented on the number of bets that can be made in a season at each value of the filter ratio but it is a familiar feature of horse betting that winning 9

11 strategies require considerable effort and data analysis while identifying frustratingly few occasions on which a bet should be placed (Crafts, 1994). Kuypers (2000), Dobson and Goddard (2001) and Goddard and Asimakopoulos (2001) follow similar procedures to Dixon and Coles in that they employ a forecasting model to identify matches where the bookmaker probability in respect of one of the possible outcomes has been underestimated to some specified degree. However, they employ ordered profit models which can encompass a variety of historic information beyond just the goal scoring record of each team. Kuypers was able to identify a limited number of profitable bets in season using a model estimated for but only when the forecasting model included not only team quality data but also odds data (i.e. he exploited a combination of fundamental and technical analysis). The research reported in Dobson-Goddard and Goddard-Asimakopoulos draws on a very rich forecasting model estimated over ten seasons to select value bets in and They find positive profit opportunities from using the model but only late in the season when the forecasting model can include much more current season information and can also benefit from the use of a dummy variable that highlights teams for whom a particular match has strong promotion or relegation significance. The positive profit depends, however, on being able to bet on single matches and on not having to pay tax. Goddard and Asimakopoulos remark that tax is now indeed abolished; but, of course, bookmakers may now for that reason alone reduce the extent to which they will offer biased odds. Dixon and Pope (1996) point to indications of a degree of inefficiency in the football betting market associated not with a failure of the odds to capture information on the 10

12 performance in the past of particular teams but with patterns in the odds across matches. Evidence related to three bookmakers and their odds in English football games over suggested that appreciably lower losses would accrue to bettors backing home wins or draws rather than away wins. Further, an examination of returns related to different ranges of odds pointed to a reverse favourite-longshot bias: wagers on outcomes with the shortest odds (i.e. the highest bookmaker probability) attracted the largest losses. Cain, Law and Peel (2000), on the other hand, however, noted superior (though still negative) returns accrued to bets on strong favourites; but their evidence was drawn from only one year, The evidence presented by Dixon and Pope in support of the proposition that odds underestimated the chances of home teams and underdogs is highly suggestive. However, we prefer to assess the existence and magnitude of these and other possible biases by multivariate regression analysis. A comparison of returns to backing short- and long-odds teams may for example generate misleading estimates of the size of any bias if odds are correlated with whether or not a team enjoys home advantage. Thus, the superior return to betting on strong favourites, noted by Cain, Law and Peel, may in fact be generated by home-away bias in betting odds since strong favourites are almost always playing at home. 9 As implied by work from Golec and Tamarkin (1991) on American football points-spread betting, the detection and estimation of favourite-longshot bias (or its reverse) requires a multivariate approach to ensure that any estimation is not contaminated by the presence of a simultaneous home-away bias. In this paper, we test for these biases and also for a possible bias linked to disparity in fan-betting volumes when opposing teams have very unequal numbers of supporters. 11

13 4. Data Our model was estimated for each of four football seasons from to For each season, we attempted to include all matches in the English professional leagues (i.e. the FA Premier League and Divisions One, Two and Three of The Football League) that were played on a Friday, Saturday or Sunday (these are the days whose fixtures are included on bookmakers weekly coupons). The outcome of each match was noted from the relevant volume of The Rothman s Football Year Book. Betting odds used in this study are for Super Soccer. Super Soccer odds are those supplied by a specialist odds-setting firm and adopted by nearly all the small, independent bookmakers of the United Kingdom (coupons carry both the name Super Soccer and the name of the local bookmaker who has subscribed to the service). The four largest national bookmaking firms quote their own individual odds. However, these are highly correlated with each other and with Super Soccer s: we collected data from all five sources for the first of our seasons, , and replicated the work reported below using, in turn, the odds from each of the other four firms. The pattern of our results was very similar irrespective of which bookmaker was used and we are therefore confident that our findings for Super Soccer are representative of the sector as a whole and Super Soccer odds were obtained directly from entry forms (coupons); a small number of coupons were missing from our set and matches for these weeks are excluded from the sample. For and , we collected Super Soccer odds from archive copies of the daily betting newspaper, The Racing Post. Both on the coupons and in the newspaper, odds against any team i winning were quoted in the traditional form, a/b. We 12

14 converted these odds so that they were expressed in the alternative form of probability-odds such that the probability-odds of a win for team i became b/(a+b). For example, with quoted odds of 4 to 1, probability-odds were 1/5 or 0.20: one would have to spend 0.20 to secure a payout of 1 in the event of team i winning would, however, not be the implicit bookmaker-probability of a win for team i because the sum of the probability-odds of the three possible outcomes (win, loss or draw for team i) is always greater than one. Therefore, the bookmaker-probability of i winning is obtained by dividing the probability-odds for team i by the sum of the probabilityodds of the three possible outcomes. A problem here is that the Racing Post, on which we relied for data from two of our four seasons, did not report odds for a draw but only for a home win and for an away win. However, in the two years for which we had coupons quoting odds on each of the three possible results, the sum of probability-odds was always close to 1.117, a figure reported by other authors for other bookmakers and one that indicates a nominal bookmaker take of 10.5%. 10 For the matches for which the data source was the Racing Post, we therefore estimated the bookmaker-probability of a team winning a particular match by dividing its probability-odds by the fixed number of Thus in our example, the 4 to 1 team would have been attributed a bookmaker-probability of 0.20/1.117 or This was then the figure used in empirical analysis. 5. Model Authors such as, most recently, Goddard and Asimakopoulos (2001) demonstrate that occasionally (in their case, in the closing weeks of the season) sophisticated forecasting models might capture more fundamental information than is encapsulated in bookmaker s odds. But, 13

15 given the growing volume of betting in recent years, bookmakers have had an increasing incentive to hire talent able accurately to process team quality and form information and it is therefore unsurprising that these authors find (Table 2) that in general bookmaker odds perform similarly as well as forecasting models in predicting on-the-field outcomes. That is not to say, however, that the fixed odds betting market may be expected to be efficient consistently. A bookmaker s function is not to provide odds that are objectively correct but to maximise profit. Accordingly odds may be biased if this allows the bookmaker to take advantage of bettors misperceptions or preferences. For example, a consistent favouritelongshot bias (positive or negative) may emerge if bettors have preferences such that they behave as if they were risk-loving or risk-averse. Sentiment (as it is termed by Avery and Chevalier (1999)) may likewise play a role in the market. Most crucially perhaps, sports betting markets differ from typical financial markets in that many investors view the expenditure not just as an investment but as a vehicle for becoming more of a stakeholder in the team for which they cheer either in the stadium or more passively; bookmakers may price bets to take account of the likelihood that the reservation price (odds) of such potential bettors will be influenced by their degree of attachment to their team as well as by their assessment of the true probability of victory. Studies of wagering markets have generally been disciplined by treating them as just financial markets which happen to be convenient to study because of such well-cited characteristics as the unambiguous termination point at which the value of the asset becomes definitively measurable. Generally, there has been no recognition that a large proportion of 14

16 participants in the market may choose between possible bets not primarily in terms of their value as financial assets but as consumer goods complementary to the sport itself. This has led to a tendency to search for, but not explain, biases in the odds and has perhaps limited the dimensions of bias for which empirical evidence has been sought. Kuypers (2000) is an exception to the extent that he (importantly) sought to build a model where a profit maximising bookmaker sets (fixed) odds in the context of a market where there is a mixture of neutral bettors (who share the (objectively correct) bookmaker assessment of the probabilities attached to each possible outcome from the match) and committed bettors (whose views are coloured by their support for a particular team and are over-optimistic with regards to its probability of a win). This appears a useful framework to represent the soccer betting market though we prefer to think of the fan bettors as capable of assessing win-probabilities objectively even though behaving as if they had misperceptions: we interpret their willingness to wager as taking account of utility obtained from supporting their team in the betting market as well as in the stadium. In his numerical example, Kuypers has ten supporters who have decided to bet and will wager on a particular outcome according to their perception of the value offered by each of the (three) possible bets. Four of the bettors are neutral and six are fans of Manchester United, Britain s most heavily supported club. Their opponents in Kuyper s hypothetical match are Liverpool. He has no committed Liverpool fans in the betting market but this is not damagingly unrealistic to the extent that the same results would have been obtained had he posited a mix of committed fans whose average assessment of a Manchester United victory was over-optimistic. 15

17 In other words, what he is really examining is the effect of net stakeholder support in favour of one of the two teams. Kuypers demonstrates in this context that the bookmaker s expected profit is increased by moving from efficient odds to inefficient odds (that implicitly overstate the probability that Manchester United will win the match). This has some intuitive appeal since the bookmaker is able to take advantage of the eagerness of Manchester United supporters to back their team by offering them a lower pay-out in the event they win. However, the formal model and the numerical example may suffer from excessive simplicity to the extent that it assumes that the number of bettors is fixed. Kuypers justifies this by modelling the bookmaker as varying the odds for each outcome within the context of a fixed over-round (which itself is realistic): he sees price as embodied in the over-round and therefore fixed across the experiment. But if Manchester United fans were modelled as interested in betting only on their team and were oddssensitive with regard to making that particular bet, it is conceivable that (depending on elasticities) it would be worthwhile for the bookmaker to offer fairer rather than less fair odds in respect of a United win. It is surprising that Kuypers empirical investigation included no test of his implied, innovative hypothesis that the probability that a bet wins will be a function of bookmaker probability and relative supporter numbers for the teams in the match. We attempt such a test here. We also test for bias in the home-away dimension; the rationale is similar to the extent that a significant proportion of potential bettors (especially on minor games) may be spectators who would consider betting (depending on value) to give themselves more of a stake in the outcome. 16

18 This should add to their excitement at the match. Bookmakers would again be expected as part of their profit-maximising decision to take account of how odds-sensitive this segment of the market (mainly home fans, betting (if at all) on home teams) is likely to be. We also test for favourite-longshot bias. The result is likely to be of interest in that evidence on favourite-longshot bias in American team sports appears to be that it is in the opposite direction from that widely observed in horse and dog racing (see Woodland and Woodland (1994 and 2001) on baseball and hockey respectively). If we were similarly to find negative favourite-longshot bias in this market, we might start to discern the beginning of a pattern in which sets of odds in team sports consistently displayed different characteristics from those in horse and dog betting. The earliest study of the fixed odds market, Pope and Peel (1989), estimated a linear probability model where the outcome of a bet (win or not) was regressed on the bookmakerprobability associated with the bet. We adopt this model and also follow their procedure of estimating by weighted least squares (with weights derived from fitted values of the dependent variable, obtained using preliminary OLS estimation) in recognition of the likelihood of a heteroscedasticity problem from ordinary least squares. However, we expand the regression to test for home-away bias and for bias associated with relative fan numbers of the teams in a match. Our model is therefore PROBWIN i = + 1 BOOKMAKER-PROBABILITY i + 2 (HOME i )*(BOOKMAKER- PROBABILITY i ) + 3 HOME i + 4 DIFFATTEND 17

19 where PROBWIN i is the probability that focus team i will win the game, BOOKMAKER- PROBABILITY i is the win probability implied by the Super Soccer odds, HOME i is a dummy variable taking the value one if team i is playing at home and DIFFATTEND, included to proxy relative numbers of committed fans, is the mean home game attendance of team i (calculated in thousands for the previous season from information in The Rothmans Football Yearbook) minus the mean home game attendance of its opposing team (similarly calculated). Each match is included once only in the estimation with the identify of the focus team i selected by random process. The test for efficiency is that the estimated values for and 2, 3 and 4 should be each zero and that for 1 should be one. The advantage of following Golec and Tamarkin (1991) in the context of odds-based soccer betting (as opposed to in their setting of points-spread NFL betting) by randomly selecting focus teams for inclusion in the sample is that this procedure permits separate evaluation of possible biases in the favourite-longshot and home-away dimensions. Suppose one always adopted the home team as focus. Ignoring DIFFATTEND, one would then have as the criterion for testing efficiency the joint hypothesis that = 0 and 1 = 1. A non-zero estimate on the intercept team would not necessarily imply home-away bias because if favourite-longshot bias were present, this would be reflected in both the slope and intercept values. A problem with Golec-Tamarkin randomisation is, however, that in addition to sampling matches from a population, it is further sampling a set of bets (those on teams picked out in the 18

20 random process). Estimating the regression equation once only would therefore be misleading because of the unreliability of the standard errors. Accordingly, we estimate each equation twenty times using twenty different randomisations. The statistical significance of the coefficients can then be examined using a procedure outlined in Snedecor and Cochran (1967). We count the number of cases in the twenty trials where a particular coefficient is significant at the 5% level. A normal approximation can be used to test the null hypothesis that the true proportion of cases where the coefficient is not equal to zero is 5%. If the null is true, the observed proportion of rejections, R 1 is distributed approximately normally with mean r and standard deviation s = (r (1-r)/n) ½, where n is the sample size, here 20. The normal deviate, with a correction for continuity, is z = ( R r - (2n) -1 )/s. The critical value for this test statistic is 2.33 at a conservative 1% significance level. If there are four significant (at 5%) coefficients out of twenty in our trials, the value of z is 2.57 which exceeds our 1% critical value, so where there are four or more significant coefficients amongst twenty trials, we conclude that the particular coefficient is significantly different from zero (or one as appropriate). 6. Results Table 1A reports our results for the whole sample in each of our four seasons. The values shown are mean coefficient estimates across twenty trials. The figures in parentheses indicate in how many trials the particular coefficient estimate was significantly different from the value specified by the null (zero or one). Where this number is four or more, we can reject the null hypothesis. 19

21 Tables 1B and 1C display disaggregated results by division. We have two subsamples, one relating to the Premier League (PL) and Division One (D1) of the Football League, the other to Divisions Two and Three (D2 and D3) of the Football League. These two groupings may fairly be represented as capturing a divide between high- and low-profile football. Nearly all the large market football clubs play in the Premier League or Division One. We review now the principal findings embodied in these Tables. Favourite-Longshot Bias There is a hint in the data of the reverse favourite-longshot bias noted for American team sports by Woodland and Woodland (1994 and 2001) and by previous researchers on soccer (Dixon and Pope (1996)) to the extent that the mean estimated coefficient on bookmakerprobability in our all matches samples is always less than one. Moreover, even when this point estimate is adjusted by the interaction term between home and bookmaker probability, this remains true. On the basis of point estimates, bookmakers appear to offer less unfair odds on longer-odds teams, home or away. However, the coefficient is only significantly different from one in a single season, The result for that particular season is consistent with a finding from Dobson and Goddard (2001, p.408). Dobson and Goddard concern themselves primarily with applying a forecasting model (including football data but not odds) to identify favourable betting opportunities. But for that one season, , they analyse in addition the profitability of purely technical trading rules (i.e. rules based only on patterns in the odds themselves). They report an advantage to betting long on away teams. That this strategy yielded superior returns would have been predicted by our regression results. Our coefficient on bookmaker-probability is significantly less than one 20

22 but adjustment for home teams from the interaction term moves the estimate substantially back towards one. Hence, a reverse favourite-longshot bias in the odds can be detected fairly confidently but only in respect of away teams. We note, however, that there is no statistically significant (reverse) favourite-longshot bias detectable in the odds in any of the other three seasons analysed here. We suspected therefore that the Dobson and Goddard trading rule that one should bet long on away teams was one whose success would have been ephemeral. This proved to be the case. In our sample for that season, , exactly 30% of away teams had posted odds longer than 3 to 1. A unit bet in support of each of these teams would have involved 459 bets and yielded a better than normal return of +2.37% (ignoring tax and restrictions against singles). Implementing the same strategy in the following season would have generated 520 bets for a return of 14.9%, which is worse than the benchmark loss suggested by the size of bookmaker over-round. Disaggregation into upper- and lower-divisions does little to modify our conclusions. In the PL/D1 results, significant bias in the favourite-longshot dimension is found only at the beginning of the period (1997-8) and then only for away teams. Indeed, by the last season (2000-1), the point estimate on bookmaker-probability is very close to one. In D2/D3, there is again only one season (1998-9) for which bias was statistically significant. We conclude that even if there were favourite-longshot bias present at the beginning of the period, it was not sufficiently stable or durable to be the basis of reliable wagering rules. 21

23 Home-away Bias That it is more financially rewarding to bet on home than on away teams has been noted before (Dixon and Pope (1996)) but the implied bias in odds setting is revealed here to have been extinguished after our first season. For that season, the coefficient on home in the full sample and in the D2/D3 sample is positive and significant. In PL/D1, home-away bias is also present once one combines the effects of the coefficients on the home dummy and the home dummy/bookmaker-probability interaction term. The extent of bias was sufficient that there would have been very divergent returns from strategies of backing all home or all away teams in A unit bet on all home teams and all away teams (in all divisions) would have yielded returns of +1.01% and 23.95% respectively (using quoted odds and ignoring tax). The comparisons for PL/D1 and D2/D3 are +0.22% versus 13.00% and +1.59% versus 32.4%. In that particular season, as for earlier times examined by Dixon and Pope (1996), home bets were much better bets than away bets. However, in neither the all divisions case nor in either of the disaggregations is there any evidence of statistically significant bias after Indeed, for the PL/D1 case in , once one combines the point estimates of coefficients on the two relevant variables, then a home team with average bookmaker probability of victory has a zero difference in winning compared with that implied by the odds. Once again, over our data period, early signs of inefficiency consistent with those noted for earlier times by other authors, disappear completely by Relative Fan Support A novel feature of this paper is that it tests the proposition that bookmakers will vary odds in response to the presence in the market of sentiment-driven demand for wagering. Kuypers (2000) proposed a model of a profit-maximising bookmaker who would exploit the backers of 22

24 well-supported teams by offering them odds more than averagely unfair. His model is restrictive, however, in that it assumes the decision to participate in the wagering market on a particular game is determined independently of relative odds. It is quite possible that if sentimental demand is elastic with respect to odds, bookmakers will aim to raise turnover by favouring larger teams in terms of the generosity of odds. In fact, the evidence here is that Kuypers is correct in expecting bookmakers to take account of the net direction of fan demand but the direction of the bias is opposite to that which he proposed. The coefficient on DIFFATTEND is positive and significant in the all-matches sample in each of the first three of our seasons. In the disaggregated results, the coefficient is positive and significant in and in PL/D1 and positive and significant in all four seasons in D2/D3. The magnitude of the coefficient is typically large. In the most recent season in the lower divisions, a difference in mean home attendance of 3,000 in favour of one team is estimated to raise the probability that a bet on that team will be successful by in excess of five percentage points relative to a given bookmaker-probability. In contrast to the findings on favourite-longshot and home-away bias, some residual inefficiency therefore remains in the market (at least for D2/D3) even at the end of our period. A betting strategy of wagering on teams with greater support than their opponents (in D2/3) would have yielded superior returns even in In that season, our data revealed 264 bets where one team had mean home attendance more than 2,000 greater than the opposition; a unit bet on each of these 264 teams at posted odds and ignoring tax would have yielded a return of 4.96% (a smaller loss than that accruing to random betting). But if the filter were raised to 23

25 3,000, one would have bet 131 times to yield a return of %; and even at a 4,000 filter, one would have found 71 bets and earned a profit of 9.15%. In the lower divisions, tax-free betting would therefore have yielded a profit from a strategy of following teams with significantly greater support providing that one could have made singles bets. Restrictions on singles may of course make it perfectly rational behaviour for bookmakers to seek to attract large groups of fans by offering better than fair odds on their team so long as they know the gain will in fact be dissipated by these bettors being forced to make more random wagers in compulsory matching bets on at least two other games. Our finding that bookmakers court or have courted large groups of fans by posting attractive odds on their teams in some sense contrasts with the findings of the paper by Avery and Chevalier (1999) on American football betting. They use proxies to identify glamorous clubs likely to have large followings (e.g. the conference to which they belong, their success in winning championships) and establish that the terms of bets on such teams overstate their likely superiority in a given match. In that context, a plausible wagering strategy would be contrarian, i.e. bet in the opposite direction to sentimental demand. That such a result holds is consistent with American bookmakers (Vegas casinos) aiming at a balanced book (equal liabilities to bettors irrespective of game outcome) on each match, i.e. with the market behaving as if parimutuel. Such behaviour appears unlikely to be expected profit maximising though Avery and Chevalier discuss reasons (such as regulatory concern that the casino should be disinterested in game outcome) why balancing the book may nevertheless be the basis of odds setting. 24

26 Our evidence specifically implies that bookmakers in Britain are not so extremely riskaverse as to balance the book on each individual game. This is unsurprising. Bookmakers offer odds on thousands of football and other sports events each year and scope for risk-pooling is therefore so great that, even if risk-averse, they would have no reason to depart from expected profit maximising behaviour in setting the odds on any individual fixture. If sentimental demand is sufficiently elastic with respect to odds, our DIFFATTEND bias will reflect appropriate behaviour by odds setters. Betting Simulations Regression-based tests for market efficiency can suggest bets which offer abnormal returns but the relevant strategies can be successful only if the biases investors observe from the pattern of odds and results in one period prove stable across time. Accordingly, in this section, we test whether a regression model estimated from one season s data is capable of being used in the following season to select a set of bets that will earn superior returns. The bets we examine are those which would have been made whenever the probability of a win as estimated from the previous season s model is more than one-fifth greater than that implied by the bookmaker odds. Table 2 displays the results. The number of bets recommended by the model in any one season may be interpreted as reflecting the degree of inefficiency embodied in the regression results for the previous year. The success of those bets will then depend on whether or not any biases continue to characterise odds setting behaviour in the current year. Both aggregate and disaggregated results for revealed the presence of several specific biases in the pattern of odds. The model accordingly generates a large number of bets 25

27 passing the 1.2 filter ratio for season The biases in proved sufficiently congruent with those for for the betting strategy to be successful. At the aggregate and for PL/D1 and D2/D3 separately, returns were sharply higher than the 10.5% implied by the level of overround. Indeed, in D2 and D3 the level of return calculated at posted odds was sufficiently high (22.14%) to cover the extra commission (17.7%) charged for the privilege of betting on a single match via the device of a half-time/full-time double. The strategy would therefore have been profitable but for tax. In any case, the levels of return in available from assuming the continuation of biases noted for may fairly be described as abnormal and the market therefore inefficient. However, we find in the regression results for particular seasons, a distinct trend towards efficiency since In PL/D1, very few bets (22) are recommended in and the impressive profit cannot be regarded as significant because it depended on a very small number of long-odds wins. In D2/D3, the pattern of odds the previous season generated 197 recommended bets in ; 162 of these were on away teams and only 35 on home teams. Here, it was still possible to make a bare positive profit (without accounting for tax and restrictions on singles bets). Some evidence of an inefficient market remained therefore in the lower divisions. The rather poor result from applying the all divisions model probably reflects that the previous season, , had seen a gap opening between the upper and lower tiers in terms of the degree of efficiency exhibited in the odds: this is evident from the regression results (Tables 1B and 1C) and therefore the application of an aggregated model was by inappropriate. 26

28 The most telling illustration of the extent to which the market was moving towards efficiency is that application of regression results to betting in yielded not one recommended bet whether one used an aggregate model or two disaggregated models. By , no specific biases (other than DIFFATTEND) appeared to be present in the odds and so value bets were hard to obtain in if relying on a regression model for guidance. As noted above however, gains still accrued to those following a simpler strategy based on only one indicator, DIFFATTEND. Overall, the betting simulations point to inefficiency in the market being close to eliminated over our four year study period. 7. Interpretation and Conclusions The literature on efficiency in wagering markets has overwhelmingly been motivated from the perspective of finance and financial economics. That has led others to expect patterns of odds to display efficiency in the sense that all bets should be equally good (or, more strictly, bad) bets. The perspective ignores that, with sufficient transactions costs to deter professional arbitrage, the wagering market in sports may be more usefully represented as a consumer good market where potential consumers have preferences on what bets they make that go beyond expected financial return. In that framework, it is easier to understand biases in the odds as reflecting price discrimination across groups of potential bettors. Price is here identified not (as is traditional) with the bookmaker over-round for the event but rather as the expected bettor loss on each possible unit bet (home, draw, away).the profit maximising price may differ in respect of each result because of customer preferences and this will manifest itself in biases in the odds. 27

29 Bookmakers have strong incentives to be (or to hire) sophisticated odds setters. This is particularly so in a market such as that on British soccer where they have eschewed the right to vary the odds during the betting period. In fact, they do display a high level of expertise. Goddard and Asimakopoulos (2001) acknowledged that there is very little difference in forecasting performance between bookmaker odds and their own very rich and detailed statistical model drawing on past team performance and many years of soccer information data. If odds setters are so skilled as to be able to match very advanced statistical forecasting models, it is reasonable to assume that biases in the odds in the dimensions studied in this paper are, if present, there by design rather than accident and that they should be viewed as a means to maximizing profit. Specific biases should be viewed not as aberrations but as examples of price discriminating strategy. Any reverse favourite-longshot bias could be interpreted merely as odds setters taking due account of bettor risk aversion. 11 The similarly familiar home-away bias could be interpreted as a response to perceived elasticity of demand amongst bettors who attend a game and consider betting to show support for their local team (they often indeed place the bet in the stadium itself). And our DIFFATTEND bias is readily interpretable along the same lines of demand being driven by the utility of a bet including non-financial elements (the fun is greater if betting on one s own team) and odds setting recognising that. Biasing the odds as a strategy may, however, be constrained by the potential presence in the market of arbitrageurs. In the absence of transactions costs, risk-neutral professional betters would be able to punish bookmakers whose odds were biased and render efficiency an essential feature of a sustainable market. 28

30 Here and in other papers, biases have been identified which (at least until 1999) created some favourable betting opportunities. But the gains here and in other earlier analyses have mostly been modest in the sense of showing a positive sign only if one assumed singles betting and no tax. Restrictions on singles and the presence of a tax have, however, been institutional features of the British market and the implied extra transactions costs have facilitated biased odds as possibly sustainable and equilibrium features of soccer betting. For illustration, we have argued that where a particular team attracts a large following, those who support it in the betting market may be offered a better price (less unfair odds) than usual. We found one betting strategy based on this idea was capable of yielding in excess of a 9% profit, a figure not untypical of the range of wagering gains cited by those who have sought to identify strategies for profitable trading. On the face of it, it appears unlikely that a bookmaker would be so eager to attract bets from fans of heavily supported teams that he would award them an expected profit of 9%. But, given the bar on singles, he would not in fact be doing so. The fan bettor would have two choices. The first would be to bet on the one game, employing the device of a series of half-time/full-time doubles; but the shading of the odds on this type of complex bet would be sufficient to convert the expected return from + 9 to 9%, a respectable profit for the bookmaker. Alternatively, the bettor could combine the bet on the match in which he had a special interest with bets on two or more other games. The fan bettor is likely to be less well informed regarding matches not involving his own team, so the bookmaker might expect to earn a normal commission on the other legs of the combination bet to again realize an expected profit for himself (and loss for the bettor). Vis a vis this group of customers, 29

31 the favourable odds offered on their own team s results is unlikely to expose the bookmaker to loss. But what of the risk-neutral professional bettor who could destroy the ability of the bookmaker to offer biased odds? This potential arbitrageur could acquire sufficient expertise to identify three favourable bets and combine them in a treble : the variance of returns would be high but the expected gain still positive. That is true but the level of tax expected in the past is equivalent to extra bookmaker commission and, in the case of the 9% prospective gain hypothesised here, the tax would be just sufficient to deter the arbitrageur from entering the market. It has, then, been customary for bookmakers to price discriminate in terms of different bets offering different degrees of value; but the restrictions on singles betting made any prospective gains unrealisable by the leisure bettors while tax deterred the professional arbitrageurs from entering the market. Inefficient odds were, in these circumstances, sustainable and indeed to be expected. The degree of price discrimination was of course constrained by the extent of the cushion offered by the tax on gaming (and this constraint may or may not have been binding). The world has, however, changed. The rapid growth of internet betting made it feasible and desirable for British bookmakers to move offshore by 1999 to cater to a wider, international market where the interest in betting on English soccer (particularly the Premier League) is very high but where competition from bookmakers in other countries was now intense. British 30

32 bookmakers would have been poorly positioned to exploit the growing international market for their product if tax had had to be levied, as it would have been if they had remained based at home. Since British households could also use offshore branches of familiar bookmakers the abolition of betting duty became inevitable and this occurred in The reduction in transactions costs inherent in the removal of tax (essentially optional since the establishment of offshore operations in 1999) severely limits the ability of bookmakers henceforth to offer biased odds. The importance of transactions costs is, of course, central to whether a market will display efficiency. If there were no transactions costs in the form of bookmaker over-round or tax, then any bias in the odds would be non-sustainable because it would create an opportunity for positive gain which arbitrageurs would be expected to eliminate. The removal of tax (and greater flexibility in taking singles bets at least on Premier League games) has greatly reduced the extent to which transactions costs impede the arbitrage process in soccer betting. It is unsurprising therefore that, while our study confirms traditional biases in the betting market up to 1999, there is a rapid move to efficiency by It is perhaps anomalous that the DIFFATTEND bias was still present in D2/D3 at the end of our period and evidently still capable of generating opportunities for positive returns. Thus far, betting on lower divisions has still been primarily a domestic market. Information is much less readily available on lower-tiers of English soccer, so transactions costs are effectively still high for potential arbitrageurs. Bookmakers have, evidently, still felt able to bias the odds for the presence in (or dominance of ) the market by fan-bettors. Equally, bookmakers still rigorously protect themselves by restrictions against singles betting for these less high-profile 31

33 games (where inside information and match fixing are larger risks). But we would not necessarily expect this lower-tier market to escape the attention of arbitrage indefinitely if taxfree gains are prospectively to be made. Globalisation, by forcing the dismantling of betting tax, appears to have triggered a strong move to market efficiency in this particular wagering market. In other jurisdictions, offshore betting is likewise a threat to any inefficiency protected up to now by tax or indeed by legal or regulatory restrictions. With a globalised gaming environment, we might in future expect fewer articles which can point to biases and favorable trading strategies in wagering markets. 32

34 FOOTNOTES 1 2 A large and convenient collection of such articles is provided by Hausch et al (1994). In the season studied in this paper, 1997/8, the English FA Premiership comprised 20 clubs while Divisions 1 to 3 of the Football League comprised 24 teams each. Each team meets a division rival twice, once home and once away. Movement between divisions is achieved by promotion and relegation. 3 Although the bulk of fixtures are played on Saturday, the number of Saturday games will occasionally be reduced because either the date is earmarked for knockout cup competition or because of international fixtures. A few games each week are moved to different days in order to be televised and betting is still possible in these cases Pools entries were (and are) made by mail or through a door-to-door collector. An exception is a televised match. Perhaps the exemption is linked to the high-profile nature of the contest chosen for television coverage. Inside information or match fixing are unlikely to present a serious problem if the teams are well-known and the betting volume high. This involves staking on the first bet, on the second and on the third. The over-round on bets is typically 11.7, i.e. one would spend to bet on all three outcomes at a level to earn 100 in pay back from the bookmaker. With a fully balanced book, the bookmaker would return of each staked to earn a commission of 10.5%. A similar model is used by Lee (1997) in examining the final ranking of teams in the 1995/6 English FA Premiership season. 9 Home advantage in soccer is very important (Clarke and Norman (1995) and Courneya and Cannon (1992)) to such an extent that, in the UK professional game, approximately twice as many wins are recorded by home teams compared with away teams. 10 Actual bookmaker return would vary from 10.5% if the book was not fully balanced, i.e. if liabilities to bettors differed accordingly to which outcome occurred. 11 Positive favourite-longshot bias, noted in home racing markets, is sometimes (tautologically) attributed to risk-loving bettor utility functions. This is not necessarily in contradiction with the analysis here. Team sports are organised so that games seldom have a genuine outsider in the horse racing sense. Unit bets on team sports are therefore very unlikely, relative to the horse racing case, to generate opportunities to lift the bettor to a different range of his utility of wealth function. 33

35 REFERENCES Avery, C., and Chevalier, J., (1999) Identifying investor sentiment from price paths : the case of football betting, Journal of Business, 72: Cain, M., Law, D., and Peel, D., (2000), The favourite-longshot bias and market efficiency in UK football betting, Scottish Journal of Political Economy, 47: Clarke, S., and Norman, J., (1995), Home advantage of individual clubs in English soccer, The Statistician, 44: Courneya, K., and Carron, A., (1992), The home advantage in sports competitions: a literature review, Journal of Sport and Exercise Psychology, 14: Crafts, N., (1985), Some evidence of insider knowledge in horse racing betting in Britain, Economica, 52: Crafts, N., (1994), Winning Systems? Some further evidence on insiders and outsiders in British horse race betting in Britain, in Hausch, Lo and Ziemba, Dixon, M., and Coles, S., (1997), Modelling association football scores and inefficiencies in the UK football betting market, Journal of the Royal Statistical Society, Series C, 46:

36 Dixon, M., and Pope, P., (1996), Inefficiency and bias in the UK association football betting market, University of Lancaster, mimeo. Dobson, S., and Goddard, J., (2001), The economics of football, Cambridge: Cambridge University Press. Global Betting and Gaming Consultants, (2001), 1 st annual review of the Global betting and gaming market, 2001, West Bromwich : Global Betting and Gaming Consultants. Goddard, J., and Asimakopoulos, I., (2001), Forecasting football results and the efficiency of fixed-odds betting, University of Wales Swansea, Department of Economics, mimeo. Golec, J., and Tamarkin, M., (1991), The degree of inefficiency in the football betting market, Journal of Financial Economics, 30: Hausch, D., Lo, V., and Ziemba, W., (1994), Efficiency of racetrack betting markets, San Diego: Academic Press. Jackson, D., (1994), Index betting on sports, The Statistican, 43:

37 Kuypers, T., (2000), Information efficiency : an empirical study of a fixed odds betting market, Applied Economics, 32: Lee, A., (1997), Modelling scores in the Premier League: is Manchester United really the best?, Chance, 10: Maher, M., (1982), Modelling association football scores, Statistica Neerlandica, 36: Mintel Intelligence Report, (2001), Online Betting, London: Mintel International Group Ltd. Pope, P., and Peel, D., (1989), Information, prices and efficiency in a fixed-odds betting market, Economica, 56: Sauer, R., (1998), The economics of wagering markets, Journal of Economic Literature, 36: Sharpe, G., (1997), Gambling on goals: a century of football betting, Edinburgh: Mainstream Publishing Company. Shin, H., (1993), Measuring the incidence of insider trading in a market for state-contingent claims, Economic Journal, 103:

38 Snedecor, G., and Cochran, W., (1967), Statistical Methods, 6 th. Edition, Ames, Iowa: The Iowa State University Press. Thaler, R., and Ziemba, W., (1988), Anomalies-parimutuel betting markets: racetracks and lotteries, Journal of Economic Perspectives, 2: Vaughan Williams, L., (1999), Information efficiency in betting markets: a survey, Bulletin of Economic Research, 51: Woodland, L., and Woodland, B., (1994), Market efficiency and the favourite-longshot bias: the baseball betting market, Journal of Finance, 49: Woodland, L., and Woodland, B., (2001), Market efficiency and profitable wagering in the National Hockey League: can bettors score on longshots?, Southern Economic Journal, 67:

39 Table 1A Regression results all matches Linear probability model estimated by weighted least squares with 20 trials Constant (0) (6) (0) (0) Bookmaker-probability [2] [11] [2] [1] (Home)(Bookmaker-probability) (0) (2) (0) (0) Home (5) (0) (0) (0) Diffattend (20) (20) (7) (0) Sample size Table shows mean coefficient estimates across twenty trials [ ] number of cases where coefficient estimate is significantly different from one ( ) number of cases where coefficient estimate is significantly different from zero 38

40 Table 1B Regression results Premier League and Division One Linear probability model estimated by weighted least squares with 20 trials Constant (2) (0) (0) (0) Bookmaker-probability [5] [1] [0] [0] (Home)(Bookmaker-probability) (4) (2) (1) (1) Home (1) (0) (1) (1) Diffattend (9) (13) (3) (0) Sample size Table shows mean coefficient estimate across twenty trials [ ] number of cases where coefficient estimate is significantly different from one ( ) number of cases where coefficient estimate is significantly different from zero 39

41 Table 1C Regression results Division Two and Three Linear probability model estimated by weighted least squares with 20 trials Constant (0) (4) (0) (0) Bookmaker-probability [1] [4] [0] [1] (Home)(Bookmaker-probability) (1) (1) (0) (0) Home (13) (2) (0) (1) Diffattend (7) (6) (7) (13) Sample size Table shows mean coefficient estimate across twenty trials [ ] number of cases where co-efficient estimate is significantly different from one ( ) number of cases where co-efficient estimate is significantly different from zero 40

42 Table 2 Results of wagering on all available to win bets where Probability of success implied by model Bookmaker probability >1.2 Number of bets, total return Model Model Used: Applied to: All Matches PL/D1 D2/D3 Season Season , % 113, % 192, % , % 22, % 197, % Calculations assume posted odds (no restriction against single bets) and ignore tax. 41

43 A large and convenient collection of such articles is provided by Hausch et al (1994). In the season studied in this paper, 1997/8, the English FA Premiership comprised 20 clubs while Divisions 1 to 3 of the Football League comprised 24 teams each. Each team meets a division rival twice, once home and once away. Movement between divisions is achieved by promotion and relegation. Although the bulk of fixtures are played on Saturday, the number of Saturday games will occasionally be reduced because either the date is earmarked for knockout cup competition or because of international fixtures. A few games each week are moved to different days in order to be televised and betting is still possible in these cases. Pools entries were (and are) made by mail or through a door-to-door collector. An exception is a televised match. Perhaps the exemption is linked to the high-profile nature of the contest chosen for television coverage. Inside information or match fixing are unlikely to present a serious problem if the teams are well-known and the betting volume high. This involves staking on the first bet, on the second and on the third. The over-round on bets is typically 11.7, i.e. one would spend to bet on all three outcomes at a level to earn 100 in pay back from the bookmaker. With a fully balanced book, the bookmaker would return of each staked to earn a commission of 10.5%. A similar model is used by Lee (1997) in examining the final ranking of teams in the 1995/6 English FA Premiership season. 9 Home advantage in soccer is very important (Clarke and Norman (1995) and Courneya and Cannon (1992)) to such an extent that, in the UK professional game, approximately twice as many wins are recorded by home teams compared with away teams. 10 Actual bookmaker return would vary from 10.5% if the book was not fully balanced, i.e. if liabilities to bettors differed accordingly to which outcome occurred. 11 Positive favourite-longshot bias, noted in home racing markets, is sometimes (tautologically) attributed to risk-loving bettor utility functions. This is not necessarily in contradiction with the analysis here. Team sports are organised so that games seldom have a genuine outsider in the horse racing sense. Unit bets on team sports are therefore very unlikely, relative to the horse racing case, to generate opportunities to lift the bettor to a different range of his utility of wealth function. 42