Forecasting horse race outcomes: New evidence on odds bias in UK betting markets

International Journal of Forecasting 26 (2010) 543–550 www.elsevier.com/locate/ijforecast

Forecasting horse race outcomes: New evidence on odds bias in UK betting markets Michael A. Smith a,∗ , Leighton Vaughan Williams b,1 a Leeds Business School, Leeds Metropolitan University, Civic Quarter, Leeds LS1 3HE, United Kingdom b Nottingham Business School, Nottingham Trent University, Burton Street, Nottingham NG1 4BU, United Kingdom

Abstract Using bookmaker odds data derived from ten seasons of Flat Racing in the UK, we confirm the existence of a favouritelongshot bias; i.e., the expected return to level stake bets placed at shorter odds exceeds that to level stake bets placed at longer odds. We find, however, that the degree of bias declines significantly over the period studied, with a lower bias being evident in the five years after 2000 than in the five preceding seasons. These findings have important implications for methods of forecasting race outcomes that include odds as an endogenous variable. In particular, we conclude that the parameters of the bias need to be re-estimated through time in order to generate the most efficient forecasts based on the prevailing odds distributions. c 2010 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

Keywords: Betting exchanges; Bookmakers; Odds; Favourite-longshot bias; Arbitrage; Quasi-arbitrage

1. Introduction In this paper, we investigate the existence and evolution over time of a favourite-longshot bias in UK betting markets, using bookmaker odds data derived from ten seasons of Flat Racing in the UK. We wish to establish whether there has been a change in the level of bias over the period, and hypothesise that the different ∗ Corresponding author. Tel.: +44 113 812 7537; fax: +44 113 812 8604. E-mail addresses: [email protected] (M.A. Smith), [email protected] (L. Vaughan Williams). 1 Tel.: +44 848 6150.

competitive forces impacting on the bookmaking industry, arising in part from Internet technologies and the advent of person-to-person betting market formats, will lead to a reduction in odds biases in bookmaker odds. The results have implications for the accuracy of forecasts of race outcomes based on models which appeal to the distribution of odds and to the degree of bias contained in those odds. The structure of the paper is as follows: Section 2 outlines the pertinent features of UK bookmaker markets; Section 3 reviews the literature on forecasting horse race outcomes; Section 4 outlines the methodology employed to analyse bookmaker odds distributions;

c 2010 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. 0169-2070/$ - see front matter doi:10.1016/j.ijforecast.2009.12.014

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Section 5 presents and discusses the results; and Section 6 concludes. 2. UK bookmaker markets Until relatively recently, high street betting shops were the main outlet for betting activities on horse racing and greyhound racing, aside from the on-course markets. At the end of the 1960s there were some 16,000 betting shops in the UK, many of which were owned by small independent bookmakers (Jones, Turner, Hillier, & Comfort, 2006), but by 2005 this fragmented ownership had become concentrated in the hands of a small number of bookmaker chains possessing considerable market power. The primary chains today are Ladbrokes, Coral, William Hill, the Tote and Betfred. Bookmakers offer prices, or “odds”, capturing the terms of the wager on an event, usually expressed on a fractional basis. For example, an outcome for which the odds are 2 to 1, or “two to one against”, indicates an implied probability of one in three. Since October 2001, when the betting tax was reformed and radically reduced in the UK, bookmakers have not made explicit deductions from winning bets. In almost all markets on horse races, however, the sum of the probabilities offered by a bookmaker will exceed one; the excess over unity is often called the over-round, and represents a proxy measure of the profit margin of the bookmaker. The average size of over-round for the 45,335 races included in this study was 21.61%. However, the over-round is not a direct indicator of margins, as bookmaker prices are not set in strict proportion to the relative amounts traded on runners. There also exists a ‘pool’ (pari-mutuel) system of betting, in which the dividend on each horse in a race is based on the relative volumes staked on runners. This is known as the Totalisator, or ‘Tote’, and resembles the pari-mutuel markets that operate at US racetracks. Unlike bookmakers, the Tote does make explicit deductions, which are currently set at 13.5% of the win pool. Since 2000, betting exchanges, a new market format based on person-to-person Internet betting, have served to provide significant additional competition to bookmakers. Betting exchanges match clients who want to bet on the outcome of an event at a given price with others who are willing to offer that price.

The person who bets on the outcome happening at a given price is the backer, while the one who offers the price is known as the layer. One advantage of this form of betting for the bettor is that it tends to reduce the margins in the odds compared to the best odds on offer with bookmakers. Exchanges also allow clients to back and lay the same event at different times during the course of the market, thus providing opportunities for hedging. 3. Odds as the basis of forecasting horse race outcomes Horse race betting markets exhibit a number of properties that appear to be contrary to the tenets of market efficiency and rationality (see Sauer, 1998 and Vaughan Williams, 1999, 2005, for comprehensive literature reviews). Most obviously, they yield negative returns in aggregate, as has been reported in numerous studies. Further, economists have observed a favourite-longshot bias, whereby favourites are relatively underbet and longshots (runners with a low probability of winning) are relatively overbet. Numerous empirical studies, dating back to Figgis (1951) and including Crafts (1985), Dowie (1976), Shin (1991) and Vaughan Williams and Paton (1997), have found evidence of a favourite-longshot bias in UK bookmaking markets, while Cain, Law, and Peel (2003) discovered a similar bias in a range of other sports where bookmakers set odds. Importantly, forecasters employing odds as an endogenous variable in models used to predict the outcomes of horse races require an accurate estimation of the bias. Shin (1991, 1992, 1993) explained the favouritelongshot bias observed in bookmaking markets as being a consequence of bookmakers’ responses to asymmetric information, where ‘insiders’ had superior information concerning the true probability of a horse winning a race. The empirical consequence was that bookmaker odds for longshots as a class were specifically depressed below the true odds in order to preserve the margins in the face of insider activity. For his dataset, Shin estimated the proportion of the turnover attributable to insiders at over 2%, a result subsequently confirmed for larger datasets using the same methodology (Law & Peel, 2002; Smith, Paton, & Vaughan Williams, 2006; Vaughan Williams & Paton, 1997). Shin and subsequent authors have used

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Shin’s measure of insider trading, known as “Shin’s z”, as a proxy measure for the favourite-longshot bias. In an empirical analysis using matched bookmaker and betting exchange odds for 700 UK horse races, Smith et al. (2006) tested the Hurley and McDonough (1995) explanation for the bias, which postulated that the higher the transaction costs of betting, the greater the favourite-longshot bias will be. They found the favourite-longshot bias in exchange markets to be significantly less pronounced than that in the corresponding bookmaker odds, a result consistent with the Hurley and McDonough hypothesis. Smith and Vaughan Williams (2008) derived a comparable estimate of the exchange bias from a much larger dataset of aggregated odds data acquired directly from Betfair, for the period 2001–2002. The favourite-longshot bias is therefore not a temporary observation confined to a few individual markets, it is a phenomenon of long standing in a range of horse race betting markets. However, the magnitude of the bias differs both from market to market and over time. Changes in the bias affect the accuracy of all forecasting models employing odds data, and it is therefore important for the bias to be estimated accurately. Indeed, a knowledge of the degree of the favourite-longshot bias is key to those classes of predictive models where nominal odds are an independent variable. There are a priori grounds for thinking that the degree of bias in bookmaker odds may have declined in recent years. Firstly, the odds offered on the betting exchanges in horse race markets are greater than bookmaker odds on average, particularly in the case of low probability runners. On these grounds, we might expect the degree of bias observed in studies of bookmaker markets prior to the advent of betting exchanges to have eroded since their inception, as bookmakers strive to remain competitive with the exchanges. Secondly, an intensification of direct competition between bookmakers themselves via online betting has also occurred during this period, which might reasonably be anticipated to have had a similar effect. Finally, as noted above, recent research suggests that there is a positive relationship between transaction costs and the degree of bias in horse race betting markets. As bookmaker deductions were abolished in 2001, we might therefore expect to see a reduction in bias as a consequence of the reduction in transaction costs. The

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main aim of this study is to substantiate this expectation by identifying significant changes in the favouritelongshot bias contained in bookmaker odds. 4. Data and methodology In order to control for the increase in competition generated by the betting exchanges, the odds data studied here were divided into two sub-samples: the first included races run during the period 1996–2000, the five years prior to the inception of betting exchanges,2 while the second covered the races run during the period 2001–2005, encompassing the first five years of betting exchange operations. The data employed were starting price (SP)3 odds for horses in UK Flat Races contested during the years in question, and were acquired from the Raceform Interactive database. Finishing positions were recorded from the results in order to compute objective probabilities for each odds value. We used a methodology adapted from Coleman (2004), and more recently employed by Peirson and Smith (2008), to measure the bias in the odds sets studied here. For the two sub-samples of races, odds values were expressed as their corresponding probability, equivalent to the price of a betting ticket to yield a return of £1 at the respective odds. The differences between the objective probabilities for each price, estimated directly from the observed ratios of winners to runners, are binomial variables, and, by the Central Limit Theorem, are normally distributed under the null hypothesis when the number of observations is sufficiently large. For each price category, a test statistic α j was calculated: p1 j − p2 j αj = √ , s1 j + s2 j

(1)

where p1 is the observed probability, derived from 2 In fact, Betfair, the world’s largest betting exchange, came into being in August 2000; however, that year was included in the first subset on the grounds that Betfair did not build significant market liquidity until 2001. Repeating the tests reported here with this period of overlap omitted has no material effect on the results. 3 The starting price, SP, is a unique odds value for each horse which is determined by official on-course odds inspectors, and at which winning bets are settled in the absence of a specified fixedodds value agreed between bookmaker and bettor.

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0

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-8

-6

-4

-2

sample race results, of winning in sub-sample 1 races (1996–2000) when the nominal ticket price is j; p2 is the corresponding probability for sub-sample 2 races (2001–2005); and s is the standard deviation of the outcome of the binomial trial associated with odds value j, where: s
pi j 1 − pi j si j = (2) ni j

-4

O=

1 − 1. j

-2

0

2

4

6

lnodds

with i denoting sub-sample (1, 2), and n i j being the number of runners at price j in sample i. Eq. (3) shows the relationship between the ticket price j and the associated odds value, O:

Insubset1 benchmark

Insubset2

Fig. 1. The favourite-longshot bias. -1

(3)

-1.2

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

-1.4 -1.6 LN p, LN j

Where one or both of the observed probabilities associated with a price in either sub-sample did not meet the large sample condition for the comparison of binomial probability means, 1 > pi j ± 3si j > 0, the category was omitted. On this basis, fifty-one of the eighty-nine original prices occurring in the data were retained for the study. The excluded categories had very few observations, mostly with one or two cases (for example odds of 1/25), all from the very high probability categories (short priced favourites). The shortest odds to pass the large sample rule were 2 to 7, with 51 and 37 observations in sub-sets 1 and 2 respectively. In contrast, the category with the most observations to fail the above rule, 38 and 33, corresponded to odds of 1 to 4. The test statistic α j in Eq. (1) is a standard normal variate. The interpretation of the statistic is that if it is negative and significant for a particular price category, this would provide evidence that the objective probability at the corresponding odds value had increased. If the relevant odds were subject to a conventional bias in the first period, this result would indicate a reduction in bias during the second period. The odds categorisation employed for this method also yielded a measure of bias in relation to individual odds categories within each sub-sample, permitting a clear picture of the distribution  of bias  over the range pi of odds. It was calculated as 1− j , which is simply the proportional difference between the subjective probabilities j (implied by the odds) and the objective probabilities p for each ticket price. We subsequently

-1.8 -2

LN p LN j

-2.2 -2.4 -2.6 -2.8

Linear (LN j) Linear (LN p)

-3 LN odds

Fig. 2. Objective and subjective probabilities by odds, sub-sample 1.

use the term “odds error” (OE) for this proportional difference in probabilities, with positive values of OE indicating that the subjective probabilities exceed the corresponding observed probabilities, i.e. j > p. The favourite-longshot bias is the tendency for the odds error to increase in magnitude, with a positive sign, at high odds values.

5. Results Our results are summarised in Table 1, which shows, for each sub-set, the subjective probability, j; the number of horses observed, N ; and the objective probability p, derived from the results, for each odds category, arranged in descending order. We begin by charting some of the more notable features of the structure of odds in the two time periods in Figs. 1–3, before discussing their statistical significance. From Table 1, with a single exception in each subsample, odds error values are positive, indicating that

M.A. Smith, L. Vaughan Williams / International Journal of Forecasting 26 (2010) 543–550 -1 -1.2

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

-1.4 LN p, LN j

-1.6 -1.8

LN p

-2 -2.2 -2.4 -2.6 -2.8

LN j Linear (LN j) Linear (LN p)

-3 LN odds

Fig. 3. Objective and subjective probabilities by odds, sub-sample 2.

the nominal odds overstate the true chances of runners across the range; this result is consistent with the aggregate negative returns observed in such markets. The favourite-longshot bias refers to the further observation that the degree to which odds overstate the true probabilities tends to be greater for longshots than for high probability winners. If the favourite-longshot bias is present in our data, therefore, we would expect the odds error values in our results to be greatest in the longshot categories. For each of the sub-samples, Fig. 1 charts p as the dependent variable, with odds as the independent variable, both of which are expressed as their natural logarithm to aid exposition. A natural benchmark for judging the degree of bias is the log of probabilities associated with the odds corresponding to the horizontal axis. Although the relationship between j and p is not monotonic, Fig. 1 confirms the presence of the classic favourite-longshot bias in each sub-sample. This is evidenced by the increasing divergence from the benchmark function as odds values increase, so that longshots’ chances are the most overstated by the odds. The divergence at high odds is more pronounced in the case of sub-sample 1, however, consistent with our hypothesis that the favourite-longshot bias will be lower in the second period. Studies of the favourite-longshot bias in parimutuel markets invariably add back track and tax deductions from the winning returns in order to establish the odds value at which the underlying normalised nominal probabilities begin to overestimate the true chances (Snyder, 1978, reviews six studies of this type). Our comparable adjustment involves scaling up p by the mean value of the over-round for each of the two datasets, 23.26% and 20.16% respectively. Figs. 2 and 3 show a truncated range of the relationships depicted in Fig. 1, for sub-samples 1 and 2

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respectively, after scaling the probabilities for overround. For the purpose of comparison, trend lines are fitted to the log jand benchmark functions. An absence of bias would require that the two lines coincide. However, in contrast, Figs. 2 and 3 suggest that a cross-over point of the log p and benchmark functions occurs at the approximate log odds of 1.9 (7/1), suggesting in both cases that odds greater than 7/1 represent increasingly poor value to bettors. Overall, the charts support a conclusion that the favourite-longshot bias is present in both datasets (all three figures), but that the magnitude of the bias in the longshot categories is less in the latter period (Fig. 1). A more formal test for a change in bias over the period surveyed was carried out by employing the methodology adapted from Coleman (2004), as outlined in Section 4. For the periods 1996–2000 and 2001–2005, the odds errors associated with the fifty-one individual odds values meeting the large sample condition were computed, along with the test statistic in Eq. (1). The arithmetic average of the fifty-one odds error values for the period 1996–2000 is 0.177. The corresponding average for 2001–2005 is 0.153, representing a 13.17% decrease in the overall error. For the ten lowest probability categories, the arithmetic average of the odds error values decreases by 17.36%, from 0.475 to 0.392. On the same basis, a 12.5% reduction in odds error over the period is also evident in the ten highest probability categories, with the average error in this odds range being 0.072 in the former period and 0.063 in the period 2001–2005. In order to establish the statistical significance of these trends, we now consider the test statistic. The statistic for the fifty-one odds categories in aggregate4 was –3.53, which is significant at the 99% confidence level, suggesting that the probabilities corresponding to the nominal odds in the later period approximated the outcomes of races more closely. Furthermore, eight of the nine individual test statistic values significant at the 90% level or better are negative, and seven of these are found in the odds categories 9/1 or more, indicating that the majority of the decrease in odds error measured on this basis is attributable to the nominal odds for low probability runners, or longshots. The

4 This was calculated as the sum of the individual values divided by the square root of the number of categories.

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Table 1 Bias and tests for a change in bias for two sub-samples. Odds (O) 66/1 50/1 40/1 33/1 25/1 20/1 16/1 14/1 12/1 11/1 10/1 9/1 8/1 15/2 7/1 13/2 6/1 11/2 5/1 9/2 4/1 7/2 100/30 3/1 11/4 5/2 9/4 2/1 15/8 7/4 13/8 6/4 11/8 5/4 6/5 11/10 1/1 10/11 5/6 4/5 8/11 4/6 8/13 4/7 8/15 1/2 4/9 2/5 4/11

J 0.015 0.020 0.024 0.029 0.038 0.048 0.059 0.067 0.077 0.083 0.091 0.100 0.111 0.118 0.125 0.133 0.143 0.154 0.167 0.182 0.200 0.222 0.231 0.250 0.267 0.286 0.308 0.333 0.348 0.364 0.381 0.400 0.421 0.444 0.455 0.476 0.500 0.524 0.545 0.556 0.579 0.600 0.619 0.636 0.652 0.667 0.692 0.714 0.733

1996–2000 N

p

Odds error

2001–2005 N

p

Odds error

Test statistic

3,219 9,661 2,934 18,349 15,329 19,142 16,242 15,476 14,656 4,121 13,626 7,190 11,023 1,829 8,971 3,739 8,109 4,928 7,334 5,493 6,245 5,012 1,628 4,163 2,417 2,745 2,418 2,265 652 1,448 793 1,035 703 680 240 472 582 401 189 359 336 290 163 188 69 184 134 99 52

0.002 0.005 0.009 0.013 0.020 0.027 0.038 0.045 0.051 0.061 0.071 0.076 0.088 0.103 0.099 0.120 0.123 0.133 0.146 0.164 0.172 0.194 0.213 0.221 0.252 0.266 0.275 0.293 0.333 0.353 0.356 0.371 0.388 0.441 0.408 0.426 0.491 0.521 0.444 0.493 0.548 0.552 0.558 0.559 0.551 0.658 0.627 0.737 0.712

0.833 0.768 0.637 0.561 0.491 0.431 0.355 0.320 0.335 0.272 0.216 0.243 0.208 0.122 0.207 0.103 0.138 0.136 0.126 0.097 0.141 0.128 0.079 0.114 0.057 0.070 0.105 0.122 0.043 0.030 0.067 0.072 0.078 0.007 0.102 0.106 0.017 0.005 0.185 0.113 0.054 0.080 0.098 0.122 0.156 0.014 0.095 −0.032 0.030

8,747 13,054 6,051 20,479 19,278 21,105 19,242 17,051 16,745 4,611 14,356 7,844 12,746 1,921 10,758 4,278 8,675 5,929 8,029 6,637 6,992 6,301 1,838 4,405 2,777 2,821 2,319 2,229 756 1,453 778 941 661 684 244 457 542 411 220 375 320 331 207 201 69 161 141 116 80

0.005 0.007 0.011 0.015 0.023 0.033 0.039 0.048 0.059 0.063 0.073 0.084 0.090 0.088 0.103 0.118 0.117 0.140 0.149 0.170 0.178 0.204 0.194 0.232 0.259 0.262 0.274 0.301 0.325 0.318 0.380 0.370 0.374 0.431 0.414 0.438 0.487 0.487 0.477 0.544 0.588 0.559 0.541 0.557 0.710 0.571 0.660 0.647 0.650

0.663 0.633 0.539 0.480 0.400 0.305 0.333 0.283 0.233 0.245 0.199 0.160 0.189 0.252 0.176 0.115 0.184 0.090 0.105 0.067 0.110 0.084 0.158 0.072 0.028 0.082 0.109 0.096 0.064 0.126 0.001 0.075 0.113 0.030 0.089 0.081 0.026 0.071 0.125 0.021 −0.015 0.068 0.126 0.124 −0.089 0.143 0.047 0.095 0.114

−2.197** −2.624*** −1.081 −1.980** −2.257** −3.504*** −0.639 −1.098 −3.041*** −0.432 −0.495 −1.890* −0.580 1.598 −0.905 0.208 1.326 −1.070 −0.605 −0.801 −0.923 −1.310 1.335 −1.164 −0.637 0.275 0.091 −0.643 0.296 1.994** −1.021 0.055 0.557 0.368 −0.125 −0.363 0.145 0.986 −0.665 −1.383 −1.032 −0.180 0.331 0.026 −1.967** 1.645 −0.566 1.449 0.747

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M.A. Smith, L. Vaughan Williams / International Journal of Forecasting 26 (2010) 543–550 Table 1 (continued) Odds (O) 1/3 2/7

J 0.750 0.778

1996–2000 N 88 51

p 0.682 0.725

Odds error

2001–2005 N

0.091 0.067

58 43

Test statistic p

Odds error

0.759 0.767

−0.011 0.013

−1.024 −0.467

Betting ticket prices are denoted by J and objective probabilities by p; N is the number of races in the sample. Odds error is measured as 1 − ( pi /j). Positive values indicate that the odds overstate the true chances of the corresponding category of runners. Of the eighty-nine original prices occurring in the data, thirty-eight were omitted due to low frequency, i.e. they did not meet the criteria for binomial large samples (see narrative). Fifty-one price categories remained. ∗ Significant at the 10% level. ∗∗ Significant at the 5% level. ∗∗∗ Significant at the 1% level.

0 -.2

probchange

.2

Lowess smoother

-.4

conclusion that odds error has decreased in the longshot categories appears to be the more robust because of the concentration of significant negative test values in the lower probability categories. A binomial sign test supports this conclusion further: of the twenty-six test values in the top half of the table (higher odds), twenty-one were negative, with a statistical probability of 0.001. In contrast, twelve of the remaining categories yielded negative test statistics which were insignificant in a sign test. Finally, the Lowess smoothing algorithm (Cleveland & Devlin, 1988) was employed to derive a visual confirmation of the above results, with the relative probability change between the two periods as the dependent variable and the associated price categories as the independent variable. Lowess is a nonparametric technique that performs a locally weighted regression and has the virtue of not assuming a specific functional form. The proximity of local values drawn into the regression is specified by the user; the degree of smoothing is depicted here with a 0.8 bandwidth. The results of the Lowess regression are shown in Fig. 4 and support the inference drawn above, that odds error in the longshot price categories has been reduced in the period associated with sub-sample 2 (2001–2005). In contrast, the relative probability change associated with higher probability categories (corresponding to favourites and near favourites) approximates zero. Collectively, the results support an inference that the favourite-longshot bias, whereby odds overstate winning chances and the odds error is greater in the longshot categories, has diminished in magnitude between the two time periods. The structural insights evident in Table 1 and Figs. 1 to 4 are valuable for forecasting models, as the varying expected values associated with the range

0

.2

.4

.6

.8

price bandwidth = .8

Fig. 4. Relative probability change between the periods 1996–2000 and 2001–2005 as a function of price. Notes: The independent variable probchange is the difference in observed probabilities for each price category between the two periods, divided by the corresponding weighted average of the two probabilities. A negative value of probchange for a price category indicates an increase in objective probability in the second period.

of odds becomes an important indicator of the degree of predictive advantage that a model’s forecasts must yield in order to beat the market. 6. Conclusions This paper presents empirical evidence of a change over time in the structure of nominal bookmaker odds in UK betting markets, notably a reduction in the favourite-longshot bias over the period surveyed. We argue that these results have clear and important implications for forecasts of race outcomes based on the employment of bookmaker odds as an explanatory variable, and specifically for models in which stated odds biases are instrumental in determining true probabilities. It is not certain, however, whether the identified trend will continue, since bookmakers have increasingly become active users of betting exchanges

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for hedging purposes, and the resulting impact of this development on odds biases remains to be seen. We believe, therefore, that further work might usefully focus on the functional formulation and estimation of these odds biases and how they are likely to change over time. References Cain, M., Law, D., & Peel, D. A. (2003). The favourite-longshot bias, bookmaker margins and insider trading in a variety of betting markets. Bulletin of Economic Research, 55, 263–273. Cleveland, W. S., & Devlin, S. J. (1988). Locally-weighted regression: An approach to regression analysis by local fitting. Journal of the American Statistical Association, 83, 596–610. Coleman, L. (2004). New light on the longshot bias. Applied Economics, 36(4), 315–326. Crafts, N. F. R. (1985). Some evidence of insider knowledge in horse race betting in Britain. Economica, 52, 295–304. Dowie, J. (1976). On the efficiency and equity of betting markets. Economica, 43, 139–150. Figgis, E. L. (1951). Focus on gambling. London: Barker. Hurley, W., & McDonough, L. (1995). A note on the Hayek hypothesis and the favourite-longshot bias in parimutuel betting. American Economic Review, 85(4), 949–955. Jones, P., Turner, D., Hillier, D., & Comfort, D. (2006). New business models and the regulatory state: A retail case study of betting exchanges. Innovative Marketing, 3, 112–119. Law, D., & Peel, D. A. (2002). Insider trading, herding behaviour, and market plungers in the British horse race betting market. Economica, 69, 327–338. Peirson, J., & Smith, M. A. (2008). Expert analysis and insider information in horserace betting: Regulating informed market behaviour. University of Kent, Department of Economics Discussion paper, KDPE 0819. Sauer, R. D. (1998). The economics of wagering markets. Journal of Economic Literature, 36, 2021–2064.

Shin, H. S. (1991). Optimal betting odds against insider traders. Economic Journal, 101, 1179–1185. Shin, H. S. (1992). Prices of state contingent claims with insider traders, and the favourite-longshot bias. Economic Journal, 102, 426–435. Shin, H. S. (1993). Measuring the incidence of insider trading in a market for state-contingent claims. Economic Journal, 103, 1141–1153. Smith, M. A., Paton, D., & Vaughan Williams, L. (2006). Market efficiency in person-to-person betting. Economica, 73, 673–689. Smith, M. A., & Vaughan Williams, L. (2008). Betting exchanges: A technological revolution in sports betting. In W. T. Ziemba (Ed.), Handbook of sports and lottery markets. New York: North Holland. Snyder, W. W. (1978). Horse racing: Testing the efficient markets model. Journal of Finance, 33(4), 1109–1118. Vaughan Williams, L., & Paton, D. (1997). Why is there a favouritelongshot bias in British racetrack betting markets? Economic Journal, 107, 150–158. Vaughan Williams, L. (1999). Information efficiency in betting markets. Bulletin of Economic Research, 51(1), 1–30. Vaughan Williams, L. (Ed.) (2005). Information efficiency in financial and betting markets. Cambridge: Cambridge University Press.

Michael Smith is Senior Lecturer in Economics at Leeds Business School. His Ph.D. was awarded by Nottingham Trent University, and was based on an empirical study of information efficiency in a range of betting markets. His current research interests centre on the structure and dynamics of betting exchanges and bookmaker markets.

Leighton Vaughan Williams is Professor of Economics and Finance and Director of the Betting Research Unit at Nottingham Business School, Nottingham Trent University. He advises the UK Government on the taxation and regulation of betting and gaming, and has published extensively in the fields of risk, asymmetric information, and financial and betting markets.