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Ausfral. J. Statist., 23 (3), 1981,287-295 ANOMALIES IN THE GAMBLING MARKET’ R. H. TUCKWELL Macquarie University S-ary Apparent irregularities between the win and the place betting markets in Australian horseracing, are examined. Win odds are used to predict win probabilities from which place probabilities are estimated and compared with the place odds on offer. It is concluded that anomalies do in fact exist and are capable, in theory at least, of profitable exploitation. 1. Introduction This paper is concerned with what appear to be anomalies be- tween the win and the place betting markets in Sydney and Melbourne horseracing. The relationship between the odds quoted for a win and those quoted for a place frequently appear to be inconsistent. For example, two horses in the same race may both be quoted at seven to one for a win but at markedly different odds for a place. One horse may be even money for a place while the other is two to one. Alternatively, two horses may have the same place odds but quite different odds for a win. It is posible that these apparent inconsisten- cies are justified by the fact that some horses are more reliable, or consistent, than others and that this superior reliability raises their chances of running a place relative to their winning chances. The paper examines whether the “anomalies” are justified or not, concludes that they are not and then proceeds to determine whether the inconsisten- cies are sufficiently large to be capable of profitable exploitation. It was found that a profitable betting strategy is possible but that its effective implementation is inhibited by one or two, not -. insuperable, practical problems. In more detail, the approach employed took the following lines. A large sample of bookmakers’ starting price odds was collected and the proportion of winners at each of a number of odds ranges noted. Estimates of the probability of a horse winning at each odds range were then derived. On the assumption that the apparent anomalies are not justified and that the win probabilities of all horses in a race are all Manuscript received August 4, 1980; revised July 20, 1981.

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Page 1: Anomalies in the Gambling Market

Ausfral. J. Statist., 23 (3), 1981, 287-295

ANOMALIES IN THE GAMBLING MARKET’

R. H. TUCKWELL Macquarie University

S-ary

Apparent irregularities between the win and the place betting markets in Australian horseracing, are examined. Win odds are used to predict win probabilities from which place probabilities are estimated and compared with the place odds on offer. It is concluded that anomalies do in fact exist and are capable, in theory at least, of profitable exploitation.

1. Introduction

This paper is concerned with what appear to be anomalies be- tween the win and the place betting markets in Sydney and Melbourne horseracing. The relationship between the odds quoted for a win and those quoted for a place frequently appear to be inconsistent. For example, two horses in the same race may both be quoted at seven to one for a win but at markedly different odds for a place. One horse may be even money for a place while the other is two to one. Alternatively, two horses may have the same place odds but quite different odds for a win. It is posible that these apparent inconsisten- cies are justified by the fact that some horses are more reliable, or consistent, than others and that this superior reliability raises their chances of running a place relative to their winning chances. The paper examines whether the “anomalies” are justified or not, concludes that they are not and then proceeds to determine whether the inconsisten- cies are sufficiently large to be capable of profitable exploitation. It was found that a profitable betting strategy is possible but that its effective implementation is inhibited by one or two, not -. insuperable, practical problems.

In more detail, the approach employed took the following lines. A large sample of bookmakers’ starting price odds was collected and the proportion of winners at each of a number of odds ranges noted. Estimates of the probability of a horse winning at each odds range were then derived. On the assumption that the apparent anomalies are not justified and that the win probabilities of all horses in a race are all

’ Manuscript received August 4, 1980; revised July 20, 1981.

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288 R. H. TUCKWEU.

that is needed to determine the probabilities of running a place, the place probabilities were computed for 3,849 horses in 286 races in Sydney and Melbourne over the period August-November 1975. These estimated place probabilities were then tested for accuracy by comparing the proportion of horses running a place with the predicted probability over a number of categories of predicted probabilities. The estimates were found to be reasonably accurate, suggesting that the apparent anomalies are in fact anomalies. The scope for profitable exploitation was examined by comparing the odds-equivalent of the estimated place probabilities with the odds for a place on the to- talisator at starting time. Of the sample of 3,849 horses, 874 (almost 23 per cent) started at place odds yielding a positive expected return. The expected rate of return on outlay was approximately 17 per cent and the actual rate of return, if the strategy had been effectively implemented over the sample period, would have been somewhat higher at approximately 20 per cent.

2. The Gambling Market

There are two betting mediums on Australian courses, bookmak- ers and the totalisator (the “tote”). Bookmakers usually only bet for a win and/or each-way, while the tote runs separate pools for win and place only betting. Describing the tote operation first, the sums wag- ered on each horse in a race are pooled, a certain percentage deducted (approximately fourteen per cent, fifteen per cent in some States) and the remainder is distributed to those who bet on the winner. In the case of place betting, the pool, net of tax, is distributed equally between the first three place-getters. The tote odds cannot be deter- mined with any certainty until all betting ceases just prior to race-start. The tote board, however, does show how the odds are progressing.

In addition to the tote, there are a large number of bookmakers who operate on course. Each bookmaker has his own board which lists his odds on offer for each horse. As in the case of the tote, these odds continually change up until starting time depending on the relative weight of money and also, in this case, on bookmakers’ opinions. Unlike the tote operation, punters betting with a bookmaker receive the odds currently quoted on his board-not the starting price odds as in the case of the tote. Most serious punters consequently bet with bookmakers as they know precisely what odds they are getting and can take advantage of favourable fluctuations in the odds during the course of betting. If, however, a punter finds himself in the position of still considering a bet very close to starting time, the odds on the tote will be known reasonably accurately, and if the bet is not of sufficient size to unduly depress the odds, he would of come bet on the tote if the odds are better than with bookmakers. This type of activity ensures that the tote odds are brought into reasonably close line with the

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ANOMALIES IN THE G&LING MARKET 289

starting price odds offered by bookmakers. In this respect, observation suggests that operators exist whose specific objective it is to take advantage of discrepancies between the two sets of odds. Similar operators also take advantage of place odds on the tote which are regarded as high in relation to the corresponding win odds. But the criterion of what is high is more ditficult to determine in this case and the conclusions reached here suggest that they are by no means exhausting all the opportunities for profit.

3. Starting Price Odds and W h g Probability

In estimating win probabilities, bookmakers’ starting price odds were used, since an examination of several hundred odds movements suggested that the final odds, as opposed to any earlier quotes, provide the more accurate indication of a horse’s chances. The published starting price odds (SPO) for all horses running in Sydney and Mel- bourne metropolitan meetings during the calendar year of 1974 were collected. In all, there were just under sixteen thousand observations. The proportion of winners at each level of SPO was noted. When there were very few runners at a particular level of SPO (usually the very short odds) and where the proportion of winners was zero, or close to it (some of the very long odds), neighbouring odds categories were aggregated. Win probabilities were estimated by first estimating the percentage loss associated with each level of SPO, or its probability equivalent (p*), and substituting these estimates into (l), which relates the probability of winning to the percentage loss.

6 =p*(l-i/lOO), (1) where

= estimated win probability, p* =probability equivalent of SPO = 1/(1 +SPO), i =estimated percentage loss.

The estimated percentage loss was derived by regressing the sample percentage loss ( L ) on p*. The sample percentage loss is the percen- tage of outlay that would have been lost by betting one unit on each runner at the particular level of SPO in the sample. L was regressed on p* using alternative degrees of polynomial and weighting each obser- vation by the standard error of L. A degree three polynomial was found to be the highest warranted. The estimated relation appears below with 1-values in parentheses. Sydney and Melbourne races were aggregated as no significant difference was found when the relationship was estimated separately for each city.

f = 50.3 - 362*7p* + 9 7 8 . 8 ~ * ~ - 777*3p*’. RZ = 0.44 (2)

(7.1) (3-1) (3.1) (2-6)

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2 90 R. H. TUCKWELL

Substituting (2) into (1) gives

6 = 0*497p* + 3*627p** - 9 * 7 8 8 ~ * ~ + 7 * 7 7 3 ~ * ~ . (3) Relation (3) was used to predict the win probabilities for the sample of 3,849 over the period August-November 1975. The estimates were adjusted, where necessary, to ensure that the win probabilities in each race summed to unity.

The predicted percentage losses at different levels of SPO (from relation (3)) are of interest for their own sake and are reproduced in Table 1.

TAEILE 1 Predicted percentage loss and starring price odds

~~

SPO 2 SPO 2

0-41 0-67 0.73 0.80 0-90 1 .oo 1.11 1.25 1.38 1.50 1.63 1.75 1.88 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.50

8.24 17.15 17.55 17.61 17.21 16.49 15.50 14-20 13.09 12.09 11.21 10.46 9.85 9.37 8-73 8.45 8.45 8.66 9.03 9.51 10.07 10-70 12.04

5.00 5-50 6.00 6.50 7.00 7.50 8.00 9-00 10-00 11.00 12.00 14.00 15-00 16.00 20.00 25.00 32.91 40-00 50.00 66-00 98.73 187.30

13.44 14.84 16.20 17.50 18.74 19.91 21.02 2344 24.84 26.43 27.84 30-24 31.27 32.20 35.17 37.76 4044 42.03 43-56 45.10 46.76 48.40

From a level of approximately 16 to 17 per cent for SPO in the odds-on categories, the percentage loss gradually falls as SPO rises to be a minimum of approximately 8.5 per cent when SPO is in the vicinity of three to one. Thereafter the percentage loss rises with SPO, until the loss is almost 50 per cent with SPO of the order of one hundred to one and above. Scott (1978), page 142, obtained a similar type of relationship using Sydney data for the 1950’s.

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ANOMALIES M THE GAMBLING MARKET 29 1

4. Predicting Place Probabilities from Win Probabilities

If there are n horses in a race, let k,, k2 and k3 represent the probabilities of the kth horse coming first, second and third, respec- tively, where k takes the values 1 , 2 , 3 , . . . , n.

The probability of the kth horse running a place is (k, + k, + k3). k , has been estimated. Estimates of k, and k, are required. Dealing with k2 first, if the jth horse wins the race, the probability that the kth horse runs second is the probability of it beating the remaining horses. This is assumed to be kl/(l-jl). The weighted sum of these prob- abilities over all j (jf k), where the weights are jl, gives the required k2.

kz = c j l ( k J ( 1 -jJ)- kl (kJ(1 - kJ). n

j-1

Turning to k,, if the ith and jth horses occupy the first two places, the probability of the kth horse coming third is the probability of it beating the remaining horses, which is assumed to be k,/(l - i, - j , ) . The probability of the ith and jth horses occupying the fust two places is assumed to be il(jl/(l - i,))+ jl(il/(l -jl)), which simplifies to i,j,/(l - i l)+ iljl(l - j ,) . The probability of the kth horse coming third is the weighted sum of kJ(1- i, -jl) over all possible pairs of i and j (i# k and J# k) where the weights are ~ ~ ~ l / ~ l - i l ) + i l j l / ( l - j ~ ) .

The term subtracted in square brackets eliminates those cases where i = k a n d j = k .

The derivation of the above formula assumes that the win prob- abilities provide sufficient information for the assignment of prob- abilities to other ranks (second and third in this case). This may not be so. For example, reliable horses may have a higher chance of running a place relative to their winning changes than unreliable horses. For the formula to be correct, it has to be assumed that the probability of a horse beating any subset of horses in the race is the same irrespective of where the other horses (those not in the subset) finish. Some idea of the extent to which this assumption is transgressed may be gained by applying the formula to the sample 3,849 horses over the August- November period 1975. The estimated place probabilities were sub- divided into ten different ranges and the average probability of placing within each range was compared with the proportion of horses running a place. This was done for all horses within each probability range and

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292 R. H. TUCKWELL

also for two classes, (1) where the oddsequivalent of the predicted place probability was exceeded by the place odds on the tote (which were noted just prior to race start at Sydney meetings) and the expected return was positive and (2) where the odds-equivalent of the estimated place probability exceeded the tote place odds and the expected return was negative. The results appear in Table 2. At the

TABLE 2 Comparison of estimated place probabilities with proportion placing

Range of estimated Positive expected Negative expected place probability return return Ail

X

0.0 and under 0-03 y n

X

0.03 and under 0-05 y n

X

0-05 and under 0-08 y n

X

0-08 and under 0.13 y n

X

0-13 and under 0.20 y n

X

0-20 and under 0.30 y n

X

0-30 and under 0.40 y n

X

0.40 and under 0.60 y n

X

0.60 and under 0.80 y n

X

0.80 and under 1-00 y n

0-019 0.0 30

0.042 0-087(0.059) 23

0.066 0*136(0.073) 22

0.100 0*114(0-048) 44

0.165 0.18 l(0.045) 72

0.240 0-231(0.044) 91

0.352 0.337(0.046) 104

0.497 0*438(0.032) 235

0-698* 0-604(0-037) 174

0.888 0-873(0.037) 79

0.017 0*025(0-006) 829

0-039 0.054(0-015) 239

0.064* 0*108(0-017) 324 0.102 0- 127(0.018) 355

0.163 0- 191(0-020) 397

0.248 0-291(0*025) 323

0-343 0-360(0-034) 200

0.484 0-456(0-033) 226

0*667* 0*452(0463) 62

0-867 0-750(0-097) 20

0.017 0~025(0-005) 859

0.040 0*057(0-014) 262

0464* 0*110(0.017) 346 0.102 0*125(0.017) 399

0.163 0-190(0-018) 469

0.248 0*278(0-022) 414

0.346 0-352(0.027) 304 0-492 0*447(0.024) 461

0-690* 0*564(0-032) 236

0-884 0*849(0-036) 99

x = average estimated place probability in range y =proportion of horses running a place (standard errors in parentheses) n =number of horses * denotes a significant ditference at the 5 per cent level

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ANOMALIES IN THE GAMBLING MARKET 293

five per cent significance level the estimated place probability differs significantly from the proportion placing in only five cases out of a possible thirty. There appears, nevertheless, to be a consistent ten- dency for the formula to under-estimate the probability of placing for the low probabilities and to overestimate it for the high probabilities. This tendency, however, appears to be shared by both of the classes and the apparent lack of difference between the results for the two classes suggests that the assumptions made in estimating the place probabilities may well be justified; that is, that differences between horses in reliability, or consistency, are relatively unimportant for present purposes, If reliability is an important factor in determining place probabilities, the formula will over-estimate the place prob- abilities of the unreliable horses and under-estimate the place prob- abilities of the more reliable ones. This will tend to push the unreliable horses into class (l), where the expected return is positive, and the reliable horses into class (2), where the expected return is negative. The fact that there is no obvious and consistent overestimation of the place probabilities in class (1) and under-estimation in class (2) lends support to the view that reliability is a relatively unimportant factor. Regressing the proportion placing on the predicted place probability, and weighting by the standard error of the proportion placing, for each of the two classes yielded the following results.

Expected Return Positive 9 =0*0052+0-955 x

(0-006)(0-029) (standard errors in parentheses)

R2 = 0-984

Expected Return Negative 9 =0-0151+0-928~ R2 = 0.752

(0*008)(0*067) The constant terms are both positive and the coefficients of x less

than unity which reflects the tendency for the low probabilities to be under-estimated and the high probabilities to be over-estimated. How- ever, at the five per cent level, neither of the constant terms differs significantly from zero and neither of the coefficients of x differs significantly from unity. Perhaps more importantly, the regressions do not d 8 e r significantly from each other. Neither the intercepts nor the slopes differ significantly.

5. FeasibiJity of Place-betbg Slrategy

One can only conclude from the above that the apparent inconsis- tencies between the win and place betting markets are not justified and that they are in effect genuine anomalies, capable, in theory at least, of

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294 R. H. TUCKWELL

profitable exploitation as evidenced by the fact that 874 (almost 23 per cent) of the sample of 3,849 horses fall in the category where the expected return is positive. The proportion of horses showing a posi- tive expected return is considerably lower than the average of 23 per cent for the low probabilities and considerably higher for the high probabilities. In the highest place probability range (0-8 to 1.0) almost 80 per cent of horses show a positive expected return which compares with only 12 per cent for horses with estimated place probabilities of less than 0.2. This may be partly due to the tendency to under- estimate the place chances for the low probabilities and to over- estimate them for the high probabilities. However, it seems more likely that it is a consequence of place bettors, not renowned for their expertise, putting relatively too much money on the long-shots in the hope of a high reward. Operating the place-betting strategy in retro- spect over: the sample period, a bet of one unit on each of the 874 selections would have yielded a profit of 177 units, equivalent to a rate of return on outlay of just over 20 per cent. The expected rate of return was somewhat lower at approximately 17 per cent.

In view of the fact that the majority of bets came from the high probability categories and there was a tendency to overestimate these probabilities, it might be expected that the actual rate of return would be below the expected rate of return, not above it. The high actual rate of return appears to have been due to some unusually favourable results in the low probability categories. Table 3 gives the details for each of the ten probability ranges.

In general, the expected rate of return exceeds the actual rate of return for the high probabilities and is below the actual rate of return for the low probabilities. In the overall figures, however, the numerical dominance of the high probabilities is outweighted by the extremely high actual returns, in relation to expected returns, for the low proba- bility categories 2, 3 and 4. The number of observations in these categories is not large and the very high returns appear to have been the result of a combination of chance occurrences, (1) an unusually high proportion of place-getters in categories 2 and 3, even allowing for some under-estimation (see Table 2) and (2) an unusually high return on the place-getters in categories 3 and 4, which is evident from a comparison of the place odds, or their equivalent probabilities, with the estimated place probabilities in Table 3.

In the longer run the actual rate of return can be expected to fall below the expected rate of return. However, the effective operation of such a place betting strategy is inhibited by two factors. First, book- makers’ starting price odds are required to estimate place probabilities. By definition, starting price odds are not known until starting time. What is required is a programmable hand-operated calculator capable of performing the calculations in a matter of seconds. At the present

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ANOMALIES IN THE GAMBLING MARKET 295

T ~ L E 3 Details of return for each probability range

Category X

1 2 3 4 5 6 7 8 9 10

0.019 0.042 0.066 0.100 0.165 0.249 0.352 0.497 0.698 0.888

N

30 23 22 44 72 91 104 235 174 79

NP ARR ERR POP P - PO,

0 2 3 5 13 21 35 103 105 69

-100.0 135.7 297.3 108.6 25-6 6.8 22.9 0.4 0.7 16.0

17.1 19.5 19-3 23-3 23.6 21.1 14.7 15-2 15.8 6.9

- 26.1 28.1 17.4 6.0 3.6 2.7 1.3 0.7 0.3

- 0.037 0.034 0-054 0-143 0.217 0-270 0.435 0.5 88 0.769

874 356 20.3 16.8

x = average estimated place probability in range N =number of horses showing a positive expected return

N , =number of place-getters ARR =actual pqcentage rate of return ERR =expected percentage rate of return PO, =average place odds for place-getters

P - PO, =probability equivalent of average place odds for place-getters

time the best of- them takes at least several minutes, but no doubt the continuing rapid technological advance in this area will soon ensure the solution to this problem.

The second factor which may restrict the effective operation of the strategy is that a bet of any size on the tote will depress the odds and reduce the rate of return. However, with present-day place pools in the vicinity of $100,000 and upwards, this should not present much of a problem if bets are restricted to less than four figure amounts.

6. Condusion Genuine inconsistencies do appear to exist between the win and

place gambling markets in metropolitan Australian horseracing. In approximately twenty to twenty-five percent of cases these inconsisten- cies are of sufficient magnitude to be capable, in theory at least, of profitable exploitations on the place tote. The continued existence of these anomalies can only be explained by a combination of ignorance on the part of everyday place punters, a lack of expertise on the part of operators attempting to exploit irregularities in the market and certain practical difficulties in the ‘fine-tuning’ of such an exploitative strategy.

References SCOTT, D. (1978). Winning. Sydney: Wentworth Press.