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We report striking evidence of semi-strong inefficiency in the UK fixed-odds football betting market using a reputable newspaper tipster which offers probabilities of match outcomes rather than simple result indicators. Betting on the Fink Tank probabilities of home wins across 10 bookmakers, when there are positive expected returns, would have generated positive returns in each of the seasons from 2006–07 to 2011–12 for a variety of different betting strategies. These returns could have been enhanced by employing the best odds from a greater number of bookmakers. However, the fact that pure arbitrage bets have existed for years and appear to last for several hours or days suggest they are in practice not exploitable to a magnitude that poses any threat to bookmakers.

It is now well recognized in the literature that betting markets are important for testing market efficiency as pointed out by Thaler and Ziemba [

There have been numerous studies of the efficiency of the fixed odds football betting market in the UK and elsewhere since the first study of the efficiency of the fixed odds betting in the UK market by Pope and Peel [

Subsequent analyses have documented the existence of mispricing as well as the apparent existence of pure arbitrage possibilities in European fixed-odds football betting markets [

Overall, the reported results from the literature on European football betting suggest that mispricing of odds has occurred over many seasons, particularly in the latter period. However Levitt [

Overall, the view of Forrest and Simmons [

Our purpose in this paper is to provide more striking and firm evidence of semi-strong inefficiency in the UK fixed odds betting market than has been previously reported. We find evidence of systematic positive returns in the English Premier League football based on the predictions of the Fink Tank, (also presented as Castrol Predictor) published weekly in the

The Fink Tank predictions are based on a statistical model that uses time-weighted shots and goals data to generate an attack and defence ranking for each club. The number of goals scored by a club in a match depends on the attack rating of the club and the defence rating of the opposition. There is also a home advantage rating, which allows for the fact that clubs score more goals when playing at home. An early version of the Fink Tank model appeared as Graham and Stott [

The Fink Tank predictions are reported in the form of the probabilities of the home, draw and away outcome. This is unusual as tipsters normally just report ‘most likely’ match outcomes. Forrest and Simmons [

We examine Fink Tank probabilities in conjunction with the odds of 10 bookmakers over the 2006–07 to 2011–12 Premier League seasons. We find systematic positive returns in each year obtained from a variety of different betting strategies based on betting on the home win. Given that the predictions of Fink Tank are readily available and positive returns have persisted for so many years, so that bettors or bookmakers have had time to learn of the value of the predictions, our findings appear to be of interest.

We also examine in more detail bets that appear to offer pure, paper, arbitrage profits

Our data set comprises the Fink Tank predicted probabilities of all possible outcomes of 1669 matches in the English Premier League, the top division of English football, over the 2006–07 to 2001–12 seasons together with the odds set by 10 bookmakers (Bet365, BWin, Gamebookers, Interwetten, Ladbrokes, Sportingbet, William Hill, StanJames, BETVICTOR, Blue Square). In

Differences in maximum and minimum odds as a ratio of minimum odds.

Statistics | Home odds | Away odds | Draw odds |
---|---|---|---|

N | 1669 | 1669 | 1669 |

Mean | 0.232 | 0.289 | 0.158 |

St. deviation | 0.115 | 0.161 | 0.099 |

We note that there is a mean difference of 23.2%, 28.9% and 15.8% between the best and worst odds posted for home, away and draw odds respectively. Clearly, placing bets with more than one bookmaker can increase expected returns or decrease losses substantially, ceteris paribus. To illustrate we randomly assumed we bet solely with either William Hills or Ladbrokes. In

William Hill’s and Ladbrokes’s odds

WH or LB versus other bookmakers | N |
---|---|

WH or LB’s home odds > other BM’s | 75 |

WH or LB’s away odds > other BM’s | 51 |

WH or LB’s draw odds > other BM’s | 87 |

LB—Ladbrokes, WH—William Hill, and BM—best bookmaker’s odds.

The expected return, for a one unit stake based on the Fink Tank probabilities is given by

Expected returns—for home win, away win and draw.

N | Mean | Median | St. dev. | Minimum | Maximum | |
---|---|---|---|---|---|---|

Home | ||||||

Expected return ≤ 0 | 837 | −0.118 | −0.096 | 0.098 | −0.674 | 0.000 |

Expected return > 0 | 832 | 0.164 | 0.110 | 0.183 | 0.000 | 1.880 |

Expected return | 1669 | 0.023 | 0.000 | 0.203 | −0.670 | 1.880 |

Away | ||||||

Expected return ≤ 0 | 746 | −0.149 | −0.120 | 0.125 | −0.890 | 0.000 |

Expected return > 0 | 923 | 0.251 | 0.170 | 0.263 | 0.000 | 2.825 |

Expected return | 1669 | 0.072 | 0.035 | 0.292 | −0.890 | 2.825 |

Draw | ||||||

Expected return ≤ 0 | 1519 | −0.157 | −0.150 | 0.089 | −0.813 | 0.000 |

Expected return > 0 | 150 | 0.125 | 0.050 | 0.277 | 0.000 | 2.650 |

Expected return | 1669 | −0.131 | −0.142 | 0.143 | −0.813 | 2.650 |

Over the sample period the Fink Tank probabilities and best bookmaker odds implied that we would bet on 832 home matches with an average expected return of 16.4%, 923 away matches with an expected return of 25.1% and 150 draws with an expected return of 12.5%. The largest expected home return of 18.8% occurred in the match Swansea

For matches where the expected return is greater than zero we consider a number of betting strategies. The first betting strategy is to stake one unit on each outcome where expected return is positive. We report the results for each season and actual returns to this betting strategy in

We observe that betting on one unit on each home team when expected return was positive would have generated a positive return in all seasons except 2008/9 with an average return of 10.75%. The actual returns to a one unit stake on away matches were negative (mean of −8.1%) but positive for draws with volatile returns across seasons. We conjecture that the Fink Tank probabilities deal with home advantage differently to bookmakers. Since home teams win in almost 50% of games, they are more likely to be favourites than away teams. Some studies have found evidence for favourite-longshot bias in European fixed odds betting markets [

Returns to a unit stake on home, away and draw when expected returns are positive and bet on home team benchmark.

Season | Stake | Winning bets | Losing bets | Wins | Profit | Return (%) |
---|---|---|---|---|---|---|

Home | ||||||

2006–07 | 132 | 65 | 67 | 85.04 | 18.04 | 13.67 |

2007–08 | 143 | 63 | 80 | 86.75 | 6.75 | 4.72 |

2008–09 | 122 | 53 | 69 | 68.59 | −0.41 | −0.34 |

2009–10 | 151 | 68 | 83 | 101.24 | 18.24 | 12.08 |

2010–11 | 150 | 74 | 76 | 100.49 | 24.49 | 16.33 |

2011–12 | 134 | 61 | 73 | 95.33 | 22.33 | 16.66 |

Total | 832 | 384 | 448 | 537.44 | 89.44 | 10.75 |

Away | ||||||

2006–07 | 130 | 31 | 99 | 75.08 | –23.92 | −18.40 |

2007–08 | 189 | 47 | 142 | 91.66 | −50.34 | −26.63 |

2008–09 | 133 | 44 | 89 | 122.05 | 33.05 | 24.85 |

2009–10 | 145 | 27 | 118 | 64.54 | −53.46 | −36.87 |

2010–11 | 161 | 34 | 127 | 132.18 | 5.18 | 3.22 |

2011–12 | 165 | 39 | 126 | 140.75 | 14.75 | 8.94 |

Total | 923 | 222 | 701 | 626.26 | −74.74 | −8.10 |

Draw | ||||||

2006–07 | 41 | 12 | 29 | 30.85 | 1.85 | 4.51 |

2007-08 | 19 | 8 | 11 | 30.90 | 19.9 | 104.74 |

2008–09 | 12 | 3 | 9 | 9.00 | 0.00 | 0.00 |

2009–10 | 25 | 3 | 22 | 9.8 | −12.2 | −48.80 |

2010–11 | 20 | 6 | 14 | 19.90 | 5.90 | 29.50 |

2011–12 | 33 | 10 | 23 | 36.65 | 13.65 | 41.36 |

Total | 150 | 42 | 108 | 137.10 | 29.10 | 19.40 |

Unit bet on home team “benchmark” | ||||||

2006–07 | 300 | 143 | 157 | 169.83 | 12.83 | 4.28 |

2007–08 | 326 | 150 | 176 | 159.38 | −16.62 | −5.10 |

2008–09 | 226 | 92 | 134 | 111.62 | −22.38 | −9.90 |

2009–10 | 271 | 134 | 137 | 161.80 | 24.80 | 9.15 |

2010–11 | 282 | 131 | 151 | 160.15 | 9.15 | 3.24 |

2011–12 | 264 | 124 | 140 | 141.22 | 1.22 | 0.46 |

Total | 1669 | 774 | 895 | 904.00 | 9.00 | 0.54 |

The bottom panel of

Of course a one unit bet on every outcome where expected return is positive does not make allowance for either the magnitude of the expected return or the probability of occurrence. A standard staking system in the betting literature is to employ the variable Kelly stake as a proportion of wealth as a solution to this problem. See Sung and Johnson [

The Kelly stake is the optimal stake for an expected utility maximiser who has a logarithmic utility function. Expected utility,

where w is the agent’s betting wealth, s is the stake, O are odds and

Our second betting strategy, is to determine returns based on staking

Returns on home, away and draw bets when using Kelly single stake.

Season | Winning bets | Losing bets | Winning stakes | Losing stakes | Wins | Profit | Return (%) |
---|---|---|---|---|---|---|---|

Home | |||||||

2006–07 | 65 | 67 | 82.95 | 82.95 | 88.36 | 5.41 | 3.26 |

2007–08 | 63 | 80 | 64.81 | 63.07 | 69.98 | 6.91 | 5.40 |

2008–09 | 53 | 69 | 58.84 | 45.26 | 63.25 | 17.99 | 17.28 |

2009–10 | 68 | 83 | 76.66 | 73.18 | 93.93 | 20.75 | 13.85 |

2010–11 | 74 | 76 | 92.43 | 74.01 | 94.16 | 20.15 | 12.11 |

2011–12 | 61 | 73 | 58.05 | 60.69 | 77.18 | 16.49 | 13.89 |

Total | 384 | 448 | 433.74 | 399.17 | 486.86 | 87.69 | 10.53 |

Away | |||||||

2006–07 | 31 | 99 | 35.91 | 79.35 | 79.97 | 0.62 | 0.54 |

2007–08 | 47 | 142 | 33.55 | 83.63 | 55.55 | −28.08 | −23.96 |

2008–09 | 44 | 89 | 36.73 | 53.21 | 76.37 | 23.17 | 25.76 |

2009–10 | 27 | 118 | 14.36 | 79.73 | 33.55 | −46.18 | −49.08 |

2010–11 | 34 | 127 | 32.38 | 97.85 | 104.64 | 6.8 | 5.22 |

2011–12 | 39 | 126 | 26.66 | 78.17 | 74.68 | −3.48 | −3.32 |

Total | 222 | 701 | 179.58 | 471.93 | 424.77 | −47.15 | −7.24 |

Draw | |||||||

2006–07 | 12 | 29 | 3.81 | 13.96 | 11.92 | −2.04 | −11.48 |

2007–08 | 8 | 11 | 1.09 | 1.56 | 6.03 | 4.46 | 168.30 |

2008–09 | 3 | 9 | 0.26 | 1.57 | 0.80 | −0.770 | −42.08 |

2009–10 | 3 | 22 | 1.58 | 14.14 | 4.34 | −9.80 | −62.34 |

2010–11 | 6 | 14 | 1.39 | 3.17 | 4.01 | 0.84 | 18.42 |

2011–12 | 10 | 23 | 1.47 | 3.66 | 6.06 | 2.40 | 46.78 |

Total | 42 | 108 | 9.61 | 38.06 | 33.15 | −4.91 | −10.30 |

If we examine the actual proportions of home away and draws outcomes and the proportions predicted obtained from best bookmaker odds and the Fink Tank we observe in

Actual Proportions of Outcomes and Predicted based on best bookmaker’s odds (BM) and Fink Tank (FT).

Home win | Home probability (BM) | Home probability (FT) | |
---|---|---|---|

N | 1669 | 1669 | 1669 |

Mean | 0.464 | 0.452 | 0.464 |

St. dev. | 0.499 | 0.183 | 0.185 |

Minimum | 0.000 | 0.052 | 0.040 |

Maximum | 1.000 | 0.869 | 0.930 |

N | 1669 | 1669 | 1669 |

Mean | 0.268 | 0.289 | 0.305 |

St. dev. | 0.443 | 0.163 | 0.162 |

Minimum | 0.000 | 0.033 | 0.020 |

Maximum | 1.000 | 0.824 | 0.850 |

N | 1669 | 1669 | 1669 |

Mean | 0.268 | 0.259 | 0.231 |

St. dev. | 0.443 | 0.045 | 0.049 |

Minimum | 0.000 | 0.097 | 0.030 |

Maximum | 1.000 | 0.315 | 0.730 |

Of course punters may not be expected utility maximizers with a logarithmic utility function or may be non-expected utility maximizers. They could, for example, be better described as expected utility maximizers with a power or exponential utility function or non-expected utility maximizers of either Tversky and Kahneman’s cumulative prospect theory [

The actual returns in

Formal evidence of the incremental value in the Fink Tank prediction of home win outcomes but not away wins or draws relative to the probability based on the best bookmaker odds is shown by the probit regressions reported in

Constant Absolute Risk Aversion.

Season | Winning bets | Losing bets | Winning stakes | Losing stakes | Wins | Profit | Return (%) |
---|---|---|---|---|---|---|---|

2006–07 | 66 | 67 | 85.12 | 85.23 | 85.60 | 0.37 | 0.22 |

2007–08 | 65 | 78 | 68.50 | 60.82 | 70.23 | 9.42 | 7.28 |

2008–09 | 53 | 70 | 61.30 | 44.66 | 62.63 | 17.97 | 16.96 |

2009–10 | 68 | 83 | 76.81 | 69.73 | 89.97 | 20.23 | 13.81 |

2010–11 | 73 | 77 | 102.53 | 73.81 | 97.38 | 23.57 | 13.37 |

2011–12 | 61 | 72 | 56.80 | 56.33 | 73.58 | 17.25 | 15.25 |

Total | 386 | 447 | 451.07 | 390.58 | 479.39 | 88.81 | 10.55 |

The Fink Tank predictions are now available online during the week preceding a match. As a consequence, this will enable a bettor employing the Fink Tank home predictions far more opportunities to bet on home wins with positive expected value as the different bookmakers odds change over the course of the week. The adjustment of betting odds on football matches by bookmakers up to kick-off is a relatively recent phenomenon in the UK. Forrest [

Returns on home, away and draw bets using Kelly stake with Ladbrokes or William Hill when expected returns is positive.

Season | Winning bets | Losing bets | Winning stakes | Losing stakes | Wins | Profit | Return (%) |
---|---|---|---|---|---|---|---|

Home | |||||||

2006–07 | 52 | 55 | 62.10 | 64.14 | 58.85 | −5.29 | −4.19 |

2007–08 | 49 | 50 | 41.01 | 37.46 | 37.66 | 0.20 | 0.25 |

2008–09 | 44 | 62 | 44.52 | 36.59 | 46.93 | 10.34 | 12.75 |

2009–10 | 57 | 80 | 61.28 | 62.32 | 76.99 | 14.67 | 11.87 |

2010–11 | 65 | 65 | 81.85 | 64.29 | 78.29 | 14.01 | 9.59 |

2011–12 | 57 | 67 | 50.50 | 53.11 | 62.26 | 9.15 | 8.83 |

Total | 324 | 379 | 341.25 | 317.91 | 360.98 | 43.08 | 6.54 |

Away | |||||||

2006–07 | 28 | 75 | 27.34 | 58.88 | 55.52 | −3.35 | −3.89 |

2007–08 | 31 | 74 | 19.76 | 44.83 | 28.32 | −16.50 | −25.55 |

2008–09 | 35 | 72 | 29.18 | 39.87 | 54.92 | 15.05 | 21.80 |

2009–10 | 22 | 108 | 11.20 | 67.61 | 25.79 | −41.82 | −53.06 |

2010–11 | 32 | 112 | 27.63 | 79.82 | 83.78 | 3.96 | 3.69 |

2011–12 | 35 | 116 | 21.61 | 63.62 | 52.94 | −10.69 | −12.54 |

Total | 183 | 557 | 136.73 | 354.63 | 301.27 | −53.36 | −10.86 |

Draw | |||||||

2006–07 | 6 | 14 | 1.21 | 9.03 | 3.11 | −5.91 | −57.71 |

2007–08 | 1 | 1 | 0.51 | 0.90 | 3.60 | 2.70 | 191.49 |

2008–09 | 1 | 5 | 0.10 | 0.78 | 0.35 | −0.43 | −48.86 |

2009–10 | 3 | 17 | 1.48 | 12.72 | 3.90 | −8.82 | −62.11 |

2010–11 | 5 | 9 | 1.27 | 1.76 | 3.67 | 1.91 | 63.04 |

2011–12 | 3 | 8 | 0.65 | 0.87 | 2.67 | 1.80 | 118.42 |

Total | 19 | 54 | 5.23 | 26.06 | 17.31 | −8.75 | −27.96 |

In our analysis we employed the best odds from 10 bookmakers available to us over our sample period. The website [

Bookmakers quote odds that are fixed for the bettor at the time of placing the wager. This means that the terms of the bettor’s wager are unaltered before the finish of the match. Significant changes in the bookmakers’ quoted odds tend to occur frequently from the first listing of odds, about three weeks before the match until the end of the match. The odds will change over the betting period in response to a number of factors. These include protection against insider trading activity [

Probit models with marginal effects for home win, away win and draw.

Home win | |||
---|---|---|---|

(1) | (2) | (3) | |

Home probability (BM) | 1.060 *** | 0.571 *** | |

(14.460) | (3.043) | ||

Home probability (FT) | 1.043 *** | 0.523 *** | |

(14.350) | (2.807) | ||

Observations | 1669 | 1669 | 1669 |

Pseudo ^{2} |
0.097 | 0.096 | 0.100 |

Away win | |||

(4) | (5) | (6) | |

Away probability (BM) | 1.006 *** | 0.873 *** | |

(15.078) | (5.284) | ||

Away probability (FT) | 0.955 *** | 0.146 | |

(14.210) | (0.872) | ||

Observations | 1669 | 1669 | 1669 |

Pseudo ^{2} |
0.121 | 0.106 | 0.121 |

Draw | |||

(7) | (8) | (9) | |

Draw probability (BM) | 1.144 *** | 0.979 *** | |

(4.484) | (2.660) | ||

Draw probability (FT) | 0.836 *** | 0.201 | |

(3.576) | (0.627) | ||

Observations | 1669 | 1669 | 1669 |

Pseudo ^{2} |
0.010 | 0.007 | 0.011 |

BM—Best bookmaker’s odds, FT—Fink Tank; *** denotes significance at 1 percent.

The over round for a particular bookmaker and the over round employing the best odds across bookmakers, from the punters perspective, each tends to fall as match day approaches. The over round of an individual bookmaker on a given match at kick-off is always above unity at approximately 8% (1.08). Previous studies have reported over rounds of around 10% to 12% [

The possibility of pure arbitrage profits was identified in a number of earlier papers and has clearly not disappeared over two decades since first noted by Pope and Peel [

An over round of below unity is necessary, but not sufficient, for a pure arbitrage opportunity. There are direct costs associated with undertaking an arbitrage. An arbitrage will not be riskless. In order to undertake arbitrage bets, it will be necessary for the bettor to have funds deposited in a number of bookmaker accounts since inspection suggests that arbitrage possibilities are not concentrated across a few bookmakers. Clearly, the less the number of bookmaker accounts covered the fewer the arbitrage opportunities available. It appears that at least 10 accounts would be needed to achieve a reasonable number of arbitrage possibilities. An arbitrage gain of £10 would appear to require an outlay typically of £500 or more so a large capital base is required. Also, there are costs and delays in depositing and withdrawing funds. While specialist methods for fund transfers do exist, such as eWallets, withdrawals are often limited to a particular amount per month or to a specific number of free monthly withdrawals. Withdrawals tend to be charged for on the eWallet side. For many bettors with medium sized stakes, these transactions costs could amount to 2% to 3% of the stake which would wipe out positive returns from arbitrage trades. For very large traders, the very existence of their accounts with large deposits and withdrawals draw the attention of bookmakers to arbitrage attempts. Bookmakers can then suddenly suspend bettor accounts imposing potentially large losses on the arbitrageur.

The formulae for the stakes on the home,

A necessary condition for a pure arbitrage gain is that

Overall, it seems scarcely credible that bookmakers are unaware of arbitrage opportunities given that they are linked to comparison web sites such as Betrescue [

We followed an apparent arbitrage possibility through from their first appearance on Betrescue [

West Bromwich Albion

Clearly, these arbitrage possibilities could have disappeared due to pure arbitrage dealing staggered over time. Arbitrage opportunities signalled by a below-unity over round may fall under the heading of ‘limits to arbitrage’ proposed by Shleifer and Vishny [

In our examples, the arbitrage possibilities did not disappear quickly. Most money has to be wagered on the favourite in a pure arbitrage bet, In the Swansea-Reading match the odds for the home outcome of the three bookmakers offering the best home odds were unchanged at 1/1. This is suggestive of a persistent arbitrage opportunity. Rather, some bettors appear to have bet on draw and away win outcomes at the more favourable odds. Similarly, in the West Bromwich Albion-Queens Park Rangers match the draw odds remained unchanged over the duration of the arbitrage possibility.

Previous literature has reported some evidence of inefficiency in bookmakers’ pricing of odds in fixed odds betting, including the potential for pure arbitrage gains betting with different bookmakers. In this paper we add to this literature and report striking evidence that betting on the Fink Tank probability of home wins across 10 bookmakers when there are positive expected returns, would have generated positive returns in each of the seasons from 2006–07 to 2011–12 for a variety of different betting strategies. These returns could have been enhanced by employing the best odds from a greater number of bookmakers. The inefficiency associated with Fink Tank match outcome probability is unlikely to be due to systematic mistakes. Bookmakers will change their subjective probability of match outcomes as they attempt to maximize their objective function. This could involve deliberately setting the “wrong” prices on some outcomes to exploit sentiment or as “loss leaders”. The extent to which the inefficiencies derived here from Fink Tank probability can be attributed to either or both of these sources of mispricing is a useful topic for further research.

We noted that paper pure arbitrage opportunities occur quite frequently, perhaps a handful a day, as bookmakers change odds in response to betting flows and news or possibly in an attempt to induce betting flows. The fact that these pure arbitrage bets have existed for years and appear to last for several hours or even days suggest they are in practice not exploitable to a magnitude that poses any threat to bookmakers. Similar remarks apply to betting strategies based on the Fink Tank probability of home win.

Bookmakers appear to set prices that are informationally (semi-strong) inefficient. However, the degree of inefficiency has clearly not been exploited to date on a scale that presents a probability problem for bookmakers. The transactions costs and risks attached to trading on the mispricing, revealed by Fink Tank probability and from other sources, appear to insulate bookmakers to a large degree from arbitrageurs.

The authors thank participants at the 2012 Gijon conference on sports economics for helpful comments.

The authors declare no conflict of interest.