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Profiting from the English Premier League: Predictive Elicitation, the Kelly Criterion, and Black Swans.

Introduction

Football (soccer) has gained increasing popularity since its contemporary introduction in 1863 in England. Today, it is the most popular sport around the world (Dunning, Joseph, & Maguire, 1993). As a consequence, betting on football through online bookmakers is by far the biggest sport in terms of turnover (Constantinou & Fenton, 2013b; Finnigan & Nordsted 2010). Therefore, the main objective of this paper is to propose a complete football betting strategy based on a simple predictive elicitation approach--the Kelly Criterion--and atypical positive returns (referred to as "black swans").

Although most of the literature on football forecasting has focused on various scoring rules to determine the performance of different methods (Constantinou & Fenton, 2012; Constantinou & Fenton, 2013a; Spann & Skiera, 2009), it is natural to determine their forecast accuracy based on the ability to generate profits against market odds (Cain, Law, & Peel, 2000; Constantinou, Fenton, & Martin, 2013; Constantinou, Fenton, & Neil, 2012; Crowder et al., 2002; Dixon & Pope, 2004; Forrest, Goddard, & Simmons, 2005; Goddard & Asimakopoulos, 2004; Graham & Stott, 2008; Hvattum & Arntzen, 2010; Kuypers, 2000b; Rue & Salvesen, 2000). However, the latter approach depends on the model's forecasting ability relative to the market odds and a betting strategy. We propose to forecast football games outcomes using a simple predictive elicitation approach (Garthwaite, Kadane, & O'Hagan, 2005; Kadane, 1980), where the hyperparameters of a Categorical Dirichlet model are elicited using betting odds from different bookmakers. Regarding the betting strategy, we use the Kelly Criterion, which defines the optimal size of a series of bets that maximizes the wealth growth rate in the long run (Kelly, 1956), and a stopping rule based on atypical positive returns (black swans), which are defined based on historical information. Therefore the novelty of our proposal is to design a complete betting strategy that includes a simple way to forecast match outcomes, determine the percentage of money that should be invested, and includes a stopping rule that might generate profits exploiting the inefficiencies in the betting market at the beginning of the season. In addition, we develop a free graphical user interface (GUI) to apply to our proposal. This is available online at https://github.com/besmarter/UIBets. This GUI is built in MATLAB, but we created an .exe version that does not require this software.

We tested our procedure in the English Premier League for the 2013-2014, 2014-2015, and 2015-2016 seasons, betting on bet365, one of the major online bookmakers (Strumbelj, 2014). Our method obtained profitable outcomes: 33.54%, 22.12%, and 49.01% returns, respectively.

The English Premier League

The English Premier League is one of the most important football leagues in the world. It was founded in 1992 after the Football League First Division members decided to break away from the Football League, which was originally founded in 1888. The EPL is composed of 20 clubs. Each club plays 38 matches in a regular season that runs from August to May, totaling 380 matches. Due to being the most watched football league in the world, and its excellent available historical records, it is very attractive for testing the ability of betting strategies to generate profits against market odds. In general, the English Football League system has been considered by many researchers (Cain, Law, & Peel, 2000; Crowder et al., 2002; Dixon & Pope, 2004; Forrest, Goddard, & Simmons, 2005; Goddard & Asimakopoulos, 2004; Kuypers, 2000b). Some methods have been shown to generate positive returns (Cain, Law, & Peel, 2000; Constantinou, Fenton, & Martin, 2013; Constantinou, Fenton, & Neil, 2012; Goddard & Asimakopoulos, 2004; Rue & Salvesen, 2000). However, others have not found that to be possible (Forrest, Goddard, & Simmons, 2005; Graham & Stott, 2008; Hvattum & Arntzen, 2010; Spann & Skiera, 2009). Many of these methods have a Bayesian flavor (Constantinou, Fenton, & Martin, 2013; Constantinou, Fenton, & Neil, 2012; Rue & Salvesen, 2000). On the other hand, there are some authors who have preferred to use frequentist approaches (Cain, Law, & Peel, 2000; Crowder et al., 2002; Goddard & Asimakopoulos, 2004; Graham & Stott, 2008; Hvattum & Arntzen, 2010; Kuypers, 2000a, 2000b). Research proposing betting strategies tries to exploit market inefficiencies (Cain, Law, & Peel, 2000).

Methodology

For convenience and facility, we propose a conjugate family for our Bayesian approach to predicting the possible outcomes of football matches. In particular, we follow Constantinou, Fenton, and Martin (2013), who propose the Categorical Dirichlet model. This model allows researchers to easily identify the prior roles and data in the posterior mean probabilities associated with each outcome. So, we assume that the likelihood is given by a categorical distribution with three possible outcomes: i = {Win, Draw, Loss}. On the other hand, the prior information is summarized in a Dirichlet distribution such that [pi](p)~D([alpha]), where p is the vector of probabilities associated with i.

Following Bayes's rule, the posterior distribution is [pi](p|Data)~D([alpha]+c), where c = ([c.sub.1],[c.sub.2],[c.sub.3]) is the vector with the number of occurrences of each category [mathematical expression not reproducible].

One question that always arises in application of Bayesian analysis is how to obtain the hyperparameters of the prior distributions. We use predictive elicitation to achieve this task. In particular, predictive elicitation helps to infer the hyperparameters from observable quantities, reversing the process between the observable quantities and the hyperparameters, whereas structural elicitation is directly based on the parameters, quantities that are not observable. Another argument for using the predictive approach is that experts have systematic biases in situations in which they are asked about many parameters. Specifically, experts over-estimate the probability of joint events and under-estimate the probability of disjoint events (Kadane & Wolfson, 1998).

So, we infer the hyperparameters of the prior Dirichlet distribution using a predictive elicitation approach based on betting odds. In particular, betting odds have good predictive power (Forrest, Goddard, & Simmons, 2005; Spann & Skiera, 2009; Strumbelj & Sikonja, 2010), bookmakers have financial incentives to correctly assess them (Strumbelj, 2016; Strumbelj & Sikonja, 2010), and there is a theoretical framework (Shin, 1991, 1993) to mitigate their implicit biases.

In particular, we take the closing line betting odds associated with each match from different online bookmakers, and transform them into outcome probabilities using the method developed by Shin (1991, 1993) but using the procedure of Jullien and Salanie (1994). Therefore, we try to mitigate the implicit biases that are present in betting odds. The reason for this decision is the well-known argument that betting odds are inherently biased due to bookmakers' profits from their service, thus they offer unfair odds (Strumbelj, 2016; Strumbelj & Sikonja, 2010). A common practice is to use a procedure called "basic normalization;" however, Strumbelj and Sikonja (2010) show that the probabilities deduced from betting odds using the Shin (1993) procedure are more accurate than the probabilities produced by basic normalization. Next, we apply maximum likelihood based on the Dirichlet models (Hizaji & Jernigan, 2009; Thomas & Jacob, 2006) using the "unbiased" probabilities from Shin's procedure to estimate the hyperparameters of the prior distributions for each match.

We use historical information until just before every football match to build the categorical likelihood. Thus, we count, for each pair of contenders, their numbers of wins, draws, and losses when they met in previous matches (this information is available at http://www.soccerbase.com/). Lastly, we obtain the posterior means from the Dirichlet distributions. These are the probabilities associated with the outcomes (Win, Draw, and Loss) for each match, and these will be used to find the Kelly fraction for betting.

Taking into account the posterior distribution, the posterior mean probability of each outcome is given by [mathematical expression not reproducible]. So, the posterior mean probability is a weighted average of the information from experts ([[alpha].sub.i]) and the historical information ([c.sub.i]).

One limitation of our approach is that the historical information is not weighted in terms of recent results being more important than older results. However, it makes sense that this fact is present in betting odds--evidence of this is that bookmakers gain predictive power in football as the season progresses (Forrest, Goddard, & Simmons, 2005; Pope & Peel, 1989), but it is precisely the discrepancies between betting odds and historical records at the beginning of the season that we exploit to identify betting opportunities.

Once we have designed a forecast method, we propose a betting strategy based on the Kelly Criterion and black swans. Regarding the former, we follow Hvattum and Arntzen (2010), who use the Kelly Criterion to define the optimal size of a series of bets that maximizes the wealth growth rate in the long run based on the following equation (Kelly, 1956):

[mathematical expression not reproducible] (1)

where [[theta].sup.b.sub.i,mt] is the betting odds for the outcome i in match m at time t from bookmaker b; [p.sub.i,mt] is the probability of the specific event, which we obtain using our Bayesian procedure for each match outcome; and [f.sup.*b.sub.i,mt] is the fraction of our budget that we should bet according to this criterion. (1)

Note that there is no restriction on [f.sup.*b.sub.i,mt], so when [p.sub.i,mt] < [1/[[theta].sup.b.sub.i,mt]], [f.sup.*b.sub.i,mt] becomes a negative fraction, but there is no possibility of betting a negative amount of money, we only bet when [p.sub.i,mt] < [1/[[theta].sup.b.sub.i,mt]]. In addition, when there is more than one positive fraction in the same game, we select the highest. So, the Kelly fraction for a specific game is [f.sup.*b.sub.i,mt] = Max{0,Max{[f.sup.*b.sub.i,mt]}}, and the bankroll of the investor is [B.sub.t]=[B.sub.t-1](1+[r.sub.t]), where [r.sub.t] = [f.sup.*b.sub.mt]([[theta].sup.b.sub.i,mt] -1) if event i in match m at time t happens, or [r.sub.t]=-[f.sup.b.sub.mt] in the other case.

We define a stopping rule for our betting strategy like those popular in finance. The principle for setting such a strategy is because some specific outcomes drive remarkable returns in the football betting market (Rue & Salvesen, 2000). Therefore, we set a stop to the betting when there is a positive percentage return on investment ([r.sub.t]=-[f.sup.*b.sub.mt]([[theta].sup.b.sub.i,mt]-1)= [[theta].sup.b.sub.i,mt], [p.sub.i,mt] -1) that is more than four standard deviations above the historical arithmetic mean percentage of positive returns associated with betting in the present season. We found that returns had a positive asymmetry, so the idea of the stopping rule is to "hunt" for those results that were abnormally positive. It is almost impossible to obtain a percentage return as high as this black swan in the present season, especially because, as we said, bookmakers gain predictive accuracy in football as the season progresses (Forrest, Goddard, & Simmons, 2005; Pope & Peel, 1989). It could be possible that the black swan is not identified immediately when it occurs early in the season. The reason is that there are not enough bets, so the black swan is not as atypical as it can be when there is more information. In this case, the betting will be stopped as soon as the black swan is identified.

Results

We tested our procedure in the English Premier League for the 2013-2014, 2014-2015, and 2015-2016 seasons. First, we gathered closing line betting odds associated with the outcome (Win, Draw, and Loss) of each match from different online bookmakers, namely: bet365, Bet&Win, Interwetten, Ladbrokes, Pinnacle Sports, William Hill, and Bet home (available at http://www.football-data.co.uk/englandm.php). Second, we implemented Shin's procedure to obtain the experts' implicit probabilities associated with each match for each bookmaker. Third, we elicited the hyperparameters of the prior Dirichlet distributions of each football match using maximum likelihood with the bookmakers' probabilities obtained in the previous stage. Finally, we calculated the posterior parameters using the hyperparameters, and all the historical records, which are available since 1893, until just before each match. So, the posterior means are the probabilities for each outcome ([p.sub.i,mt] in equation [1]).

We can see the mean squared error and mean absolute error for each season in Table 1. In particular, we compare the predictive power of our approach with the probabilities obtained from bet365 using our proposal, Shin's procedure, and a naive forecast that assigns 1/3 to each possible outcome. As we show in Table 1, there are no significant differences in the average predictive performance associated with the posterior probabilities and Shin's procedure. However, these two approaches surpass a naive forecast.

We implemented our betting strategy on bet365, one of the major online bookmakers (Strumbelj, 2014). In particular, we assume an initial capital of US$1,000, and defined the optimal bet size using the Kelly Criterion.

In Table 2 we show the outcomes of our betting strategy. In particular, we compare the arithmetic mean returns using the Categorical Dirichlet model and Shin's probabilities to calculate the Kelly fraction betting in the complete season, and until the stopping criterion is met. As Table 2 reveals, the average arithmetic returns using the Categorical Dirichlet model are higher than using Shin's procedure, except in the 2015-2016 season when betting the complete season.

We calculate the standard deviation of the positive percentage returns of betting in a moving window that is fixed at the opening match, and updated each match day. We stop betting as soon as a black swan is identified, taking into consideration that some matches take place simultaneously, and that black swans cannot be identified immediately after they occur.

As shown in Table 2, the average returns using the stopping criterion using the Categorical Dirichlet model are higher than the returns betting for the complete season. This suggests that the stopping criterion is important in our betting strategy. Another point to reinforce the relevance of our stopping criterion is that the average mean returns are positive, despite the fact that we lose the bets most of the times (i.e., the final returns are driven by some remarkable positive results).

In Table 3 we show the compound percentage profits of our betting strategy. In particular, we obtain profits that range between 22.12% and 49.01%. The black swan criterion suggests stopping betting at matches 180, 97, and 160, which imply periods between 76 and 131 days. In addition, we observe that the average odds range between 5.12 and 6.13, and the average returns between 4.07% and 6.71% in the winning cases.

In the 2013-2014 season the black swan happened in the match between Everton and Sunderland (Dec. 26, 2013; 131 days after the opening day), when Sunderland obtained a 1- 0 away win. In the 2014-2015 season, the black swan happened when Stoke City beat Manchester City at Etihad Stadium (Aug. 30, 2014; 14 days after opening day) but we only identified it as a black swan 52 days later. Regarding the 2015-2016 season, the black swan happened in the match between Tottenham and Newcastle, when the latter won 2-1. The associated probabilities with the events that actually happened can be seen in Table 4.

The odds for these events in bet365 were 9.5, 18, and 9, respectively, so using the Kelly Criterion and the proposed probability we should bet on these events. The Kelly fractions are 4.7%, 3.0%, and 2.8%, respectively, which lead to percentage returns equal to 40.7%, 50.84% and 22.4% (f*([theta]-1)) in these matches, respectively. We should take into consideration that there are negative Kelly fractions using Shin's probabilities for those matches. This creates remarkable differences in the returns despite the fact that the average predictive performance is similar using these two methods.

The differences in the probabilities associated with these matches between the two methods is the matter of history. Our approach takes into consideration two sources of information to forecast the probabilities: predictive elicitation from experts (bookmakers) and historical records, whereas Shin's procedure just takes into consideration the bookmakers. Seeing the history of these matches in Table 4, the expected result according to history was different from the odds given by bet365. In particular, in the 2013-2014 season, historical records indicate a probability of an Everton home loss equal to 0.38=67/175, whereas Shin's procedure indicates approximately 0.097, and the posterior mean from the Categorical Dirichlet model is 0.148. The posterior mean is higher than the latter due to the influence of historical records. The same pattern is present in the Manchester City versus Stoke City, and Tottenham versus Newcastle matches, where the historical records indicate probabilities equal to 36 /107=0.33 and 56/154=0.36 for the respective outcomes, whereas bookmakers' information estimates probabilities equal to 0.048 and 0.103. On the other hand, our Bayesian method estimates probabilities equal to 0.083 and 0.136 due to sample information. However, we can see from these outcomes that the influence of prior information (bookmakers) exceeds that of sample information (historical records). This is a desirable characteristic in our approach because betting odds better indicate the present performance of a football team.

Concluding Remarks

There is no academic consensus regarding the exploitation of betting market inefficiencies to obtain returns. Some authors have found profitable betting strategies whereas others have not. We proposed a novel betting strategy based on predictive elicitation, the Kelly Criterion, and black swans. The first stage is a forecast method based on a very simple natural conjugate family, the Categorical Dirichlet model, in which the hyperparameters are elicited using the betting odds from different bookmakers. The second stage defines an optimal bet size, and the third stage is a stopping rule.

Despite the fact that our method does not have a better forecast performance than that of Shin's procedure, it obtains profitable outcomes in our application due to the use of expert knowledge and sample information. This strategy allows identifying differences between these sources of information that can be exploited to obtain good betting returns in specific matches. In particular, with this simple framework we obtained a compound profit 143% ([1.3354 x 1.2212 x 1.4912]-1) betting on bet365 in the English Premier League for betting strategy between the 2013-14 and 2015-16 seasons. However, a future research agenda should consider other online bookmakers as well as other betting markets (sports). In addition, we should try other models that explicitly consider an order in football outcomes

References

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Endnotes

(1) As an anonymous referee points out, "The Kelly Criterion (staking) is optimal (in the sense that it maximizes the doubling rate) in an idealized world where we can make arbitrarily large (small) bets. The sports betting world is not like that--there is a minimum bet (that is, we can quickly go bankrupt using the Kelly Criterion) and there is a maximum bet (that is, the market does not admit exponential capital growth)." So, the Kelly fraction should be taken as a reference point. In addition, we found in our applications that the maximum Kelly fraction was 6.9%, and the minimum was 0.009%. So, if the initial amount is US$1,000, our betting fractions are compatible with bet365 limits, that is, a minimum of US$0.5 and US$25,000.

Andres Ramirez Hassan (1*) and Mateo Graciano Londono (1**)

(1) Universidad EAFIT, Medellin, Colombia

Andres Ramirez Hassan, PhD, is a professor in the Department of Economics. His research focuses on Bayesian econometrics.

Mateo Graciano Londono is student in the Mathematical Engineering Department. His research focuses on probability.
Table 1. Forecast Performance

Season   Measure             Probabilities
                  Posterior      Shin       Naive

           MSE      0.63         0.63       0.81
2013-14    MAE      1.02         1.01       1.33
           MSE      0.71         0.72       0.81
2014-15    MAE      1.15         1.15       1.33
           MSE      0.76         0.76       0.81
2015-16    MAE      1.22         1.21       1.33

Table 2. Betting Performance (*)

                             Complete season
Season            Posterior                      Shin
           Mean (SD)     #Bets (#wins)     Mean (SD)     Bets (wins)

2013-14  0.040%(0.266)      220(46)      0.020%(0.0066)    119(32)
2014-15  0.020%(0.0318)     211(43)     -0.016%(0.0041)    122(21)
2015-16  0.012%(0.0274)     188(47)      0.026%(0.0058)    109(32)

                             Stopping criteria
Season     Mean (SD)     #Bets (#wins)     Mean (SD)     Bets (wins)

2013-14  0.210%(0.0338)     113(26)     -0.030%(0.0044)    61(16)
2014-15  0.320%(0.0543)      61(12)     -0.010%(0.0035)     41(4)
2015-16  0.280%(0.0274)      87(24)      0.030%(0.0058)    56(16)

(*) Standard deviation of returns (r) in parentheses.

Table 3. Percentage Returns
Winning Cases

                                    Winning Cases
Season   Return  Games  Days  Min   Mean   Max   Average
                              Odd   Odd    Odd   Return

2013-14  33.54%   180    131  3.00  5.12  12.00   4.07%
2014-15  22.12%    97     76  3.10  6.13  18.00   6.71%
2015-16  49.01%   160    127  3.10  5.23   9      4.55%

Table 4. Black Swans (*)

Game              Posterior     Shin (SD)    Home  Draws  Away
                     (SD)                    wins         wins

Everton vs.      0.148(0.011)  0.097(0.010)   78     30    67
Sunderland
Manchester City  0.083(0.009)  0.048(0.007)   47     24    36
vs. Stoke city
Tottenham vs.    0.136(0.009)  0.103(0.009)   66     32    56
Newcastle

(*) Standard deviation in parentheses calculated from the Dirichlet
distribution.
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Author:Hassan, Andres Ramirez; Londono, Mateo Graciano
Publication:International Journal of Sport Finance
Article Type:Report
Geographic Code:4EUUK
Date:Nov 1, 2017
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