DATA MINING FOR OPTIMAL GAMBLING.

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1 DATA MINING FOR OPTIMAL GAMBLING. Gabriele Torre 1 and Fabrizio Malfanti 2 1 Dipartimento di Matematica, Università degli Studi di Genova, via Dodecaneso 35, 16146, Genova, Italy. ( torre@dima.unige.it) 2 Intelligrate s.r.l, Via XII Ottobre 2 / 92, 16121, Genova, Italy. ( fabrizio.malfanti@intelligrate.it) ABSTRACT: The main objective of this abstract is to provide an application of Data Mining on Gambling in sport, presenting a new methodology, based on the real information made available from hundreds of online bookmakers, to improve the real probability estimation for a given sportive match outcome. Moreover, the formal equivalence between the AdaBoost method and the Optimal Betting strategies is utilized to explore the application of boosting methods on the different classifiers utilized by online bookmakers. In conclusion, the main aim of the presented method is to change the gambling perspective from betting on the match outcome to betting on the best performer bookmaker. KEYWORDS: AdaBoost, optimal betting, online bookmakers. 1 Introduction Nowadays, the massive presence of online bookmakers, supplying the probability estimations for a given sport outcome, represents an accessible big database for gamblers to draw upon to. In particular, the existing variety between the estimations provided by different bookmakers, represents an opportunity for the development of new strategies for the probability estimation. Let s define as Doubling rate as: W = D(p r) D(p b) (1) where D(p r) represents the error, in the Kullback-Leibler divergence sense, committed by the bookmaker estimating the true probability distribution p as r, while D(p b) represents the error committed by the gambler, which estimate p as b. According to Kelly s criterion (Kelly, 1956), the optimal situation for gambling occurs when the Doubling rate is maximum. An hypothetic gambler, convinced about the truthfulness of b, will adopt as naive gambling strategy the one of choosing the online bookmaker whose probability estimation r maximize D(p r) in Eq.1. This strategy will lead the gambler to an almost sure

2 ruin, caused by the more reliability of the bookmakers estimates with respect to the gambler s one. From a mathematical perspective this aspect can be represented by the inequality D(p r) < D(p b) which holds true for any bookmaker. Nowadays, the role of experienced analyst in modern bookmaking is fulfilled by an extensive employment of data mining tools. Our purpose is none other than to utilize these bookmakers predictions to build a virtual bookmaker as a combination of the best performers for each sport event. This virtual bookmaker can be interpreted, from the data mining point-of-view, as an AdaBoost, i.e. a boosting method that combines weak learners (single bookmaker estimators) to build a strong classifier. The complexity of the environment and the data involved can make of AdaBoost not the best tool for building our virtual bookmaker; for this reason a comparative analysis between different boosting methods present in literature can be done, in order to find which method fits better with the real context. 2 Data A big database of the final result odds, estimated by different bookmakers for a specified sportive event, is needed in order to develop our virtual bookmaker. A suitable candidate should have the following characteristics: data have to be easily available in big volumes to better instruct the classifier and, for simpler implementation, they have to be classified just in two classes. Data regarding Football matches are easily available in big volumes from online sites, but unfortunately their classification is based in three different classes. A good compromise consists of the transformation of the three possible results Home, Draw and Away (known in Italy as 1,X,2) in Home and not Home (not good for a real implementation of the algorithm). A good database for this kind of results is the website where the odds and results are archived for the major european football leagues matches over the past fifteen years. An example of the data available in the considered database is given in table 1, where the classification is given in terms of Odds, which represent the profit factor in case of a winning bet. These quantities are inversely proportional to the occurrence probability of the respective results. In addition, we need to introduce the concept of the Book as the bookmaking percentage, i.e. the profits of the bookmaker. For this reason the sum of probabilities b in table 1 is greater that 100. A reliable comparison between different values of b onto the dataset, can be achieved normalizing the single values of b in such a way that the summation over the possible results of the relative probabilities must

3 Odds b (%) b (%) H D A H D A Team A - Team B Team C - Team D Team E - Team F Table 1. Bookmaker s odds for three sample football matches ( column 1-3 ), the resulting probabilities, for H, D and A respectively, given by Odd 100 ( b in column 4-6 ) and the summation over each the possible results of the corresponding probabilities ( b). be 1. To reduce the problem to a two class classification scheme, we highlight only the Home result assigning the value 1 if the Home probability is greater than the Draw and Away ones and 1 otherwise. The multidimensional data points, shown in table 1, are then reduced to: b N (%) Red. Data Points x H D A y Team A - Team B Team C - Team D Team E - Team F Table 2. In column 1-3 the normalized probabilities for the match outcomes b N are listed while the last column shows the corresponding reduced data points. The value 1 is set if the probability associated to the Home result is bigger then the other two (Draw and Away), 1 elsewhere. The reduced data points y in table 2, represent the optimal configuration for scouting the AdaBoost algorithm, which will be described in the following section. 3 AdaBoost and other boosting methods The first candidate algorithm for our virtual bookmaker is AdaBoost, a well known and largely used meta classifier (Freund & Shapire, 1995). The implementation adopted for this paper is not the one presented by Freund & Shapire, 1995 but the one in Rojas, 2009: in fact, the former one involves a single weak-learner instructed on differently weighted data, while the latter employs different already instructed classifiers to build a strong classifier. In order to build our virtual bookmaker, we have to implement the following three steps: a) scouting prospective weak-classifiers b) drafting them, and c) assigning a

4 weight to their contribution. Scouting is done by testing the classifiers using a training set T of N multidimensional reduced data points x i as described in table 2 with labels y i = 1 or y i = 1. We test and rank all the classifiers by charging a cost e β any time a classifier fails a result, and a cost e β every time a classifier provides the right result. With β > 0, misses are penalized more heavily than hits. This kind of error function is called as exponential loss function and it is used by AdaBoost as error criterion. When we test the L classifiers, we build a matrix S in which we record the misses (with a 1) and hits (with a zero) of each classifier. Row i in the matrix refers to the data point x i. Column j is reserved for the j-th classifier (bookmaker in our application): 1 2 L x x x x N (2) The main idea of AdaBoost is to proceed systematically by extracting one classifier in each of M iterations. The elements in the data set are weighted according to their current relevance at each iteration. At the beginning, the same weight 1/N are assigned to all elements. As the drafting progresses, the more difficult examples (those where the majority of classifiers still performs badly) are assigned larger and larger weights. The drafting process concentrates in selecting new classifiers focusing on those which can help with the still misclassified examples. Neglecting the Drafting and Weighting steps, well described in Rojas, 2009, at iteration M we obtain a virtual bookmaker given by: M V B(x i ) = α j B j (x i ). (3) j=1 4 Preliminary considerations Several improvement can be implemented in further development of the presented method. The first of them, will include the implementation of a classification algorithm that considers, as data for the scouting and drafting procedure, both the final result estimates and their statistical confidence. With this implementation the method will take advantage of the probability with which a particular outcome is expected, reflecting the goodness of the bookmaker as

5 a weak classifier, without limiting data as a mere global performance and considering information about bookmaker s different estimate strategies. Moreover, the reduced data list described in section 2 contains only the information about the Home result disregarding the probabilities estimations for Draw and Away results, which will be included in future. In conclusion, because of the complexity of the involved environment and data, an extensive study upon all the possible boosting methods is needed in order to define the one performing better with respect to the other in building our virtual bookmaker. References FREUND, YOAV, & SHAPIRE, ROBERT A decision-theoretic generalization of on-line learning and an application to boosting KELLY, JL A new interpretation of information rate. Information Theory, IRE Transactions on, 2(3), ROJAS, RAÚL AdaBoost and the super bowl of classifiers a tutorial introduction to adaptive boosting. Freie University, Berlin, Tech. Rep.