Weak Rationality in Football Results: An Analysis of the Guanabara Cup Using the Bowman and Colantoni Method

Transcription

1 Proceedings of the: 3rd IMA International Conference on Mathematics in Sport, 2011 ISBN: Weak Rationality in Football Results: An Analysis of the Guanabara Cup Using the Bowman and Colantoni Method Alessandro Martins Alves *; João Carlos C.B Soares de Mello**; Thiago Graça Ramos*** and Annibal Parracho Sant Anna **** * Universidade Federal Fluminense, Rua Passo da Pátria 156, Niterói, Brazil + address: alessandro.martinsalves@yahoo.com.br ** Universidade Federal Fluminense, Rua Passo da Pátria 156, Niterói, Brazil + address: jcsmello@pq.cnpq.br *** Universidade Federal Fluminense, Rua Passo da Pátria 156, Niterói, Brazil + address: tgramos@globo.com **** Universidade Federal Fluminense, Rua Passo da Pátria 156, Niterói, Brazil + address: tppaps@vm.uff.br Abstract. The present work uses the BC method (Bowman and Colantoni) to analyze the 2007 Guanabara Cup results. The main objective of the method is to find a coherent matrix of results that is as similar as possible to original matrix of results. A coherent matrix is obtained when the championship results with the weak rationality property, i.e., if team A defeats team B and this one draws with team C, team C can not defeat team A. Based on this coherent matrix, it is possible to identify the most incoherent results of the championship and evaluate the impact of these results in the final rank. 1. Introduction If a club wishes to develop a strategic plan, it needs to understand the factors that influence the outcome of a championship, what means to understand features and relationships that may influence the performance of each team. Improving on such features, it may get a higher ranking in the league thereafter. Football is worldwide a popular sport. This may be related to its easy rules, low cost and no need of very special equipments (Íbsan, 2006). The simplicity of the sport makes difficult to find factors that affect the performance of a team other than the talent of the players. In fact, Gelade (2007) identified that the talent available in each team directly impacts the strength of the team in the following year, in such a way that tradition may be an important explanatory variable of performance. Evaluating the teams, as well as predicting their performances has an important role on planning and making decisions about the posture and the evolution of a football team. In recent years, much effort has been devoted to predicting outcomes along a football season. See, for instance, Craig and Hall (1994), Lee (1997) who studied English football and Stefani and Clarke (1992), which studied Australian football. Another factor that has been shown to be important in explaining the results of football matches is home advantage. To avoid considering its effects, the present study analyses the results of Guanabara Cup in Guanabara Cup matches that year, as a rule, did not have a home field. Only the stadiums of Rio de Janeiro with a better structure were employed for the games involving the clubs of higher investment. This does not occur in many other leagues around the world and use of this variable becomes increasingly important, as shown in Balduck et al. (2010), where are jointly considered the effect of playing at home, the quality of the team and the final classification. Or as in Sanchez et al. (2009), where the effect of home advantage in the two main divisions of Spanish football is estimated. This study focuses on removing intransitive cycles. Changes in the results of the matches that are not coherent with other championship matches are compared in terms of its explanatory power of the clubs final ranking. Comparing matrices of consistent results makes also possible identify those teams whose inconsistency has a higher effect on the final standings of the teams. The method employed is based on finding a consistent matrix of results showing the shortest Manhattan distance to the observed results. The differences found are employed to identify matches that influenced more significantly the final outcome of the championship and the most and least likely outcomes observed. 2. The Method of Bowman and Colantoni The BC method (Bowman and Colantoni) was published in However, its popularity came with a subsequent application in the late 70's, at IBM in France, by F. Marcotorchino and P. Michaud (1979). The

2 general idea of the method is to transform the matrix of any binary relation matrix into a matrix near that but which respects the transitive property as defined by Slater (1961) and Kemeny and Snell (1962). Thus, the method seeks a consistent matrix of results showing the shortest distance to the original matrix, i. e. a set of results as close as possible to the original. The method consists in solving a linear programming problem where the function to minimize is a distance between the original results matrices (dij) and a consistent results matrix (cij). A set of constraints is responsible to assure the weak consistency of this last matrix results. Formally, this is the following linear programming problem. Objective function: Minimize for i j Subject to Cij + cjk - Cik 1 Cij + Cji = 0 Cij = 1 or -1 or 0 The first constraint expresses the concept of weak consistency of the results. For instance, if team "i" beats team "j" (cij = 1) and this team beats team "k", than, necessarily team ï beats team "k". If one of these two initial wins is replaced by a draw, the third result may be maintained or replaced by a draw. The second equation ensures that the outcomes matrix will be coherent, since the sum of the result of team "i" against team "j" with the outcome of the team "j" against time "i" must be zero. Thus, entries at opposite terms relatively to the main diagonal present opposite values. The third constraint is related to the valid values for the outcomes. In fact, for both matrices, (cij) and (dij), the only accepted values are "1", "0" and "-1" corresponding to a win, a draw or a loss for the team in row ï when playing against the team in column j. The number of equations in the first constraint depends on the number of clubs in the championship. This number of equations is the number of transitivity equations and is given by n(n-1)(n-2), for n denoting the number of clubs. In the present work, with six teams, the amount of restrictions due to the transitivity constraint is 120. Similarly, the number of different possible matrices increases with the number of teams. As the results above the diagonal determine those below the diagonal, the number of pairs to consider is n(n-1)/2. As each match may have three possible outcomes, the number of different matrices is, for the case of six clubs given by 3 15 = 14,348,907. Thus, increasing the number of teams makes it unfeasible enumerating all matrices of possible outcomes. 3. The Data Set This paper studies data from the two groups, A and B, of teams in Guanabara Cup 2007, each of them with six teams. Guanabara Cup 2007 was the 43 rd edition of the tournament and its winner was Flamengo. The results of the matches played generate the results matrix (dij). The matrix of original results of group A is in Table 1. Table 1. Observed Results in Group A Boavista Botafogo Flamengo Madureira In words, the first row of Table 1, for instance, shows that Americano has tied Boavista, has lost to Botafogo, Flamengo and Madureira and has beaten Cabofriense.

3 Table 2 presents the results for the teams in Group B. A visual analysis of this Table shows that Fluminense and Friburguense are the two teams that more strongly contradict the criterion of transitivity of results. This is true because the two last rows (or columns) of Table 2, which correspond to these clubs, present completely different results. Vasco Table 2. Observed Results in Group B Nova Iguaçu America Volta Redonda Fluminense Friburguense Vasco Nova Iguaçu America Volta Redonda Fluminense Friburguense Analysis of the Results Three matrices of consistent results for Group A of the Guanabara Cup 2007are obtained by making the minimum number of changes, which is 2, in the observed matrix. There is a consensus among these consistent matrices on the result of the confrontation Botafogo x Boavista. The three solutions point to a draw between Boavista and Botafogo, what did not occur in practice, as Boavista has beaten Botafogo. This demonstrates that the outcome of this match was the most contradictory of the other results in this group. One of the inconsistencies of Boavista's victory over Botafogo occurs because Boavista lost to Flamengo, a team that drew with Botafogo. The first matrix of consistent outcomes suggests, besides changing the result quoted above, other change related to Boavista. This other change involves the team of Madureira. This team has beaten Flamengo, which has beaten Boavista. Therefore, the results will satisfy the constraint of weak coherence, if Madureira beats Boavista. If a change were made in other matches of Madureira or Flamengo, it would require more changes, and the objective function adopted would not be minimized. Table 3 presents the results suggested by this first optimal solution. Table 3. First Transitive Solution or Group A Boavista Flamengo Madureira This matrix of outcomes results in the same two teams qualified by the matrix of real outcomes for the next step of the Cup, which are those two teams with the largest number of points earned, Madureira in the first place and Flamengo in the second. The second consistent matrix indicates a change in the outcome of the match of Flamengo and Madureira. Instead of Flamengo losing, the transitive result would be a tie as shown in Table 4. Flamengo could not loose to Madureira because it beats Boavista and Madureira, what Madureira was not able to do. Table 4.Second Transitive Solution for Group A Boavista

4 Flamengo Madureira This pair of changes, like those in the first solution, does not impact the classification of the teams of Flamengo and Madureira to the next phase, by application of the tournament untying rules. The third coherent matrix, besides changing the outcome of Boavista x Botafogo, changes the outcome of Flamengo x Boavista. Instead of Flamengo's victory over Boavista, the consistent result would be a tie between these two teams, as shown in Table 5. Boavista could not lose to Flamengo if the loss of Flamengo to Madureira is kept, because Madureira and Boavista tied. Table 5. Third Transitive Solution for Group A Boavista Flamengo Madureira The important point of this solution is that it places the teams of Madureira and Botafogo in the next step, what would alter the final outcome of the championship. In fact, Flamengo would not be the champion, since it would be out of the finals games. For a better visualization of changes, the outcome of the championship is presented in Table 6, together with the scores resulting from the three transitive solutions found. Table 6. Final Standings for Group A Real Outcome 1 st Solution 2 nd Solution 3 rd Solution Americano Boavista Botafogo Cabofriense Flamengo Madureira The analysis of Group B is slightly more complex than that of Group A, although one only solution for the linear programming problem is obtained, because this solution produces three inversions. As shown in Table 7, these changes solve the contradictions due to Friburguense beating America and losing to Fluminense as America beats Fluminense and due to Fluminense beating Friburguense and losing to Volta Redonda, which beats Friburguense. Table 7. Transitive Solution for Group B Vasco Nova Iguaçu America Volta Redonda Fluminense Friburguense Vasco Nova Iguaçu America Volta Redonda Fluminense Friburguense

5 As previously done for Group A, Table 8 shows the original final standings and those resulting from the changes in Group B. There, it can be seen that the proposed solution does not affect the classification of two teams of Group B for the nest step of the tournament. Table 8. Final Standings for Group B Real Outcome Transitive Solution Vasco Nova Iguaçu 1 1 América Volta Redonda 6 4 Fluminense 5 4 Friburguense Conclusions The BC method was used here twice to enlighten contradictions between results of four clubs in groups of six. If coherence were kept in the first group, perhaps the league champion would be different. In fact, Flamengo, the eventual champion, might not even qualify for the next step of the championship. These results call attention for the difficulty in clubs management derived from tournament rules based on cutting classes of a fixed number of two or more clubs. As pointed out by Sant`Anna et al. (2010) these cutting points are seldom clearly defined. This study did not consider a variable that is widely used by other authors in performance reviews of several teams and leagues around the world, the advantage of playing at home. In this specific study, this variable became irrelevant. A deficiency found in the model is the fact that it is computationally very intensive, to the extent of becoming impractical for championships with a large number of clubs. 6. References Balduck, A.; Prinzie, A.; Buelens, M.. The effectiveness of coach turnover and the effect on home team advantage, team quality and team ranking. Journal of Applied Statistics, 37 (4), pp Craig. L. A.; Hall, A. R. Trying out for the team: Do exhibitions matter? Evidence from the National Football League. Journal of the American Statistical Association, 89, , 1994 Marcotorchino, F. Michaud, P. Optimisation en Analyse Ordinale des Données, Masson, Paris, Gelade, G. A.; Dobsson, P. Predicting the Comparative Strengths of National Fotball Teams. Social Science Quarterly, volume 88, 1, March Íbsan, A. L.P.; Performance Evaluation of Goalkeepers of the World Cup. Journal of Science 19 (2): (2006). Kemeny, J. Snell, L. Mathematical Models in the Social Sciences, The M.I.T. Press, Cambridge U.S.A Lee, A. J.. Modeling scores in the premier league: Is Manchester United really the best? Chance, 10, 15-19, 1997 Sanchez, P. A., Garcia-Calvo, T., Leo, F.M., Pollard, R., Gomez, M.A.. An analysis of home advantage in the top two Spanish professional football leagues. Perceptual and Motor Skills, 108 (3), pp Sant Anna, A. P. Uchoa, E. and Soares de Mello, J. C. C. B. Classification of the teams in the Brazilian soccer championship by probabilistic criteria composition. Soccer and Society, 11 (3), , Slater, P. Inconsistencies in a schedule of paired comparisons. Biometrica, 48:303_312, Stefani, R. T. E, Clarke, S. R., Predictions and home advantage for Australian Rules football, Journal of Applied Statistics, 9, , 1992.