Statistical Properties of indices of competitive balance
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1 Statistical Properties of indices of competitive balance Dimitris Karlis, Vasilis Manasis and Ioannis Ntzoufras Department of Statistics, Athens University of Economics & Business AUEB sports Analytics Group Computational & Bayesian Statistics AUEB lab 2nd AUEB Sports Analytics Workshop Athens, Greece Athens University of Economics and Business November 2017 I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
2 Competitive Balance Competitive balance is a central concept in the economic analysis of professional sports leagues. It refers to the degree of equality of the playing strengths of teams. There is considerable interest in studying competitive balance across time since it affects the attendance demand and profits as well as fans welfare; the uncertainty of outcome in sport leagues. I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
3 Indices of competitive balance Various measures of competitive balance have been used in the literature Typically they measure competition within season competition across seasons There is a great variety of indices aiming to capture several characteristics of competitions, different sports, measuring effect of interventions etc I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
4 Indices Competitive balance refers to the uncertainty of the sports competition. The result is not known a priori and this makes sports very challenging BUT Uncertainty is present not only within each match but also within each season. the final standings record this uncertainty. Indices do not account for this stochasticity since They are used as single indices without the necessary variability attached to them. They are normalized with respect an improbable and unrealistic scenario of not only equal strength but also also equal result and rankings at the final standings. In this work, we account for this uncertainty and setup the baseline of a more realistic stochastic scenario of equal strength. I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
5 Motivate the study To motivate the present work we consider the following: Data from Premier League and Bundesliga Seasons: from up to We use two indices. Normalized Concentration Ratio: an index measuring the competition for the first place (NCR 1 ) The Herfindahl-Hirschman index (HHI) adjusted for use in competitive balance We want to compare the behaviour across time and between the different championships I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
6 Normalized Concentration Ratio NCR1 NCR Germany England I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
7 NCR1- Add some CIs NCR Germany England I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
8 NCR1- What if all teams of equal strength NCR Germany England I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
9 NCR1- Premier League NCR1 Premier League only black line: average index across time I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
10 Herfindahl-Hirschman index (HHI) HHI Germany England I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
11 HHI- Add some CIs HHI Germany England I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
12 HHI- What if all teams of equal strength HHI Germany England I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
13 HHI- Premier League HHI Premier League only black line: average index across time I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
14 Expected indices under equal strength assumption Most indices compare their values with the value expected under equal strength assumption: all teams are of equal strength and all results are drawed with probability one (no uncertainty). But is this a totally unrealistic assumption: How many leagues with equal ranks for all teams do you know? And even if such a league existed, why this makes the league more interesting since the result of all games will be known a-priori? Due to randomness it is improbable to observe a totally balanced final standings table even if all teams are of similar strength! So, the idea here is that we need to consider a more realistic scenario for reference for the assumption of a perfectly balanced league. I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
15 Expected Points per rank Expected points per rank (18 teams) Points rank I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
16 Story so far I need to explain how the CI s are built... Lessons that we have (hopefully) learned by now We SHOULD account for variability. European leagues may not be so far away from well balanced leagues. What about home effect under the assumption of a balanced league? I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
17 In this work We create confidence intervals based on a parametric bootstrap approach. We will also discuss (if time permits) about sensitivity on certain assumptions different scenarios different indices Our approach is model based in the sense that we replicate the leagues based on the estimated model from the true data. I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
18 Existing Related work Simulations on Brizzi (2002), Owen and King (2013,2015) for particular indices. Some results on some indices but in a very different context. Results from other disciplines do not necessarily apply here for many reasons (e.g. Gini coefficient). I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
19 Model used For ith game with home team HT i against AT i we have that the number of goals scored follow a Poisson distribution with expected counts λ 1i and λ 2i respectively. We use the following Model Structure: We have also used log λ 1i = µ + home + att HTi + def ATi log λ 2i = µ + att ATi + def HTi Negative binomial distribution with the same model structure to check about model deviations. Poisson model with different home effect for each team (interaction between home effect and home team). I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
20 Resampling approach We use a parametric bootstrap resampling approach. The methodology can be summarized by the following steps: S1: Estimate the model using the data at hand. S2: Using the model estimated, generate new data (we need to generate the responses, i.e. the number of goals per team, per game under the hypothesized model. S3: For each generated league, we calculate the indices under study. S4: Repeat steps 2 and 3 a number of time, say B. We consider B = 1000 leagues. S5: Create confidence intervals using the empirical percentiles of the generated values of the indices. We use 95% confidence intervals. I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
21 Indices used Normalized Concentration Ratio (NCR) for the 1st place (NCR 1 ). Normalized Concentration Ratio for the 2nd place (NCR 2 ). Average Concentration Ratio (ACR): it is the average for different NCRs. C 5 : measures the winning percentage of the first 5 teams. NCRI considers the competition for relegation. The SCR synthesizes competition for relegation with that for the first 5 places. The normalized version of the Herfindahl-Hirschman index that measure the share of the teams. Relative entropy. Gini concentartion coefficient. National Measure of Seasonal imbalance (NAMSI) that measures the spread in the winning percentages normalized adequately. I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
22 Results for English Premier League NCR1 NCR2 C ACR NCRI SCR I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
23 Results for English Premier League Rel. Entropy Gini NAMSI HHI I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
24 Results for Bundesliga NCR1 NCR2 C ACR NCRI SCR I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
25 Results for Bundesliga Rel. Entropy Gini NAMSI HHI I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
26 Some Comments- Findings The variability depends on the size of the league (number of teams). For some indices, variability is large and it seems that the observed values could have been generated by a fully balanced championship. Bundelsiga provides indices with clearer separation from the fully balanced scenario Taking into account the variability for all indices it changes the picture! Can we combine results from different in a single statement about competitiveness? I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
27 Robustness with respect the model- A Compare Poisson model with Negative binomial Model HHI Germany Poisson Neg. Bin I. Ntzoufras nd AUEB 2005Sports Analytics2010 Workshop , November, / 32
28 Robustness with respect the model- B Compare the model with that of varying home effect (Poisson only) HHI Germany Poisson Model A Poisson Model B I. Ntzoufras nd AUEB 2005Sports Analytics2010 Workshop , November, / 32
29 Summarizing Competitive balance indices are random variables and we need to account for their stochastic nature. Reporting plain numbers does not give the entire picture. We proposed confidence intervals based on parametric resampling. The approach seems to be robust on model assumptions. While we focused on within season indices for football, the proposed method can be extended to other indices or sports and to indices measuring balance across seasons. I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
30 Further Work Current indices make use of the final standing only. This makes use of rather limited information about the nature of the balance during the season. A natural extension is to consider indices that make use of the scores, the goal difference and/or the standings during the season (not only the final league standings). This is work in progress. I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
31 Selected bibliography Owen, P. Dorian, and Nicholas King. "Competitive balance measures in sports leagues: The effects of variation in season length." Economic Inquiry 53.1 (2015): Owen, P. Dorian, and Dorian Owen. "Measurement of competitive balance and uncertainty of outcome." Handbook on the economics of professional football (2013): Goossens, Kelly. Competitive balance in European football: Comparison by adapting measures: National measure of seasonal imbalance and top 3. University of Antwerp, Research Administration, Manasis, Vasileios, et al. "Measurement of competitive balance in professional team sports using the Normalized Concentration Ratio." Economics Bulletin 31.3 (2011): I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
32 Thanks for your attention AUEB Sports Analytics Group If you are interested please me or visit I. Ntzoufras 2nd AUEB Sports Analytics Workshop 7 8, November, / 32
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