There seems to be an upper limit to predicting the outcome of matches in (semi-)professional sports. Recent work has proposed that this is due to chance and attempts have been made to simulate the distribution of win percentages to identify the most likely proportion of matches decided by chance. We argue that the approach that has been chosen so far makes some simplifying assumptions that cause its result to be of limited practical value. Instead, we propose to use clustering of statistical team pro les and observed scheduling information to derive limits on the predictive accuracy for particular seasons, which can be used to assess the performance of predictive models on those seasons. We show that the resulting simulated distributions are much closer to the observed distributions and give higher assessments of chance and tighter limits on predictive accuracy.
In our last work on the topic of NCAA basketball [
Even if we were able to accurately describe sports teams in terms of their performance statistics, the fact remains that athletes are humans, who might make mistakes and/or have a particularly good/bad day, that matches are refereed by humans, see before, that injuries might happen during the match, that the interaction of balls with obstacles off which they ricochet quickly becomes too complex to even model etc. Each of these can affect the match outcome to varying degrees and especially if we have only static information from before the match available, it will be impossible to take them into account during prediction. 1 Others in the sports analytics community are hard at work doing just that, especially for \under-described sports such as European soccer or NFL football.
While this may be annoying from the perspective of a researcher in sports analytics, from the perspective of sports leagues and betting operators, this is a feature, not a bug. Matches of which the outcome is effectively known beforehand do not create a lot of excitement among fans, nor will they motivate bettors to take risks.
Intuitively, we would expect that chance has a stronger effect on the outcome of a match if the two opponents are roughly of the same quality, and if scoring is relatively rare: since a single goal can decide a soccer match, one (un)lucky bounce is all it needs for a weaker team to beat a stronger one. In a fast-paced basketball game, in which the total number of points can number in the two hundreds, a single basket might be the deciding event between two evenly matched teams but probably not if the skill difference is large.
For match outcome predictions, a potential question is then: \How strong is
the impact of chance for a particular league? ", in particular since quantifying
the impact of chance also allows to identify the \glass ceiling" for predictions.
The topic has been explored for the NFL in [
The actual observed distribution of win-loss records in the NFL is indistinguishable from a league in which 52.5% of the games are decided at random and not by the comparative strength of each opponent. Using the same methodology, Weissbock et al. [6] derive that 76% of matches in the NHL are decided by chance. As we will argue in the following section, however, the approach used in those works is not applicable to NCAA basketball. 2
The general idea used by Burke and Weissbock2 is the following: 1. A chance value c 2 [0; 1] is chosen. 2. Each out of a set of virtual teams is randomly assigned a strength rating. 3. For each match-up, a value v 2 [0; 1] is randomly drawn from a uniform distribution.
{ If v c, the stronger team wins.
{ Otherwise, the winner is decided by throwing an unweighted coin. 4. The simulation is re-iterated a large number of times (e.g. 10; 000) to smooth results.
Figure 1 shows the distribution of win percentages for 340 teams, 40 matches per team (roughly the settings of an NCAA basketball season including playoffs), and 10; 000 iterations for c = 0:0 (pure skill), c = 1:0 (pure chance), and c = 0:5.
By using a goodness of t test { 2 in the case of Burke's work, F-Test in the
case of Weissbock's { the c-value is identi ed for which the simulated distribution
ts the empirically observed one best, leading to the values reproduced in the
2 For details for Weissbock's work, we direct the reader to [
0.14 0.12 0.1 0.04 0.02 itrno 0.08 o p roP 0.06
Pure skill Pure chance
Chance=0.5 0 0 0.2 0.4
0.6 introduction. The identi ed c-value can then be used to calculate the upper limit on predictive accuracy in the sport: since in 1 c cases the stronger team wins, and a predictor that predicts the stronger team to win can be expected to be correct in half the remaining cases in the long run, the upper limit lies at: (1
c) + c=2, leading in the case of { the NFL to: 0:475 + 0:2625 = 0:7375, and { the NHL to: 0:24 + 0:36 = 0:62 Any predictive accuracy that lies above those limits is due to the statistical quirks of the observed season: theoretically it is possible that chance always favors the stronger team, in which case predictive accuracy would actually be 1:0. As we will argue in the following section, however, NCAA seasons (and not only they) are likely to be quirky indeed. 3
A remarkable feature of Figure 1 is the symmetry and smoothness of the resulting curves. This is an artifact of the distribution assumed to model the theoretical distribution of win percentages { the Binomial distribution { together with the large number of iterations. This can be best illustrated in the \pure skill" setting: even if the stronger team were always guaranteed to win a match, realworld sports schedules do not guarantee that any team actually plays against representative mix of teams both weaker and stronger than itself. A reasonably strong team could still lose every single match, and a weak one could win at a reasonable clip. One league where this is almost unavoidable is the NFL, which consists of 32 teams, each of which plays 16 regular season matches (plus at most 4 post-season matches), and ranking \easiest" and \hardest" schedules in the NFL is an every-season exercise. Burke himself worked with an empirical distribution that showed two peaks, one before 0:5 win percentage, one after. He argued that this is due to the small sample size ( ve seasons).
0.045 0.04 0.035 0.03 itrno 0.025 o p roP 0.02 0.015 0.01 0.005 0 0 2008, observed 2009, observed 2010, observed 2011, observed 2012, observed 2013, observed 0.4
0.5
Win percentage 0.1 0.2 0.3 0.6 0.7 0.8 0.9 1 The situation is even more pronounced in NCAA basketball, where 340+ Division I teams play at most 40 matches each. Figure 2 shows the empirical distribution for win percentages in NCAA basketball for six season (2008{2013).3 While there is a pronounced peak for a win percentage of 0:5 for 2008 and 2012, the situation is different for 2009, 2010, 2011, and 2013. Even for the former two seasons, the rest of the distribution does not have the shape of a Binomial distribution. Instead it seems to be that of a mix of distributions { e.g. \pure skill" for match-ups with large strength disparities overlaid over \pure chance" for approximately evenly matched teams.
NCAA scheduling is subject to conference memberships and teams will try to pad out their schedules with relatively easy wins, violating the implicit assumptions made for the sake of MC simulations. This also means that the \statistical quirks" mentioned above are often the norm for any given season, not the exception. Thought to its logical conclusion, the results that can be derived from the Monte Carlo simulation described above are purely theoretical: if one could observe an effectively unlimited number of seasons, during which schedules are not systematically imbalanced, the overall attainable predictive accuracy were bound by the limit than can be derived by the simulation. For a given 3 The choice of seasons is purely due to availability of data at the time of writing and we intend to extend our analysis in the future. season, however, and the question how well a learned model performed w.r.t. the speci cities of that season, this limit might be too high (or too low). 2008, observed, KS=0.0 2008, simulated, cluster bias, KS=0.0526 MC simulation, chance=0.41, KS=0.0748
MC simulation, chance=0.525, KS=0.0694 0.06 0.05 0.04 n o itrop 0.03 o r P 0.02 0.01 0 0 0.2 0.4
0.6 Win Percentage 0.8 1
As an illustration, consider Figure 3.4 The MC simulation that matches the observed proportion of teams having a win percentage of 0:5 is derived by setting c = 0:42, implying that a predictive accuracy of 0:79 should be possible. The MC simulation that ts the observed distribution best, according to the Kolmogorov-Smirnov (KS) test (overestimates the proportion of teams having a win percentage of 0:5 along the way), is derived from c = 0:525 (same as Burke's NFL analysis), setting the predictive limit to 0:7375. Both curves have visually nothing in common with the observed distribution, yet the null hypothesis { that both samples derive from the same distribution { is not rejected at the 0.001 level by the KS test for sample comparison. This hints at the weakness of using such tests to establish similarity: CDFs and standard deviations might simply not provide enough information to decide whether a distribution is appropriate. 4
The ideal case derived from the MC simulation does not help us very much in assessing how close a predictive model comes to the best possible prediction. Instead of trying to answer the theoretical question: What is the expected limit to predictive accuracy for a given league?, we therefore want to answer the practical question: Given a speci c season, what was the highest possible predictive accuracy?. 4 Other seasons show similar behavior, so we treat 2008 as a representative example.
To this end, we still need to nd a way of estimating the impact of chance on match outcomes, while taking the speci cities of scheduling into account. The problem with estimating the impact of chance stays the same, however: for any given match, we need to know the relative strength of the two teams but if we knew that, we would have no need to learn a predictive model in the rst place. If one team has a lower adjusted offense efficiency than the other (i.e. scoring less), for example, but also a lower adjusted defensive efficiency (i.e. giving up fewer points), should it be considered weaker, stronger, or of the same strength?
Learning a model for relative strength and using it to assess chance would therefore feed the models potential errors back into that estimate. What we can attempt to identify, however, is which teams are similar. 4.1
Clustering team pro les and deriving match-up settings
We describe each team in terms of their adjusted efficiencies, and their Four
Factors, adopting Ken Pomeroy's representation [
Season 2008 2009 2010 2011 2012 2013
Number of Clusters 5 4 6 7 4 3
Cluster IDs in Tournament 1,5 4 2,6 1,2,5 3,4 2 Table 2. Number of clusters per season and clusters represented in the NCAA tournament
As can be seen, depending on the season, 340 teams do not separate into many different statistical pro les. Additionally, as the third row shows, only certain clusters, representing relatively strong teams, make it into the NCAA tournament, with the chance to eventually play for the national championship (and one cluster dominates, like Cluster 5 in 2008). These are strong indications that the clustering algorithm does indeed discover similarities among teams that allow us to abstract \relative strength". Using the clustering results, we can reencode a season's matches in terms of the clusters to which the playing teams belong, capturing the speci cities of the season's schedule.
Table 3 summarizes the re-encoded schedule for 2008. The re-encoding allows us to esh out the intuition mentioned in the introduction some more: teams from the same cluster can be expected to have approximately the same strength, increasing the impact of chance on the outcome. Since we want to take all non-chance effects into account, we encode pairings in terms of which teams has home-court. The left margin indicates which team has home court in the pairing: this means, for instance, that while teams from Cluster 1 beat teams from Cluster 2 almost 80% of the time when they have home court advantage, teams from Cluster 2 prevail in almost 57% of the time if home court advantage is theirs. The effect of home court advantage is particularly pronounced on the diagonal, where unconditional winning percentages by de nition should be at approximately 50%. Instead, home court advantage pushes them always above 60%. One can also see that the majority of cases teams were matched up with a team stronger than (or as strong as) themselves. Table 3 is the empirical instantiation of our remark in Section 3: instead of a single distribution, 2008 seems to have been a weighted mixture of 25 distributions.5 None of these speci cities can be captured by the unbiased MC simulation. 4.2
Estimating chance The re-encoded schedule includes all the information we need to assess the effects of chance. The win percentage for a particular cluster pairing indicates which of the two clusters should be considered the stronger one in those circumstances, and from those matches that are lost by the stronger team, we can calculate the chance involved.
Consider, for instance, the pairing Cluster 5 { Cluster 2. When playing at home, teams from Cluster 5 win this match-up in 82.85% of the cases! This is 5 Although some might be similar enough to be merged. the practical limit to predictive accuracy in this setting for a model that always predicts the stronger team to win, and in the same way we used c to calculate that limit above, we can now inverse the process: c = 2 (1 0:8285) = 0:343. When teams from Cluster 5 welcomed teams from Cluster 2 on their home court in 2008, the overall outcome is indistinguishable from 34.3% of matches having been decided by chance.
The impact of chance for each cluster pairing, and the number of matches that have been played in particular settings, nally, allows us to calculate the effect of chance on the entire season, and using this result, the upper limit for predictive accuracy that could have been reached for a particular season. 1. Teams were described in terms of adjusted efficiencies and Four Factors { adding or removing statistics could lead to different numbers of clusters and different cluster memberships. 2. Predictive models that use additional information, e.g. experience of players, or networks models for drawing comparisons between teams that did not play each other, can exceed the limits reported in Table 4.
The table also indicates that it might be less than ideal to learn from preceding seasons to predict the current one (the approach we have chosen in our previous work): having a larger element of chance (e.g. 2008) could bias the learner against relatively stronger teams and lead it to underestimate a team's chances in a more regular season (e.g. 2010). 5
With the scheduling information and the impact of chance for different pairings, we can simulate seasons in a similar manner to the Monte Carlo simulations we have discussed above, but with results that are much closer to the distribution of observed seasons. Figure 3 shows that while the simulated distribution is not equivalent to the observed one, it shows very similar trends. In addition, while the KS test does not reject any of the three simulated distributions, the distance of the one resulting from our approach to the observed one is lower than for the two Monte Carlo simulated ones.
The gure shows the result of simulating the season 10; 000 times, leading to the stabilization of the distribution. For fewer iterations, e.g. 100 or less, distributions that diverge more from the observed season can be created. In particular, this allows the exploration of counterfactuals: if certain outcomes were due to chance, how would the model change if they came out differently? Finally, the information encoded in the different clusters { means of statistics and co-variance matrices { allows the generation of synthetic team instances that t the cluster (similar to value imputation), which in combination with scheduling information could be used to generate wholly synthetic seasons to augment the training data used for learning predictive models. We plan to explore this direction in future work. 6
In this paper, we have considered the question of the impact of chance on the outcome of (semi-)professional sports matches in more detail. In particular, we have shown that the unbiased MC simulations used to assess chance in the NFL and NHL are not applicable to the college basketball setting. We have argued that the resulting limits on predictive accuracy rest on simplifying and idealized assumptions and therefore do not help in assessing the performance of a predictive model on a particular season.
As an alternative, we propose clustering teams' statistical pro les and reencoding a season's schedule in terms of which clusters play against each other. Using this approach, we have shown that college basketball seasons violate the assumptions of the unbiased MC simulation, given higher estimates for chance, as well as tighter limits for predictive accuracy.
There are several directions that we intend to pursue in the future. First, as we have argued above, NCAA basketball is not the only setting in which imbalanced schedules occur. We would expect similar effects in the NFL, and even in the NBA, where conference membership has an effect. What is needed to explore this question is a good statistical representation of teams, something that is easier to achieve for basketball than football/soccer teams.
In addition, as we have mentioned in the preceding section, the exploration of
counterfactuals and generation of synthetic data should help in analyzing sports
better. We nd a recent paper [
Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 1 133/197 Cluster 2 210/227 Cluster 3 261/308 Cluster 4 210/211 46/182 231/352 192/357 341/374 105/272 262/374 409/663 424/448 1/45 76/247 56/261 515/818