The superstar effect and its influence on within- and between-team effort provision: the case of professional hockey

Table 6. Estimated Effort Cost Distributions

Note: Standard errors are reported in parentheses. Expected value (resp. variance) is computed as follows: and .

The (***), (**), and (*) signs correspond to statistical significance of the 0.01, 0.05, and 0.1 level, respectively.

Unfortunately, obtained point estimates are quite inaccurate as they have high standard errors. However, it can be seen in Table 6 that standard errors of parameters for bottom teams are higher than those for top teams. Moreover, in top teams they are higher for ordinary players, while in bottom teams they are almost identical. From that it can be concluded that even though the exact values may be imprecise, the difference between players' types is likely to be the same.

One should also remember that proposed partition into team groups and player types is highly coarse. Therefore, precision of point estimates can be fixed by refining division between teams and players or introducing other controls to the model to account for observed heterogeneity. One should also consider adding a team-invariant effect of the home advantage on teams' probabilities of winning a game. All of this can be done in the future to obtain better results.

4.5 Policy Experiments

This subsection uses the obtained structural estimates to run policy experiments over the optimal teams' composition. The theoretical model suggests that the optimal composition of team i is completely unaffected by the composition of the opposing team and the effort costs of its players. However, it depends on the skill difference between superstars and ordinary players, measured as the ratio, within that team (see Table 1).

Our structural point estimates imply that the ratios are 1.58 for a top team and 1.6 for a bottom team. Therefore, it can be concluded that on average, the estimated skill profiles of NHL teams fall within a range of diminishing utility of superstars. In other words, in the long-term perspective, all NHL teams are better off having homogeneous roster without superstars. This result is counter-intuitive; however, one should remember that real ratios can be bigger as obtained point estimates have high standard errors.

Nevertheless, both types of teams with such effort cost distributions can still benefit from having superstars in short-term perspective as the ratio between certain cost draws can be completely different. The dataset suggests that top NHL teams can have up to 3 superstars in their roster. This means that those of them who have 3 signed superstars can increase their probability of winning a game by including all superstars in a game roster if the ratio of their players' effort costs is bigger than 12.3. Bottom teams in the dataset can have up to 2 superstars. Hence, they can benefit from having superstars if the ratio is bigger than 19.

We estimate the probabilities of the effort cost ratios falling within each defined interval and report them in Table 7. As our calculations show, a top team with an observed maximum of 3 signed superstars can increase its chances of winning a game with a 6.68% probability. In this case, the difference in skills between superstars and ordinary players will be high enough to benefit from 3 superstars. There is only a 3.2% chance that it would be better off with 2 superstars instead of none. The top team would benefit from having only 1 superstar only with a 0.74% probability.

Analogically, a possible maximum of 2 signed superstars would benefit a bottom team with a 3.99% probability. There is only a 1.18% chance that such a team would find it profitable to include 1 superstar in its game roster.

Table 7. Probability of Effort Cost Ratios Falling within Different Intervals

To illustrate this result, we provide two examples. On December 3rd, 2019, Toronto Maple Leafs, which was considered as a top team, lost in away game against Philadelphia Flyers. Toronto's superstar Mitchell Marner missed that game due to an injury. According to the model, Toronto would have had higher probability of winning that game without its 2 other superstars Auston Matthews and John Tavares in 96.8% cases.

Another great example would be an away game for Buffalo Sabres against Philadelphia Flyers on December 19th, 2019. Sabres' coach Ralph Krueger probably decided to give a day off to team's superstar Jack Eichel as he was participated in both games on December 17th and December 21st. Buffalo lost that game; however, the model suggests that Ralph Krueger had a 2.81% chance of increasing his team's probability of winning by letting Jack Eichel play in that game. Moreover, the presence of other Buffalo's superstar, Jeff Skinner, without Jack Eichel decreased its probability of winning with a 98.82% chance.

To sum up, the model suggests with a very high probability that Philadelphia Flyers gained advantage in at least two games in December during the 2019-2020 regular season. Curiously enough, the team itself did not do a thing; it was a result of sub-optimal choice of team composition by its opponents.

Conclusion

This paper examines the influence of skill heterogeneity within a team on its performance. As our theoretical model shows, adding a privileged player with higher skills to a team has an ambiguous impact on its probability of winning. The team benefits from this only if the player is significantly better than his ordinary teammates. Meanwhile, the composition of the opponent team impacts only the equilibrium winning probability, yet it does not affect the critical ratios of players' effort costs that determine the optimal composition of a team.

Bring the model to the NHL data reveals that the presence of a superstar in a home team does not have a significant effect on its probability of winning a game. The effect of strategic absence of a superstar in away teams is the only one that was confirmed by the reduced form analysis. However, the effect is inconsistent in regard to a team composition of a home team. The positive influence of strategic absence of a superstar in away teams on a home team's probability of winning a game becomes more intense and significant if it does not have superstars.

The reduced form analysis suggests that there is a synergy between the effects related to within-team heterogeneity and home advantage. In particular, the presence of a superstar in a team increases its probability of winning a game only if combined with the home advantage.

Teams with signed superstars are much strongly affected by their absence then by their presence. It seems that coaches and ordinary teammates rely on superstars too much, so the team becomes significantly weaker if one of them is missing. This can be compensated with a home advantage; however, the problem becomes severe when a team is away.

The estimated effort cost profiles perfectly comply with the logic behind the introduced players' types. Our point estimates are likely to be inaccurate as they have high standard errors. However, the values of standard errors suggest that the true relationship between different types of players was assessed correctly.

Perhaps surprisingly, the performed policy experiments suggest that NHL teams are better off without superstars in a long-term perspective. However, one should keep in mind that the estimates are quite disperse and the actual effort cost realizations may form significantly different pattern. Nevertheless, NHL teams can still benefit from their superstars in short-term perspective as some draws from the estimated effort cost distributions give the difference in skills needed to make the presence of a superstar(s) in a team profitable.

Also, the conducted policy experiments allowed us to conclude that coaches can harm their teams and decrease their winning probability by choosing sub-optimal team composition for a given match. Multiple examples of such mistakes were found in the data. Although there is no evidence that such mistakes explicitly caused teams to lose, the impact could definitely be one of the factors that lead to an unfavorable result.

Finally, we briefly discuss how the present analysis can be improved. As we showed, the estimated difference in skills between superstars and ordinary players in the NHL is counter-intuitively low. To fix this, one can first try to modify the theoretical model and account for the fact that the environment around superstars is extremely demanding. They are always a primary focus of attention from coaches, media and fans as they are expected to make a significant difference in every game. One can account for this in the theoretical model by introducing penalty term for low effort provision in superstars' optimization program. It will reflect high demands from coaches and fans that prevent superstars from free riding. Another possible modification of the theoretical model is introduction of exclusive penalties for superstars in case of losing. This seems reasonable as the media and fans tend to blame primarily superstars during teams' unfortunate periods. One can also consider a higher benefit of winning a match for a superstar as they always attract a lot of attention during successful periods.

Alternatively, we can redefine our maximum likelihood estimator and introduce more player- and team-specific controls. This would allow us to filter observable heterogeneity out and make the estimates more precise. One can try to avoid classifying teams and players completely and represent any player's effort cost as a function of his contract cap hit. This way observable heterogeneity will be captured without introducing multiple types of athletes. Moreover, it can possibly be a unified estimation of players in the league, so there will be no need for team-specific controls.

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