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

Размещено на http://www.allbest.ru/

ФЕДЕРАЛЬНОЕ ГОСУДАРСТВЕННОЕ АВТОНОМНОЕ ОБРАЗОВАТЕЛЬНОЕ УЧРЕЖДЕНИЕ ВЫСШЕГО ОБРАЗОВАНИЯ

«НАЦИОНАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ УНИВЕРСИТЕТ

«ВЫСШАЯ ШКОЛА ЭКОНОМИКИ»

Факультет экономических наук

Выпускная квалификационная работа

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

По направлению подготовки Экономика образовательная программа «Экономика и статистика»

Выполнил:

Брыш Павел Олегович

Москва 2020



Contents

Introduction

1. Related Literature

2. Empirical Framework

3. Theoretical Framework

3.1. Model Setup

3.2 The Players' Equilibrium Behavior

3.3 Comparative Statics

4. Empirical Estimation

4.1 Data Description

4.2 Reduced Form Evidence

4.3 Structural Estimation

4.4 Estimation Results

4.5 Policy Experiments

Conclusion

References



Introduction

In order to be successful, each firm needs to constantly motivate its employees, so they would exert as much effort as possible. One conventional way to improve the overall performance of the organization is to set up a contest for promotion or any other reward that is conditional on employees' relative standing. However, the effectiveness of this incentive scheme strongly depends on heterogeneity between contest participants who generally differ in their skills. If one employee has an advantage over his rivals, this undermines their willingness to exert effort and, hence, hurts the firm.

In real life interactions, many tasks are performed by competing teams. Here, both the average skill and the composition of each team matter as they define between-team and within-team heterogeneity respectively. If the effect of between-team heterogeneity on the teams' incentives to compete is clearly negative, the role of within-team heterogeneity for effort provision stays ambiguous. On the one hand, having a strong employee (or a superstar) in a team increases its chances of winning due to the increase in its average skill. On the other hand, this triggers free riding because weaker team members may start exerting less effort or the stronger ones may spare their effort. If the latter negative effect prevails, within-team heterogeneity turns to be detrimental.

This paper studies how the two sources of heterogeneity affect the effort provision in teams. In particular, we assess, both theoretically and empirically, how the presence of a superstar(s) in a team affects its probability of winning. Moving the research on heterogeneity in contests from a between-team to a within-team level shapes the main contribution of this paper.

To answer the question of interest, we first introduce between- and within-team heterogeneity to the contest model of Tullock (2001). Two teams, consisting of 20 heterogeneous players each, compete for a single prize. The players simultaneously exert costly effort, and the team's success depends on the aggregate effort. For simplicity, the players are assumed to be of two types - superstars and ordinary employees - that differ in team-specific effort costs.

Next, we solve for a symmetric equilibrium of this game. Since it is difficult to derive and interpret a closed-form expression for effort, we focus on equilibrium winning probabilities. The equilibrium probability of winning for a team monotonously increases (resp. decreases) in the effort costs of the opponent's (resp. its own) players. Increase in the number of superstars in a team has ambiguous impact on its probability of winning depending on a difference in effort costs between its superstars and ordinary players.

Then, we bring the model to the data on the last 6 regular seasons of the National Hockey League. We suggest that a player is considered as a superstar if his average salary throughout the contract is more than 3 times bigger than the average salary for a player of his position in a respective season. We also split the teams into top and bottom ones by the median value of preseason odds of winning the Stanley Cup. This gives a 2 by 2 setting similar to the theoretical model. Reduced-form tests of the impact of within-team heterogeneity were performed by estimating linear probability and logit models. They confirm the negative effect of strategic absence of team's superstar in away games.

Using the equilibrium winning probability derived in the model, we formulate a maximum likelihood estimator and recover the skill profiles of top and bottom NHL teams. Estimates expectedly suggest that superstars in top teams are the best players in the league as they are the most stable and have the smallest effort cost. Superstars in bottom teams take the second place in regard to cost of effort slightly overtaking ordinary players from top teams, however bottom superstars are significantly more unstable than players from top teams. Ordinary players from bottom teams are the worst in the league with the lowest effort cost and highest variance. However, obtained point estimates are quite inaccurate as they have high standard errors. We briefly discuss how the precision of these estimates can be improved.

Finally, we perform policy experiments with the obtained structural estimates. They suggest that all NHL teams are better off having homogeneous rosters without superstars in the long-term perspective as their estimated skill profiles fall within a range of diminishing utility of superstars. Nevertheless, both types of teams with such effort cost distributions can still benefit from having superstars in the short-term perspective.

The paper is structured as follows. Section 2 presents the relevant literature. Section 3 introduces basic terminology and provides the necessary information about the National Hockey League. Section 4 presents a theory that models a hockey match from players' perspective and derives testable predictions. Section 5 describes the data and basic assumptions, provides evidence for the effect of within-team heterogeneity, explains the procedure of the effort costs' estimation, delivers results, and conducts policy experiments. Section 6 concludes.



1. Related Literature

Superstar Effect refers to the reduced effort provision in response to facing a significantly better skilled opponent in a rank-order competition. This notion was firstly introduced by Brown (2011) where the author aimed at assessing the performance of professional golfers at tournaments with and without a highly ranked superstar player - Tiger Woods. Using data from every PGA tournament from 1999 to 2006, she shows that the presence of Tiger Woods in a competition leads to reduced efforts of other participants. Moreover, she finds that intensity of the adverse superstar effect on scores of other players relates directly to the extent of the superstar's superiority. Competitors reduce their efforts significantly if the superstar is dominant and play as usual during his unfortunate periods.

The fact that the difference in skills between tournament participants may result in adverse incentive effects was first emphasized by Lazear and Rosen (1981). Authors conclude that heterogeneity in abilities among employees makes rank-order based remuneration an inefficient allocation of a firm's resources. Workers with lower ability tend to create sabotage in order to distort winners' selection in their favor instead of directing effort on performing their duties. Sunde (2009) and Franke (2012) find results similar to Lazear and Rosen (1981) and provide empirical evidence in sports that heterogeneity in tournaments leads to reduced effort provision. Similar to Lazear and Rosen (1981), Prendergast (1999) examines different remuneration schemes as well as tournaments; however the author addresses the issue of incentives within teams. He points out that team production assessment is crucial in situations where the output of the joint contributions is measured. However, if the reward is shared between colleagues this structure leads to the free-rider problem.

In contrast to the most of the existing tournament models this paper examines effects of two different types of heterogeneity. It focuses on teams competing with each other in environment where heterogeneity is present in both within- and between-team levels. Under certain conditions this environment triggers a non-trivial incentive effect that can reduce team's probability of winning due to a presence of a superstar.

From the economic perspective, an ideal setting for empirical studies of the superstar effect is job promotion and corporate performance. Knoeber and Thurman (1994) use data on performance of US broiler chicken producers to verify predictions from the tournament theory. Authors provide empirical evidence that tournament organizers will attempt to handicap heterogeneous players or reduce mixing to avoid the adverse effects of mixed tournaments. Levitt (1994) uses data on campaign spending in US House elections and concludes that close competition in political races results in the highest expenditure.

However, detailed data on corporate performance is extremely valuable and can rarely be found in open sources, while advanced sport statistics is widely available for free. That is why professional sport competitions have become an alternative setting to test the predictions of the tournament theory.

The presence of the superstar effect was tested in many sports settings. Sunde (2003) and Lallemand, Plasman, and Rycx (2009) use professional tennis data and study the effect of skill heterogeneity on effort provision in elimination tournaments. They conclude that lower-ranked players usually underperform in matches against more skilled opponents. Bach, Gьrtler and Prinz (2009) document less effort provided in heterogeneous competitions of the Olympic Rowing Regatta form Sydney 2000, yet they come to a different conclusion. In particular, only the favorites spare effort, while underdogs show their best performance against dominant opponents.

Lackner (2016) analyzes data on male and female top-level basketball competitions to evaluate the impact of the superstar presence on effort provision in rank-order tournaments. The empirical analysis verifies the presence of the superstar effect. Moreover, the author finds that a lack of dominance brings positive peer effects. Deutscher, Frick, Gьrtler, and Prinz (2013) turn to the problem of sabotage in tournaments with large difference in abilities between the competitors. They assume that favorites tend to exert productive effort, while underdogs are more likely to cause destruction, or sabotage. Using data from the German Bundesliga, they find that teams of higher ability win majority of tackles fairly, while weak teams more often commit fouls. Frick, Gьrtler and Prinz (2008), Nieken and Stegh (2010), Bamieh (2016), and Berger and Nieken (2016) also confirm the negative effect of heterogeneity between teams on their incentives to exert effort in different sports. It should be mentioned that regardless of the sport setting researchers tend to use betting odds as a credible proxy of skill heterogeneity between competitors.

All the studies mentioned above examine the superstar effect only in the between setting where either two players or two teams compete. However, heterogeneity within a single team can be crucial in shaping individual incentives to exert effort. In other words, it is not clear how the presence of a superstar player in a team affects his teammates and, hence, the team's winning probabilities.

On the one hand, the presence of a superstar increases the probability of winning as average ability of team's players increases, which is in line with the superstar effect one finds in the between setting. On the other hand, it can trigger free riding of players within a team. Superstar may save effort as her superior skill allows doing so, or teammates with lower skill may rely on a superstar too much. This paper aims to examine the overall effect of within-team heterogeneity, which is ambiguous.

Closest to this paper is the work of Cyrenne (2014), who takes data from National Hockey League to study the effect of uneven salary distribution within a team on team's winning percentages. Apart from that, he examines the presence of the superstar effect in the league. He derives two important observations. First, teams with higher total salaries and maximum salaries per player have higher winning percentages due to the presence of the superstar effect. Second, equal salary distribution correlates with higher winning percentages. The author concludes that to be successful teams should have no more than two superstars in order to benefit from their outstanding abilities without harming the team with high inequality in salaries distribution. This paper goes beyond the above-mentioned study as data on NHL players' salaries is used to distinguish superstars. Then the influence of their presence on winning probabilities of their team is examined.

Finally, this paper contributes to the structural literature on contests and tournaments. In particular, it aims at providing insight into the optimal team composition by structurally estimating players' costs of effort. Obtained estimations are then used to conduct policy experiments for the choice of a team's game roster. The approach is similar to the one used in the work of Degan and Merlo (2011). They studied participation and voting in US Presidential and Congressional elections. They structurally estimate unobserved parameters in the model of turnout and voting behavior using observed individual voter features from data for the 2000 elections. The unobserved parameters (citizens' ideological preferences, information about candidates and civic duty) are used to replicate various voting patterns observed in the data in order to evaluate impact of different policies on turnout and voting behavior.

The paper of Bhattacharya (2016) is an excellent example of using structural estimation in an empirical study of tournament theory. The author studies R&D procurement contests in the framework of the Department of Defense Small Business Innovation Research (SBIR) program. He analyzes social efficiency of the current competition design by modelling 3 phases of the program. The model is based on estimates of unobserved project values, research and delivery costs, and the share of surplus received by firms obtained from data on research spending and procurement contracts. This paper also recovers skill profiles of players, but the approach differs as here probability of success (winning) is a function of effort costs not vice versa. Moreover, it focuses on the teams' composition and within-team heterogeneity.



2. Empirical Framework

Before presenting a theoretical model of an ice hockey game, some basic terminology, tournament structure of the National Hockey League, and its key rules and concepts should be introduced. Clear understanding of these points is crucial, because some core assumptions of theoretical and empirical models in this paper are based on them.

The Game of Ice Hockey

Ice hockey is a contact team sport played on an ice rink with 2 teams of skaters. Players use sticks to pass and shoot a puck. The goal is to score a point by putting the puck into the opposing team's net. Match is played in 3 periods of 20 minutes, which is called a regulation time. Team with more points at the end of regulation time wins. There is no draw in ice hockey, so if teams have an equal number of points at the end of regulation time, they continue to play in a special format (either sudden death overtime or shootout) until the winner is decided.

Before the game, each team has to submit its game roster, which consists of up to 20 players (18 skaters and 2 goaltenders) and can be changed game-to-game. Therefore, we see some variation in the team composition throughout a season. Only players on this roster are eligible to participate. Each game starts with both teams at full strength, which means that they have 5 skaters (3 forwards and 2 defenders) and 1 goaltender on the ice. This format of the game is also called “5 on 5”. The other 14 players from the game roster must sit on the bench and wait for being invited to substitute an active player. During the game, every player can be substituted unlimited number of times.

National Hockey League (NHL)

NHL is a North American professional ice hockey league consisting of 31 teams Before the 2017-2018 season, there used to be 30 teams in the league. Vegas Golden Knights extended the league and began to play in that season. This fact did not affect the number of games played by one team in regular season.: 24 from the United States of America and 7 from Canada. It is considered to be the most competitive and prestigious professional ice hockey club tournament in the world. The NHL season usually starts in the first week of October and ends in the first half of June. It is divided in two competitive stages: regular season (October - April) and playoffs (April - June).

Teams in the NHL are divided into Eastern and Western Conferences. Each Conference consists of 2 Divisions. In particular, Eastern Conference includes Atlantic and Metropolitan Divisions and Western Conference - Central and Pacific Divisions. All 31 teams participate in the regular season, whilst only 16 of them continue to play in the playoffs. At the end of the season, teams' order in the upcoming NHL Entry Draft is determined depending on the results of regular season and playoffs. Some of the teams also earn prize money. More details on that will follow later.

NHL Season

NHL season includes regular season and playoffs. The regular season in the NHL is a round-robin tournament where each team plays a total of 82 games: 42 of them at the home stadium and 42 away. In every regular season game teams earn points 2 points for a victory, 1 point for a loss in overtime or shootout, and 0 points for a loss in regulation time. At the end of a regular season a team with most points in the league wins The President's Trophy.

After the end of the regular season, top-3 teams in points from each Division continue to compete in the playoffs for The Stanley Cup. On top of this, in each Conference, top-2 teams out of all left get a “Wild Card”, which allows them to enter the playoffs.

The playoffs in the NHL are an elimination tournament. It consists of 4 rounds: Round 1, Conference Semifinals, Conference Finals, and the Stanley Cup Final. The winner of the NHL playoffs is honored with The Stanley Cup. In the first round teams are split into pairs according to the Figure 1. Every round teams in each pair play a series of games against each other. First team to get 4 wins over its opponent proceeds to the next playoff round, where it plays against the winner of a neighbor pair in the bracket displayed in the Figure 1:



Figure 1. The Tournament Design of the NHL Playoffs

Figure 1 clearly displays that the higher a team finishes in its Division in a regular season the weaker opponent it gets in the first round of playoffs. A playoff series can involve from 4 to 7 matches. The line-up of games in a playoff series is shown in the Table 1:

Table 1. A Standard Calendar of the NHL Playoff Series

Team with more points in regular season	Team with less points in regular season
2 games at home	2 games away
2 games away	2 games at home
1 game at home	1 game away
1 game away	1 game at home
1 game at home	1 game away

Table 1 shows that a team, which has managed to score more points than its opponent in the regular season, gets a home ice advantage in case of proceeding to the 7th game of the series.

It should be noted that the tournament design of the NHL playoffs provides incentives for teams to score as much points as possible during the regular season. Teams would want to play against a weaker opponent in the first round of playoffs rather than starting it against a tougher one and always have a home ice advantage in case of playing the decisive 7th game of the series. This way they will have higher chances of advancing further in the tournament.

It is important to mention that unlike a regular season, there is no shootout in playoffs. If there is an even score after regulation time in a playoff game, the game continues in 20-minutes sudden death overtime periods until the decisive goal is scored. Unlike overtime during regular season, during playoffs it is played at full strength.

Remuneration Schemes, Salary Cap, and Superstars' Contracts

According to the latest Collective Bargaining Agreement (CBA) signed in 2013 between the NHL and the NHL Players Association (NHLPA), after every NHL season, the joint prize fund of more than $15 million is allocated between the teams depending on their results in this season. The winner of the Stanley Cup (resp. the President's Trophy) obtains about 29% (resp. 4%) of the prize purse. The team that has lost in the Stanley Cup Final (resp. the Conference Final) earns about 17% (resp. 9.5%). Finally, teams that were eliminated in the second round (resp. the first round) of playoffs get about 4% (resp. 2%) of the prize purse each. Individual remuneration for players such as signing It is a lump sum payment, which is transferred to a player right after the signing of a new contract. or performance It is paid for reaching certain statistical milestones or winning an individual NHL award at the end of the season. Performance bonuses are paid all at once at the end of the season. bonuses can be included in their contracts.

Regarding the distribution of money at the team level, the NHL sets a salary cap that restricts the minimum and maximum amounts of money the teams can pay their players. The salary cap aims at balancing out the financial differences between teams and its value is determined on a season-to-season basis as a 1/62 part of the NHL's forecasted revenue in the new season.

The relationships between NHL teams and players are regulated by bilateral contracts. A team can pay different amounts for a contract each year with limits on salary variance and its minimum value for a season provided that throughout its length a player will receive its total amount. That is why players' cap hits count against a salary cap in order to prevent teams from loading big amounts from new contracts in following years with more cap space. Player's cap hit is the average annual salary of a player throughout the length of a contract.

Regardless of their length, contracts of superstars compared to ordinary players usually have high cap hits as superstars have bargaining power due to high demand from other teams. Such contracts usually occupy a significant percentage of team's salary cap. Every NHL contract is guaranteed in full so a team takes high risk in signing a contact with high cap hit as it cannot be changed in any way after it is signed. Signed player cannot simply lose the contract or be excluded from the team. Therefore, a contract with a high cap hit is a strong signal that a player is valuable for a certain team. This distinction based on a cap hit of player's contract helps us to identify superstars in the empirical part of the paper.



3. Theoretical Framework

This section introduces a theoretical model that forecasts players' decisions about how much effort to exert in a game between two teams. The model is based on the contest success function from Tullock (2001). The aim of this model is to see how the presence or absence of a superstar(s) in a team affects its probability of winning a match, conditional on the opposing team's composition.

3.1 Model Setup

Consider two teams - A and B - striving to win the prize of value W>0. The benefit of losing a match is normalized to zero. Each team consists of 20 players who can be either ordinary athletes (labeled as “o”) or superstars (marked with “s”). As it was mentioned above, each ice hockey team has 20 “dressed” players in its roster. Let be a number of superstar players in team j. Player i from team j independently chooses effort and pays the cost of

,

where differs across teams and athlete types, while superstars are better skilled. In particular, we assume:

for any

Suppose that teammates always comply with the rule “one for all and all for one”. In case of winning, both teams share the prize equally among the players, i.e. each teammate gets . The outcome of the match depends on the aggregate effort exerted by all team members:



,

,

Assume all the information is public and players in both teams are risk neutral. Then, each player chooses his effort in order to maximize the expected playoff: skill team strategic superstar

,

and the solution of this optimization program depends on the teams' composition.

3.2 The Players' Equilibrium Behavior

To make the analysis tractable, we focus on a symmetric equilibrium where all players of type belonging to team j choose the same effort: We do not prove separately that this equilibrium is unique.

,

Then, we can present the probability of winning in the following form:

,

,

and specify the optimization programs each type of athletes solves:



,

,

The following proposition characterizes the equilibrium effort of each player type and the winning probability of team A.

Proposition. In a symmetric equilibrium, all players exert strictly positive effort, and the winning probability of team A is

,

Proof. Consider the system of first order conditions for every player in teams A and B:

,

where:

Combining (2) and (3), as well as (4) and (5), yields the relationship between the effort provisions and the costs for different types within the same team:

,



Then, combining (3) and (5) delivers the relationship between aggregate effort provisions of the opposing teams:

,

Plugging (6) into this equation and transforming it, yields the relationship between the effort provisions of ordinary players from the opposing teams:

,

Plugging (6) and (7) into the probability function yields the probability of winning for team A as a function of effort costs of all four types of the players:

,

The proposition specifies the probability function that is quite general. Taking makes the game equivalent to a standard Tullock contest with two players, i.e. the team setting becomes irrelevant. With , we get the case when team A has at least one superstar and team B does not, i.e. the composition turns to be important. For , superstars are present in both teams.



3.3 Comparative Statics

This subsection analyzes how the equilibrium winning probability of team A depends on the key parameters of the model (namely, , , , , , and ). Unfortunately, is highly non-linear in the parameters of interest. For this reason, we cannot characterize all the effects in a closed form and need to perform the analysis graphically.

We propose the following algorithm. In order to see how each parameter impacts the equilibrium probability of winning for team A, we randomly generate the set of , , , , , and . The effort cost parameters (namely, , , , and ) are drawn from a uniform distribution abiding the constraint of < < < . Looking ahead, the maximum number of superstars in one team observed in the used data is three. Therefore, and are random integers from 0 to 3.

Figure 2 depicts the relationship between the effort cost variables and the probability of winning for team A. It clearly shows that the probability of winning for team A monotonously increases in the effort costs of the opponent's players and monotonously decreases in the effort costs of its own players.

Result 1. The probability of winning for team A monotonously increases (resp. decreases) in the effort costs of the opponent's (resp. its own) players.

The effect of and (a number of superstars in each team) on the probability of winning for team A is ambiguous. In particular, it depends on the difference in the effort costs between a superstar and an ordinary player in the respective team. Figure 3 shows all possible relationships. If is 5 times less than the effort cost of an ordinary player (the top panel of Figure 3), then adding up to 3 superstars in team A decreases its probability of winning. If the difference is 10 times than the effort cost of an ordinary player (the second panel of Figure 3), then adding up to 2 superstars decreases its winning probability, but the 3rd superstar increases it. The 15 (resp. 50) times difference (the third (resp. bottom) panel of Figure 3) gives a positive impact from the 2nd (resp. 1st) superstar. The effect of on the probability of winning for team A is the opposite in regard to the same differences between and . The effect of and on the probability of winning for team B is symmetrical.

Figure 2. The Effect of the Effort Cost Parameters on the Winning Probability of Team A

Figure 3. The Effect of / on the Winning Probability of Team A



Result 2. The effect of on the probability of winning for team is non-monotone in the ratio. In particular, there exists such that:

achieves its minimum at , decreases in the ratio, and there exist and such that

for , for , and

otherwise

Finally, the equilibrium analysis requires examining the relationship between the probability of winning for team A and the number of superstars in the opposing team (namely, ) with constant ratio. As before, one must account for the difference between the effort costs of a superstar and an ordinary player. Figure 4 plots the probability of winning for team A as a function of the ratio.



Figure 4. The Effect of on the Winning Probability of Team A for Given /

As Figure 4 shows, the nature of the equilibrium played does not depend on the composition of the opposing team - the relationship between the ratio and the winning probability of team A is the same in all four scenarios. The only difference is the exact equilibrium probability of winning computed for each number of superstars in the opposing team. Table 2 summarizes the optimal composition of team A as a function of the ratio. Here, the ratio of 12.3 is critical for a choice between zero or three superstars, as well as between one and two superstars.

Table 2. A Number of Superstars in a Team in Descending Order of Its Winning Probability

#	,	,	,	,	,	,
1	0	0	0	3	3	3
2	1	1	3	0	2	2
3	2	3	1	2	0	1
4	3	2	2	1	1	0

To conclude, we have derived the function of equilibrium winning probability for a team that contains effort costs of superstars and ordinary athletes in the opposing teams. Graphical analysis of the function provided evidence that the winning probability for a team monotonously increases (resp. decreases) in opponent's (resp. its own) effort costs. The number of superstars in a team which provides the lowest probability of winning depends on the difference in effort costs between its ordinary players and superstars. Meanwhile, the number of superstars in the opposing team does not affect the nature of the equilibrium played.

The equilibrium probability of winning for a team will be used in the maximum likelihood estimator to recover the skill profiles of different NHL players. It will be highly useful as it helps to avoid dealing with intricate evaluation of explicit effort. We will take into account the non-monotone relationship between the number of superstars in a team and its equilibrium winning probability and will consider all possible team compositions in the maximum likelihood estimator. Finally, the ratio of effort costs within a team will be very important in policy experiments as it will allow assessing the optimality of a chosen team composition for any given game.



4. Empirical Estimation

This section describes the dataset, presents the methods of the empirical analysis, and reports the main results.

4.1 Data Description

For this study, we collect data on 6 regular NHL seasons in the period of 2014-2020. At the moment of conducting the study, the 2019-2020 regular season was paused due to the coronavirus outbreak. The NHL is the most prestigious and respected professional ice hockey league in the world. It attracts the best players, which makes the average skill level of NHL athletes very high. What is more, throughout its long history the NHL has developed multiple rules and regulations that are aimed at tackling heterogeneity between teams and providing effective growth opportunities for those who struggle. Therefore, every NHL team can afford multiple high quality players, or superstars.

The notion of a superstar is subjective, so one must come up with a formal criterion for selecting such players. To solve this issue, we focus on players' salary caps reported by spotrac.com. The main reason for that is the NHL salary cap, which restricts NHL teams in spending on their players. It forces NHL teams to carefully evaluate players before offering them a contract or taking on financial responsibilities on their current contracts while trading them from other teams. Hence, the data on players' salary cap hits aggregate all information about the payment structure of their contracts and best reflects their current value to the team.

We claim that a player is considered as a superstar if his contract's cap hit is more than 3 times bigger than the average cap hit for a player of his position in a respective season. This threshold ensures that there is enough variation for the empirical analysis and yet is high enough to select only outstanding players that are extremely valuable for their teams.

Game-by-game statistical reports for every superstar selected with the formal criterion and every NHL team are collected from the official web-site of the NHL (nhl.com/stats/player and www.nhl.com/stats/team, respectively).

From game reports, we restore the dependent dummy variable hWin, which takes the value of 1 if a home team won the game and 0 otherwise. The model developed in Section 3 focuses on binary outcome. That is why it does not matter for this study if the match has ended in regulation time, overtime or shootout. We link the game outcome to a home team in order to fix a side for which the impact of a superstar is studied. It also allows accounting for the home advantage effect, which is significant in all team sports including ice hockey. Generating all exogenous variables separately for home and away teams will make it possible to distinguish the magnitude of the effects depending on a side.

Team and individual game reports for all selected superstars are matched to create two variables - hStar and aStar. The hStar (resp. aStar) variable shows how many superstars were present in a home (h) (resp. away (a)) team's game roster. These variables are the empirical equivalents of and parameters the theoretical model presented in Section 3.

Based on hStar and aStar, we generate two other dummy variables - hHasStar and aHasStar. These indicators take the value of 1 if at least one superstar was present in the game roster of the respective team and 0 otherwise.

To measure heterogeneity between teams, we take the difference in their betting odds reported by checkbestodds.com. Bamieh (2016), Berger and Nieken (2016), and Nieken and Stegh (2010) use the same approach. For this approach to work well, the betting market must be efficient and aggregate all information available. Since it is ruled by professionals, there is no reason to reject this hypothesis. On the technical side, we calculate the hOdds variable as the ratio of the away team's odds and home team's odds. Practically, hOdds shows how much the home team's probability of winning the match exceeds the winning probability of the away team.

Unfortunately, betting odds are missing for some matches in our sample. The betting odds dataset is complete only for the two latest regular seasons (2018-2019 and 2019-2020) Nevertheless, each season still has a sufficient number of observations. One way to solve the problem of missing data is to exclude games with unavailable betting odds. However, this approach works well if and only if the observations are missing at random. Otherwise, one must control for non-random sample selection. In order to test if the gaps in the betting odds data are random, we generate the betobs dummy (1 - odds are available, 0 - odds are missing) and regress it on hHasStar, aHasStar, team and time fixed effects. As Table 3 reports, no variable has a statistically significant effect. Hence, we conclude that the odds are indeed missing at random. This allows us to exclude observations without betting odds from the dataset, which gives 4'765 matches in total.

Table 3. Testing for Non-Random Missing Betting Odds

Variable	Coefficient
const	0.6514***
	(0.0018)
hHasStar	-0.0016
	(0.0027)
aHasStar	0.0018
	(0.0020)
Team FE	YES
Time FE	YES
F(2, 6255)	0.5947
N obs	7'314

Note: this table reports the estimates of a panel OLS regression with betobs as a binary dependent variable. Standard errors are reported in parentheses. The (***), (**), and (*) signs correspond to statistical significance of the 0.01, 0.05, and 0.1 level, respectively.

We also generate the hSM and aSM variables to control for a superstar(s) strategically missing in a game roster. Here, the value of 1 (resp. 0) means strategic (resp. random) absence. These variables are calculated based on individual game reports for each selected superstar and hOdds variables. If a superstar missed only one game in a row, it was marked as a strategic miss. Most likely, he was given a day-off by the coach. It may be a recovery measure before an important game. It may be a precaution measure as a consequence of a light injury. The superstar could have played, but the coach and medical staff decided to protect this player. It also may be due to a disqualification, but this case is rare and can be ignored. If a superstar missed two games in a row, the dummy value is decided by the strength of the next opponent. If a superstar returns in a home (resp. away) game where hOdds < 0.5 (resp. hOdds > 2), all missed games are considered as a strategic decision to get the superstar well-rested for the game against a significantly stronger opponent. Otherwise, there is likely to be a serious reason that made it impossible for a superstar to participate in a game. If a superstar is absent for 3 or more games in a row, it is likely to be due to a serious injury. Therefore, all such missed games are marked as random event.

Finally, we create hSkill and aSkill binary categorical variables to display the overall skill level of a team. All teams participating in each season are arranged according to their preseason odds of winning the Stanley Cup. Here, we consider preseason odds posted by hockey-reference.com as a public expectation about teams' capabilities in the following season as a result of all changes that happened during the off-season period. These odds are assumed to be accurate enough as bookmakers have to carefully evaluate condition of each team in order to capitalize on this betting line. The hSkill variable has a value of `top' if the home team's preseason odds in the respective season were greater than or equal to the median value and the value of `bottom' otherwise. The aSkill variable is symmetrically generated for away teams.

4.2 Reduced Form Evidence

This subsection examines if the presence of a superstar player(s) in a team impacts its probability of winning a match. To assess the effect of interest, we first estimate the following linear probability model:

,

where

takes the value of 1 if a home team wins match i of season t (takes the value of 0 otherwise);

takes the value of 1 if a home team has at least one superstar in match i of season t (takes the value of 0 otherwise);

takes the value of 1 if a home team has at least one superstar strategically missing in a game roster in match i of season t (takes the value of 0 otherwise);

takes the value of 1 if an away team has at least one superstar in match i of season t (takes the value of 0 otherwise);

takes the value of 1 if an away team has at least one superstar strategically missing in a game roster in match i of season t (takes the value of 0 otherwise);

is the ratio of an away team's odds and home team's odds in match i of season t;

is the unobserved team-invariant time effect of season t;

is the error term.

Then, we estimate a logit model with the same set of controls. Table 4 reports the estimation results. As one can see from Table 4, the two models are statistically significant and give similar results. The presence of a superstar in a team's game roster came out to be insignificant. The coefficients for hHasStar and aHasStar variables are positive but statistically insignificant in both models. Precisely speaking, the LP (resp. logit) model suggests that on average, the presence of a superstar in a home team increases its average winning probability by 1.92% (resp. 1.85%), other things equal. For the logit model, we report the marginal effect at the means of hHasStar.

Another important observation is that strategic absence of a superstar in away games has much stronger impact on a team's winning probability. In the LP (resp. logit) model, strategic absence of a superstar in the away team increases the home team's average winning probability by 15.8% (resp. 16.38%), other things equal. At the same time, the coefficient for the `hSM' variable is vastly insignificant (p-value > 0.97) and close to zero in both models. Therefore, a home team is not at all affected by the strategic absence of its superstars.

Table 4. The Effect of the Superstar(s) Presence on the Winning Probability of a Home Team

	LPM	Logit
,	0.3496***	-0.6466***
	(0.0181)	(0.128)
hHasStar	0.0192	0.0744
	(0.0149)	(0.060)
hSM	-0.0014	-0.0097
	(0.0480)	(0.277)
aHasStar	0.0030	0.0133
	(0.0151)	(0.061)
aSM	0.1580***	0.6650**
	(0.0454)	(0.293)
dOdds	0.1268***	0.5560***
	(0.0094)	(0.048)
Season FE	YES	YES
F stat / Wald stat	33.356	134.09
N obs	4'765	4'765

Note: the LPM column reports the estimates of regression (8). The Logit column presents the coefficients of a logit model estimated for the set of controls specified in regression (8). Standard errors (in parentheses) are clustered by teams. The (***), (**), and (*) signs correspond to statistical significance of the 0.01, 0.05, and 0.1 level, respectively.

Even though the positive effect of a home team's superstar turns to be insignificant, one must check whether it offsets the effects that appeared to be insignificant. In order to test for this, we estimate regression (8) on two different subsamples corresponding to hHasStar = 0 and hHasStar = 1.

Table 5 contains the coefficients of interest. Both models are statistically significant, but the effect of strategic absence of a superstar in a home team turns to be consistently insignificant. In both models, the coefficients for the hSM variable are close to zero, while their p-values are greater than 0.7. The same result is obtained for the aHasStar variable.

Table 5. The Decomposed Effect of the Superstar(s) Presence on the Winning Probability of a Home Team

	LPM (hHasStar = 0)	LPM (hHasStar = 1)
	0.3470***	0.3792***
	(0.0258)	(0.0292)
hSM	-0.0225	0.0245
	(0.0613)	(0.0787)
aHasStar	-0.0010	0.0069
	(0.0217)	(0.0228)
aSM	0.1730***	0.1445
	(0.0662)	(0.1031)
dOdds	0.1308***	0.1185***
	(0.0147)	(0.0115)
Season FE	YES	YES
F stat	21.227	16.354
N obs	2'548	2'217

Note: the table reports the estimates of regression (8). Standard errors (in parentheses) are clustered by teams. The (***), (**), and (*) signs correspond to statistical significance of the 0.01, 0.05, and 0.1 level, respectively.

Apart from that, there is a fascinating change in the effect of strategic absence of a superstar in away teams. The effect is still statistically significant at all conventional levels for teams in away games against opponents without superstars (i.e. hHasStar = 0). Moreover, its magnitude increases up to 17.3% for a home team (+1.5% compared to the average). The opposite pattern is seen in away games against opponents with at least one superstar (i.e. hHasStar = 1). The magnitude of the positive effect of strategic absence of an away superstar for a home team decreases. If a home team has at least one superstar, the effect increases its probability of winning by only 14.45% (-1.35% compared to the average). Meanwhile, the coefficient for the aSM variable in the LPM estimated on the hHasStar = 1 subsample becomes insignificant. Therefore, in this setting the effect is positive but insignificant.

To sum up, the regression analysis testifies that the effect of superstar's presence in a team on its winning probability is insignificant. However, presence of a superstar in a home team erases the positive effect of strategic absence of a superstar in away team. Models suggest that this effect is significant only for a home team with homogeneous game roster.

4.3 Structural Estimation

This subsection uses the theoretical framework developed in Section 3 to construct a maximum likelihood estimator. Let subscripts A and B from the theoretical model correspond to h and a teams defined in the data. The key object one must focus on is the equilibrium probability of winning a match (1) that depends on the underlying skill profile (, , , and ) and the teams' composition ( and ):

,

This makes up six variables that are required to compute the probability of winning a given game for a team. In our sample, we observe a number of superstars in each team. This information is stored as hStar and aStar variables. However, the remaining four variables that measure effort costs of different players' types are unobservable. We formulate a maximum likelihood estimator to recover these parameters.

Ideally, superstars and ordinary players in each team should have their own unique effort costs. However, that makes up too many parameters to estimate, and the problem becomes intractable. To overcome this issue and still stay close to the theoretical framework, for each season we divide 31 teams into top and bottom groups. Such division also seems reasonable as only 16 teams out of 31 continue to compete for the Stanley Cup in playoffs after the end of a regular season. The division is based on teams' preseason betting odds on winning the Stanley Cup. Within each team type, we distinguish superstar (s) and ordinary (o) players.

Assumption 1. Effort costs of a superstar and an ordinary player depend on the type of a team they are playing for.

With this assumption, we can specify the empirical counterparts of , , , and :

,

,

where

(resp. ) for home (resp. away) teams in match i of season t that belong to the top (T) group (= 0 otherwise) and

(resp. ) corresponds to the effort cost of type in a top (resp. bottom) team.

Next, we assume that players perfectly observe , , , and , but the econometrician does not know the exact effort cost realization. This assumption does not change the solution of the game and at the same time lets us introduce randomness to the empirical setup. Then, we must specify the effort cost distributions that complies with all the requirements of the theoretical model. The lognormal distribution seems to be the best choice as it is continuous and has a positive support. Moreover, its density function features a distinct peak, which captures the base skill level for a certain type of players.

Assumption 2. Effort costs are drawn from log-normal distributions, and these draws are i.i.d. over teams' and players' types:

,

,

Therefore, for each player type, there are two parameters - and - to estimate, which gives 8 parameters in total. Finally, we normalize the value of to unity to ensure that all other parameters are identified. This step not only reduces the number of estimated parameters, but fixes units of measurements for all types of players.

Assumption 3. .

Now, we are ready to specify a maximum likelihood estimator. If effort costs are perfectly observable, the equilibrium probability of a home team winning is defined by equation (1). Since the exact realizations of , , , and are unknown to the econometrician, we must take the expectation of over these random variables, and the likelihood function take the following form:

,

where

if a home team wins match i of season t (0 otherwise);



, ;

X corresponds to the set of observables, and

N indicates a number of matches in our sample.

Taking the logarithm of this function, we obtain the optimization program to solve:

,

The variance of a maximum likelihood estimator is calculated by the inverse of the Information matrix, which is the negative of the expected value of the Hessian matrix. Thus, the variance-covariance matrix of a maximum likelihood estimator is:

,

The expected value of the winning probability function depends on the type of an opposing team and a number of superstars in them. Therefore, it must be computed for all possible scenarios. Most of these scenarios are the opposite and thus can be reversed, which makes the computational process significantly easier. First of all, multiple draws of effort costs are randomly generated for a given set of parameters and then a matrix of all unique combinations is formed. Then, the winning probability is calculated for each unique combination of draws and the mean is taken to approximate in the respective scenario.

The value of was maximized using the trust-region method with the constraint. Technical side of the trust-region method is described in Conn, Gould and Toint (2000). To avoid local solutions, the optimization was launched from four different starting points. Each time optimization terminated because the maximum change of the independent variables was less than 0.01. Obtained outputs were compared to each other by means of 100 estimations of a likelihood function. The estimates that ensured the highest value of were chosen. Hessian of the objective function was approximated numerically. All data and code are available by request.

4.4 Estimation Results

Table 6 reports the structural estimates of the effort cost distributions. The obtained coefficients comply perfectly with the basic intuition behind the defined types of players. As Table 6 shows, the best players are superstars in top teams. They have the smallest cost of effort out of all types with the expected value equal to 1.3560. Also, they are the most stable players as they display the least variance. The second smallest expected effort cost equal to 1.9319 is observed among superstars in bottom teams. They are a bit better skilled than ordinary players from top teams, who have the expected effort cost of 2.1397. However, bottom superstars are significantly more unstable than players from top teams as the variance of their effort cost is more than twice as high. It should be noted that ordinary players in top teams are almost as stable as their fellow superstars. Finally, an ordinary player in bottom teams is the worst type with the expected effort cost of 3.0890. They are also far behind everyone else in terms of stability.

...