Estimating Demand for Aggressive Play: The Case of English Premier League Football

Transcription

1 Estimating Demand for Aggressive Play: The Case of English Premier League Football R. Todd Jewell Associate Professor Department of Economics University of North Texas Denton, Texas, USA phone: (940) fax: (940) preliminary draft for presentation at the University of Granada October 2008

2 Introduction The sport of football (or soccer as it is know in the US) is played in nearly every part of the world. FIFA (Fédération Internationale de Football Association), the sport s world governing body, has over 200 member countries, a number that surpasses even the number of United Nations members. FIFA establishes the rules of the game for all competitions that it sanctions, including international matches and professional club leagues in member nations. One of the primary laws of the game (Law 12) involves fouls and misconduct and the penalties for such offenses (FIFA, 2008). Law 12 stipulates the penalties for misconduct such as kicking, tripping, or spitting on an opponent, as well as deliberately handling the ball, verbal dissent, and delaying the restart of a game. The penalties range in severity according to the seriousness of the offense, and the referee is responsible for meting out appropriate punishment with input from two assistant referees and a fourth official. The referee may award a free kick (direct or indirect) against the offending team, he may choose to caution the offending player and award a yellow card, or he may take the extreme step of sending the player off with a red card if the misconduct is judged to be egregious enough. 1 The purpose of FIFA s in-game penalties is to promote fair play, punish violent play that may lead to player injury, and generally negate any advantage that fouls and misconduct may accrue to the guilty player or team. From the perspective of an economist, the penalties also serve as an incentive mechanism; specifically, the penalty associated with a given foul is the price that must be paid for committing said infraction. 2 Under an assumption of rationality on the part of players, infractions with higher prices (greater punishments) and lower payoffs will be committed less frequently. The incentive effect of in-game penalties on player behavior has been 1 Football first used color-coded cards to differentiate a caution (yellow) from an expulsion (red) at the 1970 World Cup in Mexico. A similar system is used in other sports such as rugby, volleyball, and water polo. 2 Similar logic has led to an extensive economic literature on the impact of punishment on criminal activity.

3 studied in other sports, with researchers finding evidence that the stringency and enforcement of penalties affects player behavior. The current study attempts to estimate a demand curve for in-game misconduct in the English Premier Football League (EPL), the highest level of professional club football in England. Employing a win maximization framework in which a team chooses its level of aggressive play, fouls and misconduct are modeled as discrete observations of a team s optimum level of aggression. Within this framework, optimum aggressive play is assumed to respond to its price, where the price of aggressive play is an opportunity cost associated with the reduction in the probability of winning resulting from aggressive play. Red-card offenses are used to measure aggressive play. The results indicate that aggressive play in the EPL is indeed responsive to price, but in a way that is somewhat unexpected. Specifically, at very low prices, aggressive play responds positively to price increases, while at higher prices, aggressive play responds negatively to price increases as predicted. Data At the professional club level, European football is generally considered to be the highest quality in the world, even though national teams from South America routinely defeat European competition at tournaments such as the World Cup. Although fans and pundits argue about the relative strengths and weaknesses of a particular country s domestic league, the EPL is always in that argument. The data used in this study include observations on all 1,900 games played in the EPL over five full seasons from to Table One illustrates the level of aggressive play experienced over this time period, where aggressive play is observed as red-card offenses. Referees handed out a total of 300 red cards (60 per year, approximately one every six games). 86% of games had no red cards, 12% had one red card, and less than 2% had multiple red cards.

4 Table One: Summary Statistics for Aggressive Play Sample of 1,900 EPL Games ( to ) Frequency Percentage Red Cards = 0 1, % Red Cards = % Red Cards = % Red Cards = % This study is interested in analyzing the impact of the price of aggressive play on behavior. For this purpose, the sample of 1,900 games can be manipulated to isolate the behavior of both teams in a given match. From each match, we observe the aggressive play of two teams. Thus, the sample contains observations on 3,800 team-matches. Table Two presents summary statistics for the expanded sample. Own (Opponent) Red Card is a dummy variable equal to 1 if a team (a team s opponent) is given a red card in that match. A binary dummy variable is used rather than a categorical variable due to the small number of times a team is awarded more than one red card. Home Team is a dummy variable equal to one if the game takes place at the team s home ground. Point Differential is a continuous measure indicating the difference in points earned by the team and its opponent in the season in which the match took place; thus, it measures differences in team quality based on end-of-season points, it will be positive (negative) if the team finishes with the season with more (less) points than its opponent, and it will vary for each team by year and opponent. Table Two: Summary Statistics Estimation Sample of 3,800 EPL Team-Matches Mean Standard Deviation Minimum Maximum Own Red Card Opponent Red Card Home Team Point Differential

5 Methodology Assume that the objective of team i is to win game k. Further assume that team i maximizes the probability of winning game k over its choice of aggressiveness of play (A ik ) given the aggregate skill level of its players (S i ). In the spirit of Késanne s (1996, 2006) win maximization model, team i is constrained by its ability to produce revenue which is an increasing function of the number of games team i wins. However, team i s optimal choice of aggressive play will be influenced by the opponent in game k; specifically, optimal A ik will respond to the opponent s (indexed j) choice of aggressiveness in game k and aggregate skill level (A jk and S j respectively). At game k, team i s maximization problem is described in equation (1) below, where W(.) is a function that maps aggressive play and skill level to the probability of winning, R l>k [.] refers to future revenues up to game l, C(S i ) represents the monetary cost associated with having a team with skill level S i, and π i 0 is the amount of profit targeted by team i. 3 (1) Max W(A ik ; A jk, S i, S j ) s.t. R l>k [W(A ik ; A jk, S i, S j )] C(S i ) = π i 0 A ik The first order conditions for team i's constrained win maximization problem implicitly define team i's demand for aggressive play in game k conditional on opponent aggression (A jk ), own skill and opponent skill (S i and S j ), and own targeted profit (π i 0 ). Consider the effect that an increase in opponent aggression would have on optimal A ik. If win probability is decreasing in opponent aggression, one might assume that team i would increase its own level of aggression to offset the advantage gained by the opponent by playing more aggressively. However, this statement ignores the price of aggressive play for team i. Specifically, more aggressive play may lead to scoring opportunities for the opponent and, in the extreme, player expulsions, thus 3 See Fort and Quirk (2004) for a critique of the win maximization model. Note that although this model does not imply profit maximization, it is flexible enough to handle profit maximization (i.e., π i 0 = 0) as well as situations in which club owners or shareholders expect a positive profit.

6 potentially lowering the probability of winning game k. The decrease in the probability of winning is the opportunity cost of aggressive play for team i. Assume a linear function form for demand for aggressive play for each team i at game k: (3) D Aik = Ω ik β + ε i, where D Aik is optimum A ik. The vector Ω ik = [P A, A jk, S i, S j, π 0 i ], P A is the price of aggressive play, the vector β contains coefficients to be estimated, and ε i is the error term. As with all demand functions, demand for aggressive play is assumed to be decreasing in P A. At any given price, teams are assumed to respond to increased opponent aggression and skill with increases in own aggression. Increases in own skill should be associated with decreases in own aggression, since aggression and skill are likely substitutes in the production of wins. Continuous measures of aggressive play are not observable in football. Instead, one observes a discrete number of fouls committed, implying that D Aik is a latent variable. In the case of a dichotomous measure of aggressive play like Red Card (R ik ), the outcome is observed with the following decision rule: (4) R ik = 0 (no) if D Aik 0, and R ik = 1 (yes) if D Aik > 0. If ε is normally distributed, equation (4) implies that demand for aggressive play can be estimated using a probit model. Such estimates are presented in the next section. Estimation The main addition to the literature of this study is to estimate a demand curve for aggressive play in football. As discussed about, demand for aggressive play should be responsive to its price. However, no clear measure of the price of aggressive play exists. A solution to this missing variable problem is to estimate the price based on observable measures. The win maximization model implies that the price of aggressive behavior can be estimated as an

7 opportunity cost. That is, if the relationship between aggression and win probability is negative, then the opportunity cost of playing aggressively can be inferred from the magnitude of this negative relationship. Estimating Win Probability Given the EPL data, the relationship between the probability of winning a game and being awarded a red card can be estimated. The results from a probit estimation, in which the dependent variable is a binary indicator equal to one if the team won the game, are reported in Table Three. 4 The independent variables are Own Red, Opponent Red, Home Team, and Point Differential. Respectively, these variables measure own aggression (A ik ), opponent aggression (A jk ), any advantage a team may have playing in its own stadium, and differences in team skill levels (S i S j ). Also included are dummy variables for team, opponent, match, and year (coefficients not reported for brevity). Team and opponent dummies are included to control for team-level characteristics not measured in other variables, such as differences in the monetary return to winning or expected profit. Match dummies are included to control for differing incentives to win games at different points in a season. Year dummies are included to control for yearly differences in the probability of winning. Table Three reports marginal effects instead of coefficients, and reported standard errors are corrected for clustering on team. The results show that teams that are given a red card are almost 14% less likely to win each game. In addition, if the opponent is given a red card, the team is 11% more likely to win the game. 5 A team is 22.6% more likely to win a game played in its own stadium, and the better team (as measured by end-of-year points) is more likely to win the game. For our purposes, the most important result has to do with Own Red: the estimated negative coefficient implies that the 4 Note that Win = 0 could mean that the team lost or tied the match. Alternatively, one could estimate the model using an ordered model, with outcomes = {win, loss, tie}. 5 In other specifications, both the existence of the red card and the time the red card occurred are included. Interestingly, the time of red card does not appear to have a significant effect on win probability.

8 opportunity cost of red-card aggressive play can be measured as the difference between the predicted probability of win without a red card minus the predicted probability of win with a red card. Table Four presents details of the distribution of Red Card Price, estimated as a difference in predicted probabilities multiplied by 100. Table Three: Estimating the Probability of Winning Probit: Dependent Variable = Win N = 3,800 marginal effect (clustering-corrected standard error) Own Red (0.0334)*** Opponent Red (0.0254)*** Home Team (0.0180)*** Point Differential (0.0006)*** pseudo R 2 = ***Significant at 1% level Table Four: Deciles of Predicted Price of Aggressive Play N = 3,800 Deciles Price of Red Card Min % % % % Mean % % % % % Max Estimating Demand for Aggressive Play Having estimated the price of aggressive behavior in the previous section, we return to the equation of interest, the demand for aggressive behavior. As discussed previously, this

9 demand function can be estimated via probit, if discrete outcomes of red-card offenses are interpreted as indications of the underlying latent variable relationship. Table Five reports the results of the discrete demand estimation. The independent variables are Price of Red Card (and its square to capture nonlinearities), Opponent Red, Home Team, and Point Differential. Also included (coefficients not reported for brevity) are dummies for team, opponent, match, year, and referee. Referee dummies are included to control for any differences in individual propensity to punish offenses with a red card. Table Five reports marginal effects with standard errors corrected for clustering on team. The marginal effects in Table Five indicate that the effect of price on aggressive play is nonlinear, in that increases in price lead to increased aggressive play at low prices and to decreased aggressive play at higher prices. With an inflection point of 10.88, the implication is that the law of demand holds at prices above but not at prices less than It might be instructive to examine the entire demand function, which is presented in Table Six including estimated marginal effects (and standard errors) and estimated price elasticity of demand. Notice that only 36% (1,365 team-matches) of the sample is estimated to be on the upward-sloping portion of the demand function, with the remaining 64% (2,435 team-matches) located on the Table Five: Demand for Aggressive Play Probit: Dependent Variable = Red Card marginal effect (clustering-corrected, bootstrapped standard error) Price of Red Card (0.0047)*** Price of Red Card (0.0002)*** Opponent Red (0.0188)*** Home Team (0.0086)*** Point Differential (0.0004) pseudo R 2 = ***Significant at 1% level

10 downward-sloping portion. Approximately 25% of the sample is located in a section of the demand curve in which price elasticity is greater than one in absolute value. Table Five also reports that teams do indeed respond to opponent aggression with aggressive play; if the opponent is given a red card, a team is 6% more likely to have a red card of their own, ceteris paribus. Furthermore, teams playing at home are 3% less likely to be shown a red card than teams playing away from home. Finally, the differential in end-of season points does not significantly affect the probability of a red card. This lack of significance is likely due to the inclusion of team and opponent dummies, which may be better indicators of aggressive play outcomes. Table Six: Predicted Demand for Aggressive Play from Table Four Aggressive Play = Red Card Price Decile Pr(Red Card) Marginal Effect (t-score) Demand Elasticity (3.71)*** % (3.68)*** % (3.65)*** % (3.59)*** % (3.51)*** % (3.39)*** % (3.20)*** % (2.89)*** % (2.39)** % (1.61) % (0.57) % (0.00) (inflection point) 11 37% (-0.55) % (-1.49) % (-2.15)** % (-2.59)*** % (-2.88)*** % (-3.08)*** (-3.22)*** ** Significant at 5% level ***Significant at 1% level

11 The results with respect to the price of aggressive play are counter-intuitive for those teams that face the lowest prices in terms of the opportunity cost of winning a game; interestingly, these results are robust to different specifications of the win probability equation and the demand for aggressive play equation. For teams with low estimated prices, the opportunity cost of a win does not appear to capture the true cost of aggressive play in the EPL. Conversely, winning may not be the objective for teams with low opportunity costs. Table Seven Table Seven: Average Prices of Aggressive Play by Team Inflection Point at Price of Red Card = Team N Pr(Red Card) Mean Points Mean Price % > Year Relegated Arsenal Manchester United Liverpool Chelsea Everton Tottenham Blackburn Bolton West Ham Aston Villa Newcastle Reading Portsmouth Manchester City Middlesbrough Charlton Fulham Wigan Sheffield United Birmingham , Southampton Leeds United Crystal Palace Norwich Wolverhampton West Bromwich Sunderland Leicester Watford Derby

12 reports the average prices by team (listed highest to lowest average predicted prices), average points earned per season, the percentage of each team s prices that are above the inflection point, and year of relegation. What becomes immediately apparent from an inspection of Table Seven is that the worst teams (i.e., those that were relegated) are also the ones with the lowest average prices and the best teams have the highest prices. Of the 15 teams with the lowest average prices, only two, Fulham and Wigan, were not relegated during the period under study, but both were under threat of relegation in at one of the included seasons. Perhaps the win maximization model does not adequately capture the behavior of teams faced with the prospect of relegation. Table Seven also reflects the power structure of the league; the EPL has four teams that have dominated in recent years, and these four teams, Arsenal, Manchester United, Liverpool, and Chelsea, have the four highest average prices of aggressive behavior. Conclusion This study attempts to estimate a demand curve for aggressive play in football using data from the English Premier League and measuring aggressive play with data on red-card offenses. The results indicate that the law of demand holds for the majority of observations but not for observations in which the team has a low price of aggressive play. At low prices, teams respond to increases in price with increased aggressive behavior. The inconsistency of the results may indicate that the win maximization model is not appropriate for this application or that the estimation does not accurately reflect the opportunity cost of aggressive play for some teams, especially those that are faced with the prospect of relegation.

13 References FIFA Laws of the Game 2008/2009. FIFA, Zurich, Switzerland. July Accessed online: Fort, Rodney, and James Quirk Owner Objectives and Competitive Balance. Journal of Sports Economics 5(1): Késenne, Stephan League Management in Professional Team Sports with Win Maximizing Clubs. European Journal for Sports Management 2(2): Késenne, Stephan The Win Maximization Model Reconsidered: Flexible Talent Supply and Efficiency Wages. Journal of Sports Economics 7(4):