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Estimating demand for aggressive play: the case of English premier league football.


The sport of association football (or "soccer" as it is know in the US) is played in nearly every part of the world. FIFA (Federation Internationale de Football Association), the sport's world governing body, has over 200 member countries, a quantity 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). 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 off the offending player with a red card if the misconduct is judged to be egregious enough.

The purpose of FIFAs in-match penalties is to promote fair play, punish violent play that may lead to player injury, and generally negate any advantage that 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. When fouls and misconduct are beneficial to a team or player, penalties reduce the net benefit of such behavior. Under an assumption of rationality on the part of players and teams, infractions with higher prices (i.e., greater punishments) and lower payoffs will be committed less frequently. Such rational behavior in response to the price of misconduct in football is essentially an application of the law of demand; specifically more of a good (misconduct) will be consumed at lower prices of that good, all else equal.

The current study attempts to estimate a demand curve for in-match misconduct in the English Premier Football League (EPL), the highest level of professional club football in England. The EPL has the reputation as a league in which physically aggressive play is part of the culture, where the game is generally played "hard but fair:' Physically aggressive play is considered to be a tactic that, if employed well, can increase the ability of a team of lesser skill to compete with more highly-skilled teams. For example, the London-based club Arsenal has a reputation, whether deserved or not, of being a team composed of players that do not like playing against teams with a physically aggressive style of play, indicating that aggressive play can improve a team's chances of winning against Arsenal. However, there is a cost to physically aggressive play if it leads to fouls and misconduct. Nonetheless, the tactical use of physically aggressive play may be fairly widespread in the EPL given the league's reputation.

Employing a league-point-maximization framework in which a team chooses its level of physically aggressive play as an input, fouls and misconduct are modeled as observations of a team's optimum level of aggressive play. Within this framework, optimum aggressive play is assumed to respond to its "price," i.e., the opportunity cost associated with the reduction in the probability of a win or a tie resulting from aggressive play. Aggressive play is measured as an index of disciplinary points, which accounts for yellow cards and different types of red cards. The results indicate that aggressive play in the EPL is negatively responsive to price; the size of this response is much larger for away teams and the overall responsiveness is relatively small. The best and worst teams in the EPL are shown to be less responsive than others to changes in the opportunity cost of aggressive play. In addition, the very best teams are, on average, completely unresponsive to changes in opportunity cost. If fans of EPL football have preferences for less aggressive play, then the league may be able to increase revenues by reducing aggression through increases in opportunity cost.

Related Literature

The incentive effects of in-match penalties and enforcement of rules on player behavior have been studied extensively in both professional and amateur sports. These studies have generally been based on the Becker (1968) economic model of criminal behavior, where illegal activity is the outcome of utility maximization, in which an individual chooses legal and illegal means to accomplish his or her goals. In the Becker model, individuals make a time-allocation decision and are assumed to respond to incentives in choosing to allocate time to legal and illegal behavior. Illegal behavior has a payoff in terms of higher utility, with higher payoffs meaning more illegal behavior ceteris paribus, resulting in an illegal-behavior supply curve. The model leads to testable hypotheses concerning responsiveness to exogenous changes in the costs and benefits of illegal behavior.

One of the first studies to use the Becker model with sports data is McCormick and Tollison (1984), in which the authors use US collegiate men's basketball data to test predictions with respect to increased enforcement of laws. Using data on regular fouls, the authors find that an increase in the number of game officials leads to fewer fouls, a result the authors argue is due to "better officiating and cleaner play" (p. 224). Following the example of McCormick and Tollison, other researchers have analyzed the relationship between illegal behavior and rule enforcement in the US National Ice Hockey League (NHL) (Allen, 2002, 2005; Heckelman & Yates, 2003; Levitt, 2002; Wilson, 2005). These papers generally find that the degree of enforcement and the strength of punishment serve to limit rule violations.

Although McCormick and Tollison's main purpose is to analyze the relationship between the number of law-enforcers and illegal activity, the study also uncovers information on illegal behavior and its opportunity cost. For instance, their study finds that teams are fouled less when they shoot better, an unsurprising result since a "cost" of fouling is the points given up from free throws or additional opponent possessions. The literature on crime in the NHL also finds a negative relationship between illegal behavior and opportunity cost. Allen (2002, 2005) analyzes both violent and non-violent offenses using NHL data from the 1998-99 regular season, where violent offenses are defined as those that could potentially cause injury. Given that penalties in hockey lead to power-play opportunities for the opponent, a "cost" of fouling is the potential goal given up during the penalty period. As stated by Allen (2005, p. 505): "In the context of ice hockey, the cost of illegal activity should be greater when the opponent is more proficient at scoring power-play goals and less when a player's own team is more proficient at penalty killing or at scoring in general:' As predicted, the author's estimations indicate that non-violent offenses occur less often for teams with higher opportunity costs due to poor penalty killing and violent offenses occur less often for teams with higher opportunity costs due to low scoring potential.

The incentive effects of changes in punishments and rewards have also been analyzed in association football. Witt (2005) analyzes the impact of an increase in punishment for aggressive play in football using data from the EPL. At the beginning of the 1998-99 season, FIFA mandated that tackles from behind would be given an automatic red card. Interestingly the study finds that increasing the severity of punishment led to an increase in yellow cards but not red cards. Using data from the Spanish professional league, Garicano and Palacios-Huerta (2005) analyze the impact of changes in incentives on "sabotage" in football matches. In this case, sabotage refers to behavior that lowers the quality of opponent play. Exploiting an exogenous increase in the payoff to winning a match (from two league points to three in 1995), the authors find that the number of yellow cards increased and the number of goals scored did not change in response to the new incentives, contrary to what the new rule had intended. Furthermore, the authors show evidence that attendance decreased as a result of increased aggressive play.

Conventional wisdom suggests that if a team gets a red card, thus reducing that team to 10 rather than 11 players, the team will be less likely to win or tie against a full-strength opposing team. Given the world-wide interest in the outcome of football matches and the availability of data on in-match penalties, it comes as somewhat of a surprise that academics have not spent more time evaluating the relationship between match outcomes and in-match misconduct. Using data from the Dutch professional football league, Ridder et al. (1994) statistically model the effect of a red card on the outcome of a match. Splitting matches into 15 minute intervals, the authors find that a red card awarded to an opponent in the first 15 minutes of the match doubles the probability that the team wins the match, from 37.5% to 65%. Red cards awarded later in the match also increase the probability that the non-offending team wins, but by a smaller amount. As expected, this result suggests that the effect of a red card will be stronger the longer a team must play a man down.

Using data from the 2002 FIFA World Cup, Torgler (2004) finds that a team that had a player excluded for a red-card offense was about 30% less likely to win a match. Wright and Hirotsu (2003) model the effect of a red card on a team in the context of the "professional foul," i.e., a foul taken to negate a scoring opportunity. The authors analyze the payoff to such fouls, especially the relationship between the payoff and time of the foul. As expected, the payoff is generally lower the earlier in the match the foul is committed.

A closely-related literature involves the impact of refereeing bias in football, which is normally found to favor the home team. Given that a referee is solely responsible for in-match punishment in football, any bias in referee decisions can significantly affect how the game is played. Garicano et al. (2005) and Sutter and Kocher (2004) find evidence that referees are biased in favor of the home team in their decisions of adding playing time in excess of 90 minutes of open play and in their decisions to award penalty kicks. Dawson et al. (2007) find evidence of inconsistency in the interpretation and application of the rules among referees and that away teams are more likely to receive disciplinary sanctions by the referee. Dohmen (2008) also finds evidence that referees tend to make decisions that favor the home team.

Carmichael and Thomas (2005) analyze the home advantage in the EPL in detail, finding that the home team has other important advantages over the visiting team and that more aggressive play by the away team leads to more goals scored. The authors interpret their overall findings to mean that teams playing at home choose to be more attacking and visiting teams choose to be more defensive. Buraimo et al. (2009) evaluate the hypothesis of referee favoritism toward the home team using data from both the EPL and the German Bundesliga. Controlling for an extensive list of in-game factors, the authors find results that support previous research: It appears that EPL home teams have substantial advantages over the visiting team, and the most important is the favoritism of the referee.

Modeling Demand for Aggressive Play

In the present study, a league-point-maximization model is used to represent the behavior of EPL teams in which aggressive play is a choice input. Assume that the objective of team i is to maximize the number of league points earned in match k. Further assume that team i maximizes the probability of winning match k (and receiving three league points) or tying match k (and receiving one league point) over its choice of physical aggressiveness of play ([A.sub.ik]) given the aggregate skill level of its players ([S.sub.i]). In the spirit of Kesenne'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 league points team i earns. However, team is optimal choice of aggressive play will be influenced by their opponent in match k; specifically optimal [A.sub.ik] will respond to the opponent's (indexed j) choice of aggressiveness in match k and aggregate skill level ([A.sub.jk] and [S.sub.j] respectively).

At match 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 or tying (and, thus, the probability of earning league points), [R.sub.l>k][.] refers to revenues for future matches up to match l, C([S.sub.i]) represents the monetary cost associated with having a team with skill level [S.sub.i], and [[pi].sub.i.sup.0] is the amount of economic profit targeted by team i.

Max W([A.sub.ik]; [A.sub.jk], [S.sub.i], [S.sub.j]) s.t. [R.sub.l>k][W([A.sub.ik]; [A.sub.jk], [S.sub.i], [S.sub.j]) - C([S.sub.i]) = [[pi].sub.i.sup.0] (1) [A.sub.ik]

The first order condition for team i's constrained maximization problem implicitly defines team i's demand for physically aggressive play in match k conditional on opponent aggression ([A.sub.jk]), own skill and opponent skill ([S.sub.i] and [S.sub.j]), and own targeted profit ([pi].sub.1,sup.0]). Optimal [A.sub.ik] occurs at the level of aggressive play at which the benefit to aggression, in terms of league points generated by the marginal amount of aggression, equals the cost of that marginal amount of aggressive play.

Consider the effect that an increase in opponent aggression would have on optimal [A.sub.ik]. If win/tie 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 potentially lowering the probability of winning or tying match k, which reduces the expected return from the match in terms of the number of league points. Thus, the decrease in expected league points due to a lower probability of winning or tying can be interpreted as the opportunity cost of aggressive play for team i in match k. (1)

Assume a linear function form for demand for aggressive play for each team i at match k:

[D.sub.Aik] = [[OMEGA].sub.ik[beta]] + [[epsilon].sub.i] (2)

where [D.sub.Aik] is optimum [A.sub.ik]. The vector [[OMEGA].sub.ik] = [[P.sub.A], [A.sub.jk], [S.sub.i], [S.sub.j], [[pi].sub.i.sup.0]], PA is the opportunity cost (or price) of aggressive play, the vector [beta] contains coefficients to be estimated, and [[epsilon].sub.i] is the error term. As with all input demand functions, demand for physically aggressive play is assumed to be decreasing in [P.sub.A]. At any given price, teams are assumed to respond to increased opponent aggression and skill with increases in own physical aggression. Increases in own skill should be associated with decreases in own aggression, since physical aggression and skill are likely substitutes in the production of wins.

The present study is an application of production theory and the theory of input demand. In this context, aggressive play is technically not "illegal," except for aggressiveness that leads to red-card expulsion. As discussed above, 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 league-point-maximization model implies that the price of physically aggressive play can be estimated as an opportunity cost. That is, if the relationship between aggressive play and win/tie probability is negative, then the opportunity cost of playing aggressively can be inferred from the magnitude of this negative relationship. Given past research on win/tie probability and red-card expulsions, this study measures the opportunity cost of aggressive play in the EPL as the loss in league points associated with red-card aggressive play. The methods used to estimate opportunity cost are described in the next section.

Although the present study is similar to past research on the economics of crime in sports, this study differs from that literature in two principle ways. First, this study does not model aggressive play within Becker' paradigm, which is the basis for much of the past research. Instead, aggressive play is modeled assuming that it is an input into the production of league points. However, the models lead to similar conclusions: If the cost of illegal (or aggressive) behavior increases, then illegal (or aggressive) behavior is curtailed. In the Becker model, the relationship is due to the fact that the payoff to illegal behavior decreases when costs increase. In the model used in this paper, the relationship is derived from the team's usage of aggressive play as an input in the production of league points. The second difference between this study and past research is in its focus; this study concentrates on the relationship between the opportunity cost of aggressive play and its opportunity cost. Furthermore, the opportunity cost of aggressiveness is directly estimated, rather than inferred from exogenous variables.

Data and Estimation

At the professional club level, European association 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 FIFA World Cup. Although fans and pundits can argue about the relative strengths and weaknesses of a particular country's domestic league, the EPL is currently the most-highly ranked domestic professional league in Europe. (2) The data used in this study include observations on a112,280 matches played in the EPL by 30 different teams over six full seasons from 2002-03 to 2007-08. (3)

Aggressive play in the EPL can be measured as an index of disciplinary points associated with both yellow-card fouls and red-card fouls. The English Football Association (EFA) keeps track of the disciplinary records of players through accumulation of disciplinary points assigned to yellow-card and red-card infractions (called "penalty points" by the EFA). Red cards are assigned disciplinary points based on the nature of the infraction. Red cards for second-yellow-card fouls are assigned 10 disciplinary points, as are red cards for professional fouls and deliberate handballs. Red cards for serious foul play, violent conduct, spitting, or offensive, insulting, abusive language and/or gestures are given 12 disciplinary points, implying that the EFA feels that these are more serious offenses. (4) A yellow card is assigned four disciplinary points without regard to the specific offense, with the exception that two yellow cards for the same player are counted as a single red-card offense. Given the implicit weights assigned by the EFA, the sum of yellow cards and red cards times their weights results in an index of disciplinary points that can be used to measure aggressive play in each EPL match.

In the six seasons from 2002-03 to 2007-08, the average number of disciplinary points per EPL match was 6.6; however, the home team had 25% fewer discipline points on average than the away team. Over the sample period, EPL referees handed out a total of 371 red cards and 6,730 yellow cards. Table 1 illustrates the level of aggressive play experienced over this time period by home and away teams. In the light of research on the bias of referees toward the home team, it should be no surprise that the home team was awarded 34% fewer red cards and 25% fewer yellow cards than the away team. Another explanation could be that the difference in red and yellow cards simply reflects differences in playing style; specifically, the away team generally plays more aggressively than the home team in order to negate a home-field advantage. The most common red-card infraction is a second yellow card (45% of the total), followed by serious foul play (2%), and a professional foul (1%). (5)

Table 2 presents summary statistics for the sample separated by home and away team. During the sample period, the home team won 47% of the time, the away team won only 27% of the time, and 26% of the matches ended in a tie. The variable Disciplinary Points is a weighted index of red and yellow cards; unsurprisingly, home teams have fewer disciplinary points on average. Red Card is a dummy variable equal to one if a team is given at least one 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. (6) Red Card Time measures the number of minutes a team spends with 10 or fewer players. For example, if a team is given a red card in the 35th minute, the team will play with 10 players for 55 minutes. (7) The difference in aggressive play between the home and away team is also evident in terms of the number of yellow cards; home teams were sanctioned with 1.3 yellow cards per match on average, while away teams on average had 1.7 yellow cards.

Estimating the Price of a Red Card

The main addition to the literature of this study is to estimate a demand curve for aggressive play in football. As discussed above, the opportunity cost of playing aggressively can be inferred from the magnitude of the relationship between the probability of winning or tying a match and being awarded a red card. Demand is estimated using a two-stage method: the price of a red card is estimated in the first stage, and this estimated price is used as a regressor in a second-stage estimation of the demand for physically aggressive play. Identification is an issue in the first stage, since the probability of a red card in a given match and the price of a red card in that match are simultaneously determined by the events that occur and participants of the match. Incorporating the time a red card is given creates an interaction term between the binary variable Red Card and the continuous variable Red Card Time, which serves as an exclusion restriction to identify the first stage separately from the second stage. Furthermore, the two-stage model is technically identified since the first stage is non-linear; however, the results from this study do not hinge on the validity of identification by structural form.

Estimating the opportunity cost of a red card necessitates estimating the impact of a red card on a team's probability of winning or tying in each match. Following previous literature, the effect of a red card on the probability of winning or tying is expected to vary with the minutes a team is forced to play with 10 or fewer men: the more time a team spends a man down, the lower is the win/tie probability. As suggested by Carmichael and Thomas (2005), the number of red cards and the number of yellow cards may influence the number of goals scored, which should affect the probability of winning or tying. However, one would expect the impact of yellow and red cards to be different, since they are fundamentally different in terms of punishment. Specifically, although an individual player can only receive a maximum of two yellow cards in a match before being expelled, a team can accumulate a maximum of 11 yellow cards before having a player sent off.

As shown in Table 2, the home team is much more likely to win a match than is the away team. Past research suggests that the home and away teams in a given match choose different styles of play based on different win/tie probabilities and that the home team may have a clear advantage in terms of referee favoritism, suggesting that it would be appropriate to estimate the cost of a red card separately for home and away teams. Table 3 reports results from two multinomial logit estimations, in which the dependent variable is a trinomial, limited-dependent variable (zero = loss, one = tie, and two = win) and the sample is split into home and away teams. (8)

Own/Opponent Red Card Time (and their squares to capture any non-linearities) and Own/Opponent Yellow Cards (and squares) measure own aggression ([A.sub.ik]) and opponent aggression ([A.sub.jk]). Team skill levels ([S.sub.i] and [S.sub.j) are measured with a team's league-point total at the end of the relevant season (Own/Opponent League Points). (9) Team dummies are also included to control for team-level characteristics not measured in other variables, such as differences in the monetary return to winning or tying. Year dummies are included to control for yearly differences in the probability of winning or tying. Match dummies are included to control for differing incentives to win or tie matches at different points in a season. Matches at the end of a season often take on added significance as teams fight for the championship or against relegation to a lower division. Own/Opponent Significant Match is a dummy variable indicating whether the outcome of the end-of-season match influences the probability that a team/opponent is either relegated to the second level of English football or wins the EPL championship. (10)

The results in Table 3 show that red cards influence win/tie probability for both home and away teams. For teams that are awarded a red card, the average time spent with less than the full complement of players is 27 minutes. The estimates suggest that a home team given a red card with 27 minutes remaining is 22.5 percentage points less likely to win and 7.8 percentage points more likely to tie a match as compared to a team that was not given a red card. In comparison, an away team given a red card with 27 minutes remaining is 8.6 percentage points less likely to win and 0.5 percentage 2points more likely to tie the match. Aggressive play in terms of yellow-card fouls also affects win/tie probability, with smaller effects than for red cards. For the home team, the first yellow card decreases the probability of a win by 5.1 percentage points and increases the probability of a tie by 2.5 percentage points. For the away team, the initial yellow card lowers the probability of a win by 2.1 percentage points and increases the probability of a tie by 1.8 percentage points. (11)

Together, the results with respect to red cards and yellow cards have at least two interesting implications. First, aggressive play is more significant for the home team in terms of its effect on win/tie probability, indicating that the opportunity cost of aggressive play might be higher for the home team. Second, aggressive play appears to increase the probability of a tie and decrease the probability of a win, with the larger effect being on the probability of winning. Given that a team is assumed to have the goal of league-point maximization and that a "win" is worth three times as many league points as a "tie," the EPL data show that the marginal league-point-productivity of aggressive play is negative for the average team. However, not all teams are "average" and some teams approach a match looking for a tie rather than a win, a likely event in the case of teams with much lower skill levels than their opponents. In such teams, the league-point-maximizing choice may be to "play for a tie," which is shown to be positively related to aggressive play on average.

The results in Table 3 also suggest that team skill levels are significantly related to league-point production in the EPL. As expected, better teams have higher win/tie probability, and the marginal effect is similar for home and away teams. The effect of Significant Match varies for home and away teams. If the match is significant, a home team can expect a 21.5 percentage point increase in the probability of winning and a 14.5 percentage point decrease in the probability of a tie. The effect for the away team is consistently positive, estimated as an increase of 5.6 and 10.6 percentage points for win and tie probability respectively. Interestingly, home teams appear to "go for broke" in significant end-of-season matches, giving up the chance to tie for the chance to win. In similar matches, away teams play in a manner that increases both win and tie probability, with the largest increase associated with a tie. This result is further evidence of the variation in playing style among home and away teams and is indicative of the differences in objectives for home and away teams.

For purposes of this study, the most important result in Table 3 has to do with the negative effect of Own Red Card Time: The estimated negative effect for home and away teams implies that the opportunity cost of aggressive play can be measured as the difference between the expected number of league points without a red card and the expected number of league points with a red card. To estimate the price of a red card, the results from Table 3 are used to predict the probability of a win and the probability of a tie for each team (home and away) in the absence of a red card (Own Red Card Time and [Own Red Card Time.sub.2] = 0). Win/tie probability is then estimated at the mean number of minutes when a red card is given (Own Red Card Time = 27 and [Own Red Card Time.sub.2] = 729), under the assumption that teams react to the average expected time of playing with less than the full complement of players if awarded a red card. (12)

Given that a tie gives a team one league point and a win gives a team three league points, Red Card Price is constructed in the following manner:

Red Card Price= [Pr(tie| red=0)+3xpr(win| red=0)]-[Pr(tie| red=1)+3xPr(win| red=1)]. (3)

Note that Red Card Price will vary by team, match, and year, as well as by the values of the independent variables, implying that each team will have a different opportunity cost for each match. For home teams, the mean of Red Card Price is 0.596; thus, a red card with 27 minutes remaining reduces the expected number of league points per EPL home match by approximately 0.6. At the sample average number of league points per home match of 1.672, this corresponds to a 36% decrease in expected league points per match. For away teams, the mean of Red Card Price is 0.254, less than half that of home teams. At the sample average number of league points per away match of 1.072, this corresponds to a 24% decrease per match. As expected, the opportunity cost of aggressive play is higher for home teams. In this light, the differences in red cards among home and away teams shown in Table 1 are less surprising. (13)

Estimating Demand for Aggressive Play

Having estimated the price of physically aggressive play in the previous section, we return to the equation of interest, the demand for aggressive play in the EPL. Recalling equation (2), the vector of independent variables includes the opportunity cost of own aggressive play, opponent aggressive play, own and opponent skill levels, and expected profit. Red Card Price measures opportunity cost, and Opponent Red Card and Opponent Yellow Cards measure opponent aggressive play. Absolute team skill levels will influence aggressive play, where Own and Opponent League Points measure absolute skill levels. Team dummies are included to capture differences in tendencies toward aggressive play or expectations of economic profit among teams. Referee fixed effects are included to control for any differences in individual propensity to hand out a red card and ability or determination to enforce the rules of the game.

Relative team skill will also influence aggressive play if teams play harder in matches in which they have a greater chance of winning. Relative skill is measured with Outcome Certainty, equal to the absolute value of the standardized league-point differential of teams at any match. (14) The coefficient on Outcome Certainty is hypothesized to be negative, as teams should demand less aggressive play as the difference in skill gets larger and the outcome becomes more certain. One might also expect relative skill to influence the degree to which opportunity cost affects aggressive play. As discussed above, teams that are at a large disadvantage in relative skill may play for a tie rather than a win; given the positive relationship between tie probability and aggressive play, the opportunity cost of aggressive play for such teams should be less important. Also, for teams with a large advantage in relative skill, opportunity cost may be relatively unimportant. Thus, the negative impact of Red Card Price on aggressive play should be smaller in matches between unequally matched teams. To capture such an effect, an interaction term (Outcome Certainty*Red Card Price) is also included in the demand estimation. (15)

Table 4 reports coefficients from OLS estimations of equation (2) using Disciplinary Points to indicate aggressive play. The sample is split for home and away teams, and standard errors are corrected for clustering on team and bootstrapped due to the inclusion of an estimated independent variable. (16) The results show clear differences in demand for aggressive play among home and away teams. The coefficient on Red Card Price is negative and significant in both regressions. Thus, it appears that the opportunity cost of red-card aggression is a significant determinant of overall aggressive play in the EPL; however, the effect of price appears to be much smaller for the home team. Furthermore, the negative effect of Red Card Price on Disciplinary Points is significantly smaller in matches that pit highly mismatched teams against each other.

To illustrate the impact of Red Card Price on aggressive play, one can compute its marginal effect at different levels of Outcome Certainty. For the home team, the estimated marginal effect of Red Card Price is -1.5 at the 25th percentile of Outcome Certainty, -0.3 at the 50th percentile, and +1.3 at the 75th percentile. (17) Thus, the marginal effect of opportunity cost on aggressive play for the home team is actually positive in matches between teams of vastly different skill levels. Note that this effect applies for both high-skilled home teams and low-skilled home teams, so that a high-skilled (low-skilled) home team will play more aggressively if it plays a team with a much lower (higher) skill level ceteris paribus. For away teams, the marginal effect is consistently negative, although it is smaller in absolute value in matches with higher Outcome Certainty. Evaluated at the same percentiles of Outcome Certainty, the estimated marginal effect for away teams is -24.6 (25th), -22.3 (50th), and -19.2 (75th), showing that the away team is substantially more responsive to changes in opportunity cost than the home team.

Table 4 shows that opponent aggression is positively related to own aggression; teams may respond to opponent aggression with more of their own, or referees may simply tend to "even up" when distributing cards. Opponent Red Card moving from zero to one increases Disciplinary Points by 3.4 for the away team and 0.6 for the home team. The larger effect for away teams may reflect a change in tactics that takes place when the opponent suffers a red-card expulsion; specifically, when the home team is reduced to 10 players, an away win becomes more likely, which may lead to changing the focus from playing for the tie to aggressively playing for the win. For both the home and away teams, an additional yellow card for the opponent increases Disciplinary Points by approximately one, with the effect on the away team higher.

The results with respect to Own and Opponent League Points differ substantially for home and away teams. Home teams with higher skill have fewer Disciplinary Points, as expected if skill and aggression are substitutes in the production of wins. Home teams playing against stronger opponents have more Disciplinary Points, as expected if teams attempt to negate opponent skill with aggression. However, absolute skill levels are insignificantly related to aggressive play for the away team. The effect of skill levels for the away team appears to be in terms of relative skill; Outcome Certainty has a significantly negative coefficient for the away team as well as the home team. The marginal effect of relative skill varies with the opportunity cost of aggressive play. At the mean of Red Card Price (home = 0.6, away = 0.25), the marginal effect of Outcome Certainty is negative as expected for both home and away teams, -0.1 and -0.7 respectively.

Elasticity of Demand for Aggressive Play for EPL Teams

The results in Table 4 allow for an analysis of the responsiveness of the demand for aggressive play to changes in opportunity cost for each EPL team. Table 5 presents the estimated elasticity of Disciplinary Points with respect to changes in Red Card Price, listed by team and ranked by highest elasticity (in absolute value). Given the results in Table 4, one would expect the best and worst teams to have the lowest elasticities, and that is exactly what we find in Table 5. The bottom half of the ranking includes the four best teams in the EPL (Chelsea, Man United, Arsenal, and Liverpool) as well as all seven clubs with only one season in the EPL during the sample period (Sheffield United, Watford, Leicester, Crystal Palace, Norwich, Wolverhampton, and Derby). The remaining teams in the bottom half include a perennial candidate for relegation (Fulham) and three teams that were relegated from the EPL during the sample period (Birmingham, West Brom, and Sunderland).

In terms of magnitude, the average elasticity for each team is less than one, implying that EPL demand for aggressive play is inelastic. Furthermore, the average demand elasticity is insignificant for the six teams with the lowest estimated elasticities, which includes the four best teams in the league. It may be no surprise to those who regularly watch the EPL that the "top four" teams are unresponsive to Red Card Price on average; since these teams are much better than the other teams in the league, they do not need to rely on aggressive play to produce league points. Fans of other EPL teams might suggest that the top four teams get preferential treatment by referees and are able to get away with more aggressive play without paying a penalty, so that opportunity cost is largely irrelevant.

The teams with elasticities above the mean (-0.35) are all "mid-table teams," i.e., teams with approximately 40 to 60 league points that occupy places five through seventeen in the final ranking. For these teams, a decrease (increase) in the price of aggressive play will lead to more (less) aggressive play to a larger extent than the best or the worst teams in the league. The best teams and the worst teams generally have smaller elasticities on average, since the best teams likely do not need to use aggression to produce league points and the worst teams may be forced to use aggression due to a lack of team skill. In terms of policy, the elasticities suggest that the EPL could reduce aggressive play through an increase in opportunity cost. However, the reduction would be observed mostly among mid-table teams and would be modest in size.


This study attempts to estimate a demand curve for physically aggressive play in football using data from the English Premier League and measuring aggressive play with information on red-card and yellow-card offenses. Using a model in which teams employ aggression to maximize league points, this study gives support to the hypothesis that the demand for aggressive play is negatively related to its price, which is measured as the opportunity cost associated with a red-card expulsion. This negative relationship varies with the relative skill levels of competing teams in a given match, with teams involved in less competitive matches being less responsive to opportunity cost. In addition, demand for aggressive play is shown to be positively related to the aggressive play of the opposing team.

Garicano and Palacios-Huerta (2005) find evidence that Spanish football fans have preferences for less aggressive play. If this holds true for EPL fans, then the league can increase attendance, and possibly revenue, by decreasing the amount of aggressive play. The results presented in this study indicate that the EPL can reduce aggressive play by increasing its price, but the reduction would be relatively small. For purposes of illustrating the potential magnitude of a policy designed to reduce aggressive play, assume that the EPL increases opportunity cost by 10%. For example, the league could instruct referees to give a yellow card for fouls currently sanctioned with a normal foul. The results from this study imply that the EPL would on average see a 3.53% reduction in disciplinary points per match for each team. Over a season, disciplinary points per team would decrease by nine, translating to nine fewer yellow cards or about one less red card per team. Although the policy would reduce aggressive play, most of the reduction would be seen for mid-table teams, since the best and the worst teams are basically unresponsive to changes in opportunity cost.

The policy may also have an effect on competitive balance, although the elasticities in Table 5 suggest that such an effect would be small in magnitude. A team's win/tie probability will be influenced by the extent to which own and opponent aggressive play adjust to the price increase. Furthermore, teams who are more able to adjust skill level in response to an increase in the price of aggressive play will be made better off vis-a-vis their opponents. Any effect on competitive balance should be seen in a change in ranking among the mid-table teams, although the mid-table teams at the lower end could see themselves in the relegation zone if they are unable to adjust skill level to compensate for the change in aggressive play.


The author thanks Brittany Causey, Ashley Hanisko, and Andrea Maloy for excellent research assistance. All estimations referred to in the endnotes are available from the author.


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R. Todd Jewell

University of North Texas


(1) Recalling Allen (2002, 2005), the opportunity cost of aggressive play in football is similar to the opportunity cost of illegal behavior in ice hockey.

(2) This ranking relates to the league coefficients produced by UEFA (Union of European Football Associations), the governing body of European association football. The coefficients are calculated based on the performance of a league's teams in the UEFA Champions League and the UEFA Cup over the previous five years. In 2007, Spain had the highest-rated domestic league and the EPL was ranked second. The EPL jumped to first in the rankings in 2008 because three out of four 2007-08 UEFA Champions League semi-final contenders were English teams (Manchester United, Chelsea, and Liverpool) and the two finalists were both English (Manchester United and Chelsea).

(3) The data are gathered from the ESPN Soccernet website ( and the EPL website (

(4) If a player receives a red card, he is ejected from the current game, will be banned from future games, and may be fined. The EFA imposes different bans and fines on players for different types of red cards, with more-severe penalties accruing for more-serious offenses and for repeat offenders. For instance, players given a red card for serious foul play or violent conduct are normally banned for three games on a first offense, while red cards given for other offenses are normally accompanied by a one-game ban on a first offense. Details of EFA disciplinary regulations are given in EFA (2008). Also, the cost of an ejection is likely to be a function of the player who is ejected. Specifically, the better a player is, the more his ejection hurts his team.

(5) One could also use the number of normal fouls (those fouls which are punished by awarding the opponent a free kick without a caution) to measure aggressiveness. However, normal fouls are not easily combined with yellow cards and red cards in any meaningful way.

(6) There are only 15 matches in the sample (0.3 percent) in which a single team was given multiple red cards.

(7) Time spent a man down is calculated as 90 minutes minus the time at which the red card was awarded. No correction is made for time added by the referee since this data is not consistently available over the period under study.

(8) A Chow-type test ([chi square](22)=10,409.62, significant at 1% level) implies that it is inappropriate to pool the home and away samples. All estimations are performed using Stata Software, Version 10 (StataCorp, 2007). The estimation could also be performed using an ordered logit, under the assumption that a "win" is always a better outcome than a "tie" which is always better outcome than a "loss:' While in terms of league points per game this is true by definition, the league-point-maximization model does not necessarily imply that teams attempt to win each game. In fact, over a season some teams may be better off playing for a tie in some games (especially the away team), given team skill differences, etc.

(9) Team skill levels could also be proxied by payroll (Simmons and Forrest, 2004).

(10) Defining what constitutes an "an end-of-season significant match" is somewhat arbitrary. Estimation with several different measures resulted in a "best fit" (in terms of highest pseudo [R.sub.2]) definition of the following: Significant Match = 1 if the match was one of the last five in a season and if it was mathematically possible for a team to either win the championship or be relegated that season. For an explanation of alternative measures of match significance, see Forrest et al. (2005).

(11) When judging the significance levels of variables in an mlogit estimation, it is necessary to check the joint significance levels of variables in all categories. Red Card Time and [Red Card Time.sub.2] are jointly significant, even though these variables are not individually significant in all categories. Another model was estimated using only the linear measure Red Card Time, resulting in significant individual coefficients in all categories. However, likelihood ratio tests indicate that both the home and away mlogit estimations have significantly better fit with linear and squared terms for Red Card Time. Also, the coefficients in each category cannot be considered marginal effects. Mlogit marginal effects are estimated at sample means.

(12) Red Card Price can be evaluated at any number of minutes. Models are also estimated with Red Card Price at the median number of minutes for teams with a red card (20) and at the mean plus one standard deviation (50). Some differences exist, but the general results hold with respect to the response of red-card aggressive play to its price.

(13) Although Table 1 can be explained by differences in opportunity costs, it can also be simply explained by the tendency for referees to give more red and yellow cards to the away team. In the absence of more-detailed data on aggressive play, neither hypothesis can be refuted.

(14) Team skill difference could simply be measured as the difference in end of season league points between home and away teams. However, such a measure would be almost completely collinear with Own and Opponent League Points. Instead, Outcome Certainty is calculated as the standardized difference in league points per match at the time the match is played; thus, it more directly measures team skill differences at the time of the match. The standardization allows for comparisons over different seasons and at different points of each season. The standardization implies that Outcome Certainty is measured in standard deviations rather than in league points. Since the absolute value of the skill difference is used, Outcome Certainty is a per match indicator, not a per team indicator. Outcome Certainty could also be measured by pre-match probabilities taken from the fixed-odds betting market, but such data are difficult to obtain for earlier years in the sample.

(15) Models were estimated including year and match dummies (including Significant Match), but none of these dummy variables was found to be jointly significant. The demand estimates presented in Table 4 exclude year and match dummies for purposes of model parsimony.

(16) A Chow-type test ([chi square](7)=208.80, significant at 1% level) implies that it is inappropriate to pool the home and away samples. Since Red Card Price is an estimated value, its inclusion changes the error structure of the estimation. Instead of an explicit error-term correction, the estimations in Table 4 are bootstrapped adjusting for clustering on teams using 1,000 iterations.

(17) The 25th percentile of Outcome Certainty = 0.4, the 50th percentile = 0.9, and the 75th percentile = 1.6.

R. Todd Jewell is a professor in the Department of Economics. His research interests include the economics of sports, microfinance in Latin America, and Mexican migration to the US.
Table 1: Aggressive Play
2,280 EPL Matches (2002-03 to 2007-08)

                                      Home          Away

Disciplinary Points                    5.7           7.6
Yellow Card Infractions               2,893         3,837
Red Card Infractions                   148           223
--Second Yellow Card                   64            104
--Serious Foul Play                    39            37
--Violent Conduct                      19            39
--Professional Foul                    21            35
--Deliberate Handball                   5             5
--Offensive Language or Gestures        0             3

Table 2: Summary Statistics
2,280 EPL Matches (2002-03 to 2007-08)

                            Mean    Standard    Minimum   Maximum

Home Loss/Away Win         0.2719    0.4451        0         1
Home Tie/Away Tie          0.2561    0.4365        0         1
Home Win/Away Loss         0.4719    0.4993        0         1
Home Disciplinary Points   5.6632    5.3075        0        38
Away Disciplinary Points   7.5965    5.7494        0        44
Home Red Card              0.0601    0.2377        0         1
Away Red Card              0.0956    0.2941        0         1
Home Red Card Time         1.5868    8.5161        0        87
Away Red Card Time         2.5425    10.528        0        86
Home Yellow Cards          1.2689    1.1397        0         6
Away Yellow Cards          1.6829    1.2350        0         7
Home/Away League Points    52.133    16.632       11        95
Home Significant Match     0.0851    0.2791        0         1
Away Significant Match     0.0864    0.2810        0         1
Outcome Certainty          1.0847    0.8346        0       4.36

Table 3: Estimating the Probability of Winning or Tying
MLogit: Dependent Variable = Win/Tie/Loss
Coefficient, base category = Loss (clustering-corrected standard error)

Home (n=2,280)            Category = Tie             Category = Win

Own Red Card            -0.0118 (0.0215)           -0.0790 (0.0146) ***
  Time ###
Own Red Card            -0.0001 (0.0004)            0.0010 (0.0002) ***
Opponent Red             0.0081 (0.0145)            0.0251 (0.0178)
  Card Time ##
Opponent Red Card        0.0001 (0.0003)           -0.0001 (0.0003)
Own Yellow              -0.0159 (0.1419)           -0.3265 (0.1359) **
  Cards ###
Own Yellow              -0.0224 (0.0341)            0.0499 (0.0396)
pponent Yellow          0.1234 (0.1419)            0.0646 (0.0109)
  Cards ###
Opponent Yellow          0.0204 (0.0295)            0.0191 (0.0251)
Own League               0.0242 (0.0075) ***        0.0613 (0.0059) ***
  Points ###
Opponent League         -0.0356 (0.0045) ***       -0.0593 (0.0048) ***
  Points ###
Own Significant         -0.2773 (0.3310)            0.7520 (0.3874) **
  Match ##
Opponent                -0.3444 (0.3223)           -0.5414 (0.3128) *
  Match # pseudo
  [R.sup.2] = 0.1602

Away (n=2,280)            Category = Tie             Category = Win

Own Red Card            -0.0128 (0.0150)           -0.0321 (0.0281)
  Time #
Own Red Card             0.0002 (0.0002)            0.0002 (0.0005)
Opponent Red Card        0.0637 (0.0313) **         0.0834 (0.0256) ***
  Time ###
Opponent Red Card       -0.0010 (0.0005) *         -0.0010 (0.0004) ***
Own Yellow               0.0644 (0.1125)           -0.0826 (0.1277)
  Cards ###
Own Yellow               0.0004 (0.0225)           -0.0204 (0.0293)
Opponent Yellow          0.2947 (0.1262) **         0.3372 (0.1108) ***
  Cards ##
Opponent Yellow         -0.0674 (0.0352) *         -0.0498 (0.0243) **
Own League               0.0300 (0.0063) ***        0.0623 (0.0071) ***
  Points ###
Opponent League         -0.0280 (0.0038) ***       -0.0645 (0.0048) ***
  Points ###
Own Significant          0.6197 (0.3270) *          0.7574 (0.4637) *
  Match #
Opponent                -0.7363 (0.3004) **        -0.7496 (0.3731) **
  Match ##
  pseudo                                        Notes
  [R.sup.2] = 0.1636

* The estimated coefficients for team, match, and year fixed effects
are excluded for brevity.

* Significance of individual coefficients relative to the base
category: * = significant coefficient at the 10% level;
** = significant coefficient at the 5% level; and *** = significant
coefficient at the 1% level.

* Joint significance of variable in the entire model: # = significant
at the 10% level; ## = significant at the 5% level; and
### = significant at the 1% level.

* Overall model significance for Home equation: Own Red Card Time
and Own Red Card [Time.sup.2] jointly significant at the 1% level
(chi square](4)=47.26); Team fixed effects significant at the 1% level
([chi square](29)=19,968.65); Match fixed effects significant at the 1%
level ([chi square](29)=463.93); and Year fixed effects significant at
the 1% level ([chi square](10)=51.10).

* Overall model significance for Away equation: Own Red Card Time
and Own Red Card [Time.sup.2] jointly significant at the 10% level
(chi square](4)=8.85); Team fixed effects significant at the 1% level
([chi square](30)=62,743.35); Match fixed effects significant at
the 1% level ([chi square](29)=381.62); and Year fixed effects
significant at the 1% level ([chi square](10)=35.80).

Table 4: Estimating Demand for Aggressive Play
OLS: Dependent Variable = Number of Disciplinary Points
Coefficient (bootstrapped, cluster-corrected standard error)

Home (n=2,280)

Red Card Price                           -2.4473  (1.4497)
Opponent Red Card                         0.6422  (0.3456)
Opponent Yellow Cards                     0.9400  (0.0747)
Own League Points                        -0.0420  (0.0111)
Opponent League Points                    0.0553  (0.0133)
Outcome Certainty                        -1.5625  (0.4634)
Outcome Certainty * Red Card Price        2.3514  (0.7055)
adjusted [R.sup.2] = 0.1407

Away (n=2,280)

Red Card Price                           -26.535  (3.9061)
Opponent Red Card                         3.3858  (0.6958)
Opponent Yellow Cards                     1.1929  (0.1037)
Own League Points                         0.0406  (0.0264)
Opponent League Points                   -0.0366  (0.0303)
Outcome Certainty                        -1.8608  (0.4753)
Outcome Certainty * Red Card Price        4.5798  (1.7917)
pseudo [R.sup.2] = 0.1560


* The estimated coefficients for team and referee fixed effects
are excluded for brevity.

* Significance of individual coefficients: * = significant at
the 10% level; and *** = significant at the 1% level.

* Overall model significance for Home equation: Red Card Price
and Outcome Certainty * Red Card Price jointly significant at
5% level ([chi square](2)=8.82); Team fixed effects significant
at the 1% level ([chi square](29)=56.21); and Referee fixed
effects significant at the 1% level ([chi square](33)=577.21).

* Overall model significance for Away equation: Red Card Price
and Outcome Certainty * Red Card Price jointly significant at
1% level ([chi square](2)=68.04); Team fixed effects significant
at the 5% level ([chi square](29)=45.76); and Referee fixed
effects significant at the 1% level ([chi square](33)=304.61).

Table 5: Elasticity of Demand for Aggressive Play
Ranked by Average Predicted Elasticity

Team                    Games            Average          Average
                                        Own League        Red Card
                                      Points/Season        Price

Charlton                  190             45.80            0.455
Portsmouth                190             46.60            0.432
Everton                   228             55.33            0.476
Tottenham                 228             53.00            0.462
Manchester City           228             47.33            0.451
Wigan                     114             43.00            0.431
Reading                    76             45.50            0.460
West Ham                  152             46.75            0.450
Aston Villa               228             50.17            0.435
Blackburn                 228             53.17            0.459
Leeds United               76             40.00            0.425
Southampton               114             43.67            0.374
Middlesbrough             228             47.50            0.439
Newcastle                 228             52.17            0.437
Bolton                    228             50.67            0.437
Birmingham                190             42.40            0.402
Fulham                    228             44.17            0.434
Sheffield United           38             38.00            0.393
Watford                    38             28.00            0.238
Leicester                  38             33.00            0.271
West Bromwich             114             30.00            0.308
Crystal Palace             38             33.00            0.345
Norwich                    38             33.00            0.323
Sunderland                114             24.33            0.283
Chelsea                   228             83.33            0.421
Manchester United         228             82.33            0.461
Wolverhampton              38             33.00            0.326
Derby                      38             11.00            0.096
Arsenal                   228             78.17            0.429
Liverpool                 228             68.00            0.467
TOTAL                    4,560            52.13            0.425

Team                    Average          Average
                      Disciplinary    Elasticity of
                      Points/Game         Demand

Charlton                 5.453        -0.798 ***
Portsmouth               6.179        -0.645 ***
Everton                  6.877        -0.573 ***
Tottenham                6.368        -0.555 ***
Manchester City          6.298        -0.518 ***
Wigan                    7.684        -0.518 ***
Reading                  6.184        -0.499 ***
West Ham                 7.750        -0.475 **
Aston Villa              6.386        -0.466 ***
Blackburn                8.307        -0.455 ***
Leeds United             7.842        -0.453 **
Southampton              5.526        -0.440 ***
Middlesbrough            6.728        -0.443 ***
Newcastle                6.728        -0.443 ***
Bolton                   7.561        -0.420 ***
Birmingham               7.211        -0.388 ***
Fulham                   7.184        -0.372 ***
Sheffield United         7.737        -0.327 ***
Watford                  5.158        -0.313 *
Leicester                7.789        -0.292 **
West Bromwich            5.684        -0.225 **
Crystal Palace           6.789        -0.207 **
Norwich                  4.000        -0.163 **
Sunderland               7.456        -0.165 *
Chelsea                  6.842        -0.157
Manchester United        5.825        -0.123
Wolverhampton            7.053        -0.040
Derby                    6.684        -0.024
Arsenal                  6.123        -0.000
Liverpool                5.114        -0.000
TOTAL                    6.630        -0.353

* Average Elasticity is a predicted value for each match
(home or away), averaged over matches for each team.

* Significance of Average Elasticity is based on the average of the
standard error of each predicted value: * = significant at the 10%
level; ** = significant at the 5% level; and *** = significant
at the 1% level.
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Author:Jewell, R. Todd
Publication:International Journal of Sport Finance
Article Type:Report
Geographic Code:4EUUK
Date:Aug 1, 2009
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