Is a Red Card Ever Worth It? The Data Says Yes

Since its inception in 1970, the red card has come to signify the most brutal individual punishment a referee is capable of handing out.  Having a teammate sent off almost always forces a team to fall back into their own half, relying on counterattacks that take advantage of space left by opponents. If they get lucky, maybe they’ll be able to hold on for a draw.

And yet, despite the severe penalty, players are still sent off regularly. The decision is rarely random. Yes, red cards may sometimes be distributed in an ‘unfair’ manner, but more often than not they are the correct decision. They may be handed out when a player prevents a direct goal-scoring opportunity, injures an opponent, or deliberately starts a fight in order to get their target sent off as well. In these scenarios, red card offences are rational decisions taken upon by the player in order to produce a tangible benefit for his or her team. It just so happens these kinds of benefits (disallowing a goal, injury, etc.) are against the rules.

If red card offences were universally unproductive, then they would be exceedingly rare. Hence, the purpose of the red card is to create a negative consequence that invariably outweighs any potential benefit of breaking the rules.

But, like in most cases, the numbers tell a different story. Sometimes, getting a red card really is the best decision, and economists have figured out when that is.

Getting a Red Card – Why Earlier Is Actually Better

In 2009 Jan Vecer, an associate statistics professor at Columbia University, published a paper that analyzes a specific type of red card offense – one that prevents a direct goal scoring opportunity.

Take, for example, the infamous case of Luis Suarez, who shocked the football world when he used his hands to deny a last-minute header by Ghanaian striker Dominic Adiyah. Suarez was sent off, but his red card allowed Uruguay to push through extra-time and eventually win on penalties. Was Luis Suarez smart to play goalkeeper? His absence was certainly missed the following match, a 3-2 semifinal to the Netherlands, but it is exceedingly unlikely that Uruguay would have advanced in the first place had it not been for Suarez’s blatant disregard for the rules and Asamoah Gyan’s subsequent missed penalty.

It doesn’t take an economist to tell you that Suarez made the right choice, but the numbers explain why. Vecer’s paper expands on the findings made in two previous statistical analyses of red card impacts – Down to Ten: Estimating the Effect of a Red Card in Soccer and Consequences of players’ dismissal in professional soccer: A crisis-related analysis of group-size effects. Both studies use pre-game data such as probability of winning or probability of scoring a certain number of goals. Vecer et al. update their work by using newly available in-play betting data to look at the impact of a red card during the game itself. This kind of real-time data is becoming increasingly available to researchers, even outside of corporate partnerships and academic databanks. In-play data allows for nuance and the ability to consider rare effects, such as multiple red cards and card time intervals.

To make his calcualtions, Vecer relies on betting odds from World Cup 2006 and Euro 2008. Luckily, gambling companies suspend betting activities when there is an apparent goal or penalty, giving Vecer a natural way of delineating ‘before’ and ‘after’ betting odds. Vecer takes the difference between these before and after odds after a goal is scored in order to calculate each team’s implied scoring chances.

For example, let’s say the score between Team A and Team B is currently 0-0. Initially, the odds for either team to win can be used to calculate a baseline chance of scoring. Then if Team A scores, the new odds of drawing can actually be used to find the implied chance of Team B scoring a goal. Likewise, the odds of the Team A winning can be used to calculate the implied chance of no further goals, or the implied chance of three or more goals in a match that swing in Team A’s favor. Vecer is able to use scenarios like these to calculate the chance of scoring for both teams at any point in a match.

Now, in order to determine whether or not getting a red card is worth it, Vecer assumes all penalties have a roughly 80% chance of going in. He then compares the difference the scoring chances before and after having a player sent off, and whether or not it’s worth a penalty.

Using betting data, Vecer and his colleagues show that this kind of unsportsmanlike behavior can be optimal to achieve a victory or tie. Previous research has indicated that teams will often choose to commit an illegal offense in order to prevent goal opportunities. Vecer finds that the results depend on two factors: (a) the score-line and (b) whether or not it leads to a penalty. In the right circumstances, getting a red card can actually be desirable.

The following matrices represent the best time to deny a clear goal-scoring opportunity as a function of opponent scoring chances:

Penalty Incurred

1st table

The percentages above represent the threshold at which there *exists* an optimal time. In other words, if the game is tied, there is always an optimal time to stop a goal-scoring chance higher than 80%.When penalties are incurred, there is no single optimal time at which to commit a red card offense. While optimal times exist for individual scoring chances, these can be better thought of as ‘thresholds’ after which it is desirable to commit a red-card offense.

Granted, it’s impossible to time when these kinds of offenses actually happen in a game. But they do provide an important metric: a way of telling what period of time it’s optimal to stop a goal that has a near 100% chance of going in.

This means that in the same tied game, the optimal time threshold for stopping a 100% sure-goal is the 51st minute. For example, if Luis Suarez was going to get sent off, it would have been optimal for it to happen in the 51st minute. But it was still preferable to block Adiyah’s shot in injury time – in fact, it was preferable any time after the 51st minute. Any time before, however, and he would have done well to keep his hands to himself.

second graph

No Penalty Incurred

2nd table
For example, if you’re tied 0-0 and you have the opportunity to stop a goal without giving up a penalty (such as a last-man tackle outside the box) then the scoring threshold is 57.5%. This means that you should always take the chance if the opponent has an estimated 57.5 or higher chance of scoring. As the game progresses, it becomes increasingly optimal to make that kind of tackle at lower scoring chances. After about 60 minutes, the opponent need only a ~17% chance of scoring for it to justify a last-ditch tackle.When penalties are not incurred, things are measured a little differently. The scoring chance thresholds here show the point at which it is *always* preferable to commit a red-card offense. When the opponent chance likelihood dips below these thresholds, then there is only a specific set of times where it is preferable

first graph

So What Does This Mean?

The primary issue is that these chances are calculated by betting odds that are not directly observable in a match by players. Unfortunately, a defender has no practical way of finding out what the precise scoring chances of the opponent is before making a last-ditch tackle.

Nevertheless, the data indicates that, a lot of the time, red cards can be beneficial. Surprisingly, red cards incurred without giving up a penalty are optimal fairly early in the second half. This finding is fairly counterintuitive. After all, shouldn’t it be better to play less time with a man down than more?

As we’re about to see, that isn’t always the case.

Down To 10: Why Less Is Sometimes More

On April 27, Benfica went down to 10 men against Porto in the Portuguese Domestic Cup with sixty minutes left to play. O Clássico, as the match is referred to in Portugal, is known for being one of the most intense in Europe. To play with any sort of disadvantage is no easy task. But Benfica had done it before – ten days earlier they defeated Porto 3-1 in the semi-finals of the Portuguese League Cup after playing with ten men for an hour. And just as they hoped, Benfica went on to beat Porto again, this time winning 4-3 on penalties. Less than a week later Benfica knocked Juventus out of the Europa League at the semi-final stage.  After winning the first leg 2-1 at home, they held out for a 0-0 draw in Turin after going down to ten men in the 67th.

Is this a fluke, or is it possible that playing with 10 men gave Benfica some form of advantage? Most footballing fans would scorn the idea of having their players sent off, but what if it produced strategic benefits that could be gamed?

Arsene_Wenger

Arsenal Coach Arsene Wenger after losing to Galatasaray in the 2000 UEFA Cup Final: “It was not a huge advantage for us to have Hagi sent off, sometimes you defend better with 10 men because everybody is focused.”

Mario Mechtel, an economics PhD candidate at University of Trier, Germany, set off to investigate this very effect in 2010. He and his colleagues hypothesized that, given the potential benefits of a red card offense (an illegally stopped goal, for example), losing a player may actually be worth it in some cases.

They found that red cards impact home and away teams differently. If a home team loses a player, they are always disadvantaged. However, for away teams, if the sending-off occurs after the 70th minute of a match, it can actually positively affect the team’s performance and score.

The authors hypothesize that this is the result of several coinciding factors. These can be grouped into three separate categories: The role effect, the substitution effect, and the task effect. They find that all three are relevant to varying degrees.

  1. The Role Effect: A sending-off affects the performance of the penalized team negatively. This is how most football fans might conceive of the red card’s impact. When the 10 remaining players have to compensate for the missing player, they need to not only fulfill their role but an additional one as well. For example, if a defender is sent off, a heavier burden may fall on offensive-minded players to drop deep and defend in addition to their current task. If this effect holds, then the performance of the penalized team will be negative.
  1. The Motivational Effect: a red card affects the performance of the penalized team positively. Social theorists predict that group size is negatively correlated with outside pressure felt by group members. In other words, as a group grows larger, the perceived pressure decreases and effort per member decreases respectively. Imagine if a team is comprised of 11 players, each playing with an effort level of 6, yielding a total group effort of 66. If the team goes down to 10 men and players respond with an effort level of 7, the total group effort increases to 70. By losing a player, players may perceive a greater amount of pressure and respond with higher effort levels, in some cases yielding an improved performance.
  1. The Task Effect: A sending-off has larger negative effects on the performance of the penalized team whenever it is the home team. This hypothesis rests on the notion that away teams play more defensively than home teams. To no surprise, receiving a red card encourages a team to switch to a defensive approach, emphasizing counter-attacking as a primary means of goal scoring. Thus, the theory is that if the away team loses a player, it doesn’t significantly change their tactical approach. On the flipside, a home team whose starting lineup is built around an attacking approach will need to adjust considerably if a player is sent off.

Mechtel and his coauthors indeed observe a stronger red-card effect on home teams than on away teams, indicating the presence of the task effect. However, their results for the motivational effect and role effect are a bit more mixed. By controlling for minutes left until the end of the game, they find that the role effect only begins to offset the motivational effect after the last 20 minutes – in other words, an away team that has a player sent off between minute 71-90 actually experiences a boost to their final score.

The paper is, however, unfortunately riddled with endogeneity problems. Although they repeatedly seek to compensate for these by introducing new controls, these often fall short of being reliable estimators. In particular, Mechtel struggles with finding adequate proxies for team performances and skill. Because he does not have access to the same real-time betting data that Vecer et al. did, he and his colleagues resort to using final league positioning as a substitute. However, league positioning can be unreliable, and in the case of newly promoted teams, he uses previous season standings. To Mechtel, finishing first in the Championship is equivalent to winning the Premiere League. In another test, Mechtel attempts to use numbers of attempts on goal, corner kicks, and yellow cards as an estimator for in-game team performances. While none of these factors drastically change the results, teams that have more yellow cards may be more likely to receive a red card because second yellow offences also count. As a rule of statistics, if you are trying to use a control variable as a proxy for an independent variable (in this case, yellow cards as a proxy for team performance), it should be unrelated to all other independent variables (in this case, red card occurrence). Otherwise, the effect cannot be properly isolated.

These issues should be of concern to the reader, but they should not necessarily jeopardize our consideration of the results. They do suggest that the 70-minute cutoff for receiving a red card is not particularly accurate, but Mechtel went through the effort of repeating his tests with several data sets (one goal-based and one points-based) and found similar results. In all his tests, Mechtel finds the signs on his important coefficients remain the same at a significant level, meaning there’s consistent evidence of a positive effect generated by red cards.

Thus, the results do support a benefit for away teams who go down to 10 minutes late in the game. Ideally, Mechtel’s work could be repeated with a more precise data-set. Maybe Vecer could share his?

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Myth Busting: Does Playing the Second Leg at Home Actually Matter?

The Myth

Most football fan will tell you that when it comes to double-legged knockout tournaments, it’s better to play at home in the second leg than away. The idea is simple – the away team has the option (luxury) to play it safe (park the bus), aim for a draw, and then bank on winning the home game. Not the most elegant approach, but an effective one nonetheless.

Proponents of the strategy are not limited to fans and pundits – players and coaches can be just as guilty. Iker Casillas thinks it will give them an advantage over Schalke 04 in the Champions League this February. Dougie Freedman, former Crystal Palace striker, once attributed a victory against Charlton in the second leg of a promotion play-off semi-final to the incredible atmosphere. Patrick Kluivert, former Dutch international, believes it’s better to “make your mistakes in the first leg away from home because there is still time to put things right in the second match.”

"Maybe we have a slight advantage because we are playing the return match at home" - Carlo Ancelotti

“Maybe we have a slight advantage because we are playing the return match at home” – Carlo Ancelotti

But while the story is written by the pundits, the truth is written in the numbers. A 2007 paper by Lionel Page and Katie Page, of the University of Westminster and University of Queensland respectively, looked to find just that. The first academic research of its kind, the study made significant advancements but ultimately fell short of conducting a proper analysis of the phenomenom. It wasn’t until a follow-up study was conducted by three researchers from the Ludwig Maximilian University of Munich that the truth was uncovered: playing the second leg at home is, statistically speaking, irrelevant.

Evidence From the Champions League

It’s not hard to see why this myth has so much traction. On the surface, data from the Champions League fits the narrative: teams play conservatively in the first leg and open up in the second. This is reflected in the increase in goals scored in the return leg.

1st leg goals

2nd leg goals

1st leg goals per game

2nd leg goals per game

2012 (group)

136

148

2.83

3.08

2012 (knockout)

39

42

1.39

1.50

2011 (group)

120

134

2.50

2.79

2011 (knockout)

32

57

1.14

2.04

2010 (group)

131

145

2.73

3.02

2010 (knockout)

38

37

1.36

1.32

Total

496

563

1.99

2.29

Additional calculations by Michael Cox of Zonal Marking for the 2007-2009 Champions Leagues confirm the findings above. That is, with the exception of 2010, more goals are consistently scored in the second leg of CL knockout rounds than in the first.

This phenomenon can work for and against a team playing at home during the second leg. On one hand, the away team is less likely to sit back and park the bus. This lets the home team exploit their home advantage and take control of the deciding match. On the other hand, in a game that’s decidedly more ‘open’, a goal for the away team counts for double. Ideally these two effects would balance out and we would observe a 50-50 chance of advancing, even if more goals are scored in the second leg (e.g. if a team goes 1-0 at home and then loses 1-2 away, the away goal rule would outweigh the home field advantage).

Since we know more goals are being scored, the question becomes: for whom?

The First Study

Between 1994 and 2009, 56 percent of teams which played the second leg of a Champions League tie at home advanced to the next round. On the surface, this suggests home-field advantage frequently outweighs the benefit of the away goal rule.

But these findings are not sufficient. They do not take into consideration that the order of home-away matches for the CL round of 16 is not random.

In fact, group winners first play away, then at home. This was instituted as an added incentive for group winners, based on the very myth itself. This creates a sort of a self-fulfilling prophecy. Seeded teams play at home in the second leg, and it’s not clear if they win because they’re the better team, or because of their home-field advantage.

In their 2007 paper, Lionel Page and Katie Page look examine this specific aspect. Together, they use data from three different European Cup football competitions over 51 years in order to assess the accuracy of the second-leg advantage myth.

One solution they assess is to simply not look at data from the round of 16. Later rounds no longer seed home-away match order, randomizing the process and eliminating possible bias. But they find that this approach considerably reduces the sample size, and makes it harder to obtain meaningful results.

Page and Page instead use team-coefficients to adjust for round of 16 seeding. By controlling for team-quality based on UEFA rankings, they attempt to separate the effect of playing at home in the second leg and the effect of being a better team. Their results produce the following:

yo

This graph shows an overall decrease in second-leg home advantage, both adjusted and unadjusted for team coefficients, over time. Interestingly enough, the decrease in the late 60s in this advantage can be traced to the introduction of the away-goals rule. By increasing away-goal value, the tactic of parking the bus away and winning at home was no longer as effective. For a team playing away, losing wasn’t the end of the world, but scoring an away goal could be the difference that would send them through.

The temporary decrease in advantage between 1984 and 1990 coincides with a five-year ban on English club participation in European competitions as a result of the Heysel Stadium disaster. This suggests that, at the time, English clubs exhibited a more significant second-leg home advantage than their European counterparts. In fact, seven out of eight previous European Cup winners were English. It’s possible that part of this success was the ability to exploit the second-leg home advantage in either the form of favorable refereeing or by employing the ‘defend away, win at home’ strategy.

The Away Goal Rule

At this point, readers would be right to ask: what about the away goal rule? Why does everyone believe there’s an advantage to playing the second leg at home when the away team has the opportunity to play an extra 30 minutes where their goals count double.

Page and Page actually look into this. They analyzed 186 European cup ties which go into extra time and found that the probability of winning the second leg at home is 66.42%. These results are adjusted for team ability, regardless of the fact that ties going into extra time might reflect equally-skilled teams. Page and Page also find that matches that go to penalties on the home ground result in the home team winning 57.33% of the time.

Simply put, the advantage of playing at home for an extra 30 minutes considerably outweighs the advantage of having goals count for double for the same period of time. Note that, whether or not there is an advantage to playing at home first, home advantage is a very real phenomenon.

Arsene-Wenger1_2354548

“Statistically it is not proven. We like to think in our job that to play at home in the second leg is an advantage, but it is not proven at all in the statistics. It is 50:50” – Arsene Wenger

Busting the Myth

Page and Page make one crucial mistake that jeopardizes their conclusion that second leg at home advantage remains today. They only control for team UEFA coefficients, but this is not sufficient to isolate the advantage effect. Think about it this way: In Page and Page’s model, they attempt to separate the following effects:

chart

However, in order for their model to work, the two effects cannot be correlated outside of their impact on the dependent variable (in this case, the round of 16 performance).

If you’re a little confused right now, you should be. This is the same problem Page and Page thought they were solving. But their analysis was rudimentary: you can’t simply control for correlated effects in order to isolate them.

Suppose you have three variables A, B, and C. You hypothesize that A and B both positively affect C. The data shows as much: A, B, and C all rise together. You could jump to conclusions and assume you were right from the beginning. But really, B is positively affecting A and C, giving the illusion your initial hypothesis was correct when in fact A is completely unrelated to C.

For example, imagine that the US government introduces a new anti-smoking campaign. The campaign is a success, and raises the population’s life expectancy. The government also chooses to raise taxes to fund the campaign. One might observe taxes rising and mortality rates decreasing and falsely conclude that paying more taxes means you’ll live longer.

Likewise, Page and Page observe a lot of teams play the second leg at home because they’re good and win their group, and falsely assume that they win the next round because they played the second leg at home. In reality, a better team will have a higher coefficient going into the tournament, and will not only do better in the round of 16, but is also more likely to have the second-leg at home. This leads them to overestimate the effect match order has on the outcome.

Three researchers from the Ludwig Maximilian University of Munich took a closer look at the data, and noticed what Page and Page did. To solve for this issue, they controlled for group stage performance in order to completely isolate the home-away match order effect. Any effect team coefficients have on match order is now funneled into the group stage performance variable, giving them clear results.

The researchers found that, once they controlled for group stage performance, playing the second leg at home had no impact on a team’s chances of success. It turns out that the team that progresses is usually just better than their opponent, contrary to the superstitions of some of football’s greatest minds. Maybe this is the reason Wenger has been so successful – he knows when to listen to the numbers.