## Can İlkay Gündoğan replace Nuri Şahin?

While title celebrations at the Signal Iduna Park are still very much ongoing Dortmund fans received some bad news yesterday when Nuri Şahin announced that he will be joining Real Madrid next season in a deal reportedly worth €10-13m. The midfield playmaker played a pivotal role in Dortmund’s title triumph and at this point the Bundesliga Player of the Season award seems like a mere formality. Finding an adequate replacement for Şahin is vitally important for Dortmund and the club has been quick to react, announcing the signing of 20-year old German youth international İlkay Gündoğan last week. The 1. FC Nürnberg midfielder is seen as Şahin’s heir apparent and in this post, using statistics collected from the official Bundesliga website, I will look at whether Gündoğan can replace Şahin in Dortmund’s midfield.

For Gündoğan moving to Dortmund will be a large step up in terms of responsibility, both on and off the pitch. Moving from a midtable club, even if that club is Der Club where the players have to carry the weight of the club’s history on their shoulders, to the reigning champions is always a big step, but moving there to replace Şahin will make it doubly so.

Şahin’s role is the most important in Dortmund’s team. As the deep-lying playmaker he is the motor in the midfield and the conductor of the team’s passing game. Over the course of the season he attempted 1803 passes which even though he missed the last few matches due to injury is almost 300 passes more than the second best Hummels. Adjusted for playing minutes he contributed 14.9% of Dortmund’s passes which is comparable to the likes of Cesc Fabregas at Arsenal and Paul Scholes at Manchester United in terms of what percentage of a team’s passes a player contributes. On a per 90 minutes basis he attempted 63 passes on average.

Gündoğan’s role at Nürnberg is slightly different. He generally plays in a more advanced position and operates more in the opponent’s half than his own. In fact 61.5% of his passes were in the opponent’s half compared to 49.6% of Şahin’s. Gündoğan can and has played in a deeper role but it’s not the role he has primarily played this season.

His more advanced role to a small extent explains his lower passing numbers. Over the course of the season he attempted 745 passes or 37 passes per 90 minutes. As a percentage he contributed 10.6% of Nürnberg’s passes. You would expect those numbers to increase somewhat when playing in a deeper role. I considered going through his matches this season and separating them based on which role he played but the sample sizes are small enough as is that I don’t think we could draw any meaningful conclusions from them if we made them even smaller.

Note that in the above numbers I only included passes in open play. Since Şahin also takes corners and free kicks while Gündoğan doesn’t the 114 corners and 86 free kicks taken by Şahin would even further increase the gap between the two.

To further emphasize the difference in responsibility, if we ignore the matches the players missed due to injury Şahin started every match and played 96.1% of Dortmund’s minutes compared to Gündoğan’s 81.2%. For Gündoğan to fill Şahin’s boots he will need to be very strong mentally to not crumble under the pressure and responsibility.

There are also significant differences in the types of passes the players contribute. Şahin is a very ambitious passer. Of his passes in the opponent’s half 63.8% were forward passes, 20.9% sideways passes and 15.3% backwards passes. Of his passes in his own half 59.1% were forward passes, 20.2% sideways passes and 20.7% backwards passes. Contrast this with Gündoğan who is more conservative. Of his passes in the opponent’s half 35.2% were forward passes, 30.3% sideways passes and 34.5% backwards passes. Of his passes in his own half 42.9% were forward passes, 24.4% sideways passes and 32.7% backwards passes. While I don’t have statistics on their ratios of short to medium to long passes from watching almost all of their matches this season I can say that Şahin has a wider range of passing and is better at long range passes than Gündoğan who prefers short and medium range passes.

In terms of passing accuracy the two are very close. The success rates of Şahin’s forward, sideways and backwards passes in the opponent’s half are 63.3%, 85.6% and 96.4% while for Gündoğan they are 66.5%, 86.3% and 94.3%. In their own halves their sideways and backwards pass success rates are 92.9% and 95.2% and 90.0% and 98.9% respectively. The only real difference is in their forward passing in their own halves where Şahin has a success rate of 65.6% compared to Gündoğan’s 53.7%.

Şahin is also a more creative passer. On average he created 3.61 goal scoring chances per 90 minutes which is more than any other player in the top five European leagues. Even though Gündoğan played in a more advanced role and should therefore theoretically be in a better position to create goal scoring opportunities he created only 1.23 chances per 90 minutes.

In addition to being vitally important to Dortmund’s passing and attacking game Şahin is also a very good defender. Per 90 minutes he completed 3.26/3.92 tackles for a success rate of 83.2%. He also made 2.22 interceptions and 7.28 loose ball wins. From observation I can also say he is disciplined and has good defensive positioning and awareness.

Gündoğan’s defensive work rate is also good but in terms of easily measurable contributions he is not as good as Şahin. Per 90 minutes he completed 1.18/1.43 tackles (82.8% success rate), made 1.38 interceptions and 5.22 loose ball wins. His more advanced role partly explains the lower numbers.

So to return to the original question, can İlkay Gündoğan replace Nuri Şahin? While always keeping in mind that he is only 20 years old and still has a lot of room to improve, and that playing with better players will naturally make him better, in my opinion the answer is no, at least not directly. While they share some similarities they are still very different players. Gündoğan is more of a short passer and a dribbler (dribbling is actually the one metric where Gündoğan outperformed Şahin, both attempting more dribbles and having a slightly better success rate) who floats around midfield, receiving the ball in more advanced positions and looking to create from there. Şahin plays a deeper role and demands the ball more, always visible and always vocal in midfield, and with a wider range of passing both short and long. I think Gündoğan’s natural role is the more advanced midfield role that Kagawa and Götze have played this season. For Gündoğan to perform Şahin’s role in this current system he would need to both improve and adapt several parts of his game all at once.

If Klopp truly intends to replace Şahin with Gündoğan I suspect it may necessitate a tactical switch away from the midfield triangle with two holding players and one more advanced to an inverted triangle with just one holding player and two more advanced. This would of course result in Bender as the sole holding player having to take more responsibility in Dortmund’s passing. Dortmund would also lose the defensive solidity that comes with a duo of central holding players.

It would also mean there is a small risk that Dortmund’s attacking play could become too narrow. Dortmund’s wingers like to come into central areas to join in the short passing combinations with the midfielders, and it’s situations like this where Şahin’s long range passing is so valuable because with one pass he can quickly play the ball out to an advancing full back looking to exploit the space in the wide areas and maintain the width in their attacking play. With Gündoğan you lose some of that and I think that’s something Dortmund would have to be wary of.

Ultimately I think the transfer of Gündoğan to Dortmund is a good move for both the player and the club. It gives Klopp another midfield option and for Gündoğan it’s a good place to develop as a player and a good next step in his career. There are many question marks and it remains to be seen how Gündoğan will do at Dortmund but I think the one thing we can be sure about is that at the end of the day old Kloppo will still be laughing, and for as long as he is at Dortmund it’s a fair bet the Dortmund fans will be laughing as well.

## A Closer Look at Nuri Şahin’s Passing

Nuri Şahin was voted the best player of the first half of the Bundesliga season by his fellow players and he is the favourite to win the overall player of the year award. In this post I take a closer look at his passing using statistics gathered from the official Bundesliga website.

Şahin has been one of the most active passers in the league, as far as I can tell only Bastian Schweinsteiger and Philipp Lahm have attempted more passes. With 1803 total passes he has attempted almost 400 more passes than any other Dortmund player, and adjusted for playing minutes he has contributed 14.9% of Dortmund’s total passes, which puts him up there with the likes of Cesc Fabregas and Paul Scholes in terms of what percentage of a team’s passes a player contributes.

On average Şahin completed 47/62 passes per 90 minutes. For the above graph I took the four match rolling averages of successful and attempted passes to see how those figures have fluctuated over the season. As you can see both have stayed roughly constant, his successful passes fluctuating between 40-50 and his attempted passes between 55-70. Note that in this and in all future graphs I omitted his last match against Freiburg because he only played 28 minutes.

To get a more accurate sense of his passing I separated his passes into forward, sideways and backwards passes both in the opponent’s half and his own.

Of his passes 49.6% were in the opponent’s half, 50.4% in his own. Of his passes in the opponent’s half 63.8% were forward passes, 20.9% sideways passes and 15.3% backwards passes. Of his passes in his own half 59.1% were forward passes, 20.2% sideways passes and 20.7% backwards passes. To look at how those numbers fluctuated over the season I again took the four match rolling averages.

After an early increase in his percentage of forward passes both in the opponent’s half and his own, and a corresponding small dip in the percentage of sideways and backwards passes, his passing direction percentages stayed roughly constant until there was another small increase in forward passing in his last few matches. It is clear from these graphs that he is not content to simply pass the ball sideways and backwards, instead he looks to play the ball forward whenever possible.

The accuracy of his different passes and the total number of successful/attempted passes of each kind are summarised in the following table. The graphs show the fluctuation over the season.

Opp Half | Frwd Pass | 63.3% | 361/570

Opp Half | Side Pass | 85.6% | 160/187

Opp Half | Back Pass | 96.4% | 132/137

Own Half | Frwd Pass | 65.6% | 352/537

Own Half | Side Pass | 92.9% | 171/184

Own Half | Back Pass | 95.2% | 179/188

According to the data he misplaced about one-third of his forward passes. While this is an area he could look to improve it is not necessarily a bad thing. From watching almost every match he has played this season I can say that his first instinct is always to look for the ambitious, forward option, and for a playmaker this is a good thing because the way you create chances is not by playing it safely, it is by taking calculated risks and playing those difficult but potentially highly rewarding passes. The most creative players and the players with the highest assist counts often have below average attacking third pass completion rates and as long as Şahin is creating a lot of chances it seems to me like his lower pass completion rate is a trade-off worth making.

So does he create a lot of chances? Yes, he absolutely does. His 3.61 key passes per match is not only best in the league but according to the data from WhoScored.com also the best in any of the top five European leagues. Below are lists of the top chance creators in the Bundesliga and in all of Europe.

Nuri Şahin 3,61

Christian Tiffert 3,13

Diego 2,81

Mario Götze 2,67

Arjen Robben 2,36

Tamas Hajnal 2,33

Marco Reus 2,31

Mehmet Ekici 2,25

Franck Ribery 2,22

Juan Arango 2,05—

Nuri Şahin 3,61

Andrea Cossu 3,48

Mesut Özil 3,35

Christian Tiffert 3,13

Francesco Totti 3,04

Florent Malouda 3,03

Duda 3,00

Nene 2,90

Wesley Sneijder 2,88

Cesc Fabregas 2,83

Diego 2,81

David Pizarro 2,78

Mario Götze 2,67

So far I have only discussed Şahin’s passing in open play but he also takes corners and free kicks.

This season Şahin has taken 114 corners of which 6 were taken short. Of the 108 that were delivered straight into the penalty area 41 found a teammate, a success rate of 38.0%. He also took 86 free kicks of which 72 were crossed into the penalty area. Of those 27 found a teammate for a success rate of 37.5%. Combining his corners, free kicks and crosses from open play overall he delivered 211 crosses with a success rate of 37.4%.

In a future post I may look at his shooting, dribbling and defensive work, but for now hopefully this post has provided some insight into what kind of player, and specifically what kind of passer, Nuri Şahin is.

## Goalkeeper Performance in England, Spain and Germany

I have written in previous posts about the importance and value of shot stopping. So how have goalkeepers in the real world done so far this season?

I collected data on the current Premier League, La Liga and Bundesliga seasons from Sports Illustrated’s website and put it all together into the following graph. The data includes every match played from the start of the season to April 13th. Note that I included only goalkeepers who have played at least 1500 minutes.

In the following table I have listed every goalkeeper’s goals allowed, shots on target faced, minutes played, saves made and EV Diff, which is a measure of how much a goalkeeper has over- or underperformed relative to expectation based on how many shots on target they have faced. Ben Foster’s 18,46 goals means he has conceded 18,46 goals fewer than would have been expected based on how many shots on target he has faced. Scott Carson on the other hand has conceded 12,73 goals more than expected.

Player GA SOT Mins Saves EV Diff

Foster 43 198 2790 156 18,4555833

De Gea 41 165 2790 124 8,682483054

Hart 30 134 2880 104 8,622904031

Baumann 33 142 2250 108 8,47698894

Cech 25 119 2790 95 8,271494827

Calat’d 45 175 2689 125 8,25008919

Kameni 38 152 2430 114 7,044595077

Ricardo 37 146 2790 107 5,904031395

Gomes 31 129 2340 98 5,839100963

Alves 54 192 2790 139 5,315019622

Neuer 33 133 2610 101 5,266143418

Jääsk’n 38 147 2646 108 5,260792009

Valdes 14 79 2430 65 5,001070282

Begovic 29 119 2070 91 4,271494827

Weide’r 17 84 2520 68 3,78487335

Lopez 36 137 2790 100 3,693185872

Green 55 189 2790 135 3,244737781

M Reina 25 103 1620 79 2,563325009

Schwa’r 29 114 2250 86 2,487691759

Wetklo 18 82 1645 64 2,071352123

Toño 45 157 2790 106 1,828398145

Aranz’a 29 110 2250 81 1,060649304

Bravo 54 180 2790 125 1,033892258

AlHabsi 40 139 2520 98 0,4067071

VDS 28 105 2520 78 0,276846236

Casil’s 22 88 2612 67 0,211915804

Migno’t 24 93 1530 69 -0,004281127

Benag’o 33 118 2049 85 -0,085265787

Codina 33 117 2054 80 -0,4420264

Aouate 42 142 2700 94 -0,52301106

Robin’n 48 158 2655 110 -0,814841242

Pablo 23 86 1642 62 -1,501605423

Friedel 53 167 2880 114 -2,603995719

Iraizoz 45 144 2790 97 -2,809489832

Reina 38 124 2880 87 -2,944702105

Kessler 39 126 2048 87 -3,231180878

Ulreich 54 166 2610 113 -3,960756333

Palop 28 91 1632 61 -4,717802355

Kingson 38 119 1698 81 -4,728505173

Sippel 42 130 2250 89 -4,804138423

Howard 41 126 2880 88 -5,231180878

Nikolov 28 89 1755 61 -5,431323582

Henne’y 31 97 1620 66 -5,577238673

Adler 32 99 2430 69 -5,863717446

Rost 36 110 2203 76 -5,939350696

Starke 29 90 1800 62 -6,074562968

Schäfer 37 112 2579 75 -6,225829468

Harper 22 68 1565 46 -6,923296468

Wiese 45 131 2158 88 -7,447377809

Butt 21 63 1575 43 -7,707099536

Franco 30 84 1671 55 -9,21512665

Froml’z 34 94 1748 61 -9,647520514

Carson 46 119 2340 75 -12,72850517

Clearly this analysis isn’t perfect. For starters it considers all shots as equal, which clearly isn’t true. Some shots are more difficult to save than others, and some goalkeepers will have faced relatively more difficult shots while others will have faced relatively more easy shots. For example, has Petr Cech really performed 8,27 goals above expectation or has he just faced a disproportionately large number of easy shots which serve to make it look like he has performed better than he actually has? I don’t know. Certainly these numbers shouldn’t be taken as gospel and should instead be thought of as a supplement to your eyeball test when evaluating goalkeeper performance. But perhaps they are useful, if for no other reason than to get people to think about why they may not be an accurate representation of reality and what could be done to improve on them.

## Shot Stopping and an Elite Defense

While reading a forum discussion on the topic of Manchester United’s next goalkeeper I came across an argument that went something like this (and I’m paraphrasing):

Manchester United have a world class defense that is able to limit the opposition to only a few shots per match and therefore it is not important that United’s goalkeeper is a world class shot stopper, it is far more important that he is good at organising his defense, claiming loose balls, catching and punching crosses and distributing the ball with throws and kicks.

To investigate whether shot stopping is less valuable for a team which concedes fewer shots I did some simple calculations:

x = number of shots on target faced

z = save percentage

y = goals concededTo determine goals conceded we can use the equation:

y = x – zx

If we increased z by 1%:

y(2) = x – 1.01zx

We can then combine the two and look at the percentage change in goals conceded, %y, and ask what happens to %y when we increase z by 1%?

%y = 100(((x – 1.01zx) – (x – zx))/(x – zx))

%y = 100((x – 1.01zx – x + zx)/(x – zx))

%y = 100(-0.01zx/(x – zx))

%y = -zx/(x – zx)

%y = -zx/x(1 – z)%y = -z/(1 – z)

So %y depends on z. At z = 0.6 a 1% increase in z leads to a 1.5% decrease in y whereas at z = 0.8 a 1% increase in z leads to a 4% decrease in y. The more relevant point is that %y is independent of x, in other words the value of increasing save percentage by 1% isn’t affected by how many shots on target you face.

The reverse of the original argument, that preventing shots becomes more valuable for a team that concedes fewer shots, isn’t true either. A 1% decrease in shots on target faced always leads to a 1% decrease in goals conceded.

%y = 100(((0.99x – 0.99zx) – (x – zx))/(x – zx))

%y = 100((-0.01x + 0.01zx)/(x – zx))

%y = (-x + zx)/(x – zx)

%y = -(x – zx)/(x – zx)%y = -1

Having an elite shot stopper is extremely valuable regardless of whether you have a world class defense or not.

## Shot Stopping vs Preventing Shots

When evaluating goalkeepers there any many factors you must take into account like shot stopping ability, aerial ability, command of the penalty area, communication, kicking ability, throwing ability, athleticism and many others. Creating an exact and reliable measure of each attribute is difficult but by simplifying the question somewhat we can at least get a taste of what is valuable and what isn’t. One way of doing this is to sort a goalkeeper’s defensive actions into two categories: shot stopping and preventing shots. The first is obvious enough. By the second I mean catching and punching crosses, dealing with through balls and other loose balls, organising his defense, and others, all things which help limit the number of shots the opposition takes. If we accept this rough divide we can then ask, what is the relative value of shot stopping and preventing shots?

Assume a goalkeeper faces five shots on target per match and saves 74% of them, thus conceding 1.3 goals per match. If we decrease SoT/match by 1% while keeping save% the same we get 1.287 goals conceded per match for a gain of 0.013 goals per match. If instead we increase save% by 1% (increasing to 74.74%, not 75%) while keeping SoT/match the same we get 1.263 goals conceded per match for a gain of 0.037 goals per match. So in this example increasing save% by 1% is almost three times more valuable than decreasing SoT/match by 1%.

How much more valuable increasing save% is depends on the save%. The take home point is that at any save% above 50% increasing save% by 1% is more valuable than decreasing SoT/match by 1%. At realistic top flight save% levels (c. 60-80%) it ranges from 1.5x to 4x more valuable.

Intuitively it makes sense that shot stopping is more valuable than preventing shots. When a goalkeeper makes a save he is preventing an almost certain goal, but when a goalkeeper comes to catch a cross or deal with a through ball he is preventing a situation where some percentage of the time the opposition will get off a shot, and some percentage of that time the shot will be on target, and some percentage of that time the shot will lead to a goal.

So while we are still some way away from an accurate and complete method of mathematically evaluating goalkeeper performance, a model which values shot stopping ability above everything else is very likely going to be the place to start.

—

One thing worth noting is that in this article I used the term save percentage or save% quite often. When I use the term I mean with it a goalkeeper’s real or true save percentage, not the save percentage statistic that you can sometimes find in the papers or on the internet. I think the save percentage statistic is a poor one because it is influenced by too many external factors which may have nothing to do with a goalkeeper’s real shot stopping ability. Things like how good the defenders and midfielders are, and more specifically things like how good the midfielders are at putting pressure on the opposition shooters and how good the defenders are at forcing opposition attackers into low percentage shooting sectors, as well as countless other factors, all influence a goalkeeper’s save percentage statistic even though they have nothing to do with how good of a shot stopper a goalkeeper actually is. Goals-to-games ratio which is used in Spain when they hand out the Ricardo Zamora Trophy to the best goalkeeper of the year and number of clean sheets kept which they use in England to give out the Golden Glove award suffer from the same problems.

We need better metrics. I don’t know what those metrics would be but something that takes into account shot quality and weights saves by degree of difficulty would be a decent first step. The point I’m trying to make (in a roundabout way) is that our inability to accurately measure a goalkeeper’s true shot stopping ability doesn’t mean shot stopping as an attribute is any less valuable. Shot stopping is extremely valuable, whether we can accurately measure it or not.

## The Value of a Red Card, Part 4

*This is the fourth and final part in a series of posts investigating the value of a red card.*

Knowing the value of a red card isn’t particularly interesting in and of it’s self. If we knew that the value of a red card in a certain situation was say -0.2 goals we could say that a player getting sent off then would be hurting his team significantly, but it’s not like we need to do a long calculation to know that being sent off is generally a bad idea. However we can apply the information to answering certain in-match questions, for example when is it correct to intentionally foul someone knowing you will get sent off if you do and what is the expected value (EV) of that foul.

Imagine the following situation: two average teams are playing on neutral ground with the score at 0-0 with M minutes left in the match. By using the information from part three of this series we could use this same process for a specific situation but for the sake of simplicity I will use the average case. The opposition striker has just beaten the offside trap and is through on goal in a clear goalscoring opportunity. You are the center back and again for the sake of simplicity let’s assume that you cannot win the ball off him fairly, your only options are to let him go and hope he doesn’t score or to foul him, give away a free kick or penalty and take the red card. Knowing what we know from the previous two articles we can estimate the value of letting him go and the value of fouling him to see which is the better option.

Let’s say in this particular example there are fifteen minutes left in the match and you are outside the penalty area so by fouling you are giving away a free kick from a dangerous position. How often would a team score from the resulting free kick? It seems unlikely to me that it would be more than 10%. What if you let him go and he is 1-on-1 with the goalkeeper? He is surely a favourite to score but he won’t score every time. Let’s assume he scores 80% of the time.

If you don’t foul: 80% of the time you lose a goal and continue for 15 minutes 11vs11 (the EGD is still zero as before) and 20% of the time he doesn’t score and you continue for 15 minutes 11vs11. Thus,

EV(no foul) = -0.8 goals

If you do foul: 10% of the time they score from the free kick but 90% of the time they don’t. Either way you have to play 15 minutes 10vs11 (which as mentioned in part two of this series is worth -0.27 goals). Thus,

EV(foul) = 0.1*(-1) + (-0.27)

EV(foul) = -0.37 goals

So in this example fouling is worth 0.43 goals in EGD.

It’s worth noting that even though we are calcuting this in terms of goals, at the end of the day what we really care about are league points, and there are situations where looking at a situation in terms of goals and in terms of points can give different results.

For example if instead of there being 15 minutes left in the match let’s say there is no time left and the referee will end the match as soon as the striker scores or misses from either open play or from the free kick after you foul him. Since there is no time left being down to ten men doesn’t have any effect so we can simply say that the EV of not fouling is -0.8 goals (the striker scores 80% of the time) and the EV of fouling is -0.1 goals (the team scores from the resulting free kick 10% of the time). So fouling is worth 0.7 goals. If the match is tied then the foul is also worth the same in league points, 0.7. But what if you are leading the match 1-0? In that case if you don’t foul the player 80% of the time he will score making it 1-1 and you will get 1 point from the match while 20% of the time he won’t and you will get 3 points for a total of 1.4 points. If you do foul the player 10% of the time you get 1 point and 90% of the time you get 3 points for a total of 2.8 points. So even though the goal value of a foul is 0.7 goals irregardless of whether the score is 0-0 or 1-0, the point value changes dramatically depending on the score.

We can derive formulas for the EV of fouling when the match is tied and when you lead by one goal.

If x is how often the opposing team scores from the free kick or penalty, y is how often they score from open play if you don’t foul and v is the value of the red card in points,

EV(foul|lead) = ((1x + 3(1-x)) + v) – (1y + 3(1-y))

EV(foul|lead) = 3 – 2x + v – 3 + 2y

EV(foul|lead) = 2y – 2x + vEV(foul|lead) = 0 when y = x – (v/2)

EV(foul|lead) > 0 when y > x – (v/2)

EV(foul|lead) < 0 when y < x – (v/2)EV(foul|tied) = ((0x + 1(1-x)) + v) – (0y + 1(1-y))

EV(foul|tied) = 1 – x + v – 1 + y

EV(foul|tied) = y – x + vEV(foul|tied) = 0 when y = x – v

EV(foul|tied) > 0 when y > x – v

EV(foul|tied) < 0 when y < x – v

So for example if you lead 1-0 in a situation where the value of a red card is -0.5 points and the opposition scores the resulting free kick 10% of the time EV(foul|lead) > 0 when y > 0.35, in other words if the opposition player scores from open play more than 35% of the time fouling is always going to be the better option. As another example if the match is tied in a situation where the value of a red card is -0.1 points and opposition scores the resulting penalty kick 75% of the time EV(foul|tied) < 0 when y < 0.85, in other words unless the opposition player scores from open play more often than 85% of the time letting him go will always be better than fouling. In an extreme case where the match is tied in a situation where the value of a red card is -1 point (for example right at the beginning of a match) and the opposition scores the resulting penalty kick 75% of the time EV(foul|tied) < 0 when y < 1.75, in other words letting him go is always correct even if he scores 100% of the time.

Remember that these formulas and calculations are only applicable in the specific case where the expected point differential between the two teams is zero. To apply them to real world scenarios we need to use the general form which is applicable in any scenario:

EV(foul) = (xn + (1-x)m + v) – (yn + (1-y)m)

EV(foul) = xn + m – xm + v – yn – m + ym

EV(foul) = y(m-n) + x(n-m) + vEV(foul) = 0 when y = x – (v/(m-n))

EV(foul) > 0 when y > x – (v/(m-n))

EV(foul) < 0 when y < x – (v/(m-n))

where m is the team’s expected points if the opposition doesn’t score and n is the team’s expected points if the opposition does score. By using m = 3 and n = 1 we see that the equation is the same as EV(foul|lead) from before, and by using m = 1 and n = 0 the equation is the same as EV(foul|tied). Of course determining accurate m- and n-values in real world situations is difficult but the same basic method still applies.

These are just a few examples and you can play around with the different variables to try out different scenarios. Naturally these calculations suffer from the same limitations as the red card calculations in the previous articles and there are some things we didn’t take into account like the fact that by taking the red card the player also receives a one or three match suspension which has some negative value and that sometimes (certainly not often, but there is a nonzero chance) the referee will go easy on you and only give you a yellow card, but to again quote George E. P. Box, “essentially, all models are wrong, but some are useful.” I think this is useful.

## The Value of a Red Card, Part 3

*This is part three in a series of posts investigating the value of a red card.*

One of the problems with the work presented in part two of this series is that the calculations were done for the average case, two average Premier League teams facing off on neutral ground with both teams having an expected goal differential of zero, and as such it isn’t directly applicable to the real world. However we can improve on the results and approximate the solutions to real world problems by tinkering with the inputs.

If you recall, the original expected goal differential equation is:

Expected goal differential = ((goals scored per minute 11vs11 times the ratio of goals scored 10vs11 to 11vs11) – (goals conceded per minute 11vs11 times the ratio of goals conceded 10vs11 to 11vs11)) times the numbers of minutes left in the match

The average values for the different variables from the 03/04 to 09/10 Premier League seasons were

Goals scored per minute 11vs11: 0.0147

Ratio of goals scored 10vs11 to 11vs11: 0.72

Goals conceded per minute 11vs11: 0.0143

Ratio of goals conceded 10vs11 to 11vs11: 1.98

But what about in a specific case? Take for example Manchester United.

MU goals scored per minute 11vs11: 0.0215

MU goals conceded per minute 11vs11: 0.0081

As you would expect United were quite a bit better than average at both scoring and conceding goals. If we plug these numbers into the EGD equation, assuming for now the ratios of goals scored and conceded 10vs11 to 11vs11 stay the same, we can see that the EGD of United playing 10vs11 for 90 minutes is -0.05 goals, whereas for the average team it would be -1.62 goals. That’s if we assume the ratios of goals scored and conceded stay the same, but one could argue that United are better at playing with ten men than the average team and so the ratios should be adjusted as well. Say you think United are 10% better at scoring goals 10vs11 and 10% better at conceding goals 10vs11 so we change the ratios from 0.72 and 1.98 to 0.8 and 1.8. Using the same goals scored and conceded per minute numbers with the new ratios the EGD of United playing 10vs11 for 90 minutes would be 0.24 goals.

Now if you’re wondering how the value of a red card could be positive, remember that we aren’t calculating the value of a red card yet, we are calculating the expected goal differential of the team for the rest of the match. To determine the actual value of the red card we need to compare this figure with the team’s expected goal differential without the red card and the difference between the two is the value of the red card. I sort of glossed over this in the previous article because in the average case where the expected goal differential before the red card is zero it has no effect and the value of the red card is equal to the expected goal differential.

So what is the EGD of United in different 11vs11 situations? Well, I don’t know. If I did I would be making millions betting on sports and not writing silly articles for a blog no one reads, but we can guess.

Say United are at home against a bottom of the table side. It’s still scoreless and the second half has just begun when United have a player sent off. What is United’s EGD right before the sending off, after the sending off and what is the value of the red card?

On average over the sample United scored 0.0215 goals per minute but at home against a weak side that number should be higher, let’s say it’s 0.03 goals per minute. On average over the sample United conceded 0.0081 goals per minute but at home against a weak side that number should be lower, let’s say it’s 0.006 goals per minute. So United’s EGD(11vs11) is 0.024 goals per minute or 1.08 goals for the remaining 45 minutes. After the red card, using the ratios of 0.8 and 1.8, United’s EGD(10vs11) is (0.03*0.8 – 0.006*1.8) * 45 = 0.594 goals. The value of the red card is the difference between the two, or EGD(10vs11) – EGD(11vs11) = -0.486 goals.

We could also flip this example the other way and ask what is the value of the red card if the weaker away team receives it? Let’s say the bottom of the table team in this example is Wigan. On average over the sample Wigan scored 0.0111 and conceded 0.0160 goals per minute, but away at Old Trafford those numbers might be more like 0.0089 and 0.0192. Let’s also assume Wigan is worse than average at playing with ten men so instead of using the ratios of 0.72 and 1.98 we will use 0.6 and 2.2. Wigan’s EGD(11vs11) would then be -0.4635 goals and EGD(10vs11) would be -1.6605 goals. The value of the red card would be -1.197 goals. That a red card hurts a weaker team more than it does a stronger team shouldn’t come as much of a surprise.

Whether you agree with the exact numbers I used in the examples is irrelevant. You can play with different situations and different inputs all day long, what matters is the method and understanding how to adapt the equation to whatever scenario you want to investigate.

*In part four of this series I will use this method to investigate the issue of intentional red card fouling.*