Red Cards.What To Expect After One is Shown.

Bolton 2-2 Birmingham,EPL August 2010.

When Jussi Jaaskelainen aimed at kick at Birmingham's Roger Johnson and followed it up with a slap around the head a red card was inevitable.Bolton already trailed Birmingham by virtue of Johnson's 4th minute goal and the longterm hole they had dug for themselves got even deeper when they fell further behind just after the break.However,spurred on by a home crowd and the recipients of a couple of generous refereeing decisions they claimed an unlikely point with two late goals.

Newcastle 4-4 Arsenal.EPL February 2011.

This game has been described here.Arsenal had a 4 goal cushion when Diaby was dismissed after 50 minutes,but they weren't quite as resilient as Bolton had been in the previous example and the 10 men Gunners were grateful to return to London with a point.

We have seen that on average a red card reduces a team's in game win probability,but as the two games above demonstrates there is a wide range of possible outcomes that can materialise on the actual matchday.Taken in isolation the Bolton game could be used to justify the often repeated and totally erroneous cliche that "it is (always) harder to play against 10 men than 11".While the Arsenal game dramatically demonstrates the actual real life experience where the numerically reduced side more usually finds scoring and defending much more difficult.

To discover what happens on average it is necessary to take a large number of games and record the scoring patterns of the teams both before and after the red card was shown.I've taken each game during the 2010/11 season where a solitary red card was issued,comprising 45 games in total.The soon to be ten men scored less goals than their opponents in the time prior to the red card being shown,confirming the findings here and here that red cards are more likely to be shown to inferior sides forced to make more defensive challenges or to good sides playing badly and reacting petulantly to this unexpected dip in form.


Scoring and conceding record of red carded sides prior to the red card.EPL 2010/11.

Goals Scored by Red  Carded Side Prior to Card.	Goals Allowed by Red Carded Side Prior to Card.	Percentage of Match Goals Scored by Carded Side.
28	42	40%

In total 70 goals hit the back of the net prior to the dismissal in the 46 games and 40% of them were scored by the team about to see red.

Scoring and conceding record of red carded sides after the red card.EPL 2010/11.

Goals Scored by Red  Carded Side After Card.	Goals Allowed by Red Carded Side After Card.	Percentage of Match Goals Scored by Carded Side.
15	36	29%

The average time at which the dismissal occurred was the 64th minute and ranged from numerous injury time cards to Boyata's 5th minute departure from the Man City Arsenal game.With just 21 players on the pitch things became much more difficult for the even numbered side and they now only scored 29% of the total goals scored compared to 40% previously.So although teams who perform heroically in the face of lopsided  numerical odds tend to stick in the memory,usually as a result of inflated media interest,it is the well behaved side who much more often derive the advantage from a dismissal.

The average time of the card also allows us to see how the overall goalscoring differs pre and post dismissal.Some of the games in the sample are mis matches at the start and they will tend to be slightly more goal laden than a more usual English Premiership game.However,overall the sample of games would expect to have just slightly above 2.6 goals scored per game.We know that the average time of the dismissal is the 64th minute and we can fairly easily work out the rate at which goalscoring decays over the course of the game.In a game where an average of 2.6 goals were expected to be scored,we would expect an average of 1.05 goals to be scored after the 64th minute.If we compare the actual total number of goals scored by both teams after the red card we can not only say what proportion of goals will be scored by either team,but also how much more or less goalscoring occurs.

The 45 games produced 51 post card goals at 1.13 goals per game,slightly higher than the pre game expectation for the number of goals that would be scored after the 64th minute in a well behaved game.However,every game starts with a kickoff,but some of the red card "mini" games that lasted on average from the 64th minute onwards started unsurprisingly with a penalty kick and if we allow for these occurrences we can probably say with confidence that games blighted by red cards produce just as many goals as normal games.It's just the scoring distribution that is changed.

So if you're at a game and a red card is shown,you'll still get your money's worth in terms of goals scored,but expect the numerically superior side to do more than their pre game expected share of the scoring.

Red Cards and the Big Four.

The win probability for each team is tracked on a minute by minute basis throughout each game.At any point in a game a team will have a probability of winning the game outright and an associated probability of drawing the game.By multiplying the probability of winning the game by 3 and the probability of drawing by 1 and adding the results together you will get an expected longterm average for the number of points that team will achieve from that position or Expected Points (EP) for short.

Every football fan can remember games or incidents within games that are especially significant for their club.They usually signify that their team has reached a new high in it's development or plummeted to new and possibly permanent depths of ineptitude.Stoke fans would probably put November 1st 2008 into the former category.The initial 45 minutes of their first EPL season where they shipped a goal every 15 minutes was still a raw and recent memory,but they had partially steadied the ship with wins over Villa,Sunderland and an obliging nine man Tottenham.They occupied 15th place in the EPL,the average finishing spot for promoted teams in their first year in the higher grade,but defeats at the hands of Bolton, Middlesbro', Everton, Chelsea, Man City and Portsmouth meant that the visit of third placed Arsenal to the Britannia was a daunting prospect for the hosts.

Pre game expected points values made the Gunners huge favourites.They would expect to return with an average of 2.34 points from their visit north,while Stoke would be lucky to average half a point a game.

However,today was a day when one of the least favoured scenarios materialised.Fuller's 11th minute strike would rarely have been a decisive first goal and Stoke had to maintain that lead until after the half hour before they became the team favoured to clinch more points on average from that particular game situation.By the 70th minute Stoke still held the advantage and would be expected to gain an average of 2 points per game in such circumstances,but they tightened their grip three minutes later when Olofinjana stumbled his way into the box and bundled a trademark Delap throw over the line.Instant cult status for "Seyi".

Scorers;
1-0,Fuller,11'(Delap assist)
2-0,Olofinjana,73'(Delap assist)
Red Card,van Persie,Arsenal,76'
2-1,Clichy,90'

If Arsenal had been unaware of their underdog status from the half hour mark onwards,they certainly knew their likely fate now and three minutes later their predicament worsened even more.Stoke's keeper Thomas Sorensen deliberately dallied over picking up the ball in his box,inviting the loitering van Persie to participate in a game of cat and mouse where the Dutchman was by far the junior partner.Van Persie's half hearted lunge for the ball was never going to arrive before Sorensen claimed it for himself and what the subsequent shoulder charge lacked in violent intent it certainly made up for in stupidity.Sorensen collapsed in a heap,although he was careful to remain in front of his goal line and Stoke's favourite referring pantomime villain,Rob Styles set van Persie packing with a straight red card.The dismissal left Arsenal in a hole too deep for even a team of their attacking flair to emerge from and Clichy's 90th minute strike was merely a consolation.

The view  from the stands confirmed Stoke as a team who could compete on their own turf with the league's very best,dreams of a European Tour were a few seasons down the road,but the Potters were no longer considered to be a team out of their depth.Regarding the dismissal,it was difficult not to conclude that van Persie had reacted to the situation of trailing and most probably losing to a team who should have been dispatched with the minimum of fuss and in a manner Chelsea had achieved on the same ground a month earlier.It was a clear case of red card bourne out of frustration.

It's easy to attach too much importance to individual events,so to test the theory that the best teams see red when they fail to reach the heights they generally attain,I recorded all games involving the Big Four,Liverpool,Arsenal, ManUtd and Chelsea where a red card was shown to their players. I charted the pre game Expected Point Values and the EP for the team in the minute before their player was dismissed.Games between the big four were removed from the sample.

Games against inferior opposition where the big four received a red card.2004-2011.

Teams.	Number of Red Card Games.	Average EP at Kickoff	Average EP just before Red Card.	Average % fall in EP	% of Games where EP fell before Red Card.
Man Utd.	12	2.02	1.54	24	75
Arsenal.	15	2.09	1.73	17	67
Chelsea.	16	2.29	1.98	14	56
Liverpool.	6	1.83	1.01	45	83
Overall.	49	2.11	1.68	20	67

As can be seen the average Expected Points just prior to the red card are below the average EP at kickoff both for the big four as a group and for each individual team.The EP fell on average by 20% and in about 70% of the games it was below it's starting point.Games numbers are relatively small for each team,so conclusions are necessarily tentative.Liverpool appear to be the most bad tempered of the original big four,5 of their 6 red card games came when the Reds were performing worse than expected.Chelsea were the team who appeared least concerned with the current match position before they saw red.

The figures appear to confirm that teams who are playing badly by their expected standards are likely to compound the problem by having players dismissed.The question now becomes is the effect universal or is it more pronounced when those struggling to perform are the best teams in the division,after all they have more to lose than teams with lower pre match expectations and their under performance will be self-evident to their players.The average fall in EP at the time of the dismissal is around 20%,so in the next part of this series I'll take a representative sample of games not involving the big four and see if the red card situations remain the same.

Game of the Weekend. Birmingham v Leicester.

The win probability for each team is tracked on a minute by minute basis throughout each game.At any point in a game a team will have a probability of winning the game outright and an associated probability of drawing the game.By multiplying the probability of winning the game by 3 and the probability of drawing by 1 and adding the results together you will get an expected longterm average for the number of points that team will achieve from that position or Expected Points (EP) for short.

0-0,Referee Salisbury is replaced by Mr Wright,45'
0-0 Penalty awarded to Birmingham,50'
1-0,King(pen),50'
1-0,Red Card for Mills,Leicester,56'
2-0,Wood,84'


Substitutions can have a major impact on match results and so it proved when recently relegated Birmingham
 entertained Sven Goran Eriksson's Leicester City in the televised midday kickoff at St Andrews.

The bright start to the game had matched the weather,but like the unseasonal weather,it was destined to be short lived and the game settled down into a scrappy midfield battle with little quality on show at either ends of the field.The first half was as dull as it's Expected Points graph was featureless,but all that changed within ten minutes of the restart.

As substitutions go this game changing one was unusual.Starting referee Graham Salisbury's calf strain saw fourth official Kevin Wright take the field for the second half and he quickly found himself involved in two massive EP shifting decisions.

Firstly,he awarded Brum a 50th minute penalty following Andy King's tackle on Beausejour.On first viewing in real time it looked a harsh call,but repeated replays confirmed that it was indeed the wright call.

Mr Wright then dismissed Leicester's Mills for a two footed,partially airborne lunge just five minutes after the spot kick was converted by Marlon King.Both decisions were correct,but the travelling supporters predictably sang "1-0 to the referee" until Wood finished them off 7 minutes from time.

The penalty award reduced the Foxes EP from 1.26 to 0.65,King's successful conversion further reduced it to 0.44 and Mills' moment of madness sent it all the way down to 0.099.

EPL Referees Appear To Show Disproportionately More Red Cards To the Away Side.

A glossary of terms is in the pipeline,but for the meantime.....The win probability for each team is tracked on a minute by minute basis throughout each game.At any point in a game a team will have a probability of winning the game outright and an associated probability of drawing the game.By multiplying the probability of winning the game by 3 and the probability of drawing by 1 and adding the results together you will get an expected long term average for the number of points that team will achieve from that position or Expected Points (EP) for short.

It should come as no surprise to discover that in games where one red card is issued,then that card is more likely to be shown to the visiting side.The home side more often will have the better chance of winning the game,therefore the away side will be forced to do more defending,sometimes against superior players,which will lead to more fouls,more yellow cards and ultimately more chance of a red.In the 300+ EPL games spread over the last 7 seasons where a red card was shown over 60% of the time it was shown to the away side.To further test our idea that poorer sides are more likely to attract red cards we first need a way of quantifying pre game team quality.One very good way to represent the difference in quality between teams prior to the game starting is to compare each side's Pre Game Expected Points tallies.And once again we find that the underdog,irrespective of venue receives over 60% of dismissals in games where a single red card is shown.The average Pre Game Expected Points for the team who receives the red card is 1.29 compared to 1.45 for the team that doesn't see red.

How Red Cards Affect A Soccer Match.

After penalties and goals,red cards are the next most significant event that can take place during a soccer match.They are both dramatic and potentially game changing and they aren't quite as rare as you may imagine.Around 15% of matches feature one or more red cards,so statistically speaking you are more likely to see a red card than you are a 1-1 draw.

The team which is reduced in number suffers by scoring less goals than they would have done had they kept a full complement of players on the pitch and they also concede more goals than would have been so.

One of the most effective ways to model the expected progress of a soccer match is to calculate the average number of goals each team can expect to score at today's venue and against today's opponents.We can then use the Poisson distribution (with few tweaks) to calculate a multitude of predictions about the likely route the game will take.This is the basis for the game graphs on this site.However,if we want to carry this approach past the initial whistle we also need to know how the expected average number of goals each team will score today decays as the game progresses.Initial clues for a possible way forward lies in the distribution of goals scored in each half of play.In England approximately 45% of goals come before the interval and 55% afterwards.In fact if you plot the number of goals scored in various 10 minute segments over the course of a contest you'll find that the later segments see more goals than the earlier ones.In short the rate of goal scoring increases with time.

The best fit for goal expectancy throughout a game is;

Goal Expectancy during the Game = Initial Expectancy * (Proportion of game left)^0.85

where the initial expectancy is the average number of goals you expect the team to score in today's match up at the start of the game.If the initial expectancy is set to 1 and the proportion of the game remaining to 0.5,you'll get the answer 0.55,the observed fraction of second half goals.

This method gives us a route to determine the effect of a red card on the remaining goal expectancy of each of the two teams involved and thus the expected course of a match post that red card.As we have seen,red cards occur in about 15% of games.However they tend to occur later in matches,Kevin Pressman's red after 13 seconds and Vinnie Jones' yellow after three being exceptions.This makes collecting data for earlier game times problematical,often no data exists.What we can do is collect the average time of every red card and also record the average  goal expectancy for the team receiving the red card and their opponents immediately prior to the incident.If we then record the average number of goals the red card side and their opponents actually went on to score,we will have the average change in goal difference between the sides pre and post the dismissal.

We find that over the last 6 seasons one average dismissal occurred just after the 60th minute and the average change in goal difference amounted to 0.61 of a goal.About 64% of the 0.61 difference occurred because the side with 11 players scored more than they would have expected to score prior to the card and the rest came about by the infringing side scoring less.

We can surmise that the change in goal differences bought about by the card decays at the same rate as ordinary goal expectancy.61 minutes means that about 36% of the game remains,so by plugging the figures into the equation we can find the amount by which two teams will see their goal expectancy difference changed by if the red card is shown at the very start.Usefully we can find out how much goal expectancy a team loses overall from a red card over an entire game.

The answer is about 1.45 goals.

So if a reasonably typical EPL team starts the game a man down within seconds they can be expected to score on average 0.5 of a goal less than they would have done with 11 players and concede 0.95 more goals resulting in a swing in overall goal expectancy of 1.45 of a goal.The figure of 1.45 goals can be re plugged into the decay equation to determine the respective figures at any time throughout the course of the game.For example a dismissal as early as the 33rd minute will cost a team a goals worth of expectancy,split between goals they don't score and extra ones they do and this number can be incorporated into the in game probabilities for that particular game.   

Red Cards and Penalties

Soccer City witnessed one of the most dramatic conclusions to a World Cup game in 2010 ,when with the stalemated Uruguay Ghana game ticking down to a penalty shootout Suarez punched Adiyiah's goalbound header off the line.For Suarez the decision simple.The game had seconds to run and any goal was virtually certain to be the winner,but by taking a red card for the team Suarez had given his side around a 25% chance of taking the game to a penalty shootout.(25% is the average failure rate for spot kicks).Gyan bounced the resulting penalty off Uruguay's crossbar and twenty minutes later it was the South Americans and not the Africans who were preparing for a semi final against the Dutch.

But what if the incident hadn't happened in the game's dying seconds,would Suarez have been correct to handle the ball if the game had for example 15 minutes to run? EP graphs can help to answer this kind of scenario.

To simplify matters I've assumed the match is a normal league game,thus eliminating the possibility of extra time or penalties.A player faced with Suarez's choice can take one of two actions.He can allow the goal to be scored or he can concede the penalty.Naturally each choice will have an associated EP value that will be dependent on (principally) the time remaining,the relative strengths of both teams and the current score.By comparing the respective EP values for both scenarios we can see when it is more beneficial,longterm to the team for the player to commit the foul.When during the game it is better not to and most importantly when the crossover occurs.

Calculating the EP for the goal is simple,we use the current score,the current time and team strength.However,the concession of the penalty is slightly more complex.

Penalties are converted at around 75%,so after the red card there will be a 75% chance that the player's team still concedes the goal his handball prevented AND his team will suffer the extra disadvantage of playing with 10 men for the remainder of the game.The other 25% of the time the penalty will fail,but the team will still be playing with 10 men.

The first graph shows the how the EP changes in 10 minute increments for one of two equally match teams,currently tied depending whether a goal line defender either allows the ball to cross the line or takes the Suarez option.

The red line indicates the EP if the goal is scored,resulting in that team now trailing their opponents by a goal while the green curve plots the EP for that team if the defender prevents the goal by conceding a penalty and receives a red card.At the time the player commits the offence no one knows if the penalty will be converted or not and that uncertainty is reflected in the green curve.Allowing the goal remains the best option for around 76 minutes,the difference between the two options steadily narrowing as the game progresses.A team is more likely to gain more points by trying to retrieve the lost goal with a full compliment of players than it is relying on the shot stopping abilities of it's keeper until the game has around 15 minutes to run.Once the 76th minute is reached a player should take the illegal option to prevent the goal.

The posted graph relates to a game that is still tied when the incident occurs,if the team is currently leading by a goal,EP graphs indicate they shouldn't take the red card route until the clock has ticked towards the 80th minutes.If they are currently ahead by two they should wait until the last minute of normal time.

A more usual scenario is that the last defender can chose to foul an attacker as he bears down on goal,but before he's reached the penalty area.A red card is still the likely outcome,but the possibility of a goal resulting from the foul is much smaller compared to a penalty.The choice here is dependent on the defender being able to accurately estimate the chance of the attacker being able to convert his goalscoring opportunity.The next graph shows the changing EP with time for such a scenario where in a tied game the attacker is likely to score 1 time in 4.

Because there is no resulting penalty in this situation there is a greatly reduced possibility that the team will still find themselves trailing by a goal and reduced to 10 men.In this case the crossover point where the length of time a team plays with 10 men but retains parity is equal to the benefit of playing with 11 men but trailing by a goal is reached around the 52nd minute.Once that time has been reached the player should haul down his opponent if he wants to do what's best for his team.

Blasts from the Past.....Newcastle 4-4 Arsenal.

An extraordinary graph with Newcastle's EP spending over 80 minutes at 0.08 or below before leapfrogging Arsenal's EP for the last 7 minutes.This game more than any illustrates that the current EP merely takes you to the current position in the game and the size of that EP is an expression of what happens afterwards ON AVERAGE.

Diaby's 50th minute red card certainly made Newcastle favourites to score more than Arsenal in the remainder of the match ,but the four goal cushion should have been sufficient to enable the Gunners to hang onto the points.The dismissal advanced Newcastle's EP from 0.001 to 0.01 and Arsenal would have won the game from that position 794 times out of 800 and draw it on 5 other occasions.They'd have lost just once.
Countless Geordie fans had already made this mental calculation for themselves and left the ground at half time.

Even when Barton converted a 68th minute penalty the EP barely moved and Best's 75th minute second still saw the respective Newcastle and Arsenal EP at 0.07 and 2.879.

Crucially the clock was now favouring the 10 men visitors and just 5 minutes after Best's strike Newcastle's EP had halved,but Barton's second penalty suddenly made the impossible merely improbable.It had taken 82 breathless minutes to return each teams EP to the values they'd had after Walcott had opened the scoring after a minute.

Tiote's strike was worth just under an EP of 1 (almost 3 and a half times the worth all thee previous goals and the sending off combined) and propelled the home side to marginal EP favourites,but Arsenal held out to ensure that this game wasn't the 1 in 800 when a team gives up a 4-0 half time lead.

Eight goals in total and everyone of them integral to the final result,Tiote's strike claiming the EP plaudits.