## 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.*