## The Value of a Red Card, Part 2

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

The purpose of this article is to try and quantify the value of a red card in terms of how big an impact it has on the expected goal differential of a team for the rest of the match. Unfortunately the value of a red card isn’t static, it changes depending on many different factors including how much time is left in the match, how good the team playing with eleven men is, how good the team playing with ten men is, how good the team playing with ten men is at playing with ten men and other factors. We can however estimate the value of a red card in an average case. To do so imagine two average Premier League teams playing on neutral ground (to offset home ground advantage). To simplify the calculations we need to make a few assumptions:

1. The probability of a team scoring in any given minute is constant, in other words the probability that a team scores in the first minute is the same as the probability that a team scores in the last minute. In real life this isn’t strictly true, for example in the 09/10 Premier League season roughly 44% of all goals were scored in the first half compared to 56% in the second, but if we don’t assume it to be true the calculation quickly becomes impossible.

2. Once a red card has been given there are no more red cards for the rest of the match. Again this isn’t true but it happens so rarely that the effect it has on the final result is negligible. To include various 9vs11, 10vs10 and 9vs10 scenarios into the calculation would only add another layer of complexity with no tangible benefit.

Now back to answering the question. First we need to know a few things:

1. How many goals per minute does the average team score when playing 11vs11?

2. How many goals per minute does the average team concede when playing 11vs11?

3. How does having to play 10vs11 affect how many goals per minute a team scores?

4. How does having to play 10vs11 affect how many goals per minute a team concedes?

Once we know all of the above we can set up a formula to determine the expected goal differential of the team playing 10vs11:

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

To answer these questions I went through every match from the 03/04 to 09/10 Premier League seasons collecting data on how many minutes a team had to play 10vs11 and how many goals they scored and conceded in that time. I then used that information together with the number of goals scored in total over the sample to calculate the average goals scored by a team per minute 11vs11 and 10vs11 as well as the average goals conceded by a team per minute 11vs11 and 10vs11.

Overall the sample contained 380 red cards leading to 10094 minutes played 10vs11. According to the data the average goals

Scored per minute 11vs11: 0.0147

Scored per minute 10vs11: 0.0106

Conceded per minute 11vs11: 0.0143

Conceded per minute 10vs11: 0.0283

meaning the ratio of goals scored 10vs11 to 11vs11 was 0.72 and the ratio of goals conceded 10vs11 to 11vs11 was 1.98.

The goals conceded ratio seems fairly straightforward. According to the data teams concede roughly twice as many goals when playing 10vs11 than they would playing 11vs11, which seems reasonable. The goals scored ratio is slightly more interesting. In my first draft of the calculation, before I had collected any data, I guessed that a team playing 10vs11 would score roughly half as many goals as they would playing 11vs11. One friend suggested that the ratio should be even lower, closer to 0.2 or 0.3. I was somewhat surprised to see the data suggest a ratio as high as 0.72. This could mean that there is a flaw in the data, or it could mean that there is a disconnect between reality and our predetermined notions of how much playing 10vs11 actually affects a team’s goal scoring.

We can now plug these numbers into the expected goal differential equation from before and simplify it to get:

EGD = -0.018 * M

where M is the number of minutes remaining.

You can try different numbers yourself but for example in the extreme case of having to play 90 minutes 10vs11 the EGD would be -1.62 goals. With fifteen minutes left the EGD would be -0.27 goals. With five minutes left -0.09 goals.

To determine the value of the red card we need to calculate the difference between the EGD 10vs11 and the EGD 11vs11. Fortunately in the average case, since the EGD 11vs11 is zero for both teams, the value of the red card is equal to the EDG 10vs11 so we can use the above formula without having to worry about it.

Clearly this method isn’t perfect, but as a starting point I think it can be valuable. To quote George E. P. Box, “essentially, all models are wrong, but some are useful.” Hopefully this is useful.

*In part three of this series I will look at ways to tinker with the model to make it more real world applicable.*

I did similar research a few years ago using data from a Bundesliga season. The net gain was a little over 1 goal per 90 minutes for an 11-side versus a 10-side. In using the data did you take into account that the weaker team and the away team has higher number of sent-offs?

I didn’t, I just lumped everything together.