## 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.