Red cards have been a hot topic of discussion over the past few weeks. Those of you who have been following the Six Nations tournament are aware of its influence on Wales’ fortunes after 2 rounds of competition. If you did not see the red cards from the Six Nations competition, here they are:

As World Rugby has taken steps to promote player welfare, it can be argued that both calls were appropriately applied by “the letter of the law”. However, the purpose of this post is not to argue the appropriateness of a red card, but rather to evaluate the impact of the red cards. Both red cards occurred early enough in the game where they potentially had a major influence on the remainder of both games. Perhaps the rugby nerd in you has wondered about what the historical impact of red cards was on games, and now we’re about to find out!

**RUGBY NERD ALERT:** *I am about to go into some superficial detail about the methodology and statistics that were implemented in this study. If this is not your thing, you can scroll down and re-engage at the identified location later in the post.*

For this analysis, we elected to look only at scenarios where red cards existed in isolation – so we only wanted to look at situations where teams were playing 15 vs 14 for the duration of the game following the red card. Thus, we used the following inclusion criteria for the data to be considered in the analysis:

All games which had red cards were identified

Games with red cards followed by at least one yellow card were discarded

Games with red cards awarded within 10 minutes following a yellow card were discarded

Games with multiple red cards were discarded

When looking at historical red card data from the Six Nations tournament, one thing became evident very early in the process – there wasn’t a lot of data available. In fact, only 4 red cards existed in the data from between 2014-2020 and only 2 of them met the inclusion criteria. For the purposes of developing a more robust model, we elected to expand the analysis across many competitions with the hopes of obtaining more data points which met our inclusion criteria. Competitions that were considered in this analysis were limited to:

Super Rugby (2012-2020)

Guinness Pro 14 (2012-2020)

Gallagher Premiership (2013-2020)

Top 14 (2014-2020)

Six Nations Tournament (2014-2020)

The Rugby Championship (2014-2020)

World Rugby Spring and Fall International test matches (2012-2020)

Rugby World Cup (2015, 2019)

From this very large pool of potential data points we were able to identify 155 games which met the inclusion criteria for the study. From the data, we then identified the team that gained the player advantage from the red card and then calculated the net change in score from the time of the red card to the end of the game. We elected to calculate the net change in score rather than the points scored by the team with the player advantage because it is possible for the team playing short-handed to score, and in fact this happened more often than you would think. A red card doesn’t automatically mean that scoring will be one-way traffic in favour of the team with the player advantage. The average of the net change in score was calculated as the net Expected Points (nEP) for the red card, and these were plotted against the Time Remaining in the game (Figure 1).

**FIGURE 1:** Net Expected Points versus Time Remaining. This linear regression model provides a visual representation of the impact of an isolated red card across various top club and international competitions (2012-2020).

As you can see, even with data from 155 games the graph has a lot of scatter, which highlights why it is important to gather as much data as possible when testing any hypothesis. With more data, if there is a meaningful relationship between the dependent and independent variables we can generally reduce the amount of scatter and start to see a pattern. To avoid the risk of overfitting the data at this early stage in the development of the model, we elected to use simple linear regression to model the relationship between nEP and Time Remaining in a game. Furthermore, it is reasonable to assume that a player advantage likely has a constant effect on the rate of change of a team’s ability to score – thus a linear model makes sense. In the current iteration of the model, this relationship between nEP and Time Remaining was found to be statistically significant (*p* < 0.05). As we obtain more data from future games, we will gain a better understanding of the relationship between the variables and can revise the model as required.

If we look at the equation of the line (y = 0.241x), the slope indicates that for each minute following a red card, the team with the player advantage will score 0.241 net points. Again, this is a scenario where it may be better to trade off some accuracy for increased understanding by simply rounding the slope to 0.25 points. Thus for the sake of simplicity it may be easier to remember that the team with the red card advantage should score a quarter of a point per minute following the red card.

**NOTE: IT IS NOW SAFE FOR ANTI NERDS TO RE-ENGAGE WITH THE CONTENT AT THIS POINT**

It is clear that the impact of a red card will depend on when it occurs in a game. If it occurs with 1 minute left, it will have a much smaller impact compared to a red card called 1 minute into a game - which means that a team will have to play short-handed for the remaining 79 minutes of the game. As a general rule of thumb, the impact of a red card can be estimated by the following relationship:

**RED CARD IMPACT = (0.25) x Time remaining in the game**

The Red Card Impact is measured in net Expected Points (nEP) to account for the fact that both teams can score even though one team is playing short-handed. This nEP value can be added to/subtracted from the score of the team gaining the player advantage at the time of the red card to provide an historical estimation of the final score. Using this estimate, we can now compare the impact of Peter O’Mahony’s red card to Zander Fagerson’s red card.

**IMPACT OF PETER O’MAHONY’S RED CARD**

Immediately prior to the red card, Wales was the home team and was leading by 3 points before gaining the player advantage. There were ~67 minutes remaining in the game which converts to an estimated Red Card Impact of +16.8 nEP. We can input these game states into the win probability model to determine:

Wales’ win probability prior to red card = 53.3%

Wales’ win probability after red card = 91.3%

Thus, Peter O’Mahony’s red card increased Wales’ win probability by +38%.

**IMPACT OF ZANDER FAGERSON’S RED CARD**

Immediately prior to the red card, Wales was the away team and was losing by -2 points before gaining the player advantage. There were ~26 minutes remaining in the game which converts to an estimated Red Card Impact of +6.5 nEP. We can input these game states into the win probability model to determine:

Wales’ win probability prior to red card = 34.5%

Wales’ win probability after red card = 63.5%

Thus, Zander Fagerson’s red card increased Wales’ win probability by +29%.

**IS WALES STRUGGLING ON THE POWER PLAY?**

Those of you with inquiring analytical minds may have noticed the following:

Peter O’Mahony’s red card resulted in a Red Card Impact of +16.8 nEP. The final score margin was estimated to be +19.8 but Wales only won by 5

Zander Fagerson’s red card resulted in a Red Card Impact of +6.5 nEP. The final score margin was estimated to be +4.5 but Wales only won by 1

Does this mean that the model is crap and that we should simply give up and go back to __eating corn chips and masturbating__ on Friday nights? Well, we know that we are limited by the number of data points available so there will be some uncertainty, however as we accumulate more data over time the model will hopefully become more robust in its predictions. Another thing to consider is that maybe the model isn’t wrong so much as Wales is underperforming relative to expectations in terms of scoring with the player advantage?

As a Canadian, I couldn’t resist bringing in some kind of reference to hockey (that’s *ice hockey* to those of you who live outside of North America). Minor penalties in the National Hockey League (NHL) typically result in one team gaining a player advantage where the game is played 5 vs 4 skaters for 2 minutes (a.k.a. the power play). Historically __NHL power plays score at ~20% efficiency__ but this will vary ever so slightly from season to season. If you look at the __current league table__ you will see that there are teams with power plays functioning below 10%. Thus, these teams are underperforming in their ability to score on the power play relative to historical expectations. Perhaps Wales’ red card “power play” was underperforming in those two games?

**IMPACT OF 20 MINUTE RED CARDS**

Both Super Rugby Aotearoa and Super Rugby AU have announced that they will be trialing 20-minute red cards in their 2021 competitions. While there is little to no data available specific to either competition that will assess the effect of a 20 minute penalty, we can potentially estimate the projected impact using the Red Card Impact estimate:

RED CARD IMPACT (SR) = (0.25) x (20 minutes) = +5 net EP

Thus, for both Super Rugby competitions (Aotearoa, AU) we predict that red cards will result in a net 5 point gain to the team awarded the player advantage. We will check this prediction when enough data becomes available. Stay tuned!

**FUTURE WORK**

After this preliminary foray into evaluating isolated red cards, it’s clear that there are so many other questions left to answer such as:

What is the impact of isolated yellow cards?

What is the impact of multiple isolated or overlapping cards (red and red, red and yellow, yellow and yellow, etc.)?

Do red and/or yellow cards occur independently of one another? Or is there a tendency for match officials to unconsciously “even things up”?

Will the frequency of red cards increase in Super Rugby relative to other competitions due to the fact that they are perceived as potentially less punitive? This could be influenced by the mindset of players, coaches, and match officials?

What is the impact of red and yellow cards in 7s?

What other aspects of carded penalties would you like to explore?

Nice analysis. Would be interested to see win/loss percentage of carded team as a function of time also