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.