Penalty Shootouts

In the FIFA World Cup to date, there have been thirty penalty shootouts. The first shootout was on June 8, 1982 when West Germany eliminated France with 5-4 in a penalty shootout to advance to the final. The match was ended in 3-3, including four goals in extra time, and is best remembered because of the horrific foul by the German goalkeeper Harald Schumacher on the French player Patrick Battiston, who is still su ering the consequences of the foul that Dutch top referee Charles Corver not punished with red. To make matters worse for the French team, Schumacher was the German hero in the penalty shootout. He saved two penalty kicks before Hrubesch scored the winning penalty kick. In a penalty shootout, the two teams take turns taking a penalty kick, and after each team has made five attempts, the winner is the team with the most successfully taken penalty kicks. The rules of the game require that the penalty shootout must be stopped with less than ten penalty kicks as soon as it is clear that one team can no longer tie with the other team. If the score is still tied after each team has had five penalty kicks, the penalty shootout continues until `sudden death', in which the two teams alternate penalty kicks until one team scores and the other team misses in the same round.

Much has been written and said about the psychology and tactics of penalty kicking. The toss remains the most important in influencer of the outcome of the penalty shootouts: the Spanish professor Ignacio Palacios-Huerta of the London School of Economics - author of the bestseller - Beautiful Game Theory: How Soccer Can Help Economics - analyzed 1343 penalty kicks from 129 penalty shootouts and found that the team that started first won in 60.5% of the cases. By the way, Dutch soccer player Marco van Basten with 92.7% and English soccer player Alan Shearer with 93.5% have the highest scoring percentages of top players who have taken more than 50 penalty kicks (Messi and Ronaldo have scoring percentages of about 78% and 84%), and that while both England and the Netherlands are among the worst performing countries in taking penalty shootouts. After winning the toss for penalty the shootouts in the quarter finals of the 2008 European Championship, it was not wise of Gianluigi Buffon, the famous goalkeeper and captain of the Italian team, to leave the first penalty kick to Spain who eventually won the penalty shootout. The explanation that there is an advantage to being the first to start taking penalty kicks is that the team that starts is often ahead in the series: 1-0, 2-1, 3-2, etc. So the team that gets to shoot second feels the pressure each time to have to equalize, and that leads to more missed penalty kicks on average. Managing the stress level is not so simple when a lot is at stake.

This column focuses on a simple probability model for the calculation of the probability distribution and expected value of the number of penalty kicks in a penalty shootout. The model assumption is that the outcomes of consecutive penalty kicks are independent of each other and that each player has the same probability p of scoring in a penalty kick. For soccer matches at the level of the FIFA World Cup or the UEFA Champions League, it is reasonable to take the value  for . A penalty shootout requires at least six penalties. It ends after exactly six penalties if one team scores in each of the first three penalty kicks and the other team does not score in any of the first three penalty kicks. Denote by  the probability that in a penalty shootout  penalty kicks are needed to arrive at a winner, then combinatorial probability calculations give the following results when 

The probability that the penalty shootout ends with ‘sudden death’ (no winner after 10 penalty kicks) is

If after the five rounds with 10 penalty kicks there is still no winner, then the number of rounds to ‘sudden death’ has a geometric probability distribution with parameter , so that the expected number of penalties remaining is  So the expected value of the number of penalty kicks in a penalty shootout is equal to

All this under the model assumption that the outcomes of successive penalty kicks are independent of each other and that each player has the same success probability  to take a penalty kick. The result of an average of  penalty kicks at a success probability  can be extended to an explicit formula applicable for any value of the success probability: given the same success probability  for each player, the expected value of the number of penalty kicks needed in a penalty shootout is given by the formula

where . In this formula, the expected number of penalty kicks is symmetric in  and ; for example, for a success rate of 80% the average number of penalty kicks needed has the same value 11,353 as for a success rate of 20%. The formula shows that the average number of penalty kicks is minimal for a success probability of 50% and then has the value 10.031. Put differently, in the probability model in which each player has the same success probability, the average number of penalty kicks needed is always more than 10.

Photo

FIFA World Cup 2018, Colombia vs. England, Wikimedia Commmons by Oleg Bkhambr.

Author

Henk Tijms is honorary fellow of Tinbergen Institute and emeritus professor of operations research at Vrije Universiteit Amsterdam. He is author of various textbooks on operations research and probability theory. His most recent book is Basic Probability, What Every Math Student Should Know, World Scientific Press, 2019. Home page: personal.vu.nl/h.c.tijms/