Duel at the Penalty Spot as a Criterion for Choosing a Team

There are a lot of secrets in the history of England, but nothing touches the feelings of the British as well as one unfortunate event that happened 28 years ago and then had four unpleasant doubles. In the semifinals of the World Cup in 1990, teams from England and West Germany began to prepare for a penalty shootout. Germany could not help but win that series, in the year of the unification of the GDR and the FRG. Since then, the evil rock haunts the British every time it comes to a series of eleven-meter. Is it possible to explain such bad luck without attracting all sorts of mystical inventions?

The rules for a single strike from the penalty point in the edition of FIFA from 2012 are defined as follows:

If someone thinks that a series of penalty strokes is an unfair criterion for selecting a more worthy team, then, perhaps, we should remember what happened before. In 1954, the draw determined participation in the World Cup of the national teams of Spain and Turkey, which in the qualifying matches had a draw result on the results of two regular matches played and one additional. Fourteen-year-old Luigi Franco took an urn from the urn with a marking of Turkey.

For the first time the penalty shootout was held in the final match between the clubs “Barcelona” and “Zaragoza” on September 2, 1962. The novelty soon became widespread on different continents, and already in 1970, FIFA officially approved and fixed the rules for a series of penalty shots. Initially, these rules were as follows:

Since then, little has changed. In 2003, FIFA adopted an amendment according to which the captain, who guessed how a coin would lie, could also give way to the right to beat the first opponents.

But it is unlikely that the cap-witted in the statistics will decide to give up the championship, and that’s why. The thing is that the penalty fell to the court not only the managers of football, but also football analysts. The scientists were also enthusiastic: a series of strokes from the 11-meter mark is as close as possible to a well-placed and rigorous, according to academic standards, experiment.

For this there are all: serial regulated actions with timing, the same conditions and even the presence of a control group. In such a natural experiment, randomness is well documented and measured. The coin falls out with equal probability in one of two possible states. The speed of the ball, the movements of the beating player and the goalkeeper vary in a reliably predictable framework.

By collecting data on all known at that time series of penalties, scientists began to extract the factors of success and failure from the aggregate of numbers and variables.

A total of 10,431 penalties were recorded in the database of 1 001 games played. As can be seen from the graph, in general, the rule “more often than not, the one who struck first” is true. The odds are approximately 60% to 40%.

So far, we have not considered the goalkeeper’s actions in any way, and it’s he who is the hero of this show, which reduces the chances of a goal from 100% to 75%. Players and goalkeeper have their own strategies, and their interaction takes place according to the theory of games, once a popular discipline, originating in the military staff of the era of early nuclear-missile deterrence.

In addition, there are also psychological aspects of the duel. In the second part of this series of materials, we’ll talk about this in more detail to find out why England is constantly losing on penalties.

 It’s time to take a closer look at the actions of the goalkeeper, to find out how he sometimes gets to win a duel in a hopeless, seemingly, situation.  A goalkeeper who hits a kick from the penalty spot has two goals that complement each other:

The first ingredient of the goalkeeper’s success is based on the psychological state of his opponent. Players are nervous and make mistakes due to the very fact of having a goalkeeper standing at the gate. Even if the goalkeeper stands still, the player can be in a state of stress.

The first blow to the series of penalty scores in the match England – Portugal at the quarter finals of the 2006 World Cup: “My body was stiff at the moment when I began to approach the eleven-meter mark. God, even the enemy, I would not have wished for this walk. But still, despite all my doubts, I kept faith in my skills. All at me it will turn out. I will definitely score. I became accustomed to preparing for a strike. We put the ball exactly, like this. Do you remember how you scored 9 out of 10 in training, so what’s the matter? Ricardo is good, but I will try so hard that he will not reach. At the very place Robbo, David James and Scott Carson talked to me in training”. – “My autobiography”. Steven Gerrard

Unfortunately, things did not go as Stephen Gerard expected. Seconds before the whistle of Judge Elizondo seemed to him an eternity, he was already at the limit, and these seconds killed him. In the end, he broke the wrong way, as Robbo and his comrades taught, and Ricardo hit the ball. Frank Lampard and Jamie Carragher have not struck better, and the England team lost the football match.

After the game, the experts studied the video footage, reviewed each frame in order to decipher the body language of the penalty shooters, and it all came together. Lampard and Carragher were clearly in a panic. Putting the ball in place, they turned to the goalkeeper with his back instead of as it should be backed off. Carragher in a hurry carried out the blow too quickly and was forced to make another approach. The goalkeeper, as if knowing that those two are mistaken, and he will have time, waited a little longer, determining the direction of the strike before making the throw.

The goalkeeper, however, does not have to stand at attention. He can stand a little away from the center, provoking a blow in the most convenient for himself to the side, more often – in the right. Studies have shown that this trick works. It is enough to stand from the center for 6-10 centimeters to the side, and the number of strikes in the opposite direction will increase by 10%. This gives the goalkeeper an advantage in repelling the blow. And some goalkeepers, such as Jerzy Dudek and Bruce Grobbelar, can curl and dance, in order to distract the player’s attention.

“I often recall the 1984 European Cup final against Roma and how my freaks with my feet in the penalty shoot-out helped us take the trophy of the tournament. Many told me that I thus showed disrespect to my rivals, but I think that I just wanted to test them for resistance to stress. They could not stand the test”. – Final of the European Cup, former goalkeeper of FC Liverpool Bruce Grobbelar

Now a little bit about the strategy of the goalkeeper and the player. In mathematics there is such a discipline – game theory, which studies formal game situations and optimal strategies. Penalty is a typical zero-sum game in the language of game theory. This means that the loss of one of the players is equal to the winnings of the other.

For both participants of the duel, the optimal strategy is the one that is defined by the two inequalities listed below. Nash equilibrium in game theory is a set of strategies in which no player can increase his winnings by changing strategy. On the biography of Professor Nash filmed the movie “Mind Games” with Russell Crowe in the title role.

In order to determine whether the Nash equilibrium exists in real football, a large-scale study of the penalty shootout series was conducted: m of only 9,017 approaches to the ball in the course and 09.1995 to 06.2012. The statistics included the leading football leagues of European countries. The results of the study confirmed the correctness of the Nash theory. The choice of direction for the player’s blow and the throw of the goalkeeper is consistent with the prediction of the game theory model.

The consequence of choosing the optimal strategy is the unpredictability of the direction of the blow and the throw of the goalkeeper. This follows from mathematics. Proof of the minimax theorem. But it seems to me that this is the case, when without it is clear that the lawfulness and predictability of choice is the right way to failure.