Tag: sequential game

The round robin group stage of the World Cup was recently completed with 16 out of the 32 countries eliminated from the competition – including England, Italy and Spain. The remaining 16 countries progressed to the single game elimination section of the tournament. At the time of writing, the first round of elimination games had been completed with the remaining 8 teams proceeding to the quarter finals of the tournament. Two of these 8 elimination games ended as a draw after extra time. The winner was decided by a penalty shoot-out e.g. Brazil and Costa Rica. Are these shoot-outs just a lottery or are there any factors that significantly influence their outcome?

The penalty shoot-out was first introduced in June 1970 and has become an important part of competitions such as the World Cup and European Championships for national teams and The Champions League, UEFA Cup and FA Cup for club teams. English fans have suffered more than most with victory in only one out of the seven penalty-shoot outs they have been involved in at major tournaments. On average only three out of every five penalties taken were scored. Germany has a very different record. They have won six out of the seven shoot-outs they have participated in and have a scoring rate of 93%. The Czech Republic has an even better record as their players have not missed a single penalty in the three shoot-outs they have been involved in – including beating West Germany in 1976.

Each individual penalty can be thought of as an example of an interdependent or game theoretic situation. The penalty taker (PT) has to choose from one of three different strategies: shoot to the right, shoot to the left or shoot down the middle. The success of the penalty does not just depend on which of these strategies is chosen. It also depends on the choice made by the goalkeeper (GK) i.e. dive to the left, dive to the right or stay where they are.

In the jargon of game theory there is strategic interdependence. It can also be thought of as an example of a simultaneous game. After the ball is struck it takes approximately 0.3 seconds until it hits the back of the net!! Therefore it is impossible for the GK to observe the shot and respond. Instead they simply have to guess which way they think the PT will kick the ball and respond accordingly. The same reasoning applies to the PT. They cannot observe which way the keeper will dive before they strike the ball. A penalty shoot-out is also an example of a zero sum game. If one teams scores they are better off by one goal while the other team is worse off by one goal.

There is also a sequential element to the shoot- outs as in each round one team always follows another. Is there either a first or second mover advantage? Is there any advantage from always shooting first or second? This was a question investigated by some economists who analysed the data from 129 shoot-outs in ten different tournaments taken between June 1970 and June 2003. This cut-off was chosen because up until this point it could be argued that a penalty shoot-out was an example of a truly randomized field experiment. The team that won the coin toss was required to shoot first. Teams were not given a choice of whether to shoot first or second until the rules were amended in June 2003.

The economists found that the teams who took the first shot won in 78 (60.5%) cases while the team that shot second won in only 51 cases (39.5%). This evidence suggests that there is a significant first mover advantage. One explanation for this finding is that there is greater psychological pressure on the PTs who go second in each round of the shoot-off and this has a significantly negative effect on their performance. The researchers also found that in 19 of the 20 shoot-outs they observed after June 2003 the team that won the toss decided to kick first. They concluded that not only is there a first mover advantage, but that teams/players are aware of it.

If there is currently a first mover advantage which provides teams with an unfair advantage then is there anything that the football authorities could do to help reduce the bias? One suggestion is to change the order in which the teams shoot in each round. A similar approach could be taken to that used in tennis in order to determine the order of the server in a tie break.

Imagine a penalty shoot-out between England and Germany. The sequence below provides one possible alternative to the current structure of the contest.

Penalty 1: Germany England
Penalty 2: England Germany
Penalty 3: England Germany
Penalty 4: Germany England
Penalty 5: England Germany
Penalty 6: Germany England

This would involve increasing the number of penalties from 5 to 6 so that both teams get to shoot first in three rounds of the contest. Interestingly the authors also found any first mover advantages fell dramatically if the shoot-outs reached the sudden death stage.

It will be interesting to see if first mover advantages occur in the remaining games in the tournament.

The English Disease – How to handle pressure: lessons from penalty shoot-outs The Economist (14/6/14)
Penalty kick shootouts and the importance of shooting first Soccermetrics Research (3/1/11)
Game Theory Lesson: Man Utd v Chelsea Penalty Shootout Econfix (11/3/14)
World Cup Game Theory – What economics tells us about penalty kicks Slate (24/6/06)
Football penalty shoot-outs are unfair says new research LSE (16/12/10).

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