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Game Theory Lesson: Man Utd v Chelsea Penalty Shootout

March 11, 2014 2 comments

Game theory involves studying the alternative strategies a person may choose to adopt depending on their assumptions about their rivals’ behaviour. The most significant research into penalty kicks has been done by Steven Levitt (co-author of Freakanomics) who co-authored a paper on mixed strategies when players are diverse in their decision making and studied 459 penalties in the Spanish and Italian leagues. Another economist Ignacio Palacios-Huerta analysed 1417 penalty kicks from several European countries during the period 1995-2000.

Technically the kicker and the goalkeeper play a zero-sum game – any gain for one player is exactly offset by the loss to the other side – plus one goal for me is minus one goal for you. The situation that kickers face in a penalty kick is a simultaneous-move game where they have three alternative strategies: shooting right, left, or centre. Similarly the goalkeeper also has three alternative strategies: dive to the right, dive to the left or remaining in the centre of the goal. In defining the sides of the goal researchers use the “natural side” of the kicker (which is the goalkeeper’s right, if the kicker is right-footed, and the goalkeeper’s left, if the kicker is left-footed) and the “opposite side”. Labeled like that, the strategies of both kicker and goalkeeper will be to choose the natural side of the kicker (NS), the centre (C) or the opposite side (OS).

2008 UEFA Champions League final – Chelsea v Manchester Utd.

A mutual friend of Ignacio and Chelsea manager, Avram Grant, brought the two men together and subsequently Ignacio sent Grant some facts regarding Man Utd, in particular about their goalkeeper Van der Sar. There were 4 main points:


1. Man Utd goalkeeper (Van der Sar) tended to dive to the kicker’s natural side (i.e. GK’s right for a right footed kicker)

2. Van der Sar tends to save penalties that are hit at mid-height

3. Man Utd midfielder Cristiano Ronaldo often stops in his run-up and if he does the ball is kicked towards the right hand side of the keeper. It was important that the Chelsea goalkeeper, Petr Cech, did not move early. When goalkeepers moved early Ronaldo always scored.

4. If you win the toss you take the first penalty. 60% of teams going first win the game.

Students watch the penalty shootout between Manchester United and Chelsea and complete the following table. They have to record where the penalty taker placed the ball and which way the goalkeeper dived – NS C OS. They then answer the questions below.

Man U Chelsea Penalty Shootout

What do you notice about the Chelsea penalty takers?

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Was Ignacio correct about Cristiano Ronaldo?

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How did Man Utd goalkeeper (Van der Sar) psych out Anelka the 7th penalty taker for Chelsea? Watch the video closely.

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What did Anelka do which went against the advice of Ignacio?

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Ignacio stated in an academic paper that generally the shooter should kick to the left 38% of the time and right 62% of the time. The keeper should dive to the shooter’s left 42% of the time and to the shooter’s right 58% of the time. When both strategies are played out, the shooter will score 80% of the time. This is the best strategy for both players because any deviation from the strategy by the keeper will result in the success rate increasing while any deviation from the strategy by the shooter will result in the success rate decreasing.

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Chelsea win Champions League – Petr Cech picks the correct side everytime

May 23, 2012 Leave a comment

With the disapointment of the 2008 Champions League Final behind them, Chelsea can finally claim to be the best team in Europe. As in 2008 it was decided by another penalty shoot but this time Chelsea were the victors. As always I am particularly interested in the peanlty shoot out and the strategies, if any, that are implemented by both the penalty taker and the goalkeeper.

The situation that kickers face in a penalty kick is a simultaneous-move game where they have three alternative strategies: shooting right, left, or centre. Similarly the goalkeeper also has three alternative strategies: dive to the right, dive to the left or remaining in the centre of the goal. In defining the sides of the goal researchers use the “natural side” of the kicker (which is the goalkeeper’s right, if the kicker is right-footed, and the goalkeeper’s left, if the kicker is left-footed) and the “opposite side”. Labeled like that, the strategies of both kicker and goalkeeper will be to choose the natural side of the kicker (NS), the centre (C) or the opposite side (OS).

From data compiled by Spanish economist Ignacio Palacios-Huerta he calculated the proportion of successful penalty kicks. Below is a table that shows the success rate of penalty takers when they went to their natural side and opposite side when the goalkeeper went his natural side and opposite side. Notice that when the kicker went NS and goalkeeper OS the success rates was 95% – the remaining 5% missed the target. Similarly when the kicker went OS and goalkeeper went NS – 8% missed the target.

Here is data on the peanlty kicks from last Sunday’s Champions League final

 

 

 

 

 

 

 

 

 

Notice that Chelsea keeper Petr Cech went the correct way for all penalties (including the one taken during normal time) – seemingly on the plane over he studied all Bayern penalties since 2007. He said it took nearly 2 hours – did he have a strategy?

Also Chelsea penalty takers favoured the natural side whilst Bayern the opposite side. Below is the shoot out once again.

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