In the recent edition of econoMAX (online magazine of Tutor2u) I wrote a piece on the game theory of penalty kicks in soccer. Below is an extract from it and an example of game theory in action from the 2008 UEFA Champions League Final between Manchester Utd and Chelsea.
As we approach the business end of the football season in Europe and with the potential impact of penalty kicks deciding matches, it might be appropriate to consider the relevance of game theory – economists hold in high regard the penalty kick as a real-life example of game theory. 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).
From the data collected by Basque economist Ignacio Palacios-Huerta he calculated the proportion of successful penalty kicks. It 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 (see right). 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.
2008 UEFA Champions League final – Chelsea v Manchester Utd.
If you have read Soccernomics you will be well aware of the events that unfolded in this game. Ignacio had been recording how penalties were being taken and wrote an academic paper on strategies that players and goalkeepers employed. A mutal 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 (ie 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, does 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.
Man Utd’s Rio Ferdinand won the toss and went first – not a good omen for Chelsea. Looking at the penalties and relating it to Ignacio’s research we see the following:
1. Ballack – OS – left. Van der Sar dives left – GOAL
2. Belletti – OS – left. Van der Sar dives right – GOAL
3. Lampard – OS – left. Van der Sar dives right – GOAL
4. Cole – NS – left. Van der Sar dives left (as advised by Ignacio ball hit hard and low) – GOAL
5. Terry – OS – left. Van der Sar dives right – NO GOAL – hit the post
6. Kalou – OS – left. Van der Sar dives right – GOAL
Up to this point all Chelsea right footed players had taken on the advice of Ignacio and hit to their opposite side – Van der Sar’s left.
3. Ronaldo – paused in his run-up. Petr Cech stayed upright for as long as possible and dives right – NO GOAL – saved.
It seemed that Chelsea’s strategy of going to Van der Sar’s left had been hatched by someone on the Utd bench. As Anelka prepared to take Chelsea’s 7th penalty Van der Sar pointed to the left corner. Now Anelka had a terrible dilemma. This was game theory in its rawest form. So Anelka knew that Van der Sar knew that Anelka knew that Van der Sar tended to dive right against right footers. Instead Anelka kicked right but it was at mid-height which Ignacio warned against. – Soccernomics Page 127
7. Anelka – NS – right. Van der Sar dives right – NO GOAL – saved
If Anelka had taken Ignacio’s advice would Chelsea have won? Below you can see the drama unfold on YouTube.