Penalty – A Game of Choices
The phenomenon of penalty kick is a matter of choice – both for the penalty taker and the goalkeeper. The penalty taker needs to decide whether to shoot left, right or down the middle while the goalkeeper chooses whether to dive left, right or stay put in the center of goal. Can these choices fit a game theory model? Indranath Mukherjee explores the possibility by looking at an empirical study.
The phenomenon of penalty kick is a matter of choice – both for the penalty taker and the goalkeeper. The penalty taker needs to decide whether to shoot left, right or down the middle while the goalkeeper chooses whether to dive left, right or stay put in the center of goal. Can these choices fit a game theory model? Indranath Mukherjee explores the possibility by looking at an empirical study.Prologue
Football proposes a game involving a ball and eleven players in different positions. But the point to ponder here is that among the different positions for the eleven players on the pitch, which is the most intriguing one. The sheer dominance of two particular gentlemen in the Ballon d’Or award in the last nine years seem to suggest that goal scorers are the obvious answer to our question. The answer, however, is not that simple.
Friend and fellow writer at Goalden Times, Trinankur Banerjee, a student of Film Studies, once wrote, “When I was beginning to wander in the realm of cinema and was at an age when more than the game, football became an alternative narrative of history and philosophy, the act of goalkeeping pondered me the most. Truly, do you know who is a goalkeeper? Before one jumps onto the obvious, have you given it a thought? Are goalkeepers footballers? Or are they distant observers, aliens to the centre stage, the ideal postmodern persona of sports? Is he the Joseph K. of Football, who is always held at an irrational trial after every game?”
He went on to refer to Wim Wender’s existential masterpiece, “The Goalie’s Anxiety at the Penalty Kick” where a goalkeeper engages in some trivial activities with people on the sidelines and concedes a goal. Then, he gets himself sent off as he pushes the referee out of sheer wrath and there ends the opening sequence. The moment it ended, Trinankur had to pause and wonder, is this not the best summary of the question, “who is a goalkeeper?”
Left or Right or Left
This is indeed an intriguing question. However, during a penalty, the goalkeeper is always an active party to the game and is under the perennial dilemma of whether to dive — or not to dive. Whatever may be written about the great penalties and the goalkeepers’ geniuses, the history of penalties is actually a history of choices. And the choice is not just for the goalkeeper but for the player taking the penalty as well. In the 1994 FA Cup final against Chelsea, Eric Cantona scored twice from the spot in Manchester United’s 4-0 win. In the post-match press conference when a journalist asked Cantona about his thought before taking the second penalty kick, he responded: “Well, I first thought that I would put it in the other corner this time. But I figured the goalkeeper might know this, so I then thought about putting it in the same corner. But then again, I thought that he probably thought that I would think like this, so I decided to change again to the other corner. Then I figured that he probably thought that I was thinking that he would probably think that I would think like that, so …” He had paused, probably realizing that he is on an endless and circular loop of reasoning, and then said: “You know what? The truth is that I just kicked it.”2
Number of regular penalty takers went on record saying that even after starting their run –up to the ball to take the penalty kick, they are not sure which side they will kick the ball. I distinctly remember being stressed while thinking which side Lionel Messi would choose when he started walking up to take the first penalty kick for Argentina in the 2016 Copa América Centenario final. In the 2015 final, he took a low shot to his right and scored. Claudio Bravo, the Chilean goalkeeper, had played with Messi at Barcelona for two years by then and probably knew at least a little about Messi’s penalty-taking thought process. For a brief moment, I thought of the scenario as a classic case of game theory strategy between the two players involved. Messi went on to shoot to his right again but this time hit the ball over the crossbar, and Argentina went on to lose their third final in three years. Whether Lionel Messi will go down in the pages of football history as the greatest ever not to have won a national championship with the senior team can be debated, his Argentina team will surely not be considered as the best team to not to win a World Cup final. Hungary of 1954 is likely to be that team ahead of Holland of 1974 and the 1954 final had no penalties. Another Hungarian, John von Neumann, who died in 1957 at the age of 53 years, may not have had any interest in football, he was surely aware of his national team’s popularity and success. It will never be known whether the phenomenon of penalty kick was in his mind while working on his famous Minimax theorem3.
The Hypothesis
In his article titled “Zur Theorie der Gesellschaftsspiele” published in 1928, von Neumann proved the Minimax theorem, which essentially says that in zero-sum games with perfect information (i.e. in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses. When examining every possible strategy, a player must consider all the possible responses of the other player. The player then plays out the strategy that will result in the minimization of his maximum loss.3
Although empirical verification of strategic models of behaviour is often very difficult, the phenomenon of the penalty kick is a good candidate to test the implications of Minimax theorem in a real-life setting. The pay-off in a penalty kick is always constant-sum; the pay-off of the shot taker is exactly the negative of the pay-off of the goalkeeper. This is a case of pure conflict with no common interest between the players. Ignacio Palacios-Huerta, professor of management, economics and strategy at London School of Economics and head of talent identification at Athletic Club de Bilbao, did an extensive study which has been used in recent years to advise a number of professional football teams.2
A formal model of penalty kick may be written as follows. Players’ payoffs during the event of a penalty kick are the probabilities of success – goal for the penalty taker and no goal for the goalkeeper. The shot taker has the option of shooting either on the left or to the right. Similarly, the goalkeeper also has the option of diving to the left or to the right. Of course, there exists a third option for both the shot taker and the goalkeeper which is to shoot in the centre or stand in the middle but as we will see later in the data that those choices are statistically not significant enough and hence we keep the options as L and R in the simple model. There is another obvious dimension to any penalty kick – the height of the shot, which has been kept outside the scope of discussion for now.
Π here denotes a particular event; πLR stands for the event when the penalty taker shoots towards the left and the goalkeeper moves towards the right. It might appear that the goalkeeper has gone nuts as he is moving to the opposite direction of the shot. But one has to remember that these choices – left or right – are simultaneously taken by both the shot taker as well as the goalkeeper. It has been observed that the ball takes about 0.3 seconds to travel between the penalty spot to the goal line and this does not provide the goalkeeper with enough time to first see the shot, react, move towards the direction of the shot and complete a successful save. This means that both the shot taker and the goalkeeper must choose their strategies simultaneously. The outcome of the game is observable and in effect is decided within about 0.3 seconds after the involved players choose their strategies. Professor Palacios-Huerta has shown that the penalty kick game has a unique state of equilibrium (Nash equilibrium4) in mixed strategies when
πLR > πLL < πRL
πRL > πRR < πLR
In the above model of penalty kick, the equilibrium requires both the shot taker and the goalkeeper to use a mixed strategy. In this case, the equilibrium yields two sharp testable predictions about the behaviour of the shot taker and the goalkeeper:
- Success probabilities – the probability that a goal will be scored (not scored) for the shot taker (goalkeeper) should be the same across strategies for each player.
- Each player’s choices must be serially independent given constant payoffs across penalty kicks. This means that players must be concerned only with instantaneous payoffs and there are no intertemporal links between penalty kicks – the choices should be memoryless. In other words, the choice in one particular penalty kick must not depend on one’s own previous choices or the opponent’s previous choices or any other previous actions.
What this means is fairly simple; as the shot taker (or goalkeeper), it would be disadvantageous for you to let your opponent know your actual choice in advance. So it’s optimal for each player to choose a strategy randomly from the available pure strategies but the proportion in the mix should be such that one player cannot exploit the opponent’s choice by pursuing any particular pure strategy from all the strategies available to him or her.
Numbers and Crunching
Professor Palacios-Huerta used classical hypothesis technique and real data to test if the two hypotheses mentioned above can be rejected2. The data collected were from the leagues in Spain, England, Italy and some international games from the period between September 1995 and June 2012. Data for a total of 9,017 penalty kick records were used and the following data points were considered: name of the players (both the shot taker and the goalkeeper), name of the teams involved in the match, the date of the match, the choices made by the players (left, right or centre), time in the game when the penalty-kick was shot, the score of the match at that time, final score of the match and outcome of the penalty kick- goal or no goal (no goal included both save by the goalkeeper and penalties shot wide or to the post or to the cross-bar), each in separate categories. The data also include the shot takers kicking leg – left or right. Most of the shot takers in the sample are right-footed – as in real life. One immediate observation from the data is that right-footed shot takers shoot more often to the right-hand side of the goalkeeper and left-footed kickers shoot more often to the left hand side bowing to their basic anatomical inclinations. In the data set for example, Messi who is naturally left-footed takes about 62% of his penalties to the left side while close to 63% of Cristiano Ronaldo’s penalties are taken to the right side.
To deal with the naturally right and left-footed issue, Professor Palacios-Huerta ‘normalized’ the strategies of the game and instead of calling right and left; he renamed the choice as natural side (denoted by R) and non-natural side (L). Thus, a right-footed shot taker shooting to the goalkeeper’s right is denoted by R and the same notation is being used for a left-footed shot taker shooting to the left. For the goalkeeper, the shot taker’s natural or stronger side is R and the non-natural or the weaker side is L. So both Messi and Ronaldo’s number shows about 62% in their natural side in Table 1.
In the data set, 80.07% all the penalties resulted in goal. The rate of scoring is close to 100% when the goalkeeper’s choice doesn’t coincide with the shot taker’s choice and close to 60% when they coincide. In about half of all the penalties in the dataset, the goalkeeper’s choice coincided with the shot takers choice, mostly RR (30.5%), 16.7% are LL and 0.9% are CC. Shot takers rarely kick to the centre (6.8% in the data) and the goalkeeper’s choice of C is even rarer (3.5% in the data) perhaps because they cover part of the centre to some extent by their legs when choose R or L. The percentage of kicks where the choices don’t coincide are more or less equally divided between LR (21.6%) and RL (21.7%). Let’s now take a look at the tests of the implications of the Minimax theorem in the penalty kick game.
The Revelation
Table 1: Pearson and Runs Tests for 20 key penalty takers and goalkeepers2
Note: ** and * denote rejections at the 5% and 10% significance levels, respectively.
The table above lists the penalty taking statistics for 40 key players, top 20 of them being penalty takers and the bottom 20 being goalkeepers.
Professor Palacios-Huerta implemented standard proportions tests to test the null hypothesis that the scoring probabilities for a player (penalty taker or goalkeeper) are identical across strategies by using Pearson’s chi-squared (χ2) goodness of fit test of equality of two distributions. Put simply, he went on to verify if the actual conversion rate of penalties across strategies are identical subject to a theoretically pre-defined limit, known as p-value in the world of statistics. Thus one can reject the null hypothesis (that scoring probabilities across strategies are identical) when the p-value is less than a pre-determined significance level.
From the Pearson’s chi-squared (χ2) goodness of fit test results the null hypothesis of equal probabilities cannot be rejected for most of the players. It is rejected for just two players (David Villa and Frank Lampard) at 5% level of significance and four players (Iker Casillas and Morgan De Sanctis joins the list) at 10% level of significance. If we study deeply the way these players typically took their penalties or attempt to save them, we will probably understand why for certain players the hypothesis is rejected. For example, Frank Lampard’s technique was very different when he was up against an English goalkeeper compared to the time faced against other goalkeepers. Iker Casillas hardly stays in the middle and chose his natural side more than 50% of times against 59 penalties that he had faced until 2010 World Cup final, allowing his opponents a better chance of scoring when they chose Iker’s non-natural side.
Hence the hypothesis that scoring probabilities are identical across strategies cannot be rejected at the individual level for most players at conventional levels of significance (at 5% and 10% level as stated above). In fact, the number of rejections is identical to the theoretical predictions.
The second hypothesis was tested by applying the standard “runs test”. This is an application of the results of Gibbons and Chakraborti (1991)5. To explain this simply, let us consider the sequence of strategies chosen by a player in the order in which they occurred s = {s1, s2, …, sn} where every s will take a value of either L or R and n = nR + nL are the number of natural (nR) and non-natural side (nL) choices opted by the player. Further r (in the table) denotes the total number of consecutive identical strategies in the sequences. As for the hypothesis of serial independence, the runs test show that the hypothesis is rejected just for three players (Villa, Alvaro Negredo and Edwin van der Sar) at 5% significance level and only Jens Lehmann joins the list if we increase the level of significance to 10%. Majority of the players neither appear to switch strategies too often or too infrequently.
Curious readers may refer to Professor Palacios-Huerta’s work given in the reference section for further technical details and more insights into the results2. These tests and the results have been used to advise a number of football clubs and national teams in recent years. The most famous client being Chelsea Football Club in the UEFA Champions League final in 2008, the story of which has been narrated in details in the book Soccernomics (2012) by Simon Kuper and Stefan Szymanski6. Let’s just recall a part of the story here:
So far, the advice (of the tests) had worked very well for Chelsea (The right-footed penalty takers had obeyed it to the letter, Manchester United’s goalkeeper Edwin van der Sar had not saved a single penalty, and Chelsea’s keeper had saved Cristiano Ronaldo’s) …As Nicolas Anelka prepared to take Chelsea’s seventh penalty, the gangling keeper, standing on the goal-line, extended his arms to either side of him. Then, in what must have been a chilling moment for Anelka, the Dutchman (Van der Sar) pointed with his left to the left corner. “That’s where you’re all putting it, isn’t it?” he seemed to be saying. 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. What was Anelka to do? We all know that end. Kuper and Szymanski summarized the end as “Anelka’s decision to ignore the advice (of the tests) probably cost Chelsea the Champions League”. So that is the story of 2008 UCL final beyond John Terry’s slip.
References:
- Gonzalez E. Historias del Calcio: Una cronica de italia a traves del futbol. Barcelona: RBA Libros; 2007.
- Palacios-Huerta I. Beautiful game theory: how soccer can help economics. Princeton: Princeton University Press; 2014.
- Neumann JV. Zur Theorie der Gesellschaftsspiele. Math. Annalen. 1928;100:295–320.
- Nash JF. Equilibrium Points in N-Person Games. Proc Natl Acad Sci U S A. 1950;36(1):48-49.
- Chakraborti S, and Gibbons JD. One-Sided Nonparametric Comparison of Treatments with a Standard in the One-Way Layout. J Qual Technol. 1991;23(2):102-106.
- Kuper S, and Szymanski S. Soccernomics: why transfers fail, why Spain rule the world and other curious football phenomena explained. London: HarperSport; 2012.
