Does player specialization predict player actions? Evidence from penalty kicks at FIFA World Cup and UEFA Euro Cup

Abstract

Penalty kicks are analysed in the literature as ‘real life experiments’ for assessing the use of rational mixed strategies by professional players. However, each penalty kick cannot be considered a repetition of the same event because of the varying background conditions, in particular the heterogeneous ability of different players. Consequently, aggregate statistics over data sets composed of a large number of penalty kicks mediate the behaviour of the players in different games, and the properties of optimal mixed strategies cannot be tested directly because of aggregation bias. In this article, we model the heterogeneous ability of players. We then test the hypothesis that differently talented players randomize over different actions. To achieve this aim, we study a data set that collects penalties kicked during shoot-out series in the last editions of FIFA World Cup and UEFA Euro Cup (1994–2012) where kickers are categorized as specialists and non-specialists. The results support our theoretical predictions.

Keywords:

• mixed strategies
• penalty kicks
• aggregation bias
• players’ specialization
• action predictability

JEL Classification:

• C72
• C93
• L83

Notes

¹ A penalty shoot-out is a method used to decide which team progresses to the next stage of a tournament (or wins the tournament) following a tied game. In a nutshell, the shoot-out series is composed of five penalties for every team, but every single penalty must be kicked by a different player, so that even non-penalty-specialists are involved. The team that scores more penalties obviously wins. Details concerning specific rules and the history of shoot-outs can be found at en.wikipedia.org/wiki/Penalty_shoot-out_(association_football)

² Shoot-out penalties are kicked at the end of the match and every single kick is obviously crucial. On the contrary, with the relevant exception of the paper by Apesteguia and Palacios-Huerta (2010) discussed in section ‘Kicker’s specialization’, the data sets used in the literature refer to penalties kicked during national league matches. These penalties are usually kicked by the team specialist and can be assigned at any moment of the match. The outcome of these penalties can be more or less crucial, depending on the score and the time remaining to the end of the match.

³ A kick out of the goal is an example of such a failure, even if often reveals at least the broad intended shot direction. The same is less easy to detect for centre kicks, which could be ‘failed’ left or right kicks.

⁴ Chiappori et al. and the following literature therefore choose to consider only the ‘horizontal direction’ of the kick (or jump), deliberately ignoring other decisions, such as the vertical direction of the ball or the power of the kick. The only empirical analysis taking into account the vertical direction of a penalty kick is provided by Bar-Eli and Azar (2009). This choice can be easily understood considering that during data collection (data are usually obtained by direct inspection of the videos of soccer matches), it is difficult to identify vertical directions and almost impossible to measure the power of the kick.

⁵ Palacios-Huerta (2003, footnote 11) decides to drop the pure strategy of centre kicks/jumps because of the small proportion observed in his empirical analysis. The same applies in Baumann et al. (2011). Leininger and Ockenfels (2008), on the contrary, assign a relevant strategic role to central kicks. They observe an increase of the scoring probability in the German Football League (Bundesliga) in the period 1963–1990, and they attribute this phenomenon to a behavioural innovation that took place when kickers, during the 1970s, recognized the strategic value of occasionally kicking to the centre. On the empirical side, the decision to limit the action set of the players to just two pure strategies eliminates any ambiguity that could be introduced when defining the width of the area of the goal to be considered ‘central’.

⁶ As said, the literature usually assumes that no one player is able to anticipate the other by perceiving some clue about his choice, so that the game is indeed simultaneous; moreover, it is assumed that the game is one-shot, in that the memory of past repetitions does not induce dynamic strategies. The empirical evidence confirms these assumptions (Miller, 1998; Chiappori et al., 2002; Palacios-Huerta, 2003).

⁷ Apesteguia and Palacios-Huerta (2010) analyse a large set of penalty shoot-outs in order to understand the effect of psychological pressure on the performance of the players. Their main findings are discussed in section ‘Kicker’s specialization’. Dohmen (2008) documents that penalty kickers are more likely to choke on a penalty kick when the match takes place in their home stadium.

⁸ To be more precise, it is easy to prove that the ranking of average pay-offs respect the expected ranking for every single observation if Ps and s differ only by a constant among players. The same applies for the Fundamental Lemma.

⁹ Actually, another assumption is needed: must be lower than for every player. More properties characterizing the mixed strategy Nash equilibrium can be derived under the hypothesis that heterogeneity regards only one of the two players.

¹⁰ Bar-Eli et al. (2007), however, discuss possible nonoptimal behaviour of goalkeepers driven by ‘inaction aversion’.

¹¹ Chiappori et al. (Prop. 1) find that a ‘restricted randomization’ is possible when the scoring probability of kicking central is sufficiently low. In this case, the kicker never kicks to the centre and the goalkeeper never remains at the centre. The authors do not provide any empirical test about this prediction because of the small number of central kicks in their data set.

¹² The shoot-out series take place only at the end of play-off matches that end in a tie. It must be remarked that there need not to be 10 penalties in each series. Teams take turns to kick until each has taken five kicks. However, if one side scores more goals than the other could possibly reach with all of their remaining kicks, the shoot-out ends regardless of the number of kicks remaining. On the other hand, if the five penalty series end in a tie, further penalties are kicked until one team scores and the other team does not.

¹³ Consequently, we have assigned a left or right side even to kicks that in other papers would have been classified as ‘central’. Thus, we avoid a discretionary width of the central area of the goal to classify kicks as central. In some cases, it is difficult to determine the precise position of the penalty, but this difficulty would have been the same (or worse) with other conventions.

¹⁴ We were not able to find the video for 8 out of 12 penalties of the match Czech Rep. vs. France at the UEFA Euro Cup 1996. We know that all of them were scored and the identity of the players, but we cannot classify the underlying actions. Additionally, the videos of nine other penalties were taken from an unfortunate perspective which made it impossible to identify the half of the goal area where they were directed. These penalties were obviously rather ‘central’.

¹⁵ In particular, we are thinking of the heterogeneity of kickers, because in a shoot-out series the goalkeeper remains the same (i.e. they are all specialists), while there must be at least five kickers for every team, so that even nonspecialists must kick.

¹⁶ The mean difference is significant () only for the sample of Palacios-Huerta (2003).

¹⁷ Penalty statistics have been assembled from various public databases, in particular http://www.transfermarkt.de (for national team penalties) and http://www.worldfootball.net (for club penalties). We were able to obtain complete information concerning all penalty takers in the national team, while in 34 (out of 237) cases penalty scorers in the club were not available in the whole four-year period. Notice that available statistics include only penalties scored (not kicked) during league matches, thus introducing a (negligible) endogenity issue. In all these cases, we implicitly took a conservative approach, by assuming that no penalties have been kicked in absence of the information. In this sense, the category of nonspecialists could include some specialist (and not vice versa). We also tried different alternative specifications for this variable, taking into consideration the general goal-scoring attitude of the players or collecting the penalties kicked during other tournaments (in particular, the continental club cups). Results are not significantly different; for a better illustration of the methodology used for building the specialization variable, an example can be useful. Germany vs. Argentina on 30 June 2006, the quarter-final of the FIFA World Cup, ended after a shoot-out. Eight penalties were kicked in the series, all scored for Germany, only two scored for Argentina. Germany consequently went through to the semi-finals. The kickers for Germany were Ballack, Neuville, Podolski and Borowski. The kickers for Argentina were Cruz, Rodriguez, Cambiasso and Ayala. The observation period for assessing the specialization of the kickers is, as said, July 2004–June 2008. During this period, only Ballack – out of the eight kickers – scored at least one penalty (three, in 2005, excluding shoot-outs) for his national team. During the same interval, he also scored one penalty kick for his club team; Bayern München, in the 2004/05 season. Cruz during the same period never scored a penalty for Argentina, but he scored five penalties for his club team, Inter Milan (one, three and one, respectively, during the 2004/05, 2005/06 and 2007/08 seasons). Podolski, Neuville and Rodriguez never scored a penalty for their national team, but they scored, respectively, eight, eight and two penalty kicks for their club teams. All of these players were consequently classified as ‘specialists’, whereas Borowski, Cambiasso and Ayala never scored a penalty kick for their national or club team during those four years and were then considered as ‘nonspecialists’.

¹⁸ Specialized players could be used early in order to obtain a greater chance to put pressure on the rivals, but it could also be rational to exploit their talent in the (possible) more ‘important’ final rounds.

¹⁹ We thank a referee for attracting our attention to this fact.

²⁰ The mean difference is actually significant at the 1% level only when the team of the kicker is ahead in the score.

²¹ A safe kick, for example, has limited power and angle, but it is substantially always directed within the goal: if the goalkeeper correctly guesses the side, the probability of saving the goal is very high. On the contrary, a difficult kick can more easily miss the goal or hit the post/bar; if not, it is quite difficult to be blocked even if the goalkeeper jumps on the right side.

²² The characterizations of mixed strategy equilibria in the general case are available upon request.

²³ Because we do not claim to be able to empirically distinguish between difficult and safe kicks, a direct confirmation to our reasonable assumption is limited to _s. In this case, consider that in our entire data set, when the goalkeeper was wrong-footed, he never saved the ball.

²⁴ If no side advantage is in place, then and because .

²⁵ See again footnote 8. This difference is actually not significant at the 10% level.

²⁶ Based on Pearson’s goodness-of-fit test of equality of two distributions (see Palacios-Huerta (2003, p. 403) for details).

²⁷ Actually, those properties are verified for profiles I and III, while profile II predicts equal number of shots and jumps on natural and opposite side (see Table 8).

²⁸ However, the difference is significant () only for the action profile (O, O).

²⁹ See again footnote 26. The hypothesis that the scoring probability is identical across strategies for the kickers and the goalkeeper cannot be rejected at the 10% significance level.

³⁰ Imagine a very talented kicker who never fails the goal area even when shooting with great power and angle, especially on the natural side. This kicker is characterized by and and will always kick difficult on the natural side; thus, his action will be easily predicted by the goalkeeper. This correct prediction will be of no use for the goalkeeper because the scoring frequency will be close to one.