Advantages and High Stakes
Master Degree Thesis
Managerial Economics and Strategy
University of Amsterdam
Antonio Trucillo
11923210
Abstract
Game theory suggests that players can reach a Nash Equilibrium by randomizing their choices. However, we do not know how this randomization changes in the moment that players face an advantage by making a specific move. The aim of this research is to study the penalty kicks from various European and World Championships, making a total of 395 kicks. In addition, to have more insights from the results, we will run a framed laboratory experiment by setting up a game similar to penalty kicks. The aim of this experiment is to see if players are consistent with their strategies in the moment that they can have an advantage by making a certain decision. In the end, we will see that players tend to be more unpredictable in the moment that they are facing higher stakes. Moreover, we will investigate how players behave in the moment that they have to make a decision in a one-shot game in which they have high stakes at play. Besides observing that the players follow the theoretical predictions, we will observe that the player that change strategy is not the one who has the advantage, but its opponent. Both the experiments will be compared to the related theories.
Keywords: game theory, advantages, high stakes, penalty kicks, shoot-outs. Supervisor: Jeroen van de Ven
Statement of Originality
This document is written by the student Antonio Trucillo who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
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i. Introduction
Sometimes a game might not present an optimal and univocal winning strategy. Players, by looking at all the possible choices, can feel indifference between them and can decide to randomize their moves. By following this strategy (called mixed strategy), players can reach a mixed strategy Nash equilibrium.
The empirical results of these games are quite ambiguous. O’Neill (1987), Mookherjee and Sopher (1994) and McCabe et al. (2000) state that players can intuitively reach an equilibrium, as theoretically predicted, by playing the game repeated times, while Brown and Rosenthal (1990) give an opposite opinion about it. There are many real-life situations in where people can randomize their choices: tax authorities can randomize which citizen to inspect while taxpayers can randomize the decision to not pay their taxes; a student has to decide whether to cheat or not in an exam and the teacher has to decide whether to supervise the class or not.
Most of the empirical results come from experiments where the players can win relatively low, and usually constant among the rounds, payoffs. Moreover, the main assumption of these games is that the player is indifferent on its set of choices. There are several real-life examples where players do not have a unique dominant strategy, they have a specific preference in their moves and the potential payoffs are quite high. Penalty kicks are a common example. Chiappori et al. (2002) found out that in penalty kicks the football players do not perfectly randomize between the three choices (left, centre and right), but they tend to prefer to just randomize between left and right, thus denoting a preference in their moves. A football kicker can have a preferred leg and shooting side and might decide to shoot his penalty kick to the side where he feels stronger. However, the goalkeeper can anticipate this move and make his jumping decision according to this belief.
How do strategies change when players have an advantage in a specific move and there are high stakes at play? The main hypothesis is that a player, in a high stakes one-shot game like the penalty shoot-outs, is more likely to use his personal advantage to the detriment of his unpredictability. To answer this question, we will divide this research in two parts. Firstly, we will analyse the penalty kicks from various seasons of the FIFA World Cup, the UEFA European Cup, the UEFA Champions League, the Qualifications to the UEFA Champions League and the UEFA Europa League. Then, we will analyse a framed laboratory experiment run in an Italian high school and finally we will compare the results.
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In the results, we will have to reject our hypothesis. The players tend to not use their personal advantage when they face high stakes. In the next sections we will try to give an interpretation to these results and to give some possible explanations.
The aim of this research is to contribute to the fields of game theory and sport economics, by providing a different point of view regarding the concept of advantages and payoffs and how does it influence the strategy of a player. The results can also be useful for football trainers and football strategist, so that they can better understand how to plan their strategies for the penalties.
The first part of this paper will illustrate the previous researches regarding this topic and the related theories. Then we will discuss the results from the penalty kicks and we will illustrate the experiment design and results. The last part is dedicated to the result comparison, the discussion of the results and the conclusions.
ii. Literature Review
The literature regarding game theory and mixed strategy is quite ample, while the one specialised in football and, most specifically, in the penalty kicks is limited, mainly due to the specific nature of the topic.
The topic of this research, penalty kicks, is a typical example of mixed strategy. Introduced by John von Neumann (1928), this is widely considered as the birth of game theory. Simplified, the players only have to choose between left or right, randomly, following a distinct binomial process. This process leads to have a minimax strategy, a strategy that regardless to his opponent’s choices will allow him to obtain a certain level of expected utility from the game. In most of the games the minimax strategy corresponds to pure strategies, thus results in no mixing.
In the early 50s’ John Nash introduced the concept of equilibrium and expanded the category of games that can be theoretically predicted, by adding non-zero-sum games and games which involve more than two participants. An equilibrium is a sustainable condition in which each player has no incentive to unilaterally deviate to a different strategy. A mixed strategy equilibrium occurs when each player uses a mixed strategy; if the game only presents mixed strategy equilibria, then we will say that this is a mixed strategy game.
The most famous example of mixed strategy game is the Matching Pennies game. Each player takes a penny and has to choose between showing either heads or tails; each player then shows his choice
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simultaneously. If they make the same choice, Player B gives his penny to Player A. If they make opposite choices, Player A gives his penny to Player B. The game presents only two option and it is zero-sum. Neither choice ensures the victory, as a result each player will choose heads or tails randomly, with 50% probability.
The game can be summarised by the following payoff matrix: Player B
H T
Player A H 1, -1 -1, 1 T -1, 1 1, -1 Payoffs: (Player A, Player B)
When a player chooses heads or tails is just choosing a column (or a row) of the matrix. Each player will aim to be purely unpredictable, thus resulting in two mixed strategies (see the derivation in the Appendix, Part I).
As stated in the introduction, the empirical application of these theories resulted in some ambiguous results. O’ Neill (1987) gave support to the theory, by also stating that the past researches, which criticised the empirical application of the theory, where systematically flawed. By simplifying the specifications of the game, the results got much closer to the theoretical predictions. Most interestingly, he found out also how a repeated game can lead participants to intuitively understand the mechanism of a mixed strategy game and they will behave accordingly. This intuitive learning process finds support also by Erev and Roth (1998) and Mookherjee and Sopher (1994). In the first one, the participants, by taking part in a repeated game, tend to converge their choices to the predicted equilibrium. In the latter, the researchers found out that knowing the past history of moves influences the players’ strategies. This mechanism, called reinforcement learning, is still subject of study, especially in psychology and machine learning.
All these researches, since they were only testing the validity of mixed strategy theory, always followed the standard assumptions of the mixed strategy theory without relaxing any of them, specifically the indifference one. This assumption, which seems to be realistic and helps to simplify the analysis, does not apply to all the games. Moreover, the papers lack a link between theory, laboratory experiments and real-life cases, such as sport games.
In the past few years there have been few researches that links game theory and sport. Walker and Wooder (2001), by analysing classic tennis matches, tested whether the probability of scoring a set is equal between players who serve the ball on the right or left side of the service box. Apesteguia and
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Palacios-Huerta (2010) investigated on the psychologic and logistic factors that can influence the results of a penalty kicks, such as the turn of the team, the type of field, the position in the rankings and other minor factors. Chiappori et al. (2002) and Coloma (2007), who based his research on Chiappori’s model by expanding it, investigated on the penalty kicks settings by developing a specific model that would explain better the outcomes of the game and the strategies adopted by the players. While Apesteguia and Palacios-Huerta’s paper focused on the behavioural side of penalties, Chiappori et al. and Coloma mainly focused on the theoretical analysis of the game. There are several psychological implications that can influence the success of a match but there is no information regarding the role of the high stakes at play and the advantages that each player faces at the beginning of a game, especially in a one-shot game.
Although theory states that players have to randomise, some player can have some specific favourite tactic in their game. A goalkeeper can try to analyse the kicker’s movements and try to predict where he will kick (Savelsbergh et al. 2002), he can use some small deceptive trick to distract the player (Masters et al., 2007; van der Kamp and Masters, 2008) or he can make his choices by analysing the past kicking history of the kicker. Namely, Hans van Breukelen, former goalkeeper of the national Dutch football team, used to have a box of cards with kickers’ statistics and tactics, which he used to consult before each penalty1. These techniques, however, are not fully effective. The International Football Association Board (IFAB), in its guidelines, punishes any deceptive behaviour with a yellow card (IFAB, 2018); some kickers can foil the goalkeeper by opting for a slow shot and, most importantly, the players will decide to randomise their strategies in order to mitigate any predictability (Chiappori et al., 2002).
iii. The natural experiment
Penalties are used to solve two kinds of disputes. Either they can be used to punish a team that committed a foul in the opponent team’s penalty area or, in case of penalty shoot-outs, they can be used to declare the winner of a match (or of a turn, in case of a round-robin tournament) after a tie. They consist in a sequence of penalty kicks. The rules of the game are issued by the IFAB and are described in the Rules of the Game; although they are issued yearly, they rarely undergo huge modifications.
1 Hans van Breukelen: geïnformeerde penaltykiller (2nd November 2015). Retrieved on the 4th July 2018 from
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Penalties are a very interesting and simple game to examine, due to its simple rules and features. The shoot-outs present high payoffs at stakes, that can either be tangible, such as money, or more intangible, such as success and reputation. The penalty kicks have only two players: the kicker and the goalkeeper. The objectives of the two players are unambiguous: the kicker wants to score the goal and the goalkeeper wants to save the ball. Each turn presents exactly the same settings. At approximately 11 meters from the goal line there is the penalty spot, where to place the ball. The typical ball travels at least 100 km/h and reaches the goal in less than 0.4 seconds. This time is much less than the reaction time plus the movement time of the goalkeeper, thus we can assume this game to be a simultaneous game. This assumption is supported by Chiappori et. al. (2002), in which they find out that the hypothesis of simultaneity in choice cannot be rejected. The game presents only two outcomes: score or no score. There are no retakes, unless some specific infraction arises (i.e. a keeper’s foul, a player that is not taking part to the penalty kick is too close to the penalty area, etc.). All these features together put the basis to easily analyse the outcomes of this game.
a. Data description
The analysed championships are the FIFA World Cup 2006, 2010, and 2014, the UEFA European Cup 2008, 2012, and 2016, the UEFA Champions League 2015/2016, 2014/2015, 2011/2012, and 2008/2009 seasons, the Qualifications to the UEFA Champions League 2016/2017, 2012/2013, and the UEFA Europa League 2017/2018, 2015/2016, and 2014/2015, making a total of 395 penalty kicks (310 penalty shoot-outs and 85 penalty kicks from fouls) examined in 108 matches. These championships have been chosen due to the availability of the data. The data has been retrieved from the football websites Transfermarkt and Sky Sports, and the footages has been retrieved from YouTube and Mediaset Premium Sport, an Italian pay TV service.
The gathered data are: type of Championship and year; teams of the match; name, team and side chosen by the kicker and goal keeper; preferred foot of the kicker; foot used by the kicker; time when the penalty took place; score at the moment the penalty took place; type of penalty (either foul penalty or shootout); if the team is playing at home, as visitors or in neutral field; if the team won the match; if the team is higher to its rival in the FIFA (if national team) or UEFA (if club) rankings in the month before the match; the successfulness of the penalty kick and, in case of an unsuccessful shot, if the ball has been either saved, missed or hit the bars.
In Table I we can see a description of the dataset and in Table II a summary of all the relevant data. The concepts of left and right are based on the kicker’s point of view. As we can see, most of kicker and goalkeeper’s strategy consist in choosing either left or right. Few players choose centre,
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especially the goalkeeper, and everything relates to reputational concerns and social pressure (Chiappori et al., 2002). Few goalkeepers decide to choose centre since it would mean that they would be idle. Due to this outcome, it would be logical for kickers to choose centre more often, since they can increase the likelihood of scoring. However, this does not happen. If a kicker shoots to the centre and the goalkeeper blocks it, it would mean that the kicker literally laid the ball into the goalkeeper’s hands. Nevertheless, 75.70% of penalty kicks managed to be successful.
In the next section we will analyse the experimental results, to see if the players use their intrinsic advantage and try to give an explanation to the outcomes.
Table I: Dataset description
Competition Type Matches
N°
Kicks N°
World Cup National Teams 43 124
European Championship National Teams 18 84
Champions League Clubs 40 114
Qualification Champions League Clubs 3 27
Europa League Clubs 4 46
Total 108 395
Table II: Summary of penalty kicks data
Side chosen by goalkeeper Side chosen by kicker
Left Centre Right Left Centre Right
54.32% 2.05% 42.39% 42.03% 17.22% 40.76%
Favourite kicker’s foot Foot used by kicker
Left foot Two-footed Right foot Left Right
23.54% 15.95% 60.51% 24.81% 75.18%
Players used their favourite foot*
Matches played in
neutral field Success rate Missed balls
Ball hits the bar or the poles
96.71% 48.86% 75.70% 5.14% 2.57%
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b. Experimental results
The common sense states that, in a penalty kick, to perform a more precise and stronger shot a kicker has to shoot to the opposite side of the foot that he uses (i.e. if the kicker kicks the ball with the right foot he has to aim to the left side of the goal post) (Chiappori et al., 2002). If he does so by using his favourite foot, we will call it Natural Side. Since we do not have the past history of kicks of each player, and we do not know the favourite kicking side, we will assume that the natural side is the player’s favourite kicking side. The favourite kicking side is the side in which the player, both during matches and during training, scores the most. As we can see from our data, 382 kicks are kicked with the favourite foot (96.71%), and only 167 of them are kicked to the natural side (42.28% of the total and 43.72% of the kicks kicked with the favourite foot).
Since the main research question is about the stakes of the game, we are going to separate the data between penalty shoot-outs and foul penalty kicks. In the first case, since players face their last opportunity to shoot a goal and win the match, the stakes are much higher compared to the second case. In penalty kicks fouls, in case of failure, the team still has the rest of the match to score a new goal.
By separating the data, we have that 56.47% of foul penalty kicks go to the natural side, while for the penalty shootouts only 44.05% of kicks go to the natural side. To test if this difference is statistically significant, we will run an unpaired t-test. The F-test of equality of variances gives a f-value of 0.995 (degrees of freedom = 309, 84; p-value = 0.949) and the Shapiro-Wilk test gives for both the variables a W-value of 0.999 (p-value = 1.000). As a result we have to fail to reject at any level the null hypotheses of same variance and normality for both the samples, thus the unpaired t-test is less likely to be biased.
The null and alternative hypotheses for the test are:
𝐻0: 𝜇𝐹 = 𝜇𝑆 and 𝐻𝛼: 𝜇𝑆 > 𝜇𝐹
Where 𝜇𝐹 is average of fouls penalty kicks shot to the natural side and 𝜇𝑆 is the average of penalty kicks shoot-outs shot to the natural side. We expect that players are more likely to shoot to their natural, preferred, side when they have their last shot available and there are higher stakes at play.
Table III: Test 𝝁𝑭 = 𝝁𝑺
Shoot-outs Fouls
Mean Std. Dev. Mean Std. Dev. t-test
Penalty kicks 0.442 0.497 0.565 0.499 -2.014**
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The results are presented in Table III. By this test, we have to reject our null hypothesis on a 5% level. Most interesting, it seems that players are more likely to shoot to their favourite side on a foul penalty kick, thus when there are lower stakes at play. A possible explanation is that players want to be unpredictable on the penalty shoot-outs, since it is their last chance to score, and tend to forego this option on foul penalties and decide to exploit their personal advantage. However, we do not know if this strategy actually works and it allows to score more goals. To find an answer to this question, we will push our analysis further.
We first analyse the penalty shoot-outs. Usually, the kicker’s favourite foot is common knowledge, thus it means that also the goalkeeper is aware of it. As a result, if the player shoots to its natural side (i.e. kicks with his favourite foot to the opposite side), a clever goalkeeper can anticipate the kicker’s choice and increase his blocking probabilities. The main assumption here is that there is no correlation among the shots, thus the outcome of earlier turns does not influence the choice of the player. Moreover, as in Chiappori et al. (2002), we assume that all the goalkeepers are the same and follow the same strategy, while the kickers are more heterogeneous and differ between each other in their abilities and strategies. To test this hypothesis, we will run the following probit regression:
Pr(𝑆𝑢𝑐𝑐𝑒𝑠𝑠 = 1|𝑁𝑎𝑡𝑢𝑟𝑎𝑙) = Φ(𝛽0+ 𝛽1𝑁𝑎𝑡𝑢𝑟𝑎𝑙 + 𝛽𝑖𝑨)
The variable Success is the success rate of scoring, while the variable Natural stands for natural side. The vector A is a set of covariates used as a control: if the match has been played on a home ground or neutral field, if the kicking team is the first one to start the shootouts, if the kicking team is in a higher raking regard the saving one and if the kicker is the second, the third, the fourth, or the fifth or more of his team.
The null hypothesis is:
𝐻0: 𝛽1 = 0
or, in plain words, shooting to the natural side does not modify the probability of scoring a goal. While the type of field and position in the ranking is suggested by common sense (a team who is playing in home field is surrounded by a more supportive audience and feel less pressure, while a team higher in the ranking is supposed to be a stronger team with better players), the order of the team and order of players is suggested by Apesteguia and Palacios-Huerta (2010). In their paper they give proofs that the order is important, since the first team puts more pressure on the second one. As a result, the first kicking team has a 60% probability of winning the match. The results are presented in Table IV.
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As we can see from the table, none of the results are significant, thus we cannot reject our null hypotheses. All the variables have quite low marginal effects, thus their impact is almost ininfluent on the success rate. With these results, we can assume that shooting to the natural side, although might result in a stronger and more precise shot, does not improve the scoring probabilities. This can be explained due to the randomisation process. Football players know that if they will always make the same choice would increase the predictability of their strategy. By not choosing to use his strongest option he can counter-anticipate the goalkeeper and increase the success probability.
Table IV: Determinants of success
Probit dF/dx Probit dF/dx Probit dF/dx
Natural side -0.008 (0.643) -0.003 (0.050) -0.020 (0.157) -0.006 (0.051) -0.026 (0.160) -0.008 (0.051) First 0.176 (0.154) 0.057 (0.050) 0.181 (0.155) 0.058 (0.050) Neutral field -0.059 (0.189) -0.019 (0.061) -0.072 (0.191) -0.023 (0.061) Home field -0.024 (0.218) -0.008 (0.070) -0.035 (0.219) -0.011 (0.070) Ranking (1 if higher) 0.153 (0.157) 0.049 (0.050) 0.162 (0.157) 0.052 (0.050) Second kicker 0.010 (0.218) 0.003 (0.070) Third kicker 0.234 (0.225) 0.075 (0.712) Fourth kicker -0.234 (0.218) -0.075 (0.070)
Fifth (or more) kicker 0.100
(0.228) 0.032 (0.073) Constant 0.643*** (0.103) 0.518*** (0.207) 0.507*** (0.228) Pseudo R-squared 0.001 0.007 0.012 Observations 310 310 310 *p<10%, **p<5%, ***p<1%
Now we will run a similar analysis on the foul penalty kicks. The assumptions are the same of the earlier model. The model in this case is:
Pr(𝑆𝑢𝑐𝑐𝑒𝑠𝑠 = 1|𝑁𝑎𝑡𝑢𝑟𝑎𝑙) = Φ(𝛽0+ 𝛽1𝑁𝑎𝑡𝑢𝑟𝑎𝑙 + 𝛽𝑖𝑩)
As earlier, the variable Success is the success rate of scoring, while the variable Natural stands for natural side. The vector B presents a slightly different set of covariates, respect to the earlier model, used as a control: if the match has been played on a home ground or neutral field; if the kick takes place in the second half time or in the injury time of the first or second time (the extra time has been omitted due to collinearity); if the kicking team is in a higher ranking regarding the defending one
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and if the team, right before the kick, is winning or tying the match. In this case, since the stakes are lower, the players will feel less pressure. However, the pressure can increase as the time goes on and a penalty kick shot on the 15th minute can present a different strategy compared to a penalty kick shot on the 90th minute.
The results are presented in Table V. As we can see, also in this case none of the results are significant. Also here the effects also here are very small and do not present any relevant outcome. Each team usually has a designated kicker to kick the foul penalty kicks. This player is the strongest of the team and also has a long history of past kicks, both recorded during matches and trainings. As a result, the goalkeeper can anticipate the kickers’ choice and decide to dive to the kickers’ favourite side. In addition, there can be many unobserved variables that can influence the scoring probability.
Table V: Determinants of success
Probit dF/dx Probit dF/dx Probit dF/dx
Natural side 0.087 (0.321) 0.022 (0.083) 0.119 (0.340) 0.031 (0.087) 0.134 (0.352) 0.033 (0.088) Neutral field 0.127 (0.412) 0.033 (0.106) 0.065 (0.419) 0.016 (0.105) Home field 0.268 (0.465) 0.069 (0.119) 0.352 (0.486) 0.088 (0.120) Ranking (1 if higher) -0.089 (0.328) -0.023 (0.084) -0.251 (0.359) -0.063 (0.089) Winning -0.183 (0.465) -0.047 (0.119) -0.289 (0.484) -0.072 (0.120) Tie -0.141 (0.430) -0.361 (0.111) -0.282 (0.480) -0.070 (0.120) Second HT -0.353 (0.392) -0.088 (0.097)
First Injury Time -0.635
(0.816)
-0.159 (0.202)
Second Injury Time 0.388
(0.636) 0.097 (0.159) Constant 0.881*** (0.238) 0.899*** (0.491) 1.276** (0.621) Pseudo R-squared 0.0009 0.007 0.033 Observations 85 85 85 *p<10%, **p<5%, ***p<1%
In the next section, we will show the results of a framed laboratory experiment, conducted in a high school. This will allow us to see if the results of the natural experiment are replicable in a laboratory and, most importantly, if we can receive a more detailed insight in the results.
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vi. The experiment
The experiment has been run at the high school Liceo Scientifico Statale G. da Procida, located in Salerno (Italy). Four classes have been involved, making a total of 82 participants aged between 16 and 18 years old. The choice of a high school can be explained by several factors. First of all, the ease to find available subjects. Moreover, we can assume that the success in this game is not strongly related to the experience in the game, since the rules are quite simple, or the age.
The settings of this experiment are very similar to the penalty kicks. In the second part, players will be required to make a choice in a one-shot simultaneous game, with high stakes at play. Moreover, they will have the opportunity to make their choices by analysing the past moves of their opponents made in the first part, thus they will have the chance to have expectations regarding the predictability of their opponent.
The experiment was divided into three parts. In each section the participants received a sheet with the instructions of the relative part. In the first part, the players were anonymously and randomly matched and had to take part to a repeated constant-sum matching pennies game. This part was needed so the students could intuitively understand the mechanism of game theory and mixed strategies. The anonymity of the game is due to avoid any possible communicational and reputational bias. The payoff table was the following:
Player B
1 0
Player A 1 5, 0 0, 5 0 0, 5 5, 0 Payoffs: (Player A, Player B)
The players had to choose between 1 or 0. In case they made the same choice (i.e. either [1, 1] or [0, 0]), player A received a payoff of 5 and player B did not receive anything. In case the players made opposite choices (i.e. either [1, 0] or [0, 1]), player B received a payoff of 5 and player A did not receive anything. This part has been repeated for 10 rounds.
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The second part of the experiment had a similar setting to the first one. The players, still matched with the same participants, had to participate in a one-shot asymmetric game. The payoff table was the following:
Player B
1 0
Player A 1 10, 0 0, 5 0 0, 5 5, 0 Payoffs: (Player A, Player B)
If the players choose (1, 1), player A gets a payoff of 10 and player B does not get anything. If the players choose (0, 0), players A gets a payoff of 5 and player B does not get anything. If the players choose either (1, 1) or (0, 0), then player B gets a payoff of 5 and player A does not get anything. The whole experiment has been conducted by the only use of paper and pen, and the data has been collected by walking among the desks. Some examples had also be shown on a blackboard, to be sure that they would have understood the instructions.
After the second part the participants received a small questionnaire. The questionnaire contained the following four open questions: “Regarding Part 1, how did you manage to make your choices?”, “Regarding Part 2, how did you manage to make your choices?”, “In part 1, did you base your choices on the past moves of your opponent?” and “In part 2, did you base your choices on the past moves of your opponent?”. At the end of the experiment, per each part, a random couple was drafted from a random round and received an Amazon voucher with the conversion rate 1 point = 1€.
For further details, the instructions of the experiment are located in the Appendix.
a. The theoretical predictions
The theoretical predictions of the experiment are quite easy to calculate. In Part 1, both the players feel indifferent in choosing either 0 or 1. As a result, by choosing 1, player A will expect:
𝑈𝐴1 = 𝑝𝐵∗ 5 + (1 − 𝑝𝐵) ∗ 0
With 𝑝𝐵 denoting the probability that the Player B chooses 1 as well. Similarly, by choosing 0 player A will expect:
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Since this game does not present any dominant strategy, the player will feel indifferent in both the choices, thus:
𝑈𝐴1 = 𝑈𝐴0
𝑝𝐵∗ 5 + (1 − 𝑝𝐵) ∗ 0 = 𝑝𝐵∗ 0 + (1 − 𝑝𝐵) ∗ 5
By solving the calculations, we have:
𝑝𝐵=
1 2
By solving similar calculations for player B, we can as well derive that the probability of Player A to choose 1 is 1
2 as well.
In Part 2 the strategies change, since the game becomes asymmetrical due to the increase in the payoff for Player A. By making similar calculations as earlier, we can derive that:
𝑝𝐴 =1
2 and 𝑝𝐵 = 1 3 .
Player A will expect that Player B will be indifferent between his choices, so he will be indifferent as well. Player B will expect that Player A will be more likely to play 1, to get the payoff of 10, so he will be less likely in choosing that option.
b. Experimental results
The results of the first part of the experiment are summarised in Figure I. As we can see, the aggregate percentage of players who chose 1 per each turn fluctuates around 50%, following our theoretical prediction. To confirm this observation, that the average of aggregate percentage is equal to 0.50, we will run a one-sample student’s t-test.
The null hypothesis is:
𝐻0: 𝜇1 = 0.50
where 𝜇1 represents the aggregate average of fraction of times the players played 1 in all the turns.
We can observe the results in Table VI.
As we can see from the table, we cannot reject our null hypothesis on any level and we have to assume that the players, on average, played the option 1 50% of the times. Thus they followed the theoretical prediction.
In the second part of the experiment, we can observe a similar situation. We can see the summary on Figure II. In the last round, the total amount of players who choose 1 is equal to 43.90%. Since the
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two groups had two different payoffs, thus two different incentives in making their choices, we will now analyse two different t-tests to check if they follow the theoretical predictions.
The two null hypotheses for these tests are:
𝐻0𝐴: 𝜇1 = 0.50 and 𝐻0𝐵: 𝜇1 = 1 3
Or, in other words, the average percentage of players A who played 1 in the second part is equal to 50% and the average percentage of players B who played 1 in the second part is equal to 0.33%. By looking at Table VII, in both tests we have to fail to reject our null hypothesis on any confidence level. As a result, we can deduce that the players followed the theoretical predicted equilibrium. In both Part 1 and Part 2 the Player A was expected to choose 1 50% of the times. To see if the participants played in the same way in both the experiments, we will run a F-test of equality of variances so we can see if the results are comparable and come from the same samples.
Figure I: Summary Experiment Part 1
Table VI: Test 𝜇1 = 𝟎. 𝟓𝟎
Mean t df Sig. (2-tailed) Std. Dev.
95% Confidence Interval
Lower Upper
Fraction of times
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The null hypothesis of this test is that both the variances are the same, so that: 𝐻0: 𝜎𝐹𝑖𝑟𝑠𝑡2 = 𝜎𝑆𝑒𝑐𝑜𝑛𝑑2
The results are presented in Table VIII. As we can see, we cannot reject the null hypothesis at any level, so we have to assume that the players maintained the same behaviour during the game.
Figure II: Summary Experiment Part 2
Table VII: Test 𝝁𝟏𝑨 = 𝟎. 𝟓𝟎 and 𝝁𝟏𝑩 =𝟏
𝟑
Mean t df Sig. (2-tailed) Std. Dev.
95% Confidence Interval Lower Upper Fraction of times player A played 1 0.512 0.427 40 0.878 0.079 0.352 0.672 Fraction of times player B played 1 0.366 0.154 40 0.6717 0.076 0.212 0.520
Table VIII: Test 𝝈𝑭𝒊𝒓𝒔𝒕𝟐 = 𝝈𝑺𝒆𝒄𝒐𝒏𝒅𝟐
Part 1 Part 2
Mean Std. Dev. Mean Std. Dev. f df
Times Player
A played 1 0.556 0.025 0.521 0.079 1.035 40, 409
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To better understand the nature of these results, we will also take a look at the results of the surveys. On Figure III we can observe the results of the first two questions. In the first question “Regarding Part 1, how did you manage to make your choices?”, the participants gave similar answers so we decided to sum it up by dividing them in two groups: “Random players” and “Predictable players”. The Random players, which made up the 69.51% of the participants, are the one who said that in each turn they were making their choice randomly, thus they were choosing 1 or 0 with an equal probability (50%). The remaining 30.49% of the players was composed by the Predictable ones who, on the other hand, stated that they were following a certain strategy or patterns, e.g. always choosing the same number in all the turns, alternating the choices by following a certain pattern, and other strategies which could have been easily exposed by their opponents.
In the second question “Regarding Part 2, how did you manage to make your choices?”, we decided to divide the players in three groups. The first group, the Stubborn players (32.93% of the total), are the ones who said that they made their choice by only looking at the payoff they could have got, without taking into account the move of their opponent (i.e. a Player A chooses 1 so he can get 10). The second group, the Anticipation players (47.56%), are the ones who said that they made their choice based on what they believed their opponent would have played (i.e. a Player B chooses 0 because she expects that the Player A will choose 1). The third group, the Counter anticipation player (19.51%), are the ones who made their choice based on what they believed their opponent would have played based on the belief of their own strategy (i.e. a Player A chooses 0 because she believes that the Player B will choose 0 because he believes the Player A will choose 1). Although all the three groups were expected to arise, the relatively big size of the third group came up almost as a surprise. As for the first question, in Figure IV we can see that the players have been pretty much truthful and coherent with their answers. Most of the Random players played 1 between 40% and 60% of times, thus their strategies unpredictable, while most of the Predictable players played 1 between 0% and 40% of times and 60% and 100% of times, thus making their strategy more foreseeable.
As for the last two questions “In part 1 (part 2), did you base your choices on the past moves of your opponent?”, we can see an interesting change in answers between the two parts (Figure V). While in the first part only 59.76% of players stated that they were basing their choices on the opponent’s history, in the second part this percentage increases to 67.07%. A possible explanation is that the second part of the experiment required a slightly higher level of strategic thinking, thus the participants decided to take a look to the past history to make their decisions.
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We can conclude that players followed the theoretical predictions and they made their choices in order to anticipate and counter-anticipate their opponents. Moreover, most of the players understood the role of the first part of the experiment and, by intuitively learning the concept of mixed strategy, they made their final choice for the second part.
Figure III: Results first two questions of the survey
Figure IV: Distribution of times 1 has been played per type of player
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vii. Conclusions
Since Nash created the concept of non-cooperative games, the theory of mixed strategy equilibrium has been keeping having an increasing role in game theory. This is due the fact that most games usually do not present an unique Nash equilibrium and they have to be played by following mixed strategies.
In the past years we have seen laboratory experiment involving the concept of mixed strategies in many different variations with likewise different results. To find new interesting insights, the researchers are starting to analyse real life settings, and some of them are analysing the sports. Although penalty kicks present very simple rules and settings, they still need a lot of research in order to explain certain outcomes. As we have seen, they can be described as zero-sum games, but they have to be analysed in a different way. The kickers tend to use more often their favourite and stronger foot but tend to avoid shooting always to the strongest (natural) side, especially when high stakes are at play, in a way to maintain their unpredictability. As the chances of scoring are influenced by many unobservable variables, according to our model shooting to the natural side does not increase these chances. However, we do not exclude that the model presents wrong specifications.
With this research we have seen that, in both the experiments, as stakes increase players tend to be more unpredictable. On the other hand, their opponents can anticipate their moves by studying the past moves and analysing their patterns. By the way, in the natural experiment we assumed that a foul penalty kick has lower stakes than a penalty shoot-out. Within the same championship the players could also assign the same level of stakes to these two types of kicks. Further research should be addressed by comparing the same type of penalties on different divisions level and to verify if there is any significant difference. In addition, this part can be expanded by analysing the strategy per player, by analysing all their past penalty history and to see exactly which foot they used and which side they chose in their successful shots, thus to see exactly which strategy and which foot/side combination is their most successful and most used. Unfortunately, the data used were not enough detailed for this kind of research and the availability and quality of those data is quite limited. As for the framed laboratory experiment, we have seen that the players tend to follow the theoretical predicted outcomes. Interestingly they do not always play the move in which they have an advantage, as a way to surprise them by anticipating or counter-anticipating their opponents.
The second part can be improved first of all by executing it in a controlled laboratory. This allows to have more flexible schedules, more motivated players, more turns and can also control for any possible communicational bias. The game only had 11 turns because the teachers were willing to give
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up to just one hour of class and, even if the possible has been made to avoid any communicational bias in the class, we cannot assure that the monitoring was fully effective. The game can more accurately replicate the penalty kicks settings by being reformulated as a 3x3 constant-sum game and putting the default option to be the central one.
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Appendix
i.
Derivation of Mixed Strategy Equilibrium (Matching Pennies)
In equilibrium, a player is indifferent between the two choices when he can receive the same expected payoff from both his choices.
In this specific case, Player A’s expected utility is:
𝐸𝑈𝐴(𝐻) = 𝑝𝐵∗ 1 + (1 − 𝑝𝐵) ∗ (−1) = 2𝑝𝐵− 1 and 𝐸𝑈𝐴(𝑇) = 𝑝𝐵∗ (−1) + (1 − 𝑝𝐵) ∗ 1 = 1 − 2𝑝
With EU indicating the expected utility and 𝑝𝐵 Player A’s expectation of Player B’s strategy. In
equilibrium we have:
𝐸𝑈𝐴(𝐻)
=
𝐸𝑈𝐴(𝑇) 2𝑝𝐵− 1 = 1 − 2𝑝𝐵𝑝𝐵= 1 2 Similar calculations for Player B will lead to 𝑝𝐴 =
1 2.
ii. Instructions of the experiment (Translated from the Italian document)
Welcome to this experiment. Please do not use your mobile phones during this session and read the instructions carefully. If you have a question, raise your hand and the researcher will come to your desk to answer your question. Please do not talk to other participants during the experiment.
You can earn an Amazon voucher at the end of this experiment based on your and other participants’ decisions, with a conversion rate 1 point=€1. You have to write your decisions on the Answers Sheet. The card you picked up at the beginning of the experiment represents your ID and defines whether you’ll be Player A or Player B throughout the course of this experiment. Please write it down on your answer sheet next to “ID” and keep the card visible for the whole experiment. At the beginning of the experiment both players, A and B, will be randomly and anonymously matched together. The couple will be the same for the whole experiment. The experiment consists of three parts: Part 1 will have 10 rounds, Part 2 will have just one round and in the end there will be a small questionnaire. At the end of the experiment two couples (a player A and a player B) will be randomly picked and will collect their payoffs based on their interaction in a randomly selected round for each part.
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Please take this experiment seriously. In case someone will be caught breaking the rules (i.e. communicating with other participants, using mobile phones), they will be excluded from the final lottery and pay-out.
Part 1
At the beginning of the experiment you have been paired with another person. In each round, the amount of the voucher that you can win will depend upon a decision that you will make and upon a decision made by the other person in your pair in that period. Specifically, in each round you and your partner have to choose a number, either 0 or 1. According to your decision, you will receive a payoff according to the following table:
Player A
1 0
Player B 1 5, 0 0, 5 0 0, 5 5, 0 Payoffs: (Player A, Player B)
If both the players will make the same choice [i.e. Player A will choose 1 and player B will choose 1, that is (1, 1), or (0, 0)], Player A will receive a payoff of 5 and player B will not receive anything. If both the players will make two different choices [i.e. either they will choose (1, 0) or (0, 1)], Player B will receive a payoff of 5 and player A will not receive anything.
This part will have 10 rounds. You will have 20 seconds to make your choice. The beginning and the ending of each round will be announced by the supervisor. Write down your choice on the first page of your answers sheet under the section “Your choice” at the corresponding round. Once the end of the round is announced, you are required to put your pen on your desk. The supervisor will first walk among the desks to take note of your choices and then he will be back to write on your answers sheet your opponent’s choice and your payoff. At the end of the 10th round this section will end.
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Part 2
This part has a similar setting as Part 1. You are still matched with the same person as in Part 1 and also in this round you have the chance to win an Amazon voucher based on your and your opponent’s decision. However, this part will only have one round and the payoffs are the following:
Player A
1 0
Player B 1 10, 0 0, 5 0 0, 5 5, 0 Payoffs: (Player A, Player B)
If the players will choose (1, 1), then Player A will get a payoff of 10 and Player B will not receive anything. If the players will choose (0, 0), Player A will receive a payoff of 5 and Player B will not receive anything. If both players will make opposite choices [i.e. (1, 0) or (0, 1)], Player B will have a payoff of 5 and Player A will not receive anything.
This part will have only one round. You will have 30 seconds to make your choice. The beginning and the ending of this round will be announced by the researcher. Write down your decision on the second page of your answer sheet under the section “Your choice”. Once the end of the round is announced, you are required to put your pen on your desk. The researcher will first walk among the desks to take note of your choices and then he will be back to write on your answers sheet your opponent’s choice and your payoff. At the end, all the answers sheets will be collected and there will be the questionnaire and the final lottery.
Part 3 (questionnaire)
The experiment is finished, now you will have to answer four open questions. The researcher will walk among the desks and will distribute a sheet containing one question. You will have 3 minutes to answer. At the end of the 3 minutes, the researcher will pass to collect the answers and will distribute the new question. This process will be repeated for 4 times. The beginning and the ending of each section will be announced by the researcher. Write your ID in the related section, in the top right of the sheet and write down your answer in the answers box. Be brief and go straight to the point. At the end of this section the final lottery will take place.
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