Master’s thesis in Econometrics, specialisation: Mathematical Economics
Economics and Business, University of Amsterdam
Strategic interaction in penalty kicks
Date: 03-08-2017
Written by: Laurens van der Veer
Supervisor: dhr. dr. T.A. Makarewicz
Statement of Originality This document is written by Laurens van der Veer, who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is 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.
Contents
Abstract 41.
Introduction 5
2.
Model 7
2.1 Matching pennies game 7 2.2 Goalkeeper signals 8 2.3 Goalkeeper types and game as a whole 10 3.
Results 12
3.1 MSNE in subgames 12 3.2 Separating equilibria 15 3.3 Pooling equilibria 18 3.4 Overview of results 31 4.
Summary and Conclusion 32
References 34
Appendix A 35
Appendix B 44
Appendix C 50
Abstract
Penalty kicks in football can be analysed and modelled with game theory. Using game theory, choices of players can be predicted and optimised. Since most football matches end in a very close score, scoring a penalty kick during the match is often crucial for the outcome. Scoring a penalty kick therefore can lead to large monetary rewards for both the football club and the player in question and therefore the incentives to play this game right are immense. This paper aims to find the best strategy to score a penalty kick in football from a game theoretic perspective (and the best strategy for the goalkeeper to stop the kick taker from scoring). The common matching pennies game is extended with two factors that might be of interest. The first extension is that the goalkeeper has the possibility to make movements during the run-up of the penalty kick: with a movement, the goalkeeper can send a signal to his opponent and give information on where he will dive. The second extension is that goalkeepers can be of two types: a fast type and a slow type. The model that results is a Bayesian game. The objective of this paper is to see if the extensions influence the outcome of the game by constituting a different Mixed-Strategy Nash Equilibrium (MSNE). The keeper’s possibility to send signals of going left or right turned out not to be beneficial for the goalkeeper in any case. The extension with the fast and slow type of goalkeepers does influence the constituted equilibrium and the type that is in a vast minority benefits the most.
1. Introduction
Football is a fast-growing industry in which every year an increasing amount of money is involved. A penalty kick in football is often decisive for the outcome of the game, since the most frequent occurring end results in football are close. For this reason, the incentives for both the goalkeeper and the kick taker to apply the right strategies during a penalty kick are big. Although the incentives to play the penalty kick right are enormous, there is some research indicating that goalkeepers are biased and stay in the middle less often than they should just because the norm is to dive (Bar-Eli, Azar, Ritov, Keidar-Levin & Schein, 2007). Bar-Eli et al. continued their research on penalty kicks and the goalkeeper’s preferences and suggested that penalty kicks in the top half of the goal are the hardest to stop (Bar-Eli, Azar, 2009). Most of the research on penalty kicks is not in the field of economics but some results are interesting and worth mentioning. There is some evidence that the goalkeeper can influence the result of the penalty kick prior to the shot by either standing marginally off-center (Masters, Kamp & Jackson, 2007) or by making the kick taker alter the direction of the shot (van der Kamp, 2006). These findings are hard to capture in an economic model however, since the conventional economic models assume agents to be perfectly rational.
The current literature on penalty kicks in the field of game theory mostly focusses on finding a Mixed-Strategy Nash Equilibrium (MSNE) within a simple 3x3 matching pennies game and testing whether this equilibrium holds up in real life. In this variant of the matching pennies game the goalkeeper and the kick taker both have three strategies: left, middle and right and there is a unique MSNE in this type of game. Whether the equilibrium is actually played in real life is an empirical question and the evidence is found for the first time by Chiappori, Levitt and Groseclose (2002).
Chiappori et al. (2002) used data from 459 penalty kicks in French and Italian professional football leagues to test their model with real-world data and could not reject the hypothesis that professional football players play optimally. In 2007, Coloma used the same model and data and confirmed their results with an alternative testing method. More evidence of equilibrium play in real life is provided by Palacios-Huerta (2003) and Azar and Bar-Eli (2011) who also model the penalty kick as a 3x3 matching pennies game. The research of Baumann, Friehe and Wedow (2011) also uses game theory in penalty kicks but only in a simplified 2x2 game where kick takers either shoot left or right: their research emphasises the difference between left and right-footed kickers and the effect of the general ability of the kick taker on a penalty kick.
We can conclude that there has not been an extensive amount of research on penalty kicks in the field of game theory and the research that has been done almost exclusively focusses on the
fairly simple matching pennies game where the players have at most three strategies and act simultaneously. From real life experience however, we know that there is more going on: keepers often make feints or movements during the run up of the penalty kick to influence the kick taker. Some kick takers wait until the last moment to see if the keeper moves and then they try to shoot the ball in the opposite corner. For this reason, it will be interesting to see if a keeper’s movement during the run up will influence the kick taker’s choice and this aspect is included in our model. The proposed model for penalty kicks in this paper also considers the stochastic nature of the goalkeeper’s performance, whereas the matching pennies game assumes that both the goalkeeper and the kick taker perform at a constant level. Clearly a penalty kick in football is a one-shot interaction between two players, but the aim of this study is to find what happens if the kick taker knows that the keeper’s effectiveness is not constant but depends on his current state and if this will influence the players’ strategies. In real life, a goalkeeper will not catch the ball with a constant probability: the probability of a catch will depend on the keeper’s current form, fatigue, mental state et cetera.
For these reasons, this research proposes a more complete model for a penalty kick in football than used until now and aims to find if and how the strategies of the players are influenced by the proposed extensions.
The keeper’s possibility to send signals of going left or right turned out not to be beneficial for the goalkeeper in any case. The extension with the fast and slow type of goalkeepers does influence the constituted equilibrium and the type that is in the minority benefits the most. The key lies in the fact that the type that is in a vast minority no longer has the need to be unpredictable since the probability that the kicker this type of goalkeeper faces is so small. A further explanation and interpretation of this phenomenon will be given in the results and conclusion.
2. Model
The whole model for penalty kicks is a more complicated signalling game and therefore we start with a simpler example. The matching pennies game as described in section 2.1 is a subgame of the signalling game that is used in this research. In section 2.2 the signals a goalkeeper can send are explained as well as the effect of the signals on the payoffs of the players. In section 2.3 the types of goalkeepers and their effects on the game are explained and the chapter is concluded with an overview of the whole game. 2.1 Matching pennies game There are two players: a goalkeeper (𝐺) and a kick taker (𝑃) with three strategies each. Formally we note the set of players as follows: 𝑁 = {𝐺, 𝑃}. The goalkeeper can dive left, middle or right and the kick taker can shoot left, middle or right. The set of actions is therefore defined as: 𝐴)= 𝐴*= {𝐿, 𝑀, 𝑅}. The goalkeeper clearly wants to match the choice of the kick taker since the chance of saving the ball will then be the highest and that is why we can compare the payoff structure of the game to that of a matching pennies game. If we set the payoff of saving or scoring a penalty kick equal to 1 and the payoff of conceding or missing a penalty kick equal to −1, we can interpret the used variables both as probabilities and as expected utilities. In reality, scoring a penalty kick in a professional football match is probably worth at least a few thousand euros (of course heavily depending on the match) but estimating this value is beyond the scope of this research and determining the best strategy does not depend upon the value but only on the scoring probabilities. The game is a zero-sum game since the gain or loss in utility of the kick taker is exactly balanced by the gain or loss in utility of the goalkeeper. The Mixed-Strategy Nash Equilibrium of this game will be discussed in chapter 3, here only an overview of the game and the assumptions is provided. The payoff matrix is given by: P L M R L β, -β -𝛼, 𝛼 -𝛾, 𝛾 G M -𝛼, 𝛼 𝛾, -𝛾 -𝛼, 𝛼 R -𝛾, 𝛾 -𝛼, 𝛼 β, -β Where: 0 < 𝛽 < 𝛼 < 𝛾.
The assumptions on the parameters are justified by the following reasons:
• The probability (and thus the expected payoff) of scoring is highest when the goalkeeper moves to the opposite side as to where the player shoots the ball. The expected payoff is 𝛾. Since the penalty kick will almost certainly be scored in this case (only a total miskick can prevent this: for example, when the ball is shot wide or on the post and out). The resulting payoff will be slightly bigger than the payoff realised when the goalkeeper is half a goal away from where the ball is struck: this payoff is 𝛼. For example, when a goalkeeper dives left and the player decides to shoot through the middle the keeper can be fortunate and still save the shot with his legs. Therefore we have: 𝛼 < 𝛾. • When the goalkeeper matches the player’s choice, it is easier for the goalkeeper to catch the ball in the middle of the goal than in the corners, so we have: 𝛽 < 𝛾. • The probability of scoring when the goalkeeper is half a goal away from the shot is bigger than the probability of a save when the goalkeeper chose the correct corner (left or right), so we have: 𝛽 < 𝛼. • The final assumption is that shooting left and right is equally hard for the player (therefore the payoff matrix is symmetric). This assumption might be unfair because there is some research indicating that it is easier for a right-footed player to shoot in the left corner than in the right corner. However, we assume that this difference will be minimal or even non-existent for professional football players. 2.2 Goalkeeper signals The game introduced in this section is the same as in the previous section but with one extension: a goalkeeper is now able to send a signal. This signal is a movement of the goalkeeper during the run up of the penalty kick that he can make to influence the kick taker’s behaviour. The timing of the game is as follows: 1. First the goalkeeper sends a signal to the player about where he will dive. The signals can be left, middle or right represented by 𝜏7, 𝜏8 and 𝜏9.
2. Second the goalkeeper and kick taker simultaneously choose a corner: left, middle or right represented by L, M and R. The signal can be described in more detail as follows. The goalkeeper moves a little bit to the left of the goal just before the kick is taken as a signal of going left, this makes it harder to get back to the right pole of the goal and easier to indeed go left and reach the left pole. Conversely a signal of going right makes it harder to go left and easier to go right. Also, staying in the middle is slightly harder after giving the signal to go either left or right. The signal of staying in the middle is shown by just staying in the middle until the shot is struck and in this case, we are back to the basic matching pennies game since we can interpret this as sending no signal at all. This game is a game with imperfect information: the goalkeeper knows where he will dive while the player does not and may adjust his choice due to the signal he receives from the goalkeeper.
We introduce the parameter 𝜀 for the signal and after observing for example signal left 𝜏7 the
resulting payoff matrix is as follows: Signal 𝝉𝑳 P L M R L β + ε, -β - ε -𝛼 - ε, 𝛼 + ε -𝛾 - ε, 𝛾 + ε G M -𝛼 + ε, 𝛼 - ε 𝛾 - ε, -𝛾 + ε -𝛼 - ε, 𝛼 + ε R -𝛾 + ε, 𝛾 - ε -𝛼 - ε, 𝛼 + ε β - ε, -β + ε As we can see from the payoff matrix we assume that the difference in payoff for the goalkeeper when the player shoots left is 𝜀 and when the player shoots middle or right the difference is – 𝜀. This is because we can interpret 𝜀 as the distance that the goalkeeper moved to the left just before the shot was struck: he is now further away from the middle and the right and therefore it becomes slightly harder to catch a ball in these regions. A shot in the left corner becomes easier to catch since the goalkeeper has less distance to cover in the same amount of time and can reach the left pole easier: this way he has a bigger chance of catching a well struck shot in the left corner. Clearly when signal 𝜏9 is used it is the other way around. The signal 𝜏8 can be interpreted as no signal at all and in
that case, we are back to the payoff matrix as introduced in section 2.1 about the matching pennies game. Finally, we assume that 𝜀 > 0 is a small fixed number, that is: 0 < 𝜀 ≪ 𝛽 < 𝛼 < 𝛾. A future extension of the model could be that 𝜀 is not fixed, but that the magnitude of 𝜀 is chosen by the goalkeeper. This possible extension however is left for further research.
2.3 Goalkeeper types and game as a whole
This game is the same as described in section 2.2 with the extension that goalkeepers are of two types: they are either slow or fast and selected at random (by nature). The kick taker does not know which type of goalkeeper he faces, but does know the distribution of slow and fast goalkeepers in the population. The goalkeeper on the other hand does know which type he is. The timing of the game is as follows: 1. First Nature selects whether a goalkeeper is fast or slow. With probability 𝑝 the goalkeeper is fast and with probability 1 − 𝑝 the goalkeeper is slow. 2. Second the goalkeeper sends a signal to the player about where he will dive. The signals can be left, middle or right represented by 𝜏7, 𝜏8 and 𝜏9.
3. Third the goalkeeper and kick taker simultaneously choose a corner: left, middle or right represented by L, M and R. A slow goalkeeper has more trouble to catch a ball in the corners of the goal since his reaction time is longer due to fatigue or a lack of concentration: or the keeper just has less physical speed which is required to catch the ball in the corner. On the other hand, catching a ball in the middle is equally hard. The payoff matrix of a slow goalkeeper that sends signal 𝜏8 is defined as follows: Signal 𝝉𝑴 slow P L M R L β - x, -β + x -𝛼, 𝛼 -𝛾 - x, 𝛾 + x G M -𝛼, 𝛼 𝛾, -𝛾 -𝛼, 𝛼 R -𝛾 - x, 𝛾 + x -𝛼, 𝛼 β - x, -β + x We assume that 𝑥 < 𝛽 so that the keeper still has a positive probability to save the ball when he has chosen to dive to the correct corner of the goal. In total, we made the following assumptions on the introduced parameters: 0 < 𝜀 ≪ 𝑥 < 𝛽 < 𝛼 < 𝛾. With this last extension about the possible slowness of the keeper, our model of the penalty kick is complete and the whole game is represented in a game tree by Figure 1. We can make a distinction between the six parts of the game tree in which the goalkeeper and the kick taker simultaneously pick a corner and represent each part by a 3x3 matrix, so after the goalkeeper’s type is selected and the goalkeeper has made his signal known. We distinguish the following six subgames: fast 𝜏8, fast
𝜏7, fast 𝜏9, slow 𝜏8, slow 𝜏7 and slow 𝜏9.
3. Results
Before we consider the whole game, it is helpful to look for Nash equilibria in the six subgames. In section 3.1 the MSNE of the first subgame is discussed and an overview is given of the MSNE in all subgames. In the sections that follow, the possible equilibria in the whole game are discussed. Signalling games can have three types of equilibria: separating equilibria, pooling equilibria and hybrid equilibria. The concept of separating equilibria is explained in section 3.2 and pooling equilibria are discussed in section 3.3. We are not looking for hybrid equilibria since it is outside the scope of this research.
3.1 MSNE in subgames
Theorem 3.1.1:
The MSNE in the subgame fast 𝜏8 is given by: 𝜎)= 𝜎*= (HEFIFGEFG ,JEF IK GHEF IFG,HEFIFGEFG ). The subgame
is thus symmetric and the goalkeeper (G) and kicker (P) randomize in the same way over their set of
strategies, 𝐴) = 𝐴*= {𝐿, 𝑀, 𝑅}. The resulting expected payoffs are given by: 𝐸 𝑈) =KJE
OFIGKGO HEFIFG and 𝐸 𝑈* =JE OKIGFGO HEFIFG . Proof: To find an equilibrium in the first subgame we first look for best responses, since no best responses match, there is no Nash equilibrium in pure strategies. The payoff matrix of the first subgame: fast 𝜏8, with the best responses underlined is given by: Signal 𝝉𝑴 fast P L M R L β, -β -𝛼, 𝛼 -𝛾, 𝛾 G M -𝛼, 𝛼 𝛾, -𝛾 -𝛼, 𝛼 R -𝛾, 𝛾 -𝛼, 𝛼 β, -β Now suppose that the kick taker plays left (L), middle (M) and right (R) with chance 𝑝P, 𝑝J and 𝑝Q
respectively. Where: 𝑝P+ 𝑝J+ 𝑝Q= 1. The expected payoffs for the goalkeeper then can be
calculated as follows:
𝐸[𝑈) 𝐿 ] = 𝑝P𝛽 − 𝑝J𝛼 − 𝑝Q𝛾
𝐸 𝑈) 𝑅 = −𝑝P𝛾 − 𝑝J𝛼 + 𝑝Q𝛽 In order to mix, the goalkeeper must be indifferent between strategies L, M and R, so we have: 𝐸 𝑈) 𝐿 = 𝐸 𝑈) 𝑀 = 𝐸 𝑈) 𝑅 In addition, with the fact that 𝑝P+ 𝑝J+ 𝑝Q= 1, we have a set of 3 equations with 3 unknowns, namely: 𝑝P𝛽 − 𝑝J𝛼 − 𝑝Q𝛾 = −𝑝P𝛼 + 𝑝J𝛾 − 𝑝Q𝛼 (1) −𝑝P𝛼 + 𝑝J𝛾 − 𝑝Q𝛼 = −𝑝P𝛾 − 𝑝J𝛼 + 𝑝Q𝛽 (2) 𝑝P+ 𝑝J+ 𝑝Q= 1 (3) Solving this set of equations gives us the following solution (a stepwise solution of the system of equations is provided in Appendix A): 𝑝P = 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝑝J= 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 Now suppose that the goalkeeper plays left (L), middle (M) and right (R) with chance 𝑞P, 𝑞J and 𝑞Q respectively. Where: 𝑞P+ 𝑞J+ 𝑞Q= 1. Using the same reasoning we have the following expected payoffs for the kick taker: 𝐸 𝑈* 𝐿 = −𝑞P𝛽 + 𝑞J𝛼 + 𝑞Q𝛾 𝐸 𝑈* 𝑀 = 𝑞P𝛼 − 𝑞J𝛾 + 𝑞Q𝛼 𝐸 𝑈* 𝑅 = 𝑞P𝛾 + 𝑞J𝛼 − 𝑞Q𝛽 In order to mix, the kick taker must be indifferent between strategies L, M and R, so we have: 𝐸 𝑈* 𝐿 = 𝐸 𝑈* 𝑀 = 𝐸 𝑈* 𝑅
In addition, with the fact that 𝑞P+ 𝑞J+ 𝑞Q= 1, we have a set of 3 equations with 3 unknowns,
namely:
−𝑞P𝛽 + 𝑞J𝛼 + 𝑞Q𝛾 = 𝑞P𝛼 − 𝑞J𝛾 + 𝑞Q𝛼 (1)
𝑞P𝛼 − 𝑞J𝛾 + 𝑞Q𝛼 = 𝑞P𝛾 + 𝑞J𝛼 − 𝑞Q𝛽 (2)
𝑞P+ 𝑞J+ 𝑞Q= 1 (3)
We can easily see that this set of equations yields the exact same results as for the goalkeeper and we have that the kick taker wants to randomize if: 𝑞P= 𝑞Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝑞J= 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 Therefore, both the goalkeeper and the kick taker mix their strategies in exact the same way and this constitutes a MSNE, since it is not beneficial for any of the players to deviate from their mixed strategies. Now we calculate the expected payoffs that are generated by the mixed strategies used in the equilibrium. For the goalkeeper, we have: 𝐸 𝑈) 𝐿 = 𝑝P𝛽 − 𝑝J𝛼 − 𝑝Q𝛾 = 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾𝛽 − 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 𝛼 − 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾= −2𝛼J+ 𝛽𝛾 − 𝛾J 4𝛼 + 𝛽 + 𝛾 So: 𝐸 𝑈) = −2𝛼J+ 𝛽𝛾 − 𝛾J 4𝛼 + 𝛽 + 𝛾 For the kick taker, we have: 𝐸 𝑈* 𝐿 = −𝑞P𝛽 + 𝑞J𝛼 + 𝑞Q𝛾 = − 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾𝛽 + 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾𝛼 + 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 =2𝛼J− 𝛽𝛾 + 𝛾J 4𝛼 + 𝛽 + 𝛾 So: 𝐸 𝑈* = 2𝛼J− 𝛽𝛾 + 𝛾J 4𝛼 + 𝛽 + 𝛾 This last result finalizes the proof of this theorem.
The proofs for the other five subgames are symmetric and can be found in Appendix A. Table 1 gives an overview of the MSNE and the resulting expected payoffs of all subgames.
fast 𝝉𝑴 slow 𝝉𝑴
goalkeeper: kick taker: goalkeeper: kick taker:
L 𝑞P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝑞P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 M 𝑞J= 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 𝑝J= 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 𝑞J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝑝J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 R 𝑞Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝑞Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝐸[𝑈] −2𝛼4𝛼 + 𝛽 + 𝛾J+ 𝛽𝛾 − 𝛾J 2𝛼4𝛼 + 𝛽 + 𝛾J− 𝛽𝛾 + 𝛾J −2𝛼4𝛼 + 𝛽 + 𝛾 − 2𝑥J+ 𝛽𝛾 − 𝛾J− 2𝛾𝑥 2𝛼4𝛼 + 𝛽 + 𝛾 − 2𝑥J− 𝛽𝛾 + 𝛾J+ 2𝛾𝑥 fast 𝝉𝑳 slow 𝝉𝑳
goalkeeper: kick taker: goalkeeper: kick taker:
L 𝑞P= 𝛼 𝛽 + 𝛾 − 4𝜀 + (𝛾 − 2𝜀)(𝛽 + 𝛾) (𝛽 + 𝛾)(4𝛼 + 𝛽 + 𝛾) 𝑝P=4𝛼 + 𝛽 + 𝛾𝛼 + 𝛾 𝑞P= 𝛼 𝛽 + 𝛾 − 4𝜀 + 𝛾 − 2𝜀 𝛽 + 𝛾 + 2𝑥𝜀 (𝛽 + 𝛾)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝P=4𝛼 + 𝛽 + 𝛾 − 2𝑥𝛼 + 𝛾 M 𝑞J= 2𝛼 + 𝛽 − 𝛾 + 2𝜀 4𝛼 + 𝛽 + 𝛾 𝑝J= 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 𝑞J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 + 2𝜀 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝑝J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 R 𝑞Q= 𝛼 𝛽 + 𝛾 + 4𝜀 + 𝛾(𝛽 + 𝛾) (𝛽 + 𝛾)(4𝛼 + 𝛽 + 𝛾) 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝑞Q= 𝛼 𝛽 + 𝛾 + 4𝜀 + 𝛾 𝛽 + 𝛾 − 2𝑥𝜀 (𝛽 + 𝛾)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝐸[𝑈] −2𝛼J− 2𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾4𝛼 + 𝛽 + 𝛾 J+ 𝛾𝜀 2𝛼J+ 2𝛼𝜀 − 𝛽𝛾 + 𝛽𝜀 + 𝛾4𝛼 + 𝛽 + 𝛾 J− 𝛾𝜀 −2𝛼J− 4𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾4𝛼 + 𝛽 + 𝛾 − 2𝑥J− 𝛾𝜀 − 2𝛾𝑥 + 2𝑥𝜀 2𝛼J+ 4𝛼𝜀 − 𝛽𝛾 + 𝛽𝜀 + 𝛾4𝛼 + 𝛽 + 𝛾 − 2𝑥J+ 𝛾𝜀 + 2𝛾𝑥 − 2𝑥𝜀 fast 𝝉𝑹 slow 𝝉𝑹
goalkeeper: kick taker: goalkeeper: kick taker:
L 𝑞P=𝛼 𝛽 + 𝛾 + 4𝜀 + 𝛾(𝛽 + 𝛾)(𝛽 + 𝛾)(4𝛼 + 𝛽 + 𝛾) 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝑞P=𝛼 𝛽 + 𝛾 + 4𝜀 + 𝛾 𝛽 + 𝛾 − 2𝑥𝜀(𝛽 + 𝛾)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 M 𝑞J= 2𝛼 + 𝛽 − 𝛾 + 2𝜀 4𝛼 + 𝛽 + 𝛾 𝑝J= 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 𝑞J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 + 2𝜀 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝑝J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 R 𝑞Q= 𝛼 𝛽 + 𝛾 − 4𝜀 + (𝛾 − 2𝜀)(𝛽 + 𝛾) (𝛽 + 𝛾)(4𝛼 + 𝛽 + 𝛾) 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝑞Q= 𝛼 𝛽 + 𝛾 − 4𝜀 + 𝛾 − 2𝜀 𝛽 + 𝛾 + 2𝑥𝜀 (𝛽 + 𝛾)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝐸[𝑈] −2𝛼J− 2𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾4𝛼 + 𝛽 + 𝛾 J+ 𝛾𝜀 2𝛼J+ 2𝛼𝜀 − 𝛽𝛾 + 𝛽𝜀 + 𝛾4𝛼 + 𝛽 + 𝛾 J− 𝛾𝜀 −2𝛼J− 4𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾4𝛼 + 𝛽 + 𝛾 − 2𝑥J− 𝛾𝜀 − 2𝛾𝑥 + 2𝑥𝜀 2𝛼J+ 4𝛼𝜀 − 𝛽𝛾 + 𝛽𝜀 + 𝛾4𝛼 + 𝛽 + 𝛾 − 2𝑥J+ 𝛾𝜀 + 2𝛾𝑥 − 2𝑥𝜀 Table 1: MSNE and expected payoffs in the six subgames 3.2 Separating equilibria Now the whole signalling game is considered and first we look at separating equilibria. A separating equilibrium is an equilibrium in which the different types always send out different signals. The signal therefore reveals the sender’s type to the receiver. In a separating equilibrium, the kick taker therefore knows for sure which type of goalkeeper he faces after he has received the signal.
Theorem 3.2.1: There exist no separating equilibria if 2𝛼 + 𝛽 > 𝛾. Proof: To have a separating equilibrium we need the two types of goalkeepers to send two different signals. There are thus six possible separating equilibria we must look for: LM, LR, ML, MR, RL and RM. In these abbreviations, the first letter denotes the fast goalkeeper’s signal and the second the slow goalkeeper’s signal: for example, in the possible separating equilibrium LM the fast keeper sends
signal left and the slow keeper sends signal middle: 𝛽) 𝑓𝑎𝑠𝑡 = 𝜏7 and 𝛽) 𝑠𝑙𝑜𝑤 = 𝜏8. However, if one of the goalkeeper types benefits from a one-shot deviation, there is no equilibrium. Since the goalkeeper can choose the part of the game tree he is in, we check whether it is beneficial to play a different part of the game tree by sending out a different signal. The kicker also knows that the goalkeeper determines what part of the game tree is played (by sending out the signal), so they just play the Nash Equilibrium of the chosen subgame. For each type of goalkeeper, we therefore compare the outcomes of the subgames that follow when sending the signals left, middle and right (𝜏7, 𝜏8 and 𝜏9). First, we look at the fast goalkeeper and compare his expected payoffs obtained by the three signals: Fast 𝜏8: 𝐸 𝑈)|𝛽) 𝑓𝑎𝑠𝑡 = 𝜏8 = −2𝛼J+ 𝛽𝛾 − 𝛾J 4𝛼 + 𝛽 + 𝛾 Fast 𝜏7: 𝐸 𝑈)|𝛽) 𝑓𝑎𝑠𝑡 = 𝜏7 = −2𝛼J− 2𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾J+ 𝛾𝜀 4𝛼 + 𝛽 + 𝛾 Fast 𝜏9: 𝐸 𝑈)|𝛽) 𝑓𝑎𝑠𝑡 = 𝜏9 = −2𝛼J− 2𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾J+ 𝛾𝜀 4𝛼 + 𝛽 + 𝛾
Of course, the expected payoffs of signal 𝜏7 and 𝜏9 are the same since the game is symmetric:
shooting left and right is equally hard. If we compare the payoff of sending 𝜏7 or 𝜏9 with the payoff
of sending 𝜏8 we get that the expected payoff of 𝜏8 is higher if:
𝐸 𝑈)|𝑓𝑎𝑠𝑡, 𝜏8 =−2𝛼J+ 𝛽𝛾 − 𝛾J
4𝛼 + 𝛽 + 𝛾 > 𝐸 𝑈)|𝑓𝑎𝑠𝑡, 𝜏7 𝑜𝑟 𝜏9 =
−2𝛼J− 2𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾J+ 𝛾𝜀
Simplified this comparison yields: 2𝛼 + 𝛽 > 𝛾.
We can argue that this condition will hold in every realistic payoff matrix of a penalty kick, since the values of both 𝛼 and 𝛾 will probably be close to 1 (for example 0,8 and 1) and 𝛽 will probably lie around 0,4. Therefore, it is not profitable for a fast goalkeeper to send a signal that he will go either left or right: it is best to stay in the middle during the run up to the penalty kick. This result already excludes all candidate separating equilibria in which the fast goalkeepers sends a different signal than the signal middle (𝜏8), since it is then profitable for the fast keeper to deviate by sending 𝜏8. We can conclude that there will be no separating equilibria in the equilibrium candidates LM, LR, RL and RM since the fast goalkeeper wants to deviate by signalling middle (𝜏8) if 2𝛼 + 𝛽 > 𝛾. We can make a similar comparison for the slow goalkeeper and look at the expected payoffs he gets from sending out the three different signals: Slow 𝜏8: 𝐸 𝑈)|𝛽) 𝑠𝑙𝑜𝑤 = 𝜏8 =−2𝛼J+ 𝛽𝛾 − 𝛾J− 2𝛾𝑥 4𝛼 + 𝛽 + 𝛾 Slow 𝜏7: 𝐸 𝑈)|𝛽) 𝑠𝑙𝑜𝑤 = 𝜏7 =−2𝛼J− 4𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾J− 𝛾𝜀 − 2𝛾𝑥 + 2𝑥𝜀 4𝛼 + 𝛽 + 𝛾 − 2𝑥 Slow 𝜏9: 𝐸 𝑈)|𝛽) 𝑠𝑙𝑜𝑤 = 𝜏9 = −2𝛼J− 4𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾J− 𝛾𝜀 − 2𝛾𝑥 + 2𝑥𝜀 4𝛼 + 𝛽 + 𝛾 − 2𝑥
Of course, the expected payoffs of signals 𝜏7 and 𝜏9 are the same since the game is symmetric:
shooting left and right is equally hard. If we compare the payoff of sending 𝜏7 or 𝜏9 with the payoff
of sending 𝜏8 we get that the expected payoff of 𝜏8 is higher if:
𝐸 𝑈)|𝑠𝑙𝑜𝑤, 𝜏8 =−2𝛼J+ 𝛽𝛾 − 𝛾J− 2𝛾𝑥 4𝛼 + 𝛽 + 𝛾 > 𝐸 𝑈)|𝑠𝑙𝑜𝑤, 𝜏7 𝑜𝑟 𝜏9 =−2𝛼 J− 4𝛼𝜀 + 𝛽𝛾 − 𝛽𝜀 − 𝛾J− 𝛾𝜀 − 2𝛾𝑥 + 2𝑥𝜀 4𝛼 + 𝛽 + 𝛾 − 2𝑥 If we rewrite this expression we get: 𝑥 4𝛼J− 2𝛽𝛾 + 2𝛾J+ 4𝛾𝑥 + 𝜀 16𝛼J+ 8𝛼𝛽 + 𝛼𝛾 − 8𝛼𝑥 + 𝛽J+ 2𝛽𝛾 − 2𝛽𝑥 + 𝛾J− 2𝛾𝑥 > 0
The expression above will always be larger than zero since: 1. 𝑥 > 0 2. 4𝛼J− 2𝛽𝛾 + 2𝛾J+ 4𝛾𝑥 > 0, since 𝛾 > 𝛽 and therefore 2𝛾J> 2𝛽𝛾 3. 𝜀 > 0 4. 16𝛼J+ 8𝛼𝛽 + 𝛼𝛾 − 8𝛼𝑥 + 𝛽J+ 2𝛽𝛾 − 2𝛽𝑥 + 𝛾J− 2𝛾𝑥 > 0, since 𝛾 > 𝑥 and therefore 2𝛽𝛾 > 2𝛽𝑥 and 𝛼 > 𝑥 and therefore 16𝛼J> 8𝛼𝑥 Because of equations 1 till 4 the total expression is positive since all parts are positive and sending a signal is never beneficial for the slow goalkeeper.
This result excludes all candidate separating equilibria in which the slow goalkeepers sends a different signal than the signal middle (𝜏8), since it is then profitable for the slow keeper to deviate by sending 𝜏8. For this reason, also the equilibrium candidates ML and MR are excluded and we can conclude that there is no separating equilibrium in this game. 3.3 Pooling equilibria A pooling equilibrium is an equilibrium in which the different types always send out the same signal. The signal therefore gives no information about the sender’s type to the receiver. In a pooling equilibrium, the kick taker therefore does not know which type of goalkeeper he faces after he has received the signal. The kick taker’s posterior belief is the same as his prior: namely that he will face a fast goalkeeper with chance 𝑝 and a slow goalkeeper with chance 1 − 𝑝. Since in a pooling equilibrium both types send out the same signal we have three possible scenarios: both keepers use signal 𝜏7, 𝜏8 or 𝜏9. Theorem 3.3.1: There exist no pooling equilibria in which both goalkeepers play a pure strategy and use signal 𝜏8. Proof: In this theorem, it will be shown that there is no pooling equilibrium in which both goalkeepers play a pure strategy after sending the signal 𝜏8. Later, in Theorem 3.3.4, we will continue with looking for equilibria in which one or more of the goalkeepers plays a mixed strategy to cover all angles. Table 2 shows us the kicker’s expected payoffs in all possible scenarios (after both keepers used 𝜏8). In the first row the first letter stands for the action of the fast goalkeeper and the second for the action of the slow goalkeeper. An example: LM means that the fast goalkeeper plays a pure strategy left while the slow type plays a pure strategy middle.
𝑼𝑷 against: (𝜏8) LL LM LR ML MM L −𝛽 + 𝑥 − 𝑝𝑥 𝛼 − 𝑝𝛼 − 𝑝𝛽 𝛾 − 𝑝𝛾 − 𝑝𝛽 + 𝑥 − 𝑝𝑥 𝑝𝛼 − 𝛽 + 𝑝𝛽 + 𝑥 − 𝑝𝑥 𝛼 M 𝛼 −𝛾 + 𝑝𝛾 + 𝑝𝛼 𝛼 𝛼 − 𝑝𝛼 − 𝑝𝛾 −𝛾 R 𝛾 + 𝑥 − 𝑝𝑥 𝛼 − 𝑝𝛼 + 𝑝𝛾 −𝛽 + 𝑝𝛾 + 𝑝𝛽 + 𝑥 − 𝑝𝑥 𝑝𝛼 + 𝛾 − 𝑝𝛾 + 𝑥 − 𝑝𝑥 𝛼 𝑼𝑷 against: (𝜏8) RL RM MR RR L −𝛽 + 𝑝𝛾 + 𝑝𝛽 + 𝑥 − 𝑝𝑥 𝛼 − 𝑝𝛼 + 𝑝𝛾 𝑝𝛼 + 𝛾 − 𝑝𝛾 + 𝑥 − 𝑝𝑥 𝛾 + 𝑥 − 𝑝𝑥 M 𝛼 −𝛾 + 𝑝𝛾 + 𝑝𝛼 𝛼 − 𝑝𝛼 − 𝑝𝛾 𝛼 R 𝛾 − 𝑝𝛾 − 𝑝𝛽 + 𝑥 − 𝑝𝑥 𝛼 − 𝑝𝛼 − 𝑝𝛽 𝑝𝛼 − 𝛽 + 𝑝𝛽 + 𝑥 − 𝑝𝑥 −𝛽 + 𝑥 − 𝑝𝑥 Table 2: the expected payoffs of the kick taker when facing two goalkeeper types that both play a pure strategy after sending 𝜏8. The first letter stands for the action of the fast goalkeeper and the second for the action of the slow goalkeeper. The best responses are underlined. We start of by looking for a pooling equilibrium in pure strategies by underlining the best responses of the kick taker. Without making any assumption it is easy to see that: • 𝐵𝑅* 𝐿𝐿, 𝐿𝑀, 𝑀𝐿 = 𝑅. Both goalkeeper types then want to play right as well: 𝐵𝑅) 𝑅 = 𝑅.
Since the best responses of the kicker and the goalkeeper do not match, there is no equilibrium.
• 𝐵𝑅* 𝑅𝑅, 𝑅𝑀, 𝑀𝑅 = 𝐿. Again, both goalkeeper types want to match the kicker’s choice:
𝐵𝑅) 𝐿 = 𝐿. Since the best responses of the kicker and the goalkeeper do not match, there is no equilibrium. In the previous scenarios finding the best responses for the kick taker was straight forward, but we still must check the cases LR and RL. In these two cases, it is not clear what the kicker’s best response is straight away: it depends on the parameter values but without making assumptions on the values anything is possible. We have the following possibilities:
A) The kicker plays a pure strategy L, M or R. In all three cases the best response of both goalkeeper types is to match the kicker’s choice exactly and therefore the cases LR and RL will not occur: there is no equilibrium since at least one of the goalkeepers wants to deviate by matching the kicker’s choice.
B) The kicker mixes between two strategies. a. The kicker mixes L and M In this scenario, it is certain that there is no type of goalkeeper that wants to play R, therefore in each case one of the goalkeeper types wants to deviate. There are thus no equilibria in the cases LR and LR. b. The kicker mixes R and M In this scenario, it is certain that there is no type of goalkeeper that wants to play L, therefore in each case one of the goalkeeper types wants to deviate. There are thus no equilibria in the cases LR and LR c. The kicker mixes L and R We will show that there is no equilibrium of this kind with a proof by contradiction.
To mix between L and R the following must hold: 𝐸 𝑈* 𝐿 = 𝐸 𝑈* 𝑅 >
𝐸 𝑈* 𝑀 . Writing out the kicker’s expected utilities gives us equations (1) and (2): 𝐸 𝑈* 𝐿 = 𝛾 − 𝑝𝛾 − 𝑝𝛽 + 𝑥 − 𝑝𝑥 = 𝐸 𝑈* 𝑅 = −𝛽 + 𝑝𝛾 + 𝑝𝛽 + 𝑥 − 𝑝𝑥 (1) 𝐸 𝑈* 𝐿 = 𝛾 − 𝑝𝛾 − 𝑝𝛽 + 𝑥 − 𝑝𝑥 > 𝐸 𝑈* 𝑀 = 𝛼 (2) Solving these equations results in the following conditions: 𝑝 =P J (I) PJ 𝛾 − 𝛽 + 𝑥 > 𝛼 (II)
Next, we must check that both types of goalkeepers do not want to deviate from their strategies. Suppose the kicker plays L with chance 𝑧 and R with chance 1 − 𝑧, then we have the following payoffs for the fast and slow goalkeepers: 𝐸 𝑈) 𝐿 𝑓𝑎𝑠𝑡 = 𝑧𝛽 − (1 − 𝑧)𝛾 𝐸 𝑈) 𝑀 𝑓𝑎𝑠𝑡 = −𝑧𝛼 − 1 − 𝑧 𝛼 = −𝛼 𝐸 𝑈) 𝑅 𝑓𝑎𝑠𝑡 = −𝑧𝛾 + (1 − 𝑧)𝛽 𝐸 𝑈) 𝐿 𝑠𝑙𝑜𝑤 = 𝑧 𝛽 − 𝑥 + (1 − 𝑧)(−𝛾 − 𝑥) 𝐸 𝑈) 𝑀 𝑠𝑙𝑜𝑤 = −𝑧𝛼 − 1 − 𝑧 𝛼 = −𝛼 𝐸 𝑈) 𝑅 𝑠𝑙𝑜𝑤 = 𝑧(−𝛾 − 𝑥) + (1 − 𝑧)(𝛽 − 𝑥) First, we consider the case LR: the fast goalkeeper plays L and the slow goalkeeper plays R. The proof of the case RL is very similar because of the symmetry of the
game. The fast goalkeeper only sticks to playing L if playing L is at least as good as playing M or R. This gives us the following conditions: 𝑧𝛽 − (1 − 𝑧) ≥ −𝑧𝛼 − 1 − 𝑧 𝛼 = −𝛼 (1a) 𝑧𝛽 − (1 − 𝑧)𝛾 ≥ −𝑧𝛾 + (1 − 𝑧)𝛽 (1b) The slow goalkeeper only sticks to playing R if playing R is at least as good as playing L or M. This gives us the following conditions: 𝑧(−𝛾 − 𝑥) + (1 − 𝑧)(𝛽 − 𝑥) ≥ 𝑧 𝛽 − 𝑥 + (1 − 𝑧)(−𝛾 − 𝑥) (2a) 𝑧(−𝛾 − 𝑥) + (1 − 𝑧)(𝛽 − 𝑥) ≥ −𝑧𝛼 − 1 − 𝑧 𝛼 = −𝛼 (2b)
Rewriting (1b) gives us: 𝑧 ≥PJ and rewriting (2a) yields: 𝑧 ≤PJ, so combined we have that 𝑧 =PJ must hold to meet these conditions. Filling in 𝑧 =PJ in equations 2b gives
us: PJ 𝛽 − 𝛾 − 𝑥 ≥ −𝛼 and thus: PJ 𝛾 − 𝛽 + 𝑥 ≤ 𝛼. Now we have a contradiction
since this scenario must also satisfy: PJ 𝛾 − 𝛽 + 𝑥 > 𝛼, as we can see from condition (II). For this reason, we can conclude that there is no equilibrium of this kind.
C) The kicker mixes between all three strategies.
We will show that there is no equilibrium of this kind with a proof by contradiction. To mix
between L, M and R the following must hold: 𝐸 𝑈* 𝐿 = 𝐸 𝑈* 𝑅 = 𝐸 𝑈* 𝑀 . Writing out the kicker’s expected utilities gives us equations (1) and (2): 𝐸 𝑈* 𝐿 = 𝛾 − 𝑝𝛾 − 𝑝𝛽 + 𝑥 − 𝑝𝑥 = 𝐸 𝑈* 𝑅 = −𝛽 + 𝑝𝛾 + 𝑝𝛽 + 𝑥 − 𝑝𝑥 (1) 𝐸 𝑈* 𝐿 = 𝛾 − 𝑝𝛾 − 𝑝𝛽 + 𝑥 − 𝑝𝑥 = 𝐸 𝑈* 𝑀 = 𝛼 (2) Solving these equations results in the following conditions: 𝑝 =PJ (I) PJ 𝛾 − 𝛽 + 𝑥 = 𝛼 (II) Next, we must check that both types of goalkeepers do not want to deviate from their strategies. Suppose the kicker plays L, M and R with probabilities 𝑝P, 𝑝J and 𝑝Q, then we have the following
payoffs for the goalkeeper types: 𝐸[𝑈) 𝐿 |𝑓𝑎𝑠𝑡] = 𝑝P𝛽 − 𝑝J𝛼 − 𝑝Q𝛾 𝐸 𝑈) 𝑀 |𝑓𝑎𝑠𝑡 = −𝑝P𝛼 + 𝑝J𝛾 − 𝑝Q𝛼 𝐸 𝑈) 𝑅 |𝑓𝑎𝑠𝑡 = −𝑝P𝛾 − 𝑝J𝛼 + 𝑝Q𝛽 𝐸 𝑈) 𝐿 𝑠𝑙𝑜𝑤 = 𝑝P 𝛽 − 𝑥 − 𝑝J𝛼 + 𝑝Q(−𝛾 − 𝑥) 𝐸 𝑈 𝑀 |𝑠𝑙𝑜𝑤 = −𝑝 𝛼 + 𝑝 𝛾 − 𝑝 𝛼
𝐸 𝑈) 𝑅 |𝑠𝑙𝑜𝑤 = 𝑝P(−𝛾 − 𝑥) − 𝑝J𝛼 + 𝑝Q(𝛽 − 𝑥)
Since the game is symmetric, diving left and right only yields the same expected payoff for both goalkeeper types if the probabilities that the kicker shoots left and right are the same. To have
that neither of the goalkeepers has an incentive to deviate therefore 𝐸[𝑈) 𝐿 ] = 𝐸[𝑈)(𝑅)] and
thus 𝑝P= 𝑝Q must hold (in case LR as well as in case RL). Also, we must have that the
goalkeepers are not willing to deviate by staying in the middle, which yields the following requirements:
𝐸 𝑈) 𝐿 𝑓𝑎𝑠𝑡 = 𝑝P𝛽 − 𝑝J𝛼 − 𝑝Q𝛾 ≥ 𝐸[𝑈) 𝑀 |𝑓𝑎𝑠𝑡] = −𝑝P𝛼 + 𝑝J𝛾 − 𝑝Q𝛼
𝐸 𝑈) 𝐿 𝑠𝑙𝑜𝑤 = 𝑝P 𝛽 − 𝑥 − 𝑝J𝛼 + 𝑝Q(−𝛾 − 𝑥) ≥ 𝐸[𝑈) 𝑀 |𝑠𝑙𝑜𝑤] = −𝑝P𝛼 + 𝑝J𝛾 − 𝑝Q𝛼
The second equation is stronger since 𝑥 > 0. Together with the fact that we 𝑝P= 𝑝Q we can
simplify this equation to get that the following must hold: 𝑝J(𝛼 + 𝛾) ≤ 𝑝P(2𝛼 + 𝛽 − 𝛾 − 2𝑥) (A) The kick taker is indifferent and the easiest way for equation (A) to hold is if 𝑝J = 0, which would mean that 𝑝P = 𝑝Q=PJ. Then equation (A) simplifies to: 1 2(2𝛼 + 𝛽 − 𝛾 − 2𝑥) ≥ 0 And furthermore (A) simplifies to: 𝛼 ≥1 2(𝛾 − 𝛽 + 2𝑥) Now we have a contradiction since this scenario must also satisfy: 𝛼 =PJ 𝛾 − 𝛽 + 𝑥 , as we can see from condition (II). For this reason, we can conclude that there is no equilibrium of this kind. There are no pooling equilibria in which both goalkeepers play a pure strategy in any of the scenarios and this concludes the proof of Theorem 3.3.1. Theorem 3.3.2: There exist no pooling equilibria in which both goalkeepers play a pure strategy and use the signals 𝜏7 or 𝜏9 if 𝜀 is close to zero. Proof: The proof of this theorem is very similar to the proof of Theorem 3.3.1 and is included in Appendix C.
Theorem 3.3.3:
There exist no pooling equilibria in which one or more of the goalkeeper types mixes and the goalkeepers use the signals 𝜏7 or 𝜏9 if 2𝛼 + 𝛽 > 𝛾.
Proof: We will prove the theorem by proving it for the case in which both goalkeepers use the signals 𝜏7. The proof for the use of signals 𝜏9 is exactly the same, because of the symmetry in the game. There are two states of the world: one in which the goalkeeper is fast (occurring with chance 𝑝) and one in which the goalkeeper is slow (with chance 1 − 𝑝). The payoff matrices of the two states of the world are displayed below. State of the world 1: 𝜔P With chance 𝑝, the goalkeeper is fast: Signal 𝝉𝑳 fast P L M R L β + ε, -β - ε -𝛼 - ε, 𝛼 + ε -𝛾 - ε, 𝛾 + ε G M -𝛼 + ε, 𝛼 - ε 𝛾 - ε, -𝛾 + ε -𝛼 - ε, 𝛼 + ε R -𝛾 + ε, 𝛾 - ε -𝛼 - ε, 𝛼 + ε β - ε, -β + ε State of the world 2: 𝜔J With chance (1 − 𝑝), the goalkeeper is slow: Signal 𝝉𝑳 slow P L M R L β - x + ε, -β + x - ε -𝛼 - ε, 𝛼 + ε -𝛾 - x - ε, 𝛾 + x + ε G M -𝛼 + ε, 𝛼 - ε 𝛾 - ε, -𝛾 + ε -𝛼 - ε, 𝛼 + ε R -𝛾 - x + ε, 𝛾 + x - ε -𝛼 - ε, 𝛼 + ε β - x - ε, -β + x + ε The goalkeeper knows which type he is and therefore the fast goalkeeper only considers 𝜔P and the slow goalkeeper only considers 𝜔J. To mix, a goalkeeper must be indifferent between his strategies: playing left, middle and right must yield the same expected utility, otherwise he would prefer one strategy over the others and just play this as a pure strategy. Since the goalkeepers only consider their own state of the world, the indifference conditions that the two types of goalkeepers have are
kick taker to randomize in a different manner than the slow goalkeeper. Since both goalkeeper types are indifferent under different conditions we can conclude that there will be no pooling equilibrium in which both the fast and the slow type use mixed strategies. That is why we will look for an equilibrium in which one goalkeeper type plays a pure strategy and the other type mixes. In total, there are six possible equilibria like this: the fast type plays pure L, M or R while the slow type mixes or the slow type plays pure L, M or R while the fast type mixes.
In the first three possible equilibria, the slow goalkeeper uses mixed strategies. From Table 1 we already know under which conditions the slow goalkeeper is willing to mix and what the resulting payoff is. However, as we know from section 3.2 the payoff resulting from sending signal 𝜏7 is smaller
than the payoff in equilibrium that results from using the signal 𝜏8: 𝐸 𝑈)|𝑠𝑙𝑜𝑤, 𝜏8 >
𝐸 𝑈)|𝑠𝑙𝑜𝑤, 𝜏7 . There is one important note to make here: it could in principal be the case that the
kicker does not play an equilibrium strategy in the counterfactual cases. The kicker namely also specifies strategies in the parts of the game tree that are not actually played and these strategies can basically take any form since the chosen strategy does not affect the kicker’s payoff anyway. The subgames are all two person zero-sum games however and therefore the equilibrium strategy is the same as the minimax strategy. In other words: the equilibrium strategy is also the strategy that makes your opponent the worst of. This is the case since in a zero-sum game the gain of one player is the loss of the other, so minimizing your opponent’s payoff is equivalent to maximizing your own. For
this reason, a deviation of the goalkeeper by sending signal 𝜏8 will at least give the goalkeeper the
equilibrium payoff of the subgame 𝜏8. Since sending signal 𝜏8 gives a higher payoff than signal 𝜏7,
the slow goalkeeper wants to deviate and the equilibria in which the slow goalkeeper mixes are therefore excluded. The same reasoning can be used to show that the fast goalkeeper is not willing to stick with mixing after both goalkeepers send signal 𝜏7 if 2𝛼 + 𝛽 > 𝛾, since the payoff of sending
𝜏8 is higher: 𝐸 𝑈)|𝑓𝑎𝑠𝑡, 𝜏8 > 𝐸 𝑈)|𝑓𝑎𝑠𝑡, 𝜏7 . The fast goalkeeper also wants to deviate by sending 𝜏8: we can conclude that there is no pooling equilibrium in which the signal 𝜏7 is used and in which mixed strategies are used by the goalkeepers. Since the game is symmetric the exact same reasoning holds for the use of the signal 𝜏9.
Theorem 3.3.4:
There exist three pooling equilibria in which both goalkeeper types send the signal middle: βm fast = βm slow = τv if 2𝛼 + 𝛽 > 𝛾.
Proof: State of the world 1: 𝜔P With chance p, the goalkeeper is fast: Signal 𝝉𝑴 fast P L M R L β, -β -𝛼, 𝛼 -𝛾, 𝛾 G M -𝛼, 𝛼 𝛾, -𝛾 -𝛼, 𝛼 R -𝛾, 𝛾 -𝛼, 𝛼 β, -β State of the world 2: 𝜔J With chance (1-p), the goalkeeper is slow: Signal 𝝉𝑴 slow P L M R L β - x, -β + x -𝛼, 𝛼 -𝛾 - x, 𝛾 + x G M -𝛼, 𝛼 𝛾, -𝛾 -𝛼, 𝛼 R -𝛾 - x, 𝛾 + x -𝛼, 𝛼 β - x, -β + x
Again, the goalkeeper knows which type he is and only considers his own payoff matrix. The indifference conditions that the two types of goalkeepers have are the same as in the subgames fast
𝜏8 and slow 𝜏8. The subgames are solved in Appendix A and have different solutions as we can see
from Table 1. To be indifferent, the fast goalkeeper requires the kick taker to randomize in a different manner than the slow goalkeeper. Since both goalkeeper types are indifferent under different conditions we can conclude that there will be no pooling equilibrium in which both the fast and the slow type use mixed strategies. That is why we will look for an equilibrium in which one goalkeeper type plays a pure strategy and the other type mixes. In total, there are six possible equilibria like this: the fast type plays pure L, M or R while the slow type mixes or the slow type plays pure L, M or R while the fast type mixes.
First possible equilibrium – fast goalkeeper: L, slow goalkeeper: mix The fast goalkeeper plays a pure strategy L while the slow goalkeeper mixes, so the fast goalkeeper plays (1,0,0) and the slow goalkeeper plays 𝑞P, 𝑞J, 𝑞Q where 𝑞P+ 𝑞J+ 𝑞Q= 1. The kick taker cannot distinguish the two states and therefore considers expected payoff: 𝐸 𝑈* = 𝑝𝐸[𝑈*|𝜔P] + 1 − 𝑝 𝐸[𝑈*|𝜔J] If we use the formula above we get that the kick taker’s expected utilities are given by: 𝐸 𝑈* 𝐿 = −𝑝𝛽 + (1 − 𝑝)[𝑞P −𝛽 + 𝑥 + 𝑞J𝛼 + 𝑞Q 𝛾 + 𝑥 ] 𝐸 𝑈* 𝑀 = 𝑝𝛼 + 1 − 𝑝 [𝑞P𝛼 − 𝑞J𝛾 + 𝑞Q𝛼] 𝐸 𝑈* 𝑅 = 𝑝𝛾 + 1 − 𝑝 [𝑞P 𝛾 + 𝑥 + 𝑞J𝛼 + 𝑞Q −𝛽 + 𝑥 ] The kick taker is willing to mix if: 𝐸 𝑈* 𝐿 = 𝐸 𝑈* 𝑀 = 𝐸 𝑈* 𝑅 . These conditions give the following set of equations: −𝑝𝛽 + 1 − 𝑝 𝑞P −𝛽 + 𝑥 + 𝑞J𝛼 + 𝑞Q 𝛾 + 𝑥 = 𝑝𝛼 + 1 − 𝑝 [𝑞P𝛼 − 𝑞J𝛾 + 𝑞Q𝛼] (1) 𝑝𝛼 + 1 − 𝑝 𝑞P𝛼 − 𝑞J𝛾 + 𝑞Q𝛼 = 𝑝𝛾 + 1 − 𝑝 [𝑞P 𝛾 + 𝑥 + 𝑞J𝛼 + 𝑞Q −𝛽 + 𝑥 ] (2) 𝑞P+ 𝑞J+ 𝑞Q= 1 (3) The solution of this set of equations is given by (a stepwise solution of the system of equations is provided in Appendix B): 𝑞P= 𝛼 + 𝛾 − 𝑝𝑥 (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑞J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 + 2𝑝𝑥 (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑞Q=𝛼 + 𝛾 − 𝑝(4𝛼 + 𝛽 + 𝛾 − 𝑥) (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥)
For 𝑝 ∈ (0,HEFIFGKxEFG ) this solution is feasible:
The chance of facing a fast goalkeeper 𝑝 must be small enough, that is: 𝛼 + 𝛾 − 𝑝 4𝛼 + 𝛽 + 𝛾 − 𝑥 > 0 must hold, so that 𝑞Q> 0. Therefore, 𝑝 <HEFIFGKxEFG must hold. Since HEFIFGKxEFG < 1 this
condition also makes sure that we do not run into trouble with the denominators, since 𝑝 ≠ 1. Of course, there is also an equilibrium if 𝑝 = 0, but this would just mean that the kick taker can only
face a slow goalkeeper and we have the same equilibrium as in the subgame ‘slow 𝜏8’, which is not
really a pooling equilibrium since then there are no longer different types.
The slow goalkeeper mixes in this possible equilibrium and to be indifferent he therefore requires the kick taker to mix in the same way as in the subgame slow 𝜏8 (the solution to this subgame is given in Appendix A). The kick taker thus randomizes in the following manner: 𝑝P = 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝑝J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 Next the resulting expected payoffs can be obtained. For the slow goalkeeper, we of course have the same expected payoff as in the subgame slow 𝜏8: 𝐸(𝑈)|𝑠𝑙𝑜𝑤) = −2𝛼J+ 𝛽𝛾 − 𝛾J− 2𝛾𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥
This payoff is higher than the payoff obtained by sending signal 𝜏7 or 𝜏9 and therefore the slow
goalkeeper has no incentive to deviate (as shown in section 3.2). The fast goalkeeper plays the pure strategy left and his expected payoff is therefore given by: 𝐸 𝑈) 𝐿 𝑓𝑎𝑠𝑡 = 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥𝛽 − 2𝛼 + 𝛽 − 𝛾 − 2𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥𝛼 − 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥𝛾 =−2𝛼J+ 𝛽𝛾 − 𝛾J+ 2𝛼𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥
This payoff is higher than the payoff obtained by sending signal 𝜏7 or 𝜏9 (as shown in section 3.2) and
is higher than the payoff obtained by sending signal 𝜏8 and playing M (see Table 3). The fast
goalkeeper therefore has no incentive to deviate. The fast type of goalkeeper is very happy with this equilibrium: the existence of the slow goalkeeper allows him to get an even higher payoff than he would be able to get without the slow goalkeeper!
For the kick taker, we fill in the found probabilities for 𝑞P, 𝑞J and 𝑞Q in one of the formulas (since L,
M and R give the same utility): 𝐸 𝑈* 𝐿 = −𝑝𝛽 + (1 − 𝑝)[𝑞P −𝛽 + 𝑥 + 𝑞J𝛼 + 𝑞Q 𝛾 + 𝑥 ] = −𝑝𝛽 + (1 − 𝑝)[ 𝛼 + 𝛾 − 𝑝𝑥 1 − 𝑝 4𝛼 + 𝛽 + 𝛾 − 2𝑥 −𝛽 + 𝑥 + 2𝛼 + 𝛽 − 𝛾 − 2𝑥 + 2𝑝𝑥 1 − 𝑝 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝛼 + 𝛼 + 𝛾 − 𝑝 4𝛼 + 𝛽 + 𝛾 − 𝑥 1 − 𝑝 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝛾 + 𝑥 =2𝛼 J− 𝛽𝛾 + 𝛾J+ 2𝛾𝑥 − 2𝑝(𝛼𝑥 + 𝛾𝑥) 4𝛼 + 𝛽 + 𝛾 − 2𝑥
The kicker’s expected payoff is thus given by: 𝐸 𝑈* =2𝛼J− 𝛽𝛾 + 𝛾J+ 2𝛾𝑥 − 2𝑝(𝛼𝑥 + 𝛾𝑥) 4𝛼 + 𝛽 + 𝛾 − 2𝑥 Since the slow goalkeeper randomizes in such a way that shooting left, middle and right all give the same payoff, the kicker has no incentive to deviate either and this makes the equilibrium complete. Taking a closer look at the kicker’s expected payoff shows us that he gets the same expected payoff as in subgame slow 𝜏8 in the case that there are no fast goalkeepers (𝑝 = 0). If 𝑝 > 0 however, the
kicker receives a penalty in expected payoff for meeting the fast type of goalkeeper.
In Table 3 and Table 4 are the results from all the equilibrium candidates displayed. To be sure that these candidates are indeed equilibria we need to make sure that none of the players wants to deviate. We will start of by discussing the three possible equilibria in which the slow goalkeeper mixes. As just showed in the previous paragraph is the scenario in which the fast goalkeeper plays left an equilibrium since no player has an incentive to deviate. Playing right as a fast goalkeeper is also an equilibrium since the game is perfectly symmetric. The scenario in which the fast goalkeeper plays middle is not an equilibrium however, because the fast goalkeeper can do better in this case by playing left or right: 𝐸 𝑈* 𝐿 𝑜𝑟 𝑅 𝑓𝑎𝑠𝑡 =KJE OFIGKGOFJEx HEFIFGKJx > 𝐸 𝑈* 𝑀 𝑓𝑎𝑠𝑡 KJEOFIGKGOKJGx HEFIFGKJx . Now we will look at the possible equilibria in which the fast goalkeeper mixes (see Table 4). The fast goalkeeper is not willing to deviate in any of the three possible equilibria since sending signal 𝜏7 or 𝜏9
gives him a worse payoff than sending signal 𝜏8 if 2𝛼 + 𝛽 > 𝛾, as shown in section 3.2. The kick
taker has no incentive to deviate in any of the three cases either since shooting left, middle and right all result in the same payoff (the kicker mixes because he is indifferent). If we compare the payoffs of the slow goalkeeper we can see that the slow goalkeeper wants to deviate in the cases that he plays
left or right since playing middle gives him a higher expected payoff: 𝐸 𝑈* 𝑀 𝑠𝑙𝑜𝑤 =
KJEOFIGKGO
HEFIFG > 𝐸 𝑈* 𝐿 𝑜𝑟 𝑅 𝑠𝑙𝑜𝑤 =
KJEOKJExFIGKGOKJGx
HEFIFG . In conclusion, we have three pooling
equilibria in total: two in which the slow goalkeeper mixes and the fast goalkeeper plays either left or right and one in which the fast goalkeeper mixes and the slow goalkeeper plays middle.
fast L, slow mix
fast goalkeeper: slow goalkeeper: kick taker:
L 1 𝑞P= 𝛼 + 𝛾 − 𝑝𝑥 (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 M 0 𝑞J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 + 2𝑝𝑥 (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 R 0 𝑞Q= 𝛼 + 𝛾 − 𝑝(4𝛼 + 𝛽 + 𝛾 − 𝑥) (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝐸[𝑈] −2𝛼J+ 𝛽𝛾 − 𝛾J+ 2𝛼𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 −2𝛼J+ 𝛽𝛾 − 𝛾J− 2𝛾𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 2𝛼J− 𝛽𝛾 + 𝛾J+ 2𝛾𝑥 − 2𝑝(𝛼𝑥 + 𝛾𝑥) 4𝛼 + 𝛽 + 𝛾 − 2𝑥 condition on p 𝟎 < 𝒑 < 𝜶 + 𝜸 𝟒𝜶 + 𝜷 + 𝜸 − 𝒙 fast M, slow mix
fast goalkeeper: slow goalkeeper: kick taker:
L 0 𝑞P= 𝛼 + 𝛾 (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 M 1 𝑞J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 − 𝑝(4𝛼 + 𝛽 + 𝛾 − 2𝑥) (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 R 0 𝑞Q= 𝛼 + 𝛾 (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝐸[𝑈] −2𝛼4𝛼 + 𝛽 + 𝛾 − 2𝑥J+ 𝛽𝛾 − 𝛾J− 2𝛾𝑥 −2𝛼 J+ 𝛽𝛾 − 𝛾J− 2𝛾𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 2𝛼J− 𝛽𝛾 + 𝛾J+ 2𝛾𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 condition on p 𝟎 < 𝒑 < 𝟐𝜶 + 𝜷 − 𝜸 − 𝟐𝒙 𝟒𝜶 + 𝜷 + 𝜸 − 𝟐𝒙 fast R, slow mix
fast goalkeeper: slow goalkeeper: kick taker:
L 0 𝑞P= 𝛼 + 𝛾 − 𝑝(4𝛼 + 𝛽 + 𝛾 − 𝑥) (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 M 0 𝑞J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 + 2𝑝𝑥 (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 R 1 𝑞Q= 𝛼 + 𝛾 − 𝑝𝑥 (1 − 𝑝)(4𝛼 + 𝛽 + 𝛾 − 2𝑥) 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 − 2𝑥 𝐸[𝑈] −2𝛼4𝛼 + 𝛽 + 𝛾 − 2𝑥J+ 𝛽𝛾 − 𝛾J+ 2𝛼𝑥 −2𝛼 J+ 𝛽𝛾 − 𝛾J− 2𝛾𝑥 4𝛼 + 𝛽 + 𝛾 − 2𝑥 2𝛼J− 𝛽𝛾 + 𝛾J+ 2𝛾𝑥 − 2𝑝(𝛼𝑥 + 𝛾𝑥) 4𝛼 + 𝛽 + 𝛾 − 2𝑥 condition on p 𝟎 < 𝒑 < 𝜶 + 𝜸 𝟒𝜶 + 𝜷 + 𝜸 − 𝒙 Table 3: Pooling equilibrium candidates in which the fast goalkeeper plays a pure strategy
slow L, fast mix
fast goalkeeper: slow goalkeeper: kick taker: L 𝑞P= −3𝛼 − 𝛽 + 𝑥 + 𝑝(4𝛼 + 𝛽 + 𝛾 − 𝑥) 𝑝(4𝛼 + 𝛽 + 𝛾) 1 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 M 𝑞J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 + 2𝑝𝑥 𝑝(4𝛼 + 𝛽 + 𝛾) 0 𝑝J= 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 R 𝑞Q= 𝛼 + 𝛾 + 𝑥 − 𝑝𝑥 𝑝(4𝛼 + 𝛽 + 𝛾) 0 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝐸[𝑈] −2𝛼J+ 𝛽𝛾 − 𝛾J 4𝛼 + 𝛽 + 𝛾 −2𝛼J− 2𝛼𝑥 + 𝛽𝛾 − 𝛾J− 2𝛾𝑥 4𝛼 + 𝛽 + 𝛾 2𝛼J− 𝛽𝛾 + 𝛾J+ 2(1 − 𝑝)(𝛼𝑥 + 𝛾𝑥) 4𝛼 + 𝛽 + 𝛾 − 2𝑥 condition on p 𝟑𝜶 + 𝜷 − 𝒙 𝟒𝜶 + 𝜷 + 𝜸 − 𝒙< 𝒑 < 𝟏 slow M, fast mix
fast goalkeeper: slow goalkeeper: kick taker:
L 𝑞P= 𝛼 + 𝛾 𝑝(4𝛼 + 𝛽 + 𝛾) 0 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 M 𝑞J= −2𝛼 − 2𝛾 + 𝑝(4𝛼 + 𝛽 + 𝛾) 𝑝(4𝛼 + 𝛽 + 𝛾) 1 𝑝J= 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 R 𝑞Q= 𝛼 + 𝛾 𝑝(4𝛼 + 𝛽 + 𝛾) 0 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝐸[𝑈] −2𝛼4𝛼 + 𝛽 + 𝛾J+ 𝛽𝛾 − 𝛾J −2𝛼 J+ 𝛽𝛾 − 𝛾J 4𝛼 + 𝛽 + 𝛾 2𝛼J− 𝛽𝛾 + 𝛾J 4𝛼 + 𝛽 + 𝛾 condition on p 𝟐𝜶 + 𝟐𝜸 𝟒𝜶 + 𝜷 + 𝜸< 𝒑 < 𝟏 slow R, fast mix
fast goalkeeper: slow goalkeeper: kick taker:
L 𝑞P= 𝛼 + 𝛾 + 𝑥 − 𝑝𝑥 𝑝(4𝛼 + 𝛽 + 𝛾) 0 𝑝P= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 M 𝑞J= 2𝛼 + 𝛽 − 𝛾 − 2𝑥 + 2𝑝𝑥 𝑝(4𝛼 + 𝛽 + 𝛾) 0 𝑝J= 2𝛼 + 𝛽 − 𝛾 4𝛼 + 𝛽 + 𝛾 R 𝑞Q= −3𝛼 − 𝛽 + 𝑥 + 𝑝(4𝛼 + 𝛽 + 𝛾 − 𝑥) 𝑝(4𝛼 + 𝛽 + 𝛾) 1 𝑝Q= 𝛼 + 𝛾 4𝛼 + 𝛽 + 𝛾 𝐸[𝑈] −2𝛼4𝛼 + 𝛽 + 𝛾J+ 𝛽𝛾 − 𝛾J −2𝛼 J− 2𝛼𝑥 + 𝛽𝛾 − 𝛾J− 2𝛾𝑥 4𝛼 + 𝛽 + 𝛾 2𝛼J− 𝛽𝛾 + 𝛾J+ 2(1 − 𝑝)(𝛼𝑥 + 𝛾𝑥) 4𝛼 + 𝛽 + 𝛾 − 2𝑥 condition on p 𝟑𝜶 + 𝜷 − 𝒙 𝟒𝜶 + 𝜷 + 𝜸 − 𝒙< 𝒑 < 𝟏 Table 4: Pooling equilibrium candidates in which the slow goalkeeper plays a pure strategy
3.4 Overview of results
We have seen that the MSNE of the subgames are informative for the equilibria in the whole game. There are no separating equilibria since sending out signals 𝝉𝑳 and 𝝉𝑹 turned out not be
beneficial for either of the goalkeeper types. In retrospect, we could have expected this result since
sending out the signal 𝝉𝑳 only has a positive effect on the goalkeeper’s payoff when the goalkeeper
ultimately plays left and a negative effect on both middle and right. The kicker knows this and makes use of it by shooting more frequently in the middle and the right to obtain a higher payoff on
average. The same reasoning holds for the signal 𝝉𝑹 because of the symmetry of the game.
As previously shown in this chapter coincide the mixed strategies used in the MSNE in the subgames with the strategies suggested by the minimax rule. This result, together with the result
that the goalkeepers prefer sending the signal 𝜏8, makes sure that there are no pooling equilibria in
which the goalkeepers use the signals 𝜏7 and 𝜏9.
Finally, we obtained three pooling equilibria and in all of them the type that is in the minority benefits. There is one equilibrium in which the fast type mixes and two in which the slow goalkeeper mixes. If the slow goalkeeper is in a vast minority he can benefit from this by playing a pure strategy middle. The kicker knows that the slow goalkeeper in this case will always stay in the middle, but frankly he does not care because the probability of meeting the slow goalkeeper is so small. The slow goalkeeper in his turn prefers to stay in the middle because he finds it harder to dive to the corners. Then we have two equilibria in which the fast goalkeeper is in the minority and either plays left or right while the slow goalkeeper mixes. The fast goalkeeper excels in this scenario because he no longer has the need to be surprising: the kicker knows that the fast goalkeeper will play a pure strategy in these equilibria but since the chance of facing the fast goalkeeper is very small this does not affect the way he wants to play. It can easily be seen though that the kicker’s payoff is a decreasing function of 𝑝 (the probability of facing a fast goalkeeper). All together we can conclude that the results are intuitive and have a clear interpretation.
4. Summary and Conclusion
The aim of this research was to propose a more realistic model for penalty kicks and find out if the proposed extensions are a valuable addition to the matching pennies game that is used thus far in the field of game theory. If we compare the proposed model for penalty kicks with the basic matching pennies model we can draw two important conclusions. First, sending a signal of going left or right as a goalkeeper is never beneficial in the discussed model. A signal can only be beneficial if it influences the kick taker to shoot less accurate by making the kick taker nervous. This research however assumes that players are perfectly rational and therefore nerves or other irrational behaviour is ruled out. A good extension to the proposed model is a model where the goalkeeper can not only choose to send a signal, but where he can also choose the strength of the signal. However, if the payoffs are structured in the same way as in this model, sending a signal will never be beneficial. This extension could be interesting, but only if there is proof that sending a signal influences the quality with which the kick taker strikes the ball and this is also considered in the model.
Second, the extension that goalkeepers can be of different types does have an interesting interpretation: the pooling equilibria that we found tell us that it can be beneficial to be a minority. Clearly it is better to be the fast goalkeeper than the slow goalkeeper, but if there are a lot of fast goalkeepers, the slow goalkeeper can get the same expected payoff as the fast type by just staying in the middle. The expected payoff the slow goalkeeper then generates is higher than the payoff he would get if there were only slow goalkeepers. This means that it does not matter as a goalkeeper that you are slower than usual if the kick taker you are facing does not know about it or at least assigns a small probability to it. But it also works the other way around: a fast goalkeeper in a world full of slow goalkeepers excels even more than in a world with only fast goalkeepers. Since the kick taker usually faces a slow goalkeeper, he shoots more frequently in the corners than he would if there were only fast goalkeepers. The fast goalkeeper therefore likes to do what he does best: dive to one of the corners and profit from the fact that he is in the minority.
Because of the scope of this research hybrid equilibria were not considered but this could be interesting for further research. Another extension that could be made to the proposed model is dropping the assumption that shooting left and right is equally hard, since some players might find it easier to shoot in a certain corner. Finally, it might be interesting to look at a repeated version of the game, which occurs in a penalty shootout. It is very likely that players will behave differently based on the penalty kicks taken before. For example: due to bounded rationality, the kick takers might try
to avoid shooting left multiple times in a row since they may think this outcome is too unlikely to occur at random. With minor adjustments, the developed model for penalty kicks can be used for economic purposes too. An example is the strategic interaction between a company and a tax inspector. There are two players: the inspector and the company. First, the company fills in a tax report that reports either a high income, a middle income or a low income: this is a signal and we can compare it with sending the signals 𝜏7, 𝜏8 and 𝜏9 in our original model. Next the tax inspector can decide on three
levels of inspection: a full investigation, a regular check-up or no inspection at all. Where a full investigation has a higher cost than a regular check-up and executing no inspection does not bear any costs. In this game, we have one player (the inspector) who only wants to monitor the other player (the company) when there is tax fraud to be detected, while the company only wants to be inspected if they did not commit fraud. This game is similar to a matching pennies game since the inspector wants to ‘match’ the inspection to the fraud while the company wants to do the opposite. If the company is inspected after they truthfully filled in their tax report it is good for their reputation and if they committed fraud they obviously want to hide it because of the punishment the inspector can address. Of course, this is just one specific example, we could also use the model for other economic applications where monitoring is of interest, such as in principal-agent theory.
