Consider the following (simplified) situation of a penalty kick in a football match. The kicker (from...

  

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Consider the following (simplified) situation of a penalty kick in a football match. The kicker (from her point of view) can either aim for the left or the right corner of the goal. The goalkeeper can either jump left or right (from the point of view of the kicker). The utility of scoring a goal is +1 for the kicker, and 0 else, while the utility of the goalkeeper is -1 if conceding a goal, and 0 else. If the goalkeeper jumps to the opposite direction of the kick, then a goal is scored for sure. The goalkeeper is slightly better covering the left corner such that if the kicker aims left and the goalkeeper jumps into the left corner, then the shot gets blocked for sure. However, if the kicker aims for the right corner and the goalkeeper jumps to the right corner, then the shot gets blocked only with probability p such that 0 < p < 1, i.e. with probability 1-p a goal is scored even if the goalkeeper and the kicker both aim for the right corner. Both players are expected utility maximisers. (a) Write down the game in strategic form. (b) Determine the unique Nash equilibrium. (c) Suppose the goalkeeper improves covering the right corner of the goal such that he manages to block the kick with probability p such that p < p < 1 if both aim for the right corner. In the unique Nash equilibrium, how will the goalkeeper change his behaviour in response to his improvement? How will the kicker react to the goalkeeper's improvement? Interpret and explain your answer. [4] [6] [6] (d) Suppose now p = 1 but the kicker discovered that there is also an option to just kick into the centre (hoping that the keeper will select one of the corners). However, when aiming for the centre, the keeper blocks the penalty for sure if he just chooses to cover the centre, and even if he jumps into either corner, there is a chance of o that he blocks the penalty, where 0 <o</ What is the unique Nash equilibrium here? [9] Consider the following (simplified) situation of a penalty kick in a football match. The kicker (from her point of view) can either aim for the left or the right corner of the goal. The goalkeeper can either jump left or right (from the point of view of the kicker). The utility of scoring a goal is +1 for the kicker, and 0 else, while the utility of the goalkeeper is -1 if conceding a goal, and 0 else. If the goalkeeper jumps to the opposite direction of the kick, then a goal is scored for sure. The goalkeeper is slightly better covering the left corner such that if the kicker aims left and the goalkeeper jumps into the left corner, then the shot gets blocked for sure. However, if the kicker aims for the right corner and the goalkeeper jumps to the right corner, then the shot gets blocked only with probability p such that 0 < p < 1, i.e. with probability 1-p a goal is scored even if the goalkeeper and the kicker both aim for the right corner. Both players are expected utility maximisers. (a) Write down the game in strategic form. (b) Determine the unique Nash equilibrium. (c) Suppose the goalkeeper improves covering the right corner of the goal such that he manages to block the kick with probability p such that p < p < 1 if both aim for the right corner. In the unique Nash equilibrium, how will the goalkeeper change his behaviour in response to his improvement? How will the kicker react to the goalkeeper's improvement? Interpret and explain your answer. [4] [6] [6] (d) Suppose now p = 1 but the kicker discovered that there is also an option to just kick into the centre (hoping that the keeper will select one of the corners). However, when aiming for the centre, the keeper blocks the penalty for sure if he just chooses to cover the centre, and even if he jumps into either corner, there is a chance of o that he blocks the penalty, where 0 <o</ What is the unique Nash equilibrium here? [9]

Answer rating: 100% (QA)

a Kicker Uleft corner left 1 Uleft corner right p Uright corner left 1 Uright corner right 0 Goalkee... View the full answer