BCS 66 In soccer, when a match is about to end in a draw, it comes down to penalty
kicks to decide on the winner. This was the case for instance in the match between
France and Italy in the 2006 World Cup Final in Germany.
In penalty kicks, the kicker tries to kick the ball into the goal 12 yards away and the
goalkeeper of the opposing team tries to prevent it. The kicker has to decide to kick it
either into the left corner, the center, or the right corner of the goal. The goalkeeper has
to decide whether to dive in to the left corner, the center, or the right corner of the goal.
Since the ball is extremely fast (70 to 105 miles per hour), the decisions of the kicker and
goalkeeper are essentially taken simultaneously.
In the below payo matrix, the left entry shows the frequencies of scoring a goal, and
the right entry is the frequency of not scoring a goal. We can take these frequencies as
payo since the objective of the kicker is to maximize the probability of scoring while
the objective of the goalkeeper is to minimize it.
Derive the Nash equilibrium. Suppose there is a kicker who can kick much better into
the left corner of the goal and that's why he decides always to kick into the left corner.
Assume further that the opposing team knows about it because they observed the kicker
in training. What is the best response of the goalkeeper.
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