Consider the penalty kick game below.

Striker goes East Striker goes West

Keeper goes East 1 , 0 0 , 1

Keeper goes West 0 , 1 1 , 0

a. Find all Nash (mixed strategy and pure strategy) equilibria of this game.

b. Now say that there is a Sports Illustrated photographer on the west side of the stadium. The striker really likes having their photo taken. So if the striker scores by kicking in the west side of the goal, then the striker not only gets the goal but gets an additional "bonus" of 1. The keeper's payoffs do not change. Model this as a strategic form game. Find all Nash (mixed strategy and pure strategy) equilibria of this game. Note that the coaches of the two teams do not care at all about photographs, and only care about whether a goal is scored. Does the existence of the photographer change the overall probability that a goal is scored?

c. Now say that both keeper and striker really like having their photo taken. Again, if the striker scores by kicking in the west side of the goal, then the striker not only gets the goal but gets an additional "bonus" of 1. Also, if the keeper does a successful save by blocking the shot on the west side of the goal, the keeper gets the save and also an additional "bonus" of 1. Model this as a strategic form game. Find all Nash (mixed strategy and pure strategy) equilibria of this game. Does the existence of the photographer change the overall probability that a goal is scored?

d. Now say that instead of a bonus of 1, both keeper and striker get a bonus of b, where b is a number which is greater than zero (for example, part c. above corresponds to b = 1). As before, if the striker scores by kicking in the west side of the goal, then the striker not only gets the goal but gets an additional "bonus" of b. If the keeper does a successful save by blocking the shot on the west side of the goal, the keeper gets the save and also an additional "bonus" of b. Find all Nash (mixed strategy and pure strategy) equilibria of this game. As b increases (in other words, as keeper and striker become more vain), does the overall probability that a goal is scored increase, decrease, or stay the same?