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If a soccer game ends in a tie it goes

If a soccer game ends in a tie, it goes into a penalty-kick shootout in which each team chooses five players to take penalty kicks. The team that makes the most subsequent penalty kicks wins the game. In a penalty-kick shootout, the shooter and the keeper each decide simultaneously on a direction to move. They can choose left, right, or middle.

These strategies yield the following game theory table, where the first value is the shooter’s probability of scoring and the second value is the keeper’s probability of stopping the shot:

This is an example of a constant-sum game since each pair of entries in the game theory table sums to 1. This can be analyzed in the same manner as a zero-sum game.

a. Use dominance to reduce the game to a 2 × 2 game. Which strategies are dominated?

b. What is the solution to this game for the shooter and for the keeper?

c. What is the shooter’s expected probability ofscoring?

These strategies yield the following game theory table, where the first value is the shooter’s probability of scoring and the second value is the keeper’s probability of stopping the shot:

This is an example of a constant-sum game since each pair of entries in the game theory table sums to 1. This can be analyzed in the same manner as a zero-sum game.

a. Use dominance to reduce the game to a 2 × 2 game. Which strategies are dominated?

b. What is the solution to this game for the shooter and for the keeper?

c. What is the shooter’s expected probability ofscoring?

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