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?

Keeper Center Left Right Left 0.35, 0.65 0.90, 0.10 0.85. 0.15 Shooter Center 0.30, 0.70 0.25, 0.75 0.45, 0.55 Right 0.95, 0.05 0.90, 0.10 0.30, 0.70