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.

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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 payoff 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 payoff since the objective of the kicker is to maximize the probability of scoring while the objective of the goalkeeper is to minimize it.

Goalkeeper Left Center Right Left 0.65, 0.35 0.95, 0.05 0.95, 0.05 Kicker Center 0.95, 0.05 0.00, 1.00 0.95, 0.05 Right 0.95, 0.05 0.95, 0.05 0.65, 0.35