Question

Game theory problem:

Penalty kicks in soccer.

Let's consider a situation where a football player has to faceoff the goalkeeper in a penalty kickoff.

Standing infront of the goalpost, the Kicker (player 1) has several angle which he could kick the ball to the goalpost. Let's say he could kick the ball in the Left corner of the goalpost, Right corner of the goalpost or shoot straight through the Center. And, same as the player, the goalkeeper (player 2) also has three options to predict which direction the player would kick the ball and try to stop it.

This game can be represented using the following 3×3 matrix:

Left Center Right
Left (61/100,39/100) (93/100,7/100) (92/100,8/100)
Center (93/100,7/100) (0/100,100/100) (93/100,7/100)
Right (90/100,10/100) (90/100,10/100) (65/100,35/100)
In the above matrix, the payoff of the kicker is the probability that he scores and the payoff of the goalkeeper is the probability that the kicker doesn't score. We know that the total probability of an event is 1, therefore, all the payoffs sum up to 1.

Find a mixed strategy σ1=(p1,p2,(1−p1−p2)) for Player 1 that will make Player 2 indifferent about playing Left, Center or Right.
Find a mixed strategy σ2=(q1,q2,(1−q1−q2)) for Player 2 that will make Player 1 indifferent about playing Left, Center or Right.
What is the probability that the Kicker(player 1) scored a goal, or in other words what is the expected utility for player 1.

Answer 1

Answer:

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