Question

Consider the following game for a penalty kick in a game of soccer. Player 1 is the kicker. They can kick the ball towards the left side of the goal or the right side. Player 2 is the goalie and can anticipate and position to block the left side of the goal or the right side.

The kicker has a more powerful kick when kicking towards the right side of the goal. So when kicking to the right side, the odds of making the goal are greater. However, the goalie can reduce the odd's of a goal by anticipating the side that the kicker chooses.
The players's choices happen simultaneously, so when kicking, the kicker does not yet see which side the goalie is favoring, and the goalie when choosing a side to defend does not yet see where the kicker is aiming. The strategic grid is given below where the payoffs are the probability of scoring for the Kicker and the probability of blocking the kick for the goalie.
Goalie
Defend Left Defend Right
Kicker Aim Left 0.1 , 0.9 0.7 , 0.3
Aim Right .85 , .15 0.2 , 0.8

Let 'p' be the probability that the kicker aims left and 1-p is the probability that the kicker aims right.
Let ' q ' be the probability that the goalie defends left and 1- q is the probability that the goalie defends right.

find the mixed strategy nash equilibrium.
in the mixed strategy NE, what is the probability that the kicker aims left?
p=?

Answer 1

To find the mixed strategy Nash Equilibrium in the soccer penalty kick scenario, simultaneous equations are set up for the kicker and goalie's expected payoffs, ensuring both strategies are equally favorable. These equations are then solved to find the probabilities 'p' and 'q' which indicate the likelihood of each player choosing to aim or defend left.

Step-by-step explanation:

To find the mixed strategy Nash Equilibrium (NE) in this soccer penalty kick game, we set up equations representing the expected payoffs for the kicker and goalie based on their probabilities of aiming/defending left or right. The kicker's expected payoff for aiming left (EL) and aiming right (ER) should be equal at equilibrium, as should the goalie's expected payoffs for defending left (DL) and defending right (DR).

For the kicker, we have:

• EL = 0.1q + 0.7(1 - q)
• ER = 0.85q + 0.2(1 - q)

Setting EL = ER to find q:

0.1q + 0.7(1 - q) = 0.85q + 0.2(1 - q)

Solving this equation for q gives the goalie's probability of defending left. The kicker's p is then found by equating the goalie's expected payoffs for defending left and right:

• DL = 0.9p + 0.15(1 - p)
• DR = 0.3p + 0.8(1 - p)

Setting DL = DR to find p:

0.9p + 0.15(1 - p) = 0.3p + 0.8(1 - p)

Solving this equation for p gives the kicker's mixed strategy NE probability of aiming left.