Asked by Michael Frank on May 02, 2024
Verified
Consider the goalie's anxiety at the penalty kick.Let the kicker's payoffs below represent the kicker's probability of success and the goalie's payoffs the probability of failure. 
The goalie should defend left with probability
A) .60.
B) .20.
C) .40.
D) .80.
E) 1.
Penalty Kick
A penalty kick in economics might refer metaphorically to an opportunity to score or gain significantly with little opposition, though it's primarily a sports term.
Probability of Success
A quantitative measure expressing the likelihood of achieving a specified outcome or reaching a goal.
Probability of Failure
The likelihood that a system or component will fail to perform its intended function within a specified period.
- Understand the concepts of probability and strategy in decision-making.
- Grasp the concept of Nash equilibrium in game theory.
Verified Answer
ZK
Zybrea KnightMay 07, 2024
Final Answer :
B
Explanation :
The best strategy for the goalie is to randomize between left and right with different probabilities. Let p be the probability that the goalie defends left, and thus 1-p is the probability that the goalie defends right. The kicker's payoffs are represented by the first number in each box and the goalie's payoffs are represented by the second number in each box.
If the goalie defends left:
- If the kicker shoots left, the goalie saves the goal with probability 0.6 and gains 1 point.
- If the kicker shoots right, the goalie saves the goal with probability 0.2 and gains 2 points.
If the goalie defends right:
- If the kicker shoots left, the goalie saves the goal with probability 0.2 and gains 2 points.
- If the kicker shoots right, the goalie saves the goal with probability 0.4 and gains 1 point.
The goalie's expected payoff from defending left is:
0.6*(1) + 0.4*(2) = 1.4
The goalie's expected payoff from defending right is:
0.2*(2) + 0.4*(1) = 0.8
Therefore, the goalie should defend left with probability p=0.2 to maximize their expected payoff.
If the goalie defends left:
- If the kicker shoots left, the goalie saves the goal with probability 0.6 and gains 1 point.
- If the kicker shoots right, the goalie saves the goal with probability 0.2 and gains 2 points.
If the goalie defends right:
- If the kicker shoots left, the goalie saves the goal with probability 0.2 and gains 2 points.
- If the kicker shoots right, the goalie saves the goal with probability 0.4 and gains 1 point.
The goalie's expected payoff from defending left is:
0.6*(1) + 0.4*(2) = 1.4
The goalie's expected payoff from defending right is:
0.2*(2) + 0.4*(1) = 0.8
Therefore, the goalie should defend left with probability p=0.2 to maximize their expected payoff.
Learning Objectives
- Understand the concepts of probability and strategy in decision-making.
- Grasp the concept of Nash equilibrium in game theory.