Consider the goa lie'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).38.
B).63.
C).13.
D).88.
E)1.

Question

Consider the goa lie'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).38. 
B).63. 
C).13. 
D).88. 
E)1.

Quizplus · Accepted Answer

The optimal strategy for the goalie is to mix their actions, making the kicker uncertain about where they will defend. Let x be the probability of defending left, and (1-x) be the probability of defending right. Then, the expected payoff for the kicker for shooting left is:
0.8x + 0.2(1-x) = 0.6x+0.2
And the expected payoff for shooting right is:
0.3x + 0.7(1-x) = 0.7 - 0.4x
The kicker will choose to shoot left if 0.6x+0.2>0.7-0.4x, or equivalently, if x>0.45. 
The optimal strategy for the goalie is to set x=0.45, so that the kicker is indifferent between shooting left and shooting right:
0.6(0.45)+0.2 = 0.7-0.4(0.45)
0.32 = 0.32
Therefore, the goalie should defend left with probability 0.45, and right with probability (1-0.45) = 0.55. 
Converted to percentages, the goalie should defend left with probability 45% and right with probability 55%, which is closest to option B) 63%.

Consider the Goa Lie's Anxiety at the Penalty Kick