Let€™s have another look at the soccer example that was discussed in the text. But this time, we will generalize the payoff matrix just a little bit. Suppose the payoff matrix is as follows.

Now the probability that the kicker will score if he kicks to the left and the goalie jumps to the right is p. We will want to see how the equilibrium probabilities change as p changes.
(a) If the goalie jumps left with probability πG, then if the kicker kicks right, his probability of scoring is _____________
(b) If the goalie jumps left with probability πG, then if the kicker kicks left, his probability of scoring is ___________
(c) Find the probability πG that makes kicking left and kicking right lead to the same probability of scoring for the kicker. (Your answer will be a function of p.) ____________
(d) If the kicker kicks left with probability πK, then if the goalie jumps left, the probability that the kicker will not score is ________
(e) If the kicker kicks left with probability πK, then if the goalie jumps right, the probability that the kicker will not score is __________
(f ) Find the probability πK that makes the payoff to the goalie equal from jumping left or jumping right. _____________
(g) The variable p tells us how good the kicker is at kicking the ball into the left side of the goal when it is undefended. As p increases, does the equilibrium probability that the kicker kicks to the left increase or decrease? ___________ Explain why this happens in a way that even a TV sports announcer might understand _________________________________

Transcribed Image Text:

The Free Kick Kicker Kick Left Kick Right Jump Left Goalie Jump Right I, 0 1, 0