1- 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. The Free Kick Kicker Kick Left Kick Right JumpLe Jump Right 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 asp changes. (a) If the goalie jumps left with probability :rG , then if the kicker kicks right, his probability of scoring is (b) If the goalie jumps left with probability arc , then if the kicker kicks left, his probability of scoring is (C) Find the probability :rG 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 :rK , then if the goalie jumps left, the probability that the kicker will not score is (e) If the kicker kicks left with probability 3K , then if the goalie jumps right, the probability that the kicker will not score is (I) Find the probability x that makes the payoff to the goalie equal from jumping left or jumping right. (9) 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. 2. Two software companies sell competing products. These products are substitutes, so that the number of units that either company sells is a decreasing function of its own price and an increasing function of the other product's price. Let P1 be the price and X1 the quantity sold of product 1 and let p2 and X2 be the price and quantity sold of product 2. 1 1 1 1 Then x1 =1000(90- Ep1 +Zp2) and x2 =1000(90- .5132 +Zp1} Each company has incurred a fixed cost for designing their software and writing the programs, but the cost of selling to an extra user is zero. Therefore each company will maximize its profits by choosing the price that maximizes its total revenue