Suppose that the soccer penalty-kick game of Section 7.A in this chapter is expanded to include a total of six distinct strategies for the kicker: to shoot high and to the left (HL), low and to the left (LL), high and in the center (HC), low and in the center (LC), high right (HR), and low right (LR). The goalkeeper continues to have three strategies: to move to the kicker’s left (L) or right (R) or to stay in the center (C). The players’ success percentages are shown in the following table:
In this problem, you will verify that the mixed-strategy equilibrium of this game entails the goalie using L and R each 42.2% of the time and C 15.6% of the time, while the kicker uses LL and LR each 37.8% of the time and HC 24.4% of the time.
(a) Given the goalie’s proposed mixed strategy, compute the expected payoff to the kicker for each of her six pure strategies. (Use only three significant digits to keep things simple.)
(b) Use your answer to part (a) to explain why the kicker’s proposed mixed strategy is a best response to the goalie’s proposed mixed strategy.
(c) Given the kicker’s proposed mixed strategy, compute the expected payoff to the goalie for each of her three pure strategies. (Again, use only three significant digits to keep things simple.)
(d) Use your answer to part (a) to explain why the goalie’s proposed mixed strategy is a best response to the kicker’s proposed mixed strategy.
(e) Using your previous answers, explain why the proposed strategies are indeed a Nash equilibrium.
(f) Compute the equilibrium payoff to the kicker.