Exercise S7 in Chapter 6 introduced a simplified version of baseball, and part (e) pointed out that the simultaneous-move game has no Nash equilibrium in pure strategies. This is because pitchers and...

Exercise S7 in Chapter 6 introduced a simplified version of baseball, and part (e) pointed out that the simultaneous-move game has no Nash equilibrium in pure strategies. This is because pitchers and batters have conflicting goals. Pitchers want to get the ball past batters, but batters want to connect with pitched balls. The game table is as follows:

(a) Find the mixed-strategy Nash equilibrium to this simplified baseball game.

(b) What is each player’s expected payoff for the game?

(c) Now suppose that the pitcher tries to improve his expected payoff in the mixed-strategy equilibrium by slowing down his fastball, thereby making it more similar to a curve ball. This changes the payoff to the hitter in the “anticipate fastball/throw fastball” cell from 0.30 to 0.25, and the pitcher’s payoff adjusts accordingly. Can this modification improve the pitcher’s expected payoff as desired? Explain your answer carefully and show your work. Also, explain why slowing the fastball can or cannot improve the pitcher’s expected payoff in the game.