In baseball, pitchers and batters have conflicting goals. Pitchers want to get the ball past batters but batters want to connect with pitched balls. The following ta ble shows the payoffs for a hypothetical game between a pitcher and a batter. The payoffs take into account the fact that fastballs, when anticipated and hit by a batter, yield better hits (more bases) than do similarly anticipated and hit curve balls.
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PITCHER
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Throw curve
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Throw fastball
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BATIER
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Antici pate curlle
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2
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-1
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Anticipate fastball
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-1
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4
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(a) Verify that there is no pure-strategy equilibrium in this game, and then find the mixed-strategy equilibrium.
(b) Draw the best-response curves for both players, and indicate the mixed strategy equilibrium on your diagram.
(c) Can the pitcher improve his expected payoff in the mixed-strategy equi librium of this game by slowing down his fastball (thereby making it more similar to a curve ball)? Assume that slowing down the pitched fastball changes the payoff to the hitter in the "anticipate fastball/ throw fastball" cell from 4 to 3. Explain carefully how you determine the an swer here and show your work. Also explain why slowing the fastball can or cannot improve the pitcher's expected payoff in the game.