# Consider a simplified baseball game played between a pitcher and a batter. The pitcher chooses…

Consider a simplified baseball game played between a
pitcher and a batter. The pitcher chooses between throwing a fastball or a
curve, while the batter chooses which pitch to anticipate. The batter has an
advantage if he correctly anticipates the type of pitch. In this constant-sum
game, the batter’s payoff is the probability that the batter will get a base
hit. The pitcher’s payoff is the probability that the batter fails to get a
base hit, which is simply one minus the payoff of the batter. There are four
potential outcomes:

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(i) If a pitcher throws a fastball, and the batter
guesses fastball, the probability of a hit is 0.300.

(ii) If the pitcher throws a fastball, and the batter
guesses curve, the probability of a hit is 0.200.

(iii) If the pitcher throws a curve, and the batter
guesses curve, the probability of a hit is 0.350.

(iv) If the pitcher throws a curve, and the batter
guesses fastball, the probability of a hit is 0.150.

Suppose that the pitcher is “tipping” his pitches. This
means that the pitcher is holding the ball, positioning his body, or doing
something else in a way that reveals to the batter which pitch he is going to
throw. For our purposes, this means that the pitcher-batter game is a
sequential game in which the pitcher announces his pitch choice before the
batter has to choose his strategy.

(a) Draw this situation, using a game tree.

(b) Suppose that the pitcher knows he is tipping his
pitches but can’t stop himself from doing so. Thus, the pitcher and batter are
playing the game you just drew. Find the rollback equilibrium of this game.

(c) Now change the timing of the game, so that the batter
has to reveal his action (perhaps by altering his batting stance) before the
pitcher chooses which pitch to throw. Draw the game tree for this situation,
and find the rollback equilibrium.

Now assume that the tips of each player occur so quickly
that neither opponent can react to them, so that the game is in fact
simultaneous.

(d) Draw a game tree to represent this simultaneous game,
indicating information sets where appropriate.

(e) Draw the game table for the simultaneous game. Is
there a Nash equilibrium in pure strategies? If so, what is it?