Problem: Consider a (simplified) game played between a pitcher (who chooses between throwing a fastball or a curve) and a batter (who chooses which pitch to expect). The batter has an advantage if he guesses the pitch that is actually thrown. This is a constant-sum game, where the payoff is measured by the probability that the batter will get a base hit (a high probability of a hit is good for the batter and bad for the pitcher).

If a pitcher throws a fastball, and the batter guesses fastball, the probability of a hit is 0.300.
If the pitcher throws a fastball, and the batter guesses curve, the probability of a hit is 0.200.
If the pitcher throws a curve, and the batter guesses curve, the probability of a hit is 0.350.
If the pitcher throws a curve, and the batter guesses fastball, the probability of a hit is 0.150.

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

(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. find the rollback equilibrium.

Solution:

 (b) Pitcher throws a fastball; batter guesses fastball.

(c) Batter guesses curve; pitcher throws a fastball