Game theroy

Consider a game of baseball. The pitcher can throw either a fastball or a curveball; the batter can either swing at the pitch or take (not swing). These choices are simultaneous for each pitch. On the first pitch, if the batter swings at a curveball or takes a fastball, he strikes out and gets 0. If the batter swings at a fastball, he has a probability of 0.75 of hitting a home run and getting a payoff of 1 and a probability of 0.25 of hitting a flyball and getting 0. If the batter takes a curveball, there is a second pitch.

On the second pitch, the first three combinations (swing at a curveball, take a fastball, and swing at a fastball) work as before. If a batter takes a curveball second pitch, he walks and gets a payoff of 0.25.

This is a zero-sum game. The two pitches constitute a sequential move game while each individual pitch is a simultaneous move game.

Solve this game. Find the pitcher's and batter's equilibrium strategies and expected payoffs for the first and second pitch in the subgame perfect equilibrium.

Why is the batter's probability of swinging so small?