In a baseball game, a pitcher throws a ball towards a batter who tries to hit it. In simplified game, the pitcher can pitch high or pitch low, and the batter can swing high or swing low. The ball moves so fast that the battar has to commit to swinging high or swinging low before the ball is released.

If the pitcher throw high and the batter swing low, or the pitcher throw low and the batter swing high, the batter misses the ball, so the pitcher wins.

If pitcher throw high and the batter swing high, the batter always connect. If the picther throws low and the batter swings high, batter will connect only half the time.

This story lead us to following payoff matrix, where if the batter hits the ball, he gets a payoff of 1 and the pitcher gets 0, and the batter misses, the pitcher gets a payoff of 1 and the batter gets 0

Batter

Swing high Swing low

Pitcher Pitch high. 0;1. 1;0 Pitch Low 1;0. 0.5 ; 0.5

a) does this game have a pure-strategy nash equilibrium? if so find it

b) does this game has a mixed-strategy equilibrium? if so find it