Question: In baseball, a pitcher faces a batter. If hit, the coming pitch will produce the gamewinning run, if not the team loses. There are two types of pitch-a fastball and a curve. If the batter swings at the fastball he will hit it out of the park, but if he does not swing it will be strike three. If the batter swings at a curve ball he will likely miss it, but if he lets it go by he will walk and the winning run will come home. (Disregard foul balls.) Construct a game that portrays this situation and show that it has a solution in mixed strategies. Then ask how the pitcher's and batter's choices will change if

(1) instead of the ninth inning it is the first, and

(2) if there is only one out, but the next to bat (as in the National League) will be the pitcher. (Assume that a pinch hitter is unavailable.)