Baseball game theory, part 1

While with family in North Carolina I got my first chance to play a Wii, the newest Nintendo system.  It uses the motion-sensing Wiimote to translate your arm and hand movements into game movements, so, for example, you can swing the Wiimote like a tennis racket to have your game player swing his racket.  My brother and I played a game of Wii Sports baseball.  The game is vastly simplified compared to most baseball video games:  There are only three innings, it only allows you to bat and pitch (the computer takes care of the fielding and baserunning) and, most importantly, a pitch goes right down the middle of the strike zone no matter what arm motion you use, with only one exception that we found.  I guess this last property is a result of Nintendo trying to make the system too forgiving.

I quickly figured out that my most effective pitches were a fastball (down the middle) and a splitter in the dirt—the other two available pitches, a curveball and a screwball, were relatively easy to hit as they were slower and always ended up right down the middle anyway.  The fastball was most effective with a fast wrist snap; the splitter was only unhittable when thrown in the dirt with a slow wrist snap (it was the only way to throw a pitch that wasn't right down the middle).  The two were hard to tell apart on the screen until it was too late, so the pitcher and the batter were essentially trying to outguess the other.

The pitchers dominated, and the game was scoreless into the bottom of the third (and last) inning when my brother got a hit.  I sensed that he was reading my wrist speed to determine which pitch I was throwing, so after I got two strikes on him I tried to cross him up by throwing a fastball but with a slow wrist snap, effectively throwing him a changeup.  I expected him to take it for an unexpected third strike, but he outguessed me and hit a game-winning homer.  If only I'd thought ahead one more step and thrown a splitter in the dirt I might have had him.  (I also regret not trying a splitter with a faster wrist snap instead . . .)

My defeat got me thinking about applying game theory to baseball.  For example, consider the following situation:  The score is tied and the bases are loaded with two out and a full count in the bottom of the ninth.  For the sake of simplicity, let's assume the batter has two options, swing at or take the pitch, and the pitcher has two, throw a fastball for a strike or a splitter in the dirt.  The pitcher's splitter is so effective that the batter can't tell it from a fastball until it's too late.  A ball will walk in the winning run for the home team and a strike will send the game into extra innings, presumably giving each team a 50% chance of winning.  Let's further assume that if the batter swings at a fastball strike, he has a 50% chance of getting a hit and winning the game and a 50% chance of getting out.  Each player must decide what to do without knowing what the other will do.  So the resulting probabilities that the home team will win are

		pitcher
		strike	ball
batter	swing	75%	50%
	take	50%	100%

In this oversimplified situation, what should each of the two players decide to do?  If both players use their optimal strategies, what's the resulting probability that the home team will win?  (Hint:  It's somewhere between 50% and 75%.)