# Home runs, spin, and optimal pitching strategy

The WSJ's Science Journal column (\$) has an interesting discussion of topspin, backspin, and home runs. It turns out that, ceteris paribus (speed of pitch, bat, and accuracy of contact), a curve ball is more likely to be hit out of the park than a fastball. Why? According to Mont Hubbard of UC Davis, it's in the spin.

When a curveball leaves the pitcher's fingers it has topspin, which means the top of the ball rotates in the direction of flight (toward the plate). Fastballs, in contrast, have backspin, with the bottom of the ball rotating in the direction of flight. Topspin causes a ball to experience a downward force, because the rotation changes the distribution of air pressure around the ball so there is more pressing down on the ball than up. Hence curveballs' habit of suddenly plunging, to batters' dismay. Backspin, in contrast, generates an upward force, somewhat like the one that keeps an airplane aloft, which is why a fastball rises unless the pitcher gives it a countervailing spin.

When the bat makes contact, the most obvious thing it does is reverse the ball's direction, so it heads toward the field rather than the plate. But contact also changes the ball's spin. Assuming good contact in each case, a fastball that arrived with backspin therefore leaves with topspin, while a curveball arriving with topspin leaves with additional backspin and thus more home run potential.

"A curveball already has batted backspin," says Prof. Hubbard. "With a fastball, in order to give it backspin and let it benefit from aerodynamics, you have to reverse the spin," which is tough to do. The well-hit curveball heads for the field with more of the kind of spin that gives it fence-clearing lift and distance.

Let's put this in the context of optimal pitching strategy. Used sparingly enough, a curve ball is more likely to fool the batter, reducing the accuracy of contact. This offsets the effect of spin on the likelihood of hitting the ball out of the park. Thinking about this using economic logic yields some interesting conclusions. First, assume that only home runs matter, or at least that the probability of hitting a home run summarizes the (in)effectiveness of a pitcher. Second, assume that as a given pitch is thrown with greater frequency, it is more likely to be expected by the batter, and thus more likely to be hit out of the park. That is, there is declining marginal effectiveness of each pitch type. Third (and this is false but useful for the moment), assume that the effectiveness of one pitch type has no impact on the effectiveness of another.

Under these conditions, optimal pitching strategy implies that the probability of hitting one out of the park is the same for all pitches thrown by a given pitcher. If one pitch had a lower probability at all times, that would be the only pitch ever thrown - what economists call a "corner solution." This come close to describing some modern day closers. They throw fastball after fastball for a period of short duration (an inning). Where the simple analysis above goes wrong is in the assumption of independence across pitch types. A curve may be thrown despite the greater likelihood that it's blasted out of the park, if it makes the fastball more effective, reducing that probability. But again, the economic prediction is that the higher probability of the curve being hit is offset by the reduction in probability for the fastball - the effects

balance each other in an optimal strategy.

The physics and the economics are both interesting. I wonder what the data say?