I've heard Angels broadcaster Mark Gubicza tell batters they must "focus on the fastball" and pitchers to be "careful of the breaking ball" when there is a runner on third far, far too many times. His reasoning is straightforward: a wild pitch on a slider in the dirt scores the runner from third, leading the pitcher to want to throw the fastball more frequently, in turn encouraging the batter to prepare for more fastballs. But rather than trusting Mark Gubicza, I did some investigating of my own. Turns out he was only partially right...

I'm a game theorist, but I will try to keep this as intuitive as possible. There's a simple game at work here: batters can either guess fastball or slider (in a simple, two-pitch setup) and pitchers can throw fastball or slider. If the batter guesses correctly, he is more likely to record a hit. And if the pitcher throws a slider, there is some chance the runner from third base will score on a wild pitch.

You can read the math behind the solution in the manuscript under review at SABR here (http://wjspaniel.wordpress.com/about/) or watch a video solving it:

Regardless, if the players are playing optimally, the batter guesses fastball more frequently than with the bases empty, but the pitcher throws sliders at the same frequency as with the bases empty. Essentially, the batter displays risk-averse behavior; he knows a slider will score the runner with some probability anyway, so he maximizes his minimum payoff by more heavily favoring fastball.

But despite the runner on third, if the pitcher throws fastballs more frequently than he did before, then the batter can always focus on a fastball and do better than he could when he was merely guessing fastball more frequently. If the batter would be doing better, then the pitcher would be doing worse, so the pitcher maintains the same fastball/slider ratio.

One other interesting tidbit: because the pitcher maintains the same fastball/slider ratio, the batter does not guess the correct pitch more frequently than he did with the bases loaded. This implies that the risk of a wild pitch does improve the batter's batting average.