On June 14, the Pirates were ahead 2-0 over the Dodgers in the bottom of the third inning. With one out, Andrew McCutchen broke for home on a ground ball towards shortstop Hanley Ramirez. Although the throw was slightly to the first base side of the plate, catcher AJ Ellis got the swipe tag on McCutchen, and the gamethread erupted into complaints about what a stupid decision it was to send Cutch on contact.
But, as always, we have to be careful to not confuse results with process. Was having McCutchen run on contact a bad decision, or was it a good decision with a bad outcome?
In the third inning, run expectancy is a good tool to use to evaluate decisions*. At this point in the game, the approach should be to maximize runs scored, as opposed to playing for a single run.
First of all, the only situation that matters to the analysis is what happens on a ground ball hit within the reach of an infielder. If a ground ball is hit through the infield, the run will score regardless of whether he runs on contact or waits for it to go through. On a fly ball, the runner will wait to see where it goes - tagging up if it's caught or scoring if it's not, regardless of what he would have done had the ball been hit on the ground.
So even though the runner on third doesn't wait to see if a ground ball is within reach of an infielder, the analysis only evaluates those cases.
If the runner on third doesn't go on contact and the batter hits a ground ball within reach of an infielder, we can assume that the batter will be thrown out at first, and the runner will remain on third. The result is a runner on third and two out, which has a run expectancy of 0.3634.
If the runner on third does go on contact and the batter hits a ground ball within reach of an infielder, we can assume two outcomes: 1, the runner is out, with probability p. 2, the runner is safe, with probability (1-p).**
In outcome 1, the result is a runner on first and two out, which has a run expectancy of 0.2213. In outcome 2, the result is a runner on first and one out, which has a run expectancy of 0.5115, plus 1 for the run that scored, or 1.5115.
The run expectancy of the contact play is therefore p * (0.2213) + (1-p) * (1.5115).
We're playing to maximize run expectancy, so the runner should go on contact if the run expectancy of the contact play is greater than the run expectancy of not going on contact - or p * (0.2213) + (1-p) * (1.5115) > .3634.
Algebraic rearrangement gives p < .89.
In other words, the runner should go on contact if the chance of him getting thrown out at the plate on a ball hit within reach of an infielder is less than 89%. The runner should not go on contact only if the chance of him getting thrown out at the plate on a ball hit within reach of an infielder is more than 89%.
So: What's the chance of McCutchen getting thrown out at the plate if an infielder is able to field a ground ball? Honestly, I don't know where to look to try to estimate that, beyond taking some guesses. I'm thinking with Cutch running it's maybe around 50/50. I'd be comfortable saying it's somewhere between a 25% chance and a 75% chance of throwing him out. I'm very comfortable saying that if Cutch goes on contact 10 times, he is not going to get thrown out 9 - which is the rate that would be required for the decision to run on contact to be a bad one.
Bottom line - good decision, bad outcome.
* Run expectancy taken from the 2012 full-season values on BB-Pro.
** There are more possible outcomes, of course. The fielder could throw the ball away, allowing the runner to score and the batter to advance to second. The runner could stop midway between third and home and get caught in a rundown if he sees the throw will beat him easily, allowing the batter to go to second. These outcomes are low probability and don't affect the outcome significantly, so we'll ignore them.