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I was watching Fox Saturday Baseball and for what felt like the tenth thousandth time in the five years I am watching MLB, the guy in the booth said something like „bla bla bla, you should not make the first or third out at third base, bla bla bla“. This not-making-the-first-or-third-out-at-third-thing seems to be one of the ten baseball commandment along with „playing the game the right way“ and „with two outs you have to be hacking“ etc.
Unlike that hacking-thing, the out-at-third rule seems to make some sense. Making the first or third out at third certainly hurts your team, but then again, so does making the second out at third. Why is the rule not „never make an out at third“ or, even better, „never make an out on the basepaths“? Conventionell wisdom says that with no outs, you might as well stay at second because a ground ball to the right side or a long fly ball will still get you to third with less than two outs. On the other hand, that ground ball or long fly ball could score you if you already were on third. With two outs, the logic is that you are already in scoring position and will score on a hit anyway. But on third, you could also score on a passed ball or an infield hit.
I think it is pretty clear that you are always better off on third than on second and that making an out while trying for third is bad, no matter how many outs there are. What the rule really implies is that the advantage of getting to third with only one out is so great that it is worth the risk, but the advantage of getting there with zero or two outs is not worth the same risk. This, of course, is something that we can easily test using the run expectancy matrix from tangotiger and a little decision theory. The matrix applies for the years 1999 – 2002 and needs to be adjusted to the current run scoring environment, but I think those four years are good enough to get idea.
With a runner on second and no outs, 1.189 runs are expected to be scored in this inning. If the runner gets to third with no outs, this increases to 1.482, while it drops to 0.297 when the runner gets thrown out and the bases are empty with one out. Now we can calculate the probability of making is safely to third that yields the same expected runs for staying at second and trying for third: 1.482 * p + 0.297 * (1-p) = 1.189 is solved by p = 75.27%. So if you think the chances of making it to third safely with no outs are a little better than three in four, you should go for it.
Now with one out, being on second will yield 0.725 expected runs, while being on third is worth 0.983 runs. Bases empty with two down only leads to 0.117 runs on average. Again, we solve 0.983 * p + 0.117 * (1-p) = 0.725 for p and get p = 70.21%. And with two down, being at second is worth 0.344, while getting a base further increases that to 0.387. Getting thrown out at third obviously makes it impossible to score any more runs, since the inning ends immediately. Solving 0.387 * p + 0 * (1-p) = 0.344 gives us p = 88.89%.
So if you are heading for third with two down, you should to be pretty sure to make it there safely, just like the rule tells us. On the other hand, contrariwise to what conventional wisdom tells us, you can be a little more willing to take a risk with no outs than with one out. I guess one of the reasons for this surprising (?) result is that getting from second to third with no outs seems easy enough, but actually is not that common as one might think. While there are only two „productive outs“ in this situation, namely a groundball to the right side and a long fly out (if we ignore the bunt), there are plenty of outs that do not get the job done: ground ball to the left side, dribbler back to the pitcher, pop-up, line-out, strike-out and short fly all keep you stranded at second. So get to third while you can, especially with no outs.
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