# Run For Third!

Imagine that it’s a tie game in the bottom of the ninth inning and there is one out. You are a base runner on first base. The outfield is playing back to prevent a double and the hitter hits a shallow fly ball into right field. There is no way to tell if the right fielder will catch it. You must wait between first base and second base to see if the ball will be caught. This is to guarantee that if the ball is caught, you may return to first base without an embarrassing double play, ending the inning.

Except that this strategy lowers your chances of scoring a run. We can prove it, and we did so in the April 2016 edition of Math Horizons, a publication of the Mathematical Association of America (MAA). (Our article, Run for Third! A Defense of Aggressive Base Running is free to members of the MAA, and most university libraries will subscribe to the journal Math Horizons).

The fact is that at the major league level, your chances of scoring from third base with one out are so much higher than your chances of scoring from second base with one out, that you should run even if you don’t know whether or not the ball will be caught.

To be more specific, if I (an aggressive base runner) see the ball hit to right field, and don’t know if the outfielder will get there on time, I will run and I might make it to third base instead of second base. Sometimes this aggressiveness will result in a double play. If you play the conventional way, you may end up at second base. Sometimes (if the ball is caught) you will end up at first base with TWO outs.

The analysis is therefore simple. Compare the advantage of my way versus the advantage of your way. If the ball falls in for a hit, I have the advantage. I’m at third base, where major leaguers score (with one out) more than 60% of the time. You are at second, where major leaguers score (with one out) less than 40% of the time.

If the ball is caught, you have the advantage. But it turns out this advantage is small. At the major league level, a base runner scores from first base with two outs about 14% of the time (only 13% in high leverage situations, like a tie game in the bottom of the ninth).

Each strategy has an advantage—mine when the ball falls in for a hit, and yours when the ball is caught. We can compare these two advantages by using a little bit of undergraduate mathematics (hence our choice of a famously student-accessible journal). It turns out that my advantage is greater than yours. It’s not close. As long as the game situation suggests that playing for one run is the correct strategy, then this version of aggressive base running is also the correct strategy.

This has been missed by modern statisticians simply because it hasn’t been tried. So please try it! *Please* let me know if you see a game where it was used, or a game where it should have been used and wasn’t.

Thanks to my coauthors, Peter Macdonald and Ian McQuillan. We’ve been talking about this for some time. We’ve tried posts before (like this one), as well as other preliminary preprint versions of the paper that were not as clear as this publication. So please read this article for the details!