Watching Baseball with the Run Expectancy Matrix

Note: This post first appeared on the sonsofsamhorn.com web site in May 2015 and is re-posted here with permission.

During Thursday night’s Red Sox-Mariners game, in the top of the second with none out and no score, Xander Bogaerts attempted to advance from first to second on a ball in the dirt. Mike Zunino made a good throw to Robinson Cano at second and Bogaerts was out. 

Given the challenges the Red Sox have been having in scoring runs recently, it wasn’t surprising to see Xander force the Mariners to make a good play here. During the NESN broadcast, Steve Lyons went out of his way to emphasize that he liked the “aggressiveness” displayed by Bogaerts, despite the outcome.

In the interest of full disclosure, I have known Steve for many years and respect him immensely. In my opinion, he has great baseball instincts, borne of decades of experience both as a player and as a broadcaster.

And as a generalization, my observation is that baserunners (and base coaches) usually err on the side of being too conservative in attempting to advance. This is a natural human tendency – in uncertain situations, a player or coach who chooses caution cannot be proven wrong. Whereas one who takes the more aggressive path runs the risk of making an out that all can criticize.

In applied statistics, this is referred to as the Type I/Type II error dilemma. A Type I error is to do something that turns out not to be correct. Here’s an example from a different field – business recruiting. A Type I error would be to hire a candidate that does not work out. In this case, the error is highly visible and exposed to criticism. In contrast, a Type II error would be to fail to hire a candidate that would have been an excellent choice. In this case, the error is by definition unobservable, and as a result, immune to criticism. So, it is human nature to lean toward making Type II errors and choose not to take action, thereby avoiding possible criticism, in order to avoid risking Type I errors and the possible criticism that could come with the decision if it was wrong.

Given my trust in Steve’s baseball acumen, coupled with my belief that baserunners tend to be too conservative, my initial reaction was “Good observation, Steve. Too bad for Xander that it did not work out but good to see him being aggressive.”

But there’s a somewhat more objective way to assess these situations, with the aid of what is known as the Run Expectancy Matrix. This insightful tool was created by well-known sabremetrician Tom Tango and made popular in “The Book: Playing the Percentages in Baseball”.

This handy analytical aid looks at the 24 possible states of a game situation – eight different baserunner combinations, times three different “number of outs” (none, one, or two). Each cell in the resulting matrix shows the expected number of runs that result from that point to the end of that inning, based on empirical analysis of actual MLB results.

Here’s an example, based on all MLB results from games played in 2013:

The cell reflecting the game state before the pitch was thrown on which Xander attempted to advance is circled. On average across all game situations where a team had no one out and a runner at first, they scored 0.8262 runs.

(Note that while these results vary slightly from year to year, they are generally remarkably stable – with the exception of some significant run-scoring inflation during the years that correlate with the most egregious alleged usage of steroids, peaking in 1998. The matrix can be found online here).

Now let’s consider this particular play. When the ball hits the dirt, Xander can either stay put (with an expected result that inning of 0.8262 runs), or attempt to advance. If he attempts to advance, two outcomes are possible. He is out, and the game state changes to none on and one out, or he makes it and the game state becomes runner at second and none out. These two outcomes and the resulting gain or loss in expected runs is shown here:

So we see that if Xander advances, he creates an additional 0.2237 expected runs. But if he fails in his attempt to move up (as happened on Thursday), he destroys 0.5773 expected runs.

Now we can calculate his breakeven probability of success – the probability at which staying put or trying for the extra base have the same expected value . This is pretty straightforward – if we call the probability of success X, then this breakeven point occurs when:

0.2237 * X – 0.5773 * (1-X) = 0

And a bit of high school algebra yields X = 0.7207

Let’s consider what this means. It says that Xander needs to be 72% sure he is going to make it to second before it is a good idea to try to advance.

While no generally accepted definitions of “aggressive” and “conservative” baserunning exist, I would offer that aggressive baserunning is attempting to advance when the likelihood of success is less than 50%, and conservative baserunning is attempting to advance only when the likelihood of success is greater than 50%.

In this situation then, the run expectancy matrix suggests that a baserunner should be a bit conservative. Interestingly, both Steve’s and my instinctive reactions ran counter to this analytical insight.

Now, please do not conclude that this is an exact science. Other factors certainly influence the decision making here, such as who is coming up next, who is on the mound and how important that run is given the score and inning. Advanced sabremetricians also adjust this matrix for the “run scoring environment” which attempts to account for how much easier or more difficult it is to score runs in this particular park compared to the league average. But I think we can all agree it represents an improvement over going with your gut.

If you are so inclined, I suggest you use this matrix to test some baseball adages. For instance the old saw that goes “Never make the first or third out at third base.” Or refer to it the next time a third base coach sends a runner who ends up getting thrown out at home – how sure did he need to be that the runner would score for the send to be the “right” decision (regardless of the outcome)?

There is a world of insight and wisdom embedded in this simple yet powerful tool.