Sabermetric Overview Series Part III: The Matrix and Smallball

Today we're going to be looking at a lot of numbers, but I promise it won't be that bad. When sabermetricians talk about the Matrix they're referring to a table that shows the average number of runs a team scores in various baseball situations. Here's one such matrix:

Run Expectancy Matrix 2004

RUNNERS       0       1       2
    ---  0.5379  0.2866  0.1135
    1--  0.9259  0.5496  0.2460
    -2-  1.1596  0.7104  0.3359
    12-  1.4669  0.9577  0.4605
    --3  1.4535  0.9722  0.3623
    1-3  1.8540  1.2236  0.5219
    -23  2.1343  1.4717  0.6179
    123  2.2548  1.5946  0.8082

Stay with me! I promise it's not as bad as it looks. See, in the bottom left hand corner it shows that with no outs and the bases load a team scored an average of 2.2548 runs in 2004. Looking at the top right we see that with the bases empty and 2 outs we can "expect" a team to score only 0.1135 runs (or usually zero). These 24 possibilities, 0-2 outs, bases empty to loaded and every possibility in between, comprise every situation a team might find itself in.


It's important to note that these numbers were reached by looking at the results of real baseball games involving humans. We're not just pulling them out of our saberasses. We're looking at 2004, but any full year is enough of a sample and it doesn't change to any great degree. Here is a table showing 1999-2002. It's basically the same.

Now let's talk sacrifice bunts. Say we have a runner on first with no outs. Consulting the table above we see we can expect to score 0.9259 runs this inning. Not too shabby. But say we have the next batter lay down a bunt and selfishly sacrifice him to second. Now we have a runner on second and one out... and our run expectancy has fallen to 0.7104.

Sacrificing a runner from 2nd with no outs to 3rd with one is similarly hurtful (1.1596 to 0.9722). This gets to one of the core discoveries of the sabermetric movement: the importance of outs. In all but the most specific situations outs are more valuable than bases. In 1984 in their classic book The Hidden Game of Baseball Pete Palmer and John Thorn flatly stated, "The sacrifice bunt ... is a bad play. ...With the introduction of the lively ball, the sacrifice bunt should have vanished..." Basically a whole lot of what we learned about "good baseball", what we're constantly reminded of by broadcasters is wrong.


When someone talks about the batter in the second spot of the lineup needing to have bat control to hit behind the runner, they're essentially advocating sacrificing a runner over, thus hurting their scoring chances in the first inning. In 2007 this is not at all a good move. Earl Weaver once famously said, "If you play for one run, that's all you'll get."

But what if you only need one run? What if the score is tied in the 9th inning? It's important to remember that the run expectancy matrix takes an average. It includes all those big 6-run innings that sacrificing cuts down on. But if we only need to score one run we don't care about that. Here's a table that shows the chances of scoring 1 run:

2005 Scoring Expectation (percentage)
Runners   0 Out   1 Out   2 Outs
empty     28.0    16.5     7.1
1st       41.7    27.2    12.7
2nd       62.5    41.0    22.9
3rd       82.7    66.1    25.4
1+2       61.6    41.4    22.8
1+3       84.6    64.5    26.8
2+3       86.1    67.4    26.6
loaded    85.6    65.4    30.7

You should know now right where to look now. Runner on first no outs scores a run 41.7% of the time, runner on second 1 out scores 41% of the time. It's still not a good play. However moving a runner from second to third with the first out does increase your chances of scoring one run (62.5 to 66.1).

So far we've been assuming the sacrifice attempt will succeed. But what if it doesn't? Also doesn't bunting put pressure on the defense, which could lead to errors? Who's ready for another superfun table?!

Sacrifice Bunt Outcomes 2000-2005 (percentage)
Situation       Sacrifice  Success  Failure  DP
1st, 0 Out       72.9       13.6     11.2   2.0  
1st, 1 Out       71.6       10.6     14.4   3.2
2nd, 0 Out       69.5       18.6     11.4   0.5
1st+2nd, 0 Out   67.0       17.1     13.4   2.3

In this table, "Sacrifice" means it went according to plan--the runner advanced and an out was made. "Success" means an error, no outs were made, and/or the runner got an extra base--unexpected good stuff. "Failure" means the defense got the lead runner or an out was made and the runner couldn't advance. "DP" is a type of failure but is listed seperately because it's so devastating--a double play.

What we see here is that the secondary effects of the sacrifice--putting pressure on the defense, making stuff happen--are basically completely mitigated by the bad things that can happen. In the cases with a runner on first the failures and double plays occur more often than the successes. Bunting from second to third has more success.

Now remember when we were looking at the decreased scoring resulting from sacrificing a runner from first to second (whether overall or one run)? That came from comparing the before-and-after situations of a successful sacrifice, essentially assuming it would work. But we see here that it only works out aobut 85% of the time (adding the sacrifice and success columns) so it's even a worse play than we thought.

Before we end let's briefly talk about stealing bases. Stealing is all about weighing the benefit and the cost. Looking back at our matrix above we see that stealing second with no outs (going from 1-- 0 out to -2- O out) increases our run expectancy from 0.9259 to 1.1596. That's good. But if the runner got thrown out (empty, 1 out) our run expectancy would drop to 0.2866. That's bad. In fact that's way more bad than the successful stolen base was good.

Really smart guys with computers looked at this and many other factors and concluded that if you don't successfully steal at least 75% of the time you're better off not trying to steal at all. Previously conventional wisdom had it that if you succeeded over 50% of the time you were doing well (after all it's more good than bad, right? Wrong, because the cost of the bad far outweighs the benefit of the good).

The secondary effects of stealing bases--distracting the pitcher, pressuring the defense--don't add up to anything. Studies show that batters tend to fare worse with an agressive basestealer on first, possibly because they're taking good pitches or swinging at bad ones to protect the runner.

This is the basics of why saber-types are against "smallball." Outs are precious and giving them up leads to less scoring.

Further reading: The Book: Playing the Percentages in Baseball Chapter 9: Sacrifice Bunt, Chapter 11: Base Stealing

Baseball Between the Numbers: Why Everything You Know About the Game is Wrong Chapter 4-2: When Is One Run Worth More Than Two?

Baseball Prospectus Basics: Stolen Bases and How to Use Them

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