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May 6, 2004

Taking One for the Team

When Does it Make Sense to Sacrifice?, Part 1

by James Click

One of the most striking discoveries of much of the statistical research done in baseball over the last 20 years is that outs are more valuable than bases. This breakthrough means that stolen bases are only good when the stolen base percentage is above a certain break-even point. Furthermore, it means that "sacrifices" are an extremely bad idea if you're trying to score runs, which we'd like to assume everyone is trying to do--even that team in Los Angeles.

This line of thinking derives almost entirely from the grid of expected runs in an inning based on the outs and runner situation, originally developed by John Thorn and Pete Palmer in The Hidden Game of Baseball:

                          Runners
Outs   None   1st    2nd    3rd   1st&2nd  1st&3rd  2nd&3rd  Loaded
 0    0.454  0.783  1.068  1.277   1.380    1.639    1.946   2.254
 1    0.249  0.478  0.699  0.897   0.888    1.088    1.371   1.546
 2    0.095  0.209  0.348  0.382   0.457    0.494    0.661   0.798

Looking at the table, the most obvious feature is that expected runs decrease much more quickly going down (as outs increase) than they increase going across (as runners move around the bases).

Where do these numbers come from? Originally, this grid was the summation of years of data for the seasons up to and including 1983. These antiquated numbers are the first problem with basing conclusions on this chart. It should be news to no one that things have changed a great deal since then; players are bigger, parks are smaller, and run scoring is much higher than it was 20 years ago. Thus, it's important to use information from more recent seasons. Here are the numbers from 2003:

                          Runners
Outs   None   1st    2nd    3rd   1st&2nd  1st&3rd  2nd&3rd  Loaded
 0    0.531  0.919  1.177  1.380   1.551    1.869    2.023   2.474
 1    0.282  0.535  0.706  1.032   0.909    1.211    1.428   1.544
 2    0.109  0.237  0.341  0.384   0.454    0.518    0.541   0.797

As offensive levels have increased over the last 20 years, the cost of an out has moved accordingly--and as a result, sacrifices appear to make less sense, from a tactical perspective, than ever before. Looking specifically at sacrifice situations--a runner on first and no outs, first and one out, or second and no outs--it's possible to calculate exactly how detrimental sacrificing is to the offense. In the original table, the cost of sacrificing can be determined by subtracting the run expectation from the situation before the sacrifice from the situation after. Specifically, with a runner on first and no one out, 0.783 runs are expected; subtract a runner on second and one out, 0.699, and we can see that the sacrifice "costs" 0.084 runs. Repeating this calculation with the 2003 data, a sacrifice in this particular situation now costs 0.213 runs (0.919 - 0.706), or more than twice as much. For other sacrifice situations--such as a runner on 1st and one out or a runner on 2nd and no outs--the cost has increased from 0.130 to 0.193 and, interestingly, decreased from 0.171 to 0.145. Regardless, sacrificing still looks like a bad idea.

Of course, it's possible that in some situations, sacrificing does make sense. The information above is simply the league average for the year, but when making the decision to sacrifice, each manager is presented with much more specific information than the table provides. In particular, depending on the particular batters who are due up in the lineup, the manager can adjust his run expectation in the inning and better determine if sacrificing would increase that expectation, overall. Our objective now is to find those points in the lineup where sacrificing begins to pay dividends.