Anyone who follows Baseball Prospectus or other sabermetrically inspired baseball websites can tell you that the sacrifice bunt is usually the wrong play in games that are not close and not in the later innings. However, these arguments rest on the fact that the pitcher, catcher, and corner infielders are agile and skilled enough to convert the sacrifice bunt into a routine out. What if the amateur players mishandle a bunt more frequently than their MLB analog? What I’d like to do is see how much higher the error rate would have to be on bunted balls to make sacrifice bunting the right play.
For the purposes of this discussion, I will only be focusing on sacrificing with runners on 1st/2nd with 0 outs.
First, let’s figure out what the value of various gamestates are, We can see the amount of “expected runs” scored year-by-year, but we’ll use the years 1999-2002 as found here on TangoTiger.
Runners on 1st/2nd with 0 outs is worth: 1.573
Runners on 2nd/3rd with 1 out is worth: 1.467
If the sacrifice is executed 100% of the time with no errors, the loss of expected runs can be easily calculated like this:
Ex2nd_3rd_1 - Ex1st_2nd_0 = Value
1.467 - 1.573 = -0.106
Therefore, if the fielders can always turn the sacrifice bunt into an out at first, this play has a value of -0.106 runs.
However, what happens when the throw is botched?
Runners on 2nd/3rd with 0 outs is worth: 2.052 + 1 = 3.052
This is the most likely scenario when a sacrifice bunt is thrown away - the runners advance to 2nd/3rd on a bunt, the throw goes wild, the runner going to third scores, the runner going to second goes to third, and the guy who bunted goes to second.
Also, for the purposes of this discussion, we’re going to assume that the manager will always call for a sacrifice bunt with runners on 1st/2nd with 0 outs and not worry about other types of batted balls. Other assumptions include that the error is always a throwing error that results in 2nd/3rd + a run with 0 outs, that the sacrifice is never recorded as a base hit, and that the sacrifice never leads to a double play. That’ll be for another article - this is just the groundwork.
Let’s add in some errors, shall we? Our Error Rate modifier formula looks like this:
ExpTotRuns = Ex2nd_3rd_1(SuccessRate) - Ex1st_2nd_0 + [(ErrorRate)(Ex2nd_3rd_0 + 1)]
ErrorRate is the % of times an error is made on the play. SuccessRate is the % of times the play is successfully executed.
Expected Total Runs equals Expected runs with runners on 2nd/3rd with 1 out minus Expected runs with runners on 1st/2nd with 1 out (this is multiplied by the success rate) plus the sum of Error Rate times Expected runs scored with runners on 2nd/3rd and 0 outs plus 1 run.
Make sense? Try reading it a few more times before commenting for clarification.
Here’s some sample calculations:
Error rate of 0%: 1.467(1.0) - 1.573 + [(0)(2.052 + 1)] = -0.106 runs (that third variable is 0)
Error rate of 1%: 1.467(0.99) - 1.573 + [(0.01)(2.052 + 1)] = -0.09015 runs
Error rate of 5%: 1.467(0.95) - 1.573 + [(0.05)(2.052 + 1)] = -0.0239 runs
Ah-hah, getting closer! Of course, we can simply change the formula around to solve for the end result we want…
Break Even Point (careful, 9th grade algebra ahead!):
ExpTotRuns = Ex2nd_3rd_1(SuccessRate) - Ex1st_2nd_0 + [(ErrorRate)(Ex2nd_3rd_0 + 1)]
0 runs = 1.467(1.0-x) - 1.573 + [(x)(2.052 + 1)]
0 = 1.467 - 1.467x - 1.573 + 2.052x + 1x
0 = 1.585x - 0.106
0.106 = 1.585x
x = 0.06688
It seems as though the breakeven error rate is about 6.69%.
Do I think that amateur players in the leagues I play in commit overthrowing errors on sacrifice bunts at least 7% of the time? Yes, I do.
Conclusions:
First of all, I think my math is right, though I am not 100% sure on using ErrorRate and SuccessRate. I ran it by a few people and they all thought it was necessary, but it could be counting the error rate twice unnecessarily.
The work above has a lot of assumptions which makes the formula simple. We assume that the error is always an overthrow that causes a specific gamestate, we assume that the sacrifice bunt is never a hit, we assume that the runner is smart enough to go home on small overthrows, we assume that the hitter is always going to sacrifice, and that the hitter never fails to make the bunt contact. Since there are all these assumptions, the model’s not perfect (obviously). However, it’s a step in the right direction for this type of research.
