In discussing the A’s/Twins ALDS Game 1 matchup last week, I suggested that Kotsay could have broken even attempting to steal second base if he’d had as little as a 45% chance of success.  It was surprising to me that only one person offered feedback on that assertion, but not at all surprising that it came from an A’s fan.  While stopping short of suggesting that I’d taken leave of my senses, he did say that he didn’t understand how I arrived at it.

Looking at the compiled season stats from this year in MLB (from www.baseballprospectus.com), we can see that having a runner on 1^st base with 1 out is worth - in the abstract - 0.5675 runs.  Moving him up to 2^nd base without making an out (e.g. via a stolen base) increases that to 0.73631 runs.  But the “cost” of losing that runner (e.g. if he’s caught trying) is that the run expectancy drops all the way to .10907.  In general, if you want to figure out how often to attempt a play in baseball to maximize run scoring, you can use this formula:

(cost)
------------------ = break-even success chance.
(cost+benefit)

In this example, that means that in order to avoid losing runs, a team would need to have a:

(.5675 - .10907 = .45834) <-- cost of getting caught stealing
--------------------------------------------------------------------------

(.45834 + (.73631-.5675 = .16881) <-- benefit if successful)

or about a 73.1% chance of success. 

I wish I had better data available, but I’ll lean on some of the great work done by Tangotiger (www.tangotiger.com), and his “Run Frequency Matrix, 1999-2002” for some numbers that should be very near the actual values for a game in 2006.  From the data he collected, the following numbers apply to the situation under discussion:

·        1^st base, 1 out: 0.283 chance to score.  0.161 chance for 2+ runs.
·        2^nd base, 1 out: 0.406 chance to score. 0.175 chance for 2+ runs.
·        Bases empty, 2 out: 0.077 chance to score.  0.025 chance for 2+ runs.

Using the same formula as above, these numbers show that a team desiring to increase their chance of scoring (not overall expected runs scored) would attempt to steal with a 62% chance.  More later about 1-run strategies vs. run-maximizing strategies.

True.  And to be honest, I’m not entirely sure what all the factors are that play into getting the percentage down to 45% (see previous article where I use win% numbers from The Win Expectancy Finder which Chris Shea put together, mining the invaluable data from Dave Smith and the folks at www.retrosheet.org). 

First, there’s a hidden bias in the “Run Frequency Matrix”.  This comes from the fact that teams attempt to steal when the success rate is about 66% or more, and so a large number of the “man on first, 1 out” situations will be followed by a stolen base attempt.  Since, on these stolen base attempts, the average success rate will be high enough to significantly improve the chance of scoring on these plays, the “chance to score” on plays where there is no stolen base attempt goes down to about .250-.260.  This lowers the run-expectancy break-even point to about 55%.

Next, there are other situational factors which enter into play here.  Teams are much more likely to employ 1-run strategies with their weaker hitters, largely to avoid the GDP.  That means that a “man on first, 1 out” situation will be turned into a “man on second, 2 out” situation even by hitters who normally strike out or GDP a lot.  And this also implies that most of the “man on first, 1 out” situations will involve hitters who are among the league’s better hitters, or at least are adept at moving runners over with their outs.

To be honest, I’m always a bit suspicious of factors I cannot measure, and while I do trust the Win Expectancy data, I’ll back off the 45% and go with 55%, which is still much less conservative than more managers would even consider.

Um, right, but that’s only a 55% chance to improve the chance to score one run.  Don’t I care about the drop off in expected runs?

Here’s the catch about runs.  In the abstract, a run is worth about 0.1 wins.  So, if you send two teams against each other 100 times, and they each score 500 runs, that’s 50 wins apiece, and it all averages out in the end (I know, I know, it’s a terrible example).  That’s great for large sample sizes.  But we’re talking about a particular situation - the 6^th inning of a game where the visiting team is ahead by 2 runs, and gets a guy on first base with 1 out.  Let’s look at the situation from two vantage points:

First, Win Expectancy, the metric I used last week - teams historically have a 0.767 chance of winning the game in the initial situation.  At the end of the inning, here are some possible outcomes, in terms of winning percentages:

·        0.730 if 0 runs scored (2-0 going into bottom of 6^th).
·        0.838 if 1 runs scored (3-0 going into bottom of 6^th).
·        0.902 if 2 runs scored (4-0 going into bottom of 6^th).
·        0.945 if 3 runs scored (5-0 going into bottom of 6^th).

So, in terms of wins, the first run scored is worth 0.108 wins (about the average Runs: Wins ratio, statistically speaking).  But after that, the incremental value of each run goes down quickly (0.064 for the 2^nd run, 0.043 for the 3^rd run).

Roughly the same pattern shows up using the Pythagorean Theorem (win%=RS^2/(RS^2+RA^2)), and assuming a constant 0.45 runs scored/half inning the remainder of the game:

·        0.776 if 0 runs scored (2-0 going into bottom of 6^th).
·        0.854 if 1 runs scored (3-0 going into bottom of 6^th).
·        0.898 if 2 runs scored (4-0 going into bottom of 6^th).
·        0.926 if 3 runs scored (5-0 going into bottom of 6^th).

The extreme example of this is in the bottom of the ninth inning in a close game, where 1 more run can become worth approximately half a win (if it allows a team to win or tie).  This is a big part of why the ability to “do the little things” is so important for winning playoff baseball - since the teams are supposedly much more evenly matched than during the regular season, the chance for a situation to arise when 1 run makes a world of difference goes up a lot.  Think Dave Roberts in 2004!