THE BOOK--Playing The Percentages In Baseball

One problem with this analysis is you seem to be assuming the pitcher to bat will be an average hitter. 

The Rockies had the choice of having a reliever bat, which would i suspect have a good deal lower non-out percentage than 18%, or a starter who didn’t start on that day. 

And I suspect having a starter hit may also have been lower than 18%. 

So I think you’re overestimating the danger of the walk.

There is no problem in the analysis.

1. I’m giving you the baseline pitcher numbers
2. I’m giving you the methodology
3. I said this:
“The approach would have to be refined to see how many weak and good singles the pitcher in question can hit”

You can say that I was too lazy to finish the job.  But, there’s no problem in the analysis based on your point.

I’m thinking there’s a few factors that need to be added:

1) The pitcher is almost certainly a reliever who bats rarely and hits worse than your average pitcher.  Actually, I’d be interested in what the batting stats are for all relievers, since I’m making an assumption here.  The average pitcher is a pretty bad batter, relievers may not be worse than that.

2) If you force the pitcher to hit, he might be lifted for another pitcher who is a better hitter, but who is presumably a worse pitcher in subsequent innings.  There’s an advantage to be gained by this, but it would have to be looked at on a case-by-case basis.

3) If you get out of the inning, in subsequent innings the batting order is advanced.  You now have to face the top of the order with no outs, instead of pitching to the pitcher with no outs in the next inning.

My guess is that of these, (3) is the most important, then (1), then (2).  But I don’t know.  The easiest data to find would be hitting stats for relievers.  I tried a query to b-ref with G > 40, PA in game = 1, but I’m not a subscriber.

I’ll try the math for (3) in my next post, since this is getting long.

Excellent point about #3.  The idea that in extra innings that both side have a 50-50 chance of winning is predicated on the assumption that both sides are equals.  Obviously the home team has a slight advantage (*), but if the home team has to start the inning with the pitcher, then it’s not 50-50.

Not to mention that if the batter gets a regular walk or a single, that now brings up the pitcher and not an average batter. 

Good job highlighting those issues.

These things are easier handled via a sim of course.  I was more interested in laying out the considerations and how you can try to calculate it yourself without a sim.  Even so, I still fell short here.

(*) I was racking my brain why the Markov was consistently lower in win% than the empirical.  And it finally dawned on me that the Markov presumes 50/50, while the empirical naturally includes the home side advantage.

You aren’t kidding with how bad relief pitchers are at hitting.

I looked for 2010 NL pitchers who appeared in at least 40 games but had fewer than 10 AB.  The totals for this group, with 133 total PA:

133 PA, 118 AB
.068 BA, .120 OBP, .102 SLG
Striking out 61% of the time.

I have the math below, but first a better thought.  What’s happening here is that the visiting team is trying to distribute a high leverage situation to a poor hitter.  The problem is that there are actually TWO equivalent high-LI situations to distribute.  1__/2 (or 12_/2) and ___/0, which will be required in the next inning in order to win.  In fact, ___/0 is a bit HIGHER leverage (unless the visiting team scores a bunch of runs in the top of the inning).  The pitcher will bat in one of these regardless.  The choice is who else will, the guy currently up, or the guy after the pitcher (or possibly the guy after him).  Focusing on the pitcher is a red herring, it seems to me.  That said, I promised I’d do this.

Assuming an average pitcher batting:

OBA = 0.82
XB% = 0.02 (Effect is bounded by all 2B & all HR)
1B% = 0.16 (includes BB, HBP, 1B ROE)

Looking at the bottom of the next inning, the worst case is that the game is tied.  Either the pitcher is up with no outs, or the 1 or 2 batter is up with no outs.

For an average batter coming up (WE for home, shown with state/outs)

WE XXX/0 (Tied) = 0.634
WE XXX/1 (Tied) = 0.577
WE 1XX/0 (Tied) = 0.715
WE X2X/0 (Tied) = 0.807

WE XXX/0 (H Down 1) = 0.194
WE XXX/1 (H Down 1) = 0.108
WE 1XX/0 (H Down 1) = 0.208
WE X2X/0 (H Down 1) = 0.282

Tied in bottom of next inning:
Walk: hWE = 0.634
Got hitter out:
hWE = 0.577*0.82 + 0.715*0.16 + 0.807*0.02
= 0.604

Advantage to getting the hitter out is 0.030.  If the extra base hits are all HRs, it goes down to 0.026.

V up by 1 in bottom of next:
Walked 8-hitter: hWE = 0.194
Got 8-hitter out:
hWE = 0.108*0.82 + 0.208*0.16 + 0.282*0.02
= 0.0942

Advantage to getting the hitter out is 0.100.  It goes down to .093 if all the XBH are HRs. 

I’d look up more situations, but I think this gives an idea of the situation.  These differences only matter if you get to the next inning, which happens more than 2/3, but less than all the time.  So, they should be discounted to .02 and .07, respectively fot their overall effect.  That’s more than the gains Tango shows above.  But I think the LI argument above makes this math unnecessary.

The Leverage Index (LI) in the bottom of a tied game, with 2 outs and…

... runner on 1B is 2.4
... runner on 1B, 2B is 4.4

With 0 outs, tied, bases empty, it’s 2.3.  If down by 1 run, it’s 3.6.  If down by 2 runs, it’s 2.0.

So, getting the pitcher up to bat by forcing the leverage to go up would seem to be a good thing.

If the batter gets an extra base hit, the runner will score, and the game is over.

Are doubles that don’t score a runner from first (including automatic and “ground rule” doubles) rare enough that they don’t make much difference here?

A tangential question: I always wondered just how much difference the “prevent defense” we typically see in these situations (IFers guarding the lines, OFers deep) really has on the rate of singles, doubles, outs, and base-runner advancement.  Has this been studied?

Rally - Relief pitchers aren’t very good as hitters but I can’t quite get them to be as bad as your numbers say they are.  I have all relief pitchers in 2010 getting 28 hits in 253 ABs for a BA of .111.  The rest of the slash line is OBP of .169 and SLG of .145.  At first I thought the difference was your requirement of 10 or fewer ABs but I checked and none of the relief pitchers had more than that.  The slash line for 2010 is very similar to what I have for 2000-2010 of BA .121, OBP of .166 and SLG of .154.  The only difference between our methods is the 40 appearences you require.  Perhaps long relief pitchers are much better hitters than closers and set up men.  So how one does the math is very dependent on what group that you assign the pitcher to. 

The best solution is to have a very good sim that can incorporate all the correct variables and see what the outcomes are for 100,000 games or so.

Tango/8:

That seems wrong to me, the leverage is increasing because you’ve given away WPA.  I think you need to include that you’ve given the initial 2.4 LI PA to a batter with 1.000 OBA (or somewhere around 0.720 wOBA - guaranteed BB).

So the choices are:

LI Batter wOBA State
--- ------------- ---------
2.4 0.720 1__/2
4.4 Very Low 12_/2
(6.4) 0.340 (assume avg) 123/2
2.3/3.6/2.0 0.340 (assume avg) ___/0

OR

LI Batter wOBA State
--- ------------- ---------
2.4 0.340 1__/2
2.3/3.6/2.0 Very Low ___/0

There are other possibilities, the first situation includes the chance of a 123/2 situation, which I’ve shown.  The second includes the possibility of reaching the same sequence as the first if the initial batter reaches base without ending the game.  But, I think this describes the situation better.  You need to account for the initial walk, you’ve chosen to give the 2.4 LI situation to a very good batter, a guy who walks all the time.

Are doubles that don’t score a runner from first (including automatic and “ground rule” doubles) rare enough that they don’t make much difference here?

Michael K - You are correct.  Even with the score tied and 2 outs a significant number of the runners are held.  I have the runner scoring 98 times in 167 opportunities for that situation from 2000-2010.  Runner was out 7 times and held 62 times.  So Tango’s numbers should reflect that the runner will score around 59% of the time and the inning will end on a baserunning out 4% of the time.  Doubles make up 77% of the extra base hits of relief pitchers.

Forget my last sentence in the previous post. I forgot that the relief pitcher would be batting with men on 1st and 2nd.

Peter, thanks for that.  Those numbers are pretty consistent with overall league numbers:

http://tangotiger.net/destmob2.html

I have the runner holding up at 3B at 40% of the time in all situations, while you have it at 37%, in this end-of-game situation.

I have the runner thrown out 4.5% in all situations, while you have it at 4.2% in walk-off-situations.

Given the extremely small number of opps that you have, and that those numbers are not appreciably different from a random situation, then we may as well just use the overall averages.

Garik: I am sorry that I wrote as strongly as I did.  I should have been more charitable to your use of the word “problem”.

***
The best solution is to have a very good sim that can incorporate all the correct variables and see what the outcomes are for 100,000 games or so.
***

Yeah, but sims just don’t grow on trees.
vr, Xei

What about the fact that even in the 26% of the time when the runner gets on you still have the pitcher up to bat. I confess I may have missed that part in the analysis.

CJE: good job.  Right, my baseline presumed average batter for rest of the inning, when in fact, that’s not true.

So, when I said this:
“Remember, chance with an average batter batting with a runner on 1B, and it’s a 56 - 58% chance of winning.”
That presumed the on deck hitter was going to be an average batter.  Since it’s the pitcher, that drops the odds down a bit.