THE BOOK--Playing The Percentages In Baseball

An average pitcher as a starter is a .470 pitcher, right? So an average pitcher gets $8M. I guess that sounds about right.

I know BPro has in the past mapped VORP against free agent salaries...has anyone done that with WAR? As long as you’re making these estimates for free agents, it would probably be worthwhile to put them in a spreadsheet and chart them against actual contracts...see if the linear relationship holds.

Yes, an average pitcher as a starter is .470 and as a reliever he’s .560.

I’m not expert enough to know if the salaries are reasonable ... but if you say they are, then I say that this has convinced me. 

Great post.

I’m not sure if you meant “exponentially” in its proper sense as being subject to a polynomial equation (i.e., an equation that has an exponent in it), although a little bit of the salary modeling that I’ve done (not much) showed that trying to fit the regression line to a quadratic (X-squared) or cubic (X-to-the-third) model didn’t do much to help the fit.  I’d have to go back and look at it a little more closely though.

If you look at what Silver does with MORP and JC does with his Sabernomics calculator, they have something like:
sal = a*value^2 + b*value + c
or some such.

Whatever improvement to the fit this has (and this has not even proven to have a better fit to begin with), will be very tiny.  And, since a linear equation is far easier to describe, why in the world go with some sort of quadratic equation?

Fun, profit.  Plus, there is evidence that in the Real World, power-law distributions tend to hold for things like income.  Why not start with those results in baseball, too?

A-rod is going to be the test for this. And the good news is that we’ll find out soon enough.

WCW—two words: Occam’s razor.

When I did the quadratic vs. the linear, there was a small uptick in R-squared by about 0.1% The problem was that the R-squared wasn’t all that great to begin with (17%?).  It’s not a matter of linearity vs. exponentiality.  The market isn’t efficient in evaluation of players.  However, I think those analyses might have been on everyone in the league, both free agents and rookie/arbitration types.

I find them fairly efficient (or at least consistent).  Clearly, you must have extra parameters if free agent or not, and whether he’s an arb of 5+ to 6, 4+ to 5, 3+ to 4, and super 2s.

Don’t blame the market’s inefficiency because your model doesn’t support years of service!

That should be 3 to 3+, 4 to 4+, 5 to 5+.

Again, please tell me EXACTLY how you get .375 starter and .470 and .560 as an average pitcher in a starter/reliever mode.

.470 of what?

All I do, for example, if I have a premier starter like Webb/Peavy is say that he is 1.2 runs better than an average starter.  Now, how do you convert that into wp?  Doesn’t it matter how many IP per start?  Am I being dense here?

I then say that if he pitches 6.75 ip per start (.75 of a 9 IP game), he allows .9 runs per game better than an average pitcher.  I assume that the rest of the game is pitched by average relievers.  That means that my premier (best in baseball, other than the Santana and the like) pitchers allow 3.8 runs and their team scores 4.7 rpg (in a 4.7 rpg environment).  Using pythag, that gives me a wp or like .603.  If he pitches 220 ip per season, that is around 33.3 starts or around 3.5 wins above average.  I think that is the correct way to do it.

But again, if he pitches the same number of innings but pitches a different ip per game, I think you would come up with a (slightly) different number of wins above avrerage.

And for relievers, I would have to do it the same way.  Figure that reliever pitches 1 inning per game (or whatever) and then figure out how many runs above or below average he allows in that one inning as compared to an average reliever or average pitcher (whatever my base line is) and then use pythag and multipy by his number of games appeared (based on his numbre of IP per season).  That also gives me his wins above average.  And of course, I want to multiply that by his average leverage per inning. 

So again, how do you get your .470, .560, and .375 numbers and what are the assumptions?

One of the problems using my method (and I don’t know how you accomodate this in your method) is that if I want to know the wp or WAA of a replacement starter, I need to know how many ip per game he pitches, and certainly how many ip per season he pitches.  If I am comparing a starter to a replacement starter, do I always assume the same ip per season?  Is that fair? 

For example, if we compare my premier pitcher to a replacement pitcher (lets say I wanted to see exactly how much pitcher A is going to add to my team’s wins as opposed to someone on the scrap heap), I would say that my replacement starter were 1 run worse per 9 than an average starter (remember I said that my Peavy’s, Webb’s, Holliday’s, etc. were 1.2 runs better).  I think that is a true replacement starter.  Now, how would you translate that into wp??

Again, it seems to me that it depends on how many ip per game you are going to allow him to pitch.  If I just assume that he will also pitch the same 6.75 ip per game and 220 IP as my premier starter, then we now have a pitcher who allows .675 run per game worse than an avreage starter, for a per game wp of .433, far above your .375.  And I think that 1 run above an average pitcher is conservative.  The average worst starters in baseball in true talent are only .8 runs worse than average according to my numbers.

That replacement pitcher, if he threw 6.75 ip per game for 220 ip per season would be worth 2.2 wins below average.  So our premier pitchers (he .603 guys) would be 5.7 WAR or so.

But in reality, if I decided not to use one of these premier pitchers, and instead use a replacement starter, who I am still claiming is only 1 run worse than an average starter, he would obviously only pitch around 5 ip per game and then I would bring in a reliever.  Do we assume that those extra 1.75 ip of reliever use is from a replacement reliever?  I guess so although it is not all that clear that I cannot use my regular bullpen guys who are good, for more innings than I would use them for if I had one of those premier starters rather than the replacement starter.

Now even if I have to use a replacement reliever for those extra 1.75 ip, is that the same thing as using my replacement starter for 6.75 ip?  It can’t be, otherwise why limit him to only 5 ip?  If I take him out after 5 ip on the average, it must be because I have better relievers (and I am assuming that I am not buying any more reliever talent to absorb those extra 1.75 ip times 33.3 starts, or 58 ip).

Now, if we assume that our replacement starter only pitches 5 ip per game, now he only gives up an extra .56 runs per game rather than .675.  If we assume that the slack is taken up by average relievers, which may be a bad asumption, then our replacement starter is now a “allow 5.26 and score 4.7” run pitcher, or a .443 pitcher, even better than before.  A .443 pitcher (5 ip per game for 33.3 starts) is a 2 win below average pitcher.

We can go even further and say that our replacement does not pitch so many games (33.3 starts) but then we have chaining problems I guess.

So again, one of my questions is how can we define what wp a replacement starter (or reliever) is without assuming a certain ip per game and then how can we determine how many WAA without determining an ip per season?

And again, why so low for a replacement starter?  In order to get a .375 pitcher, it has to be one who allows like 3 runs a game more than an average starter and pitches 70-75% of his starts!

When you say a replacement starter is a .375 pitcher (or any pitcher is a .xxx pitcher), do you mean if he pitches a whole game??  I guess you do and then you just multiply that by his ip/9 to get his WAA.  Even then, you are saying that a replacement starter is like 1.4 runs above average (scores 4.7 and allows 6.1?  There is no pitcher I know of with a true talent like that as a starter!  Take the cheapest FA pitchers on all the rosters and see what their collective RA are.  I will bet that it is around 1 run worse than average.  That has to be close to replacement level.

Anyway, if you are simply saying that a replacement pitcher who is 1.4 runs worse than average (per 9) is a .375 pitcher (using pythag - O.K. that is a .372 pitcher, let’s use 1.3 for a .380 pitcher), and just multiplying by, say 22 9 ip starts, for a minus 2.64 win pitcher (as compared to average), that ain’t right.  You have to do it on a game by game basis for those 198 ip, asuming that he pitches, say 6 ip per game (or 5).  Per game, if he pitches 6 ip, his team is a score 4.7 and allow 5.57 (4.7 + 1.3*.67), or a .416 team.  Now for those same 198 ip, he now pitches 33 games.  33 games at a .416 clip is now a - 2.77 win pitcher, rather than 2.64, not a lot of difference, but some.

O.K., I see where you are, but I think there are 2 fundamental mistakes.  One, as I said, I don’t think there can be any way that a replacement starter is .375 which means he is almost 1.4 runs worse than an average starter.  Two, I don’t think you can take the difference between replacement wp (whatever that is) and your pitcher in question’s wp and multiply by his IP/9.  The reason is what I said before.

Your pitcher’s value over replacement is literally how much you would lose of you got rid of that pitcher and replaced him with a real life replacement pitcher and we tracked what happen.  What would happen is that your replacement pitcher would pitch 5 ip per game and the difference in ip per season or per game between your replacement pitcher and the pitcher you got rid of would be eaten up by replacement relievers (at worst).  Replacement relievers are better than our replacement starter, as you admit yourself.  If you do the math, you will see that the win difference for the season between your original pitcher and a replacement starter is NOT their difference in wp multiplied by your original pitcher’s number of ip/9.  Please address that.

We can debate the first thing (what replacement level is for a starter), but I don’t see a debate for the second one, although I could be wrong.

You are right that technically, we want to use the starter’s expected IP/GS, and then add in a reliever’s ERA to the rest of the game to get a win% for the team.  And for a reliever, it’s even more technically sound to do that.  His impact is alot more linear to his team, than a starter.

So, I should update my process to do it the right way.  Let me work a bit on this at the office, and see what kind of differences I get.

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As for the actual replacement level, I start with a team replacement level of .300, and work my way up to a .380 nonpitcher and .410 pitcher replacement level.  The .410 pitcher level implies .380 for a starter and .470 for a reliever (since a pitcher will gain .090 wins per 9IP going from starter to reliever).

You on the other hand work the other way.  You start with a .410 or whatever nonpitcher replacement level, and .430 or whatever pitcher replacement level (which you work out to say .410 for a starter and .480 for a reliever or whatever).  Your team replacement level if .350.

I took a team that scores 5 and allows 4 runs.  PythagenPat says this team will win .6018 games.

What if I had a pitcher that pitched in 6 innings per game, giving up runs at at the rate of 4 per 9, and another pitcher that pitched in 3 innings, giving up runs at the rate of 5 per 9.  That team, scoring 5 and allowing 4.3333, will win .5665 games.

If this was all linear, then we’d expect two-thirds of .6018 and one-third of .5000 to be our win percentage, or .5678.

So, there is a slight difference in the approaches.  We need to treat our 4runs per 9IP pitcher as being worth +.0665 wins per 6 IP (or +.0997 per 9 IP), and not +.0678 wins per 6 IP (or +.1018 per 9 IP).

Similarly, scoring 5 and allowing 3.538 runs would give you a .6526 win% (+.1526 wins above average).

But, if a pitcher pitches 7 innings at that rate, and the other two innings are by .500 pitchers, his team will score 5 and allow 3.863, and win .6166 games.  That means our pitcher is worth +.1166 wins per 7 IP, or +.1500 wins per 9 IP.

So, that’s the extent of the impact, that if you calculate a pitcher’s “win%” based on him throwing 9IP per start, that you will overcredit him by something like .002 or .003 wins per 9IP, at the extreme.

That works out to less than 0.1 wins per full season.  I can live with that.

As for relievers, taking that same 3.538 runs allowed per 9IP, and presuming that only gets done 1 inning, that reliever will have an impact of +.0157 wins per 1 IP (in a 9 inning game), or a rate of +.1409 wins.

If we treated the reliever as being a 9-inning reliever, he’s at +.1526 wins.

So, the gap is much larger, a total overcrediting of +.0117 wins per 9 IP.  But, since relievers only pitch 75 innings, that works out to 0.1 wins being overcredited.  Same as a starter.

Again, I’ll live with that.

That wasn’t my principal beef other than the level of the replacement pitcher or replacement starter.  It was the assumption that the ip by a non-replacement starter would be completely takent up by a replacement starter in figuring his (the non-replacement starter) WAR.  I am contending that part of those ip’s will be taken up by a replacement reliever, a better pitcher for those ip, since a replacement starterr will only pitch 5 ip a game or so.  Given that, the WAR for ALL srarters is going to be too high.  Not that we really care though.  We really only care about the value of one starter as compared to another, and as Tango and I have said before, the better baseline (than replacement) is average, since that is easily defined.  OTOH, to some teams it is important to know how much they are paying players as opposed to having a replacement player at that position/role.  Because if the extra revenue that team earns from those extra wins is not less than the salary paid to that player (minus min salary), then it is not worth having that player (as opposed to a replacement player), at least from a profit perspective.

In essence it just changes the level of the replacement starter.  His damage gets mitigated by the fact that they only let him pitch 5 ip per game and he gets replaced by a replacement reliever.

Tango, do the same calcs with a replacement reliever and assume that in once case he pitches 7 ip per game and gets replaced by an average reliever.  In the other case, have him pitch 5 ip per game and assume that he gets replaced by a replacement reliever for 2 ip and then an average reliever for the last 2 ip.  I think you will get quite a bit of difference in value for the season (figure 150 ip or so).

Your post 16 seems to be the “chaining” issue that we’ve talked about in the past (with Patriot et al at Fanhome).

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The average reliever is a .525 pitcher, while the replacement pitcher as a reliever is a .475 pitcher, more or less.  In runs terms, one allows 5% less than the league average, while the other allows 5% more (5.25 runs allowed).

***

As for your point in #17: if you consider that the average starter is a .486 pitcher (allows 5.15 runs in a league that score 5.00), then figure that his RA each time through the order would be: 4.90, 5.15, 5.40.

It’s to the team’s benefit to pull out the average starter in favor of a replacement pitcher who will only come in to face the order once.

Well, one of these days, a team is going to take us up on our (yours originally) suggestion of never letting your 4th and 5th starters bat.  It kills 2 birds with one stone.  Just so it doesn’t look so ridiculous and it is not no insulting to the starter, I would suggest to a team that they wait until the first decently high leverage and non-bunting situaton to take out the starter for a pinch hitter.