THE BOOK--Playing The Percentages In Baseball

Hey Tom,

Two questions:

1) Do you think there’s a cost to bringing up a player who is replacement level now but has a non-trivial amount of upside? 

I ask, because in my sim leagues I find myself skipping over a more talented prospect for a slightly less talented minor-league journeyman when a major-league starter goes down with an injury. 

My reasoning is that while the prospect is better than the journeyman, the prospect will benefit from spending next developing in the minors, and it’s not worth paying him $400k next year if he’s not going to play in the bigs.  The journeyman, on the other hand, can be cut at the end of the season and then easily replaced with another journeyman on a minor-league contract.

Is this a roundabout way of asking whether value is measured solely in terms of wins even when we’re talking about a cup of coffee?

2) When you and Dave talk about paying for wins linearly, I always wonder if your equation holds when looking at individual free agent signings - and whether that matters.  In other words, would a simple regression of wins on salary produce an extremely good fit?  If it doesn’t, do you assume teams are simply mis-estimating the win values of their signings?

* next year developing (my bad)

The linear model holds well.

***

You have a fair point regarding keeping a prospect down to give him guaranteed playing time as opposed to possibly using him in the bigs in a backup role. 

Just look at the M’s, who kept Michael Saunders down after Endy Chavez got the terrible injury.  They instead went after Ryan Langerhans.  Just a short time after that, Saunders was brought up into the starting role, even though Langerhans fulfilled the Endy role quite nicely.

http://www.baseball-reference.com/teams/SEA/2009-lineups.shtml

I would guess that there is a cost to bringing up someone like Saunders if you don’t at least platoon him.

What about using the repl level concept for HOF discussions. Why should that not have its own baseline. How many years would, say, an avg player have to play before he should be a HOFer? More than they actually do play, I would say. So, maybe the minimum should be a .550 player, which translates into a +10 runs per 162 G, or so, for 20 years. I’m gonna apply this to Rally’s all-time WAR list, to see what it would look like.

Sure, that could have its own baseline.

The repl-level for GM purposes is the zero-cost level, the “b” intercept, in the linear equation.

"Are wins bought linearly?  Yes, pretty much.”

I don’t disagree at all. My point in the article was to demonstrate that the foundations of the replacement level concept are in the economic concept of opportunity cost. And replacement level is a handy placeholder for opportunity cost, but nevertheless it is an approximation. And that’s tremendous to have, because it’s much easier to use a commonly accepted approximation, like 3.14 for pi, than to use the full expansion. But that doesn’t mean that, strictly speaking, we aren’t still using an approximation.

y = mx + b
, where y is the salary, m is the slope (multiplier), x is the wins above average, and b is the intercept (i.e., replacement level)

This statement doesn’t seem correct to me.  I would think replacement level, in wins below average, would be the value of x when y equals the minimum major league salary.  b, the y intercept, would seem to be the salary of an average player.  The x intercept, or absolute zero cost level, has no meaning that I can see.

Peter, typically when Tango or I talk about salary, we mean salary - minimum. So we should be in agreement here.

Colin - No we are not.  I await Tango’s response since the comment was directed to the formula he presented.

First, Colin is correct that I should have said salary above minimum as “x”.

Ok, now that I’m writing at the office and not at home, I can actually think about what I’m saying.  Let’s see if I made another mistake.  I said:
y = mx + b
and “x” is the wins above average.

The average payroll (above minimum) is around 77MM.  That gets split as 44MM for nonpitchers and 33MM for pitchers.  There is roughly 8.65 full-time equivalents (FTE for your corporate americans) among nonpitchers, when you consider PH, DH, etc.  So, that means each nonpitcher makes 5.1MM per 162G above the minimum.

Ok, so first I need to make a clarification.  “x” is wins above average, per 162G.  y is salary pro-rated to 162G.

That means that to get a player’s real salary (above minimum), you have to take the mx+b equation and multiply it by his percentage of playing time (expected, deserved, or whatnot).

Anyway, let’s go back:
y = mx + b

If you have a team of average nonpitchers, then y=b for each of the 8.65 players.  We already said that y=5.1MM, so that becomes b=5.1MM as well.

y = mx + 5.1

At what point is y = 0?  If we presume that replacement level is -2.25 wins per 162G, then m, the slope is 5.1/2.25 = 2.26

So, each marginal win is worth 2.26MM marginal dollars.

As I noted, we have 44MM marginal dollars for our nonpitchers at work.  44/2.26 is 19.5 wins above replacement for our nonpitchers.

***

For pitchers, we have 33MM to distribute over 162 full games, or $0.204MM per 9IP.
y = mx + b

As noted, mx is 0 when you have an average pitcher, so y = b = 0.204.

y = mx + 0.204

Remembering that m=2.26, then x = -0.090

That is, when a player is .090 wins per 9IP below average (i.e., .410 win%), then y = 0, i.e., replacement-level.

***

Now, you don’t have to call it “replacement-level”.  y is 0 when the pitcher is a .410 win% pitcher, or the nonpitcher is 2.25 wins per 162G below average.  y is the salary above the minimum.

I call that replacement-level, but you can call it whatever you want.

***

Let’s go back to the pitcher equation:

y = 2.26x + .204

Which we can rewrite as:

y = 2.26 (x + .090)

Remember, x is wins above average per 9IP. 

y = 2.26 (win% - .500 + .090)
y = 2.26 (win% - .410)

That .410 is the replacement level (or the point where y is zero).

***

Does everyone follow?

Tango,

I like your model, and I think you show rather nicely how marginal wins can be bought more or less linearly.

However, I do believe that because of information asymmetry (not all teams are equally capable of predicting value, and no team can do so perfectly, even discounting bad luck) it can take some time to find replacement level.

Compounding that factor is that some players are especially unproven. Let’s imagine a player who has a 50% probability of being worth 1 WAR and a 50% probability of being worth -1 WAR. (Or if you prefer, a normal distribution around 0 WAR, I don’t believe it matters.) His expected value would be 0 WAR. However, we might not know for several months which was his true talent level--and even then we would not be 100% certain!

So yes, upgrades are upgrades and replacement level is exactly as you and others have described it. But in specific cases where tradeoffs are known, we have to think in terms of opportunity costs.

Tommy, let’s use real examples because all these academic arguments should be settled empirically.  After all, economic theory needs to somehow be validated, otherwise you can make reasonable claims on both sides.

Use the case of the Mariners.

They lost Endy Chavez for a year.  Balentien and Junior are on the roster.  They have Michael Saunders (OF) in the minors (never played MLB).  They have former MLBer Mike Morse, infielder, in the minors.  And out there in AA on the Nationals is once-MLBer Ryan Langerhans.

Go.

If you don’t like that one, use some other reasonable illustration (though I have to say, it’s going to be pretty hard to top that one!).

I don’t know how much development issues affect whether you’d bring a top prospect up or go with less talented minor league filler.  You can make a case for learning by playing everyday in the minors, but also a case for learning by being around actual big leaguers.

The key issue, which is obvious to me, is the service time.  You’ve got 6 years of relatively cheap labor from this prospect.  I’d rather have 6 years with him as a regular than 4 as a regular and 2 as a sub.  Even if my bench is hurt a bit by playing Scrubby McScrub ahead of him in the short run where I need an emergency bench guy.

A great example!

Let’s start with Zduriencik’s appetite for risk. Since he traded away Clement and others for Jack Wilson and Ian Snell, I’m going to assume he’d rather have certain production than a chance at higher production. (This may or may not be the right call for the Mariners at this point in the season, but it seems to be his disposition.)

Michael Saunders was rated as the 25th best prospect in the minors immediately prior to his call up (see: http://www.baseballamerica.com/blog/prospects/?p=5456 ). So his expected value is relatively high, albeit combined with lots of risk. He put up a .384 wOBA in Double-A last year and a .400 wOBA in Triple-A this year. His MLEs from Triple-A (courtesy Sackmann) were .277/.330/.449, which I translate to be

(1.8*.330+.449)/3 = .348 wOBA

*side note: I had been using 2*OBP but my colleagues at BtB helpfully informed me that I was using the outdated coefficient from The Book and that 1.8 was more accurate

In any event, let’s say Saunders is a .348 wOBA guy. With Fangraphs pegging average wOBA at .329, converting to runs, and assuming Saunders could get 200 PA over the rest of the season:

[(.348-.328)/1.15]*200 = 3.5 RAA over the rest of the season

I don’t have good enough data to make any conclusions about his defense. So we’ll say he’s an average fielder, give him an extra ~7 runs for replacement and ding him 2.5 runs for position adjustment, giving:

3.5 + 7 - 2.5 = 8 RAR for the rest of the season

Let’s say Zduriencik would discount this production by 20% because it is uncertain (again, I’m not sure this is wise but I’m taking it as a given). So that means Saunders gives Zduriencik the same utility as a guaranteed 6.2 RAR player.

Now, the alternatives. I take Morse’s MLE’s (.277/.322/.412 at Tacoma) and covert to wOBA and get .331. Morse was an infielder but not an especially good one (I once had a chance to ask Bill Bavasi what in the world he was thinking putting him at short--the answer was not satisfying). So we’ll say he’d be average in a corner too. That gives us about a third of a run above average and about 4.8 for the remainder of the season. He might be a little more certain than Saunders but I bet Zduriencik would apply some discount. Let’s say it’s 10% which puts Morse the value of a guaranteed 4.3 player.

Wlademir Balentien had a .274 wOBA in 170 PA, which is pretty terrible, but his MLEs in Triple-A in 2008 are equivalent to a .340 wOBA. Even still, he appears to have overstayed his welcome and is probably worse than Saunders with just as much risk.

Langerhans has a career .314 wOBA, which is terrible for a corner outfielder. I don’t think I need to say more than that.

So to use your example, even assuming Zduriencik has a small appetite for risk, Saunders was the right call.

However, a funny thing happened on the way to Safeco. Before Saunders was called up, Langerhans was IN FACT almost exactly league average (.329 wOBA and all). That was worth about 0.9 WAR (per Fangraphs). Since Saunders was called up, he’s gotten at least two PA in every game, nearly always at the expense of Langerhans. However his production has been worse: .290 wOBA.

Now, I don’t expect Saunders to be a .290 wOBA guy the rest of the year (and it’s a ridiculously small sample, just 50 PA). However, if Zduriencik thinks this M’s team is still hanging tough in the WC (4.5 GB going into today), the option of having the more certain level of production represented by Langerhans might have value. It might be helpful to think of the value of having a put option on Saunders.

The only trick is, Zduriencik won’t know for a while whether the .290 wOBA is Saunders’ true talent or not.

In this case, the opportunity cost to Zduriencik of playing Saunders is not replacement but rather the more “certain” value he gets from Langerhans and his .314+ (+ if you think he’s better than his career numbers suggest) wOBA.

After all that, how does that change your final evaluation from my WAR methodology?

That is, after you do all that work, do you get anything different from a simple WAR application?

Yes, I’m sorry if that was not clear.

Langerhans would have been expected to be a 0.2 WAR player using the above methodology, Saunders to have the value of a 0.6 WAR player. However, Saunders has yet to reach that level and has in fact been below replacement thus far.

If Zduriencik (or Wakamatsu) tires of Saunders, he has a more stable commodity (who is still above replacement, albeit not by much) that he can slot in the outfield instead.

The other options would have carried similar risk as Saunders, but with lower expected values. So essentially, I’m saying if you’re going to have two guys the ones to have are Langerhans and Saunders not Saunders and any of the others.

It seems to me what you are saying is that you are applying a larger discount value to Saunders because of his limited MLB playing time.

So that, while I would argue that the replacement level wins per 162 is 2.25 wins below average based on my expectation of the “AAAA” player, you are saying that maybe it should be 2.50 wins below average because we are not very efficient at separating the real “AAAA” from the “AA” player.

I don’t necessarily disagree with you.  All that does is value the bench players more.

But then again, what if my 2.25 is actually the discounted value?  After all, MGL uses something closer to 2 wins below average per 162 G.  And so, if you used his forecasts, you might set the replacement level at 2.25 because of the uncertainty of separating the AA from the AAAA player.

In the end, we are pretty much in agreement on the overall approach, and are disagreeing on teh periphery.  I can live with that…

"All that does is value the bench players more.”

It’s the least I can do to help people appreciate Jim Eisenreich.

I never intended to strike at the foundations of replacement level analysis, but rather to ensure that people were adequately valuing (especially mid-season) player acquisitions.

If you’re satisfied, so am I.

What’s the objection - and Im sure there will be a few - to the following argument:

If WAR is considered linear, then that implies that a 4 WAR player is worth twice as much as a 2 WAR player, or is worth two 2 WAR players. Surely, however, risk has to come into play when evaluating the actual expectation of value. That is, if your 4 WAR player is injured (and for this whole argument I mean DL-type injury, not nagging injury played through), he needs to be replaced by a bench player who, for the sake of argument, is simply average at 0 WAR [I realize this actual value could change my argument, but it would take a particularly strong bench player to cancel out my argument as it continues, I believe] and you therefore lose 4 WAR in any given game. If one of your 2 WAR players is injured, however, you only stand to lose 2 WAR, as the other is still in the lineup. Given that it is significantly more likely (twice as likely?) that your one 4 WAR player will be injured than that both of your 2 WAR players will be injured, there is greater risk associated.

That said, the same type of risk calculation must be applied, to a degree, as to whether these players will play to their WAR potential. A team will need, arguably, twice the luck for its 2 WAR players to both play to their full potential and for just one 4 WAR player to play to his. However: simply because DL-type injuries always mean a 100% loss of value (again, assuming 0 WAR bench player) as opposed to the, perhaps, 50% loss of value if both players are having off-years, the risk must be greater for one 4 WAR player versus two 2 WAR players. 

Thoughts?

Let’s see. Suppose you lose him a player for 10% of the games in a season.

You have your one 4.44 WAR guy, and 90% of the time he gives you 4.44 WAR performance, and 10% he gives you 0 WAR, for an overall average of 4 WAR.

You have your one 2.22 WAR guy, and same deal: he’s 2.22 90% of the time and 0 WAR 10% of the time.

The objection is that… if you have 2 guys at 2.22, the chance of both of them going down at the same time is less than the 4.44 guy?  Yes,that’s true.  At the same time, the chance that one of the two 2.22 guy going down is greater.

Regardless, the comparison point is that you have two players: 4.44 and 0 on one team, and 2.22 and 2.22 on the other team.  Which would you rather have?  They’re equivalent in terms of how much they are getting paid, how much chance of missing games to injuries, etc.

You could argue that it’s tough to lose the 4.44 guy, but it’s easy to replace the 0 guy.

All that gets rolled into the WAR equation.  If you don’t think that a .340 OBP / .410 SLG guy with 150 games is a 2 WAR player, then change the baseline to make this guy a 2.5 WAR player.

All of a sudden, instead of having a 4 WAR and 0 WAR player compared to a 2 and 2, you have a 4.5 and 0.5 WAR players compared to 2.5 and 2.5.

So, yes, in this case, the two 2.5 are better than the 4.5.  But, I don’t see the point.  Since the old 4.5 guy is now a 5.0 guy, and he is worth these old 2, now new 2.5 guys.  You’ll always have some equivalency somewhere. 

Just a matter of setting the baseline.