One Last Word on Shawn Hill: He's STILL Underpaid

I've been startled at the reaction on blogs and chatrooms about Shawn Hill's arbitration victory. People seem actually disappointed that Hill won. Or if not disappointed, then at least somewhat bemused.

I always think it's weird when fans root for the owners in contract disputes, and I don't see why any Nationals fan would be cheering for Ted Lerner to get a tiny bit richer at Shawn Hill's expense. It's not like the $100k or $200k difference between Hill winning and losing is going to allow us to sign John Lackey next year.

But the big disconnect is that, even after winning a one-year deal of $775k for 2009, by my estimate, Hill is underpaid.

Some quick background for those who aren't yet totally familiar with how things work: MLB players for their first three seasons have essentially no say in the terms of their employment. They have to play for the team that drafts them, and they have to take whatever salary they are offered. They can quit Major League Baseball and play for the Fort Worth Cats, but otherwise they have no rights whatsoever and almost always make something close to the league minimum ($390k in 2008).

In years 4-6 players can begin to negotiate, but they still can't negotiate with any other teams, and ultimately a panel arbitrators decides what they deserve to earn. It's a system that gives the players something closer to their real market value, but not really, because the owners win about two-thirds of the time in arbitration. Players aren't paid their true market value until they reach their seventh major league season when they qualify for free agency and can peddle their services to whatever employer they choose (a right most Americans take for granted every day).

Back to Hill and why I think he's underpaid. It's actually pretty hard to say what would be fair market value for a player without actually allowing the market to set that price, but Dave Cameron made a solid attempt at calculating "deserved" player value under the "Value" stats category on the player pages at Fangraphs.

Dave did a lengthy series of posts explaining the methodology, but the quick summary for pitchers is that he used Fielding Independent Pitching (or FIP, which is a stat that projects ERA based on a pitcher's rates for HRs allowed, Ks, and BBs, and HBPs--four stats a pitcher has a great deal of control over) to quantify pitchers' run prevention value above replacement level, neutralized those stats for park factor and other variables, then converted those runs prevented into wins above replacement at a rate of about 10 runs per win, and then multiplied that by the average amount of money per win that teams were willing to spend in free agency. That was a wicked run-on sentence, so if you didn't follow all that, just think of this as a system for figuring out what a guy would get if he was a free agent.

Dave's analysis tells us that Shawn Hill in 2008 pitched 63.1 innings with a 4.06 FIP, which is worth a skosh less than one win over replacement. Teams the previous off-season spent on average $4.6 million per additional marginal win in 2008. Therefore, Hill's fair market valuein 2008 by this metric was $4.4 million. In 2007, when teams spent $4.1 per marginal win, and Hill pitched 97 innings with a 4.03 FIP, he was worth 1.5 wins over replacement and about $6 million.

Hill's actual salary for those two seasons was $800k. So that means going into 2009, he's at least $5.2 million short of what old school hip-hop geniuses Eric B. and Rakim would call "paid in full."

Now some of you I'm sure are thinking, "wait, I'm looking at his baseball card right now and it says his ERA in 2008 was 5.83. How can that be worth anything over replacement?" The answer is that ERA is a wildly flawed stat that lumps in the whole team's overall run prevention ability and lays it all at the feet of the starter. Factors like fielding range, the bullpen's effectiveness stranding inherited runners, and pure sample-size flukiness all skew the pitcher's ERA, and it's not at all unusual to see a pitcher with an ERA a run or more above or below what he really "deserves." There are two stats out there that do the best job of isolating pitcher skill and rendering that as an ERA-like stat--FIP and tRA*. Both give you pretty similar results, so no need to get into that raging debate here.

In Hill's case, he was a bit lucky in 2007 and a quite a lot unlucky in 2008. His FIPs were pretty much the same--4.03 and 4.06, while his ERA swung from 3.42 to 5.83. How does that happen? His BABIP against rose from .269 to .373 (something around .305 is probably the mean for a groundballer like Hill), and his strand rate sunk from 71.6% to 61.9% (70% is typical). So those of you looking at his one win and ERA and thinking he stunk up the joint are looking at the wrong stats and drawing wrong conclusions.

Now back to the value per win calculation. I actually have a little issue with Cameron's method, which is that it implicitly assumes that if every player was a free agent that teams would keep spending the same amount per marginal win. Obviously, if every player was a free agent, the market would be flooded and simple supply and demand laws tell you that the marginal value per win will fall, and by quite a bit.

I prefer the method used by FiveThirtyEight's Nate Silver in his contribution to Baseball Between the Numbers, an essay titled "Is Alex Rodriguez Overpaid?" In it, Nate looks at the effect of each additional marginal win on team revenue, and figures what I think is a better measure of player value--not what owners will pay to acquire players, but what fans will pay to see them. Looking at all a team's revenue sources, from gate receipts to merchandise sales to revenue-sharing, he was able to use a bunch of fancy math regression analyses to conclude that in 2004 an average team generated about $1.1 million in additional revenue for each win. MLB total revenues increased 50% from 2004-2007, however, so we should raise that number to at least $1.65. (I haven't seen total revenue numbers for 2008; if anyone has them lemme know.)

Even with this more conservative definition of player value, fair compensation for Hill's 2.5 wins above replacement for 2007-2008 would amount to $4.1 million total for the two years, $3.3 million over what he was actually paid.

Of course, this method actually understates what players would get in a competitive bidding environment. The concept of the "winner's curse" in economics says that in a competitive bidding environment (like an auction or the MLB free agent market) the winning bid almost always overestimates the true value of the product. It's probably most fair to say that Hill's true fair market value for 2007-2008 is somewhere between $4.1 million and $6 million, or something on the order of $2.5 million a year, which coincidentally is what Daniel Cabrera got for 2009. And I'd rather have Hill than Cabrera, so that all sounds about right.

So y'all can root for the owners in arb if you want, but as far as I'm concerned, I'm with Jay-Z when he says, "Pay us like you owe us for all the years that you hold us / We can talk, but money talks so talk mo' bucks."

• Update: Scott Olsen and the Nationals settled on a one-year, $2.8 million contract. Applying the same process to Olsen that we did for Hill, we find that Olsen's 2007 176.2 IP and 5.33 FIP was worth 0.3 wins above replacement, while his 2008 201.2 innings and 5.02 FIP was worth 0.6. (If it seems odd to you that the durable Olsen hasn't been as valuable as the fragile Hill, but the fact is that 170 innings of pretty good pitching is more valuable than 370 innings of lousy pitching.) Total, the 0.9 win Olsen gave would be worth as much as $2 million per year using the free agent market to establish value and as little as $750k per year. I'll again split the difference and call him a $1.4 million pitcher--in other words we're paying him roughly twice as much as he's been worth the last two seasons. I don't begrudge Olsen--you gotta overcharge the game for what they did to the Cold Crush. But let's not act like Olsen's millions are fair and Hill's thousands aren't, ok?