More Musings on What Teams Are Paying for a Win in Free Agency

Earlier this week, I wrote about the cost of a win in free agency. I loved seeing the discussion of that article online and in the comments section, so I thought I’d set aside some time to consider a few of the questions readers had. Here are my answers to those questions.

What if We Used More Tiers?
If three tiers is good, would four be better? Five? Six? In my initial analysis, I ran all these variations in the background and decided that three was optimal for presentation and clarity. I also determined that the sample sizes would get vanishingly small as we expanded to more and more tiers. But as several readers asked for more granular looks, why not? Here is a four-tier version:

And a five-tier version:

Why didn’t I use those? They didn’t add all that much information. Per a log likelihood test, the four-tier model reduces error by around 1-2% relative to the three-tier model, a minor upgrade. If you’ll recall from the previous article, going from one to three tiers was worth a 13% reduction in error. By the time I divided things into five tiers, log likelihood saw no improvement relative to the three-tier model. That’s the cost of increasing complexity with small sample sizes. The free agent market is split into multiple levels depending on player caliber. But there are diminishing marginal returns as you slice the data more and more finely. A three-level model is a huge improvement on a one-level model, but past that, the data is less clear.

What About Pitchers Versus Hitters?
I modified my script to run tiers for pitchers and hitters separately. Here’s how those look:

The differences in the top two bands aren’t statistically meaningful. But that 0-1 WAR tier of pitchers is fascinating. Of note, there are far more hitters in the 0-1 WAR group than there are pitchers; since we’ve separated out relievers from our data, the populations don’t quite match. Still, it’s a pretty interesting effect.

I have one main theory. That bucket definitely has a lot of pitchers returning from injury. Imagine a pitcher with a 0.5-WAR projection in their first year back from elbow surgery signing a two-year deal. My rule-based system for defining WAR mechanically assigns them 0.5 WAR in year one and 0.1 WAR in year two. But that year one projection was low not because of a lack of talent, but a lack of availability. When you’re working with big sample sizes like this, it’s important to think about weird corner cases your rules might create, and this seems like a good one to point out.

In any case, there’s something interesting here: Pitchers don’t have as clean of a talent-based split in compensation compared to hitters. Teams always need more pitching, and there’s probably something to the fact that pitchers aren’t getting any younger, which means that some of these contracts might be more about availability than talent. We talk about the lack of availability of good pitching all the time; maybe that’s leading to a dislocation in the lower reaches of the free agent market. It’s something to keep an eye on, in any case.

Is Offensive WAR Better Compensated Than Fielding WAR?
The way I collected the data means there’s no way to separate out this particular facet of player value, but I have thoughts on it anyway. The answer is a clear yes. When I make individual contract predictions rather than looking at broad league-wide trends, I include that effect every time. There are any number of reasons this could be, but I have a few favorites.

First, it’s a supply and demand thing. Good fielders aren’t actually that monetarily expensive for most teams, because many good fielders are young and cost-controlled. There are more plus center field defenders in the minors than plus center field hitters. Given that, why would teams pay the same rate for good fielders and good hitters on the open market? WAR is a nice summary statistic given a neutral context, but this is clearly not a neutral context.

Could it be a positional adjustment issue? Sure. The DH and first base adjustments might be a bit harsh, something we’re looking into in the medium term. But the effect isn’t limited to those positions, at least in my experience in contract predictions. Offense-first players are prioritized across the board; it’s just most evident in the two most offensively-focused positions.

Could it be a stability of defensive projections issue? Eh, I’m skeptical. I haven’t seen any evidence for it, but it’s the kind of thing that I don’t know how to prove or disprove mathematically. For now, I’ll just say this: As a rule of thumb, bat-first players get a little more than their raw projected WAR would suggest, and glove-first players get a little less. Going into any more depth is outside the confines of this type of analysis.

What if Replacement Level Is Higher?
This was the main argument I wanted to write about today. This is one obvious interpretation of the fact that teams don’t want to pay much for “one win above replacement.” If replacement level is one win higher than we thought, of course that first “win” isn’t very valuable.

Here’s a simple thought experiment: Imagine two full-time players. Player A is projected for 3.0 WAR and receives a one-year, $40 million contract, $13.33 million per win. Player B is projected for 1.5 WAR and gets a one-year, $15 million deal, $10 million per win. Now, let’s raise replacement level by a win and re-evaluate. Now we’ve got 2.0 WAR and $40 million, or $20 million per “new win,” for Player A, and 0.5 WAR and $15 million, or $30 million per “new win,” for Player B. A higher replacement level makes dollars-per-WAR go up for everyone, but it makes that number go up the most for the players who are closest to the new replacement level.

In the past, I’ve shied away from this method because of one problem: Teams still do pay for that first win above the current replacement level. By moving the bar higher, we introduce a system where teams pay for wins below replacement. That sounds conceptually weird to me. But I’ve spent some time brainstorming this over the years, and I came up with a workaround that I think lets me do the math anyway.

First, I reset replacement level by reducing the number of WAR available to the league as a whole. Right now, we define replacement level so that there’s exactly 1,000 WAR every year, which implies that a replacement-level team would play at a .294 winning percentage clip. I reduced the amount of WAR handed out sequentially, anywhere from 50 to 300 wins (a .356 replacement level). That involves changing the per-PA and per-IP replacement level and then recalculating each player’s WAR at each breakpoint. Luckily, I could just tell my computer to do that.

Next, I had to sort out the below-replacement-level pool. If a player is projected for negative WAR, but a team signs him for $2 million, they’re paying $2 million for some base competence level that they don’t feel they could get for free. I defined this mathematically by taking the amount of money teams guaranteed to below-replacement-level players, dividing by their playing time, and then adding that up into full-time equivalents (160 IP, 600 PA). That gave me the base amount that teams are paying free agents for their talents. I then subtracted that base amount from every player’s contract, and did the same WAR tiering as before. A methodological note: I excluded 2020 for playing-time-projection reasons.

Let’s do a quick example of how this works, because I know that’s a lot of math words all slammed together. If I redefine replacement level so that the league only has 750 WAR available, there have been 93 contracts handed out to below-replacement players in my sample set. They got an average of $3.9 million per full-season playing time equivalent. In other words, the cost of finding a capable, full-time veteran is $3.9 million, whether they clear the replacement level bar or not.

From here, I separated the remaining players into the same tiers as before: 0-1, 1-2, and 2-plus WAR. I also subtracted out the base contract amount from above, pro-rated by the playing time that Steamer projected for each player. I got 377 0-1 WAR players, 158 1-2 WAR players, and 87 2-plus WAR players. In addition to the base amount, teams paid $7.7 million per win to the 0-1 WAR group, $11.1 million per win to the 1-2 WAR group, and $13.9 million per win to the 2-plus WAR group. In other words, the tiered structure remains a good description of the market.

That’s the example, but here are all the numbers:

The higher I set replacement level, the smaller the gap between the 0-1 WAR and 2-plus WAR tiers got. It never disappeared, though. Even at higher replacement levels, teams still pay up for stars.

For good measure, I tried another, simpler way of moving replacement level. Instead of looking at the contracts handed out to below-replacement-level players to calculate a base amount for a free agent contract, I simply threw them out. Then I did my original tier-by-tier study the same way: calculating dollars per WAR for each group separately. Here’s how that looks:

In other words, no matter how you slice it, the tiered nature of free agent compensation is about more than just where replacement level is. Sure, setting a higher replacement level compresses the spread. It has to, mechanically. But the data still show that teams are paying the best players more per win, even if you assume replacement level is very high.

I had dreams of building another version of this replacement level study where I create multiple replacement levels for different buckets of teams. The Dodgers and White Sox don’t have the same replacement level, of course. I might still get that math done one day, in fact. But I wasn’t quite able to make it work for this article, and I wanted to get some data out, so it will have to wait. The takeaway here is that while changing replacement level might explain some of this behavior, the salary structure remains non-linear. Teams pay more per win for the very best free agents.