Is the Cost of a Win in Free Agency Still Linear?

It’s no secret that free agency has changed over the last decade. As more teams have embraced analytics by focusing on paying for future, rather than past, performance, and owners have pinched pennies, we’ve seen slower winters, and in the case of last offseason, teams paying significantly less for a win on the open market. This offseason has seen a welcome return of activity, with good players receiving top-dollar contracts. When we consider the health of free agency for players, the big deals seem to grab a lot of attention, as with Gerrit Cole, Anthony Rendon, and Stephen Strasburg‘s this season, and Manny Machado and Bryce Harper’s a year ago. Mega-deals create the impression that all is well, and the size of those deals can have an outsized affect when calculating dollars per win, as in my piece yesterday on the cost of a win in free agency. But the players who don’t receive those big contracts deserve a bit more attention because it is possible that as free agent spending has shifted, the money teams are paying for wins may no longer be linear.

When we talk about the linear cost of a win, we’re talking about there being a uniform amount teams are generally willing to pay per win on the free agent market; if the cost of a win is $9 million, a three-win player gets $27 million, a two-win player gets $18 million, and a one-win player receives $9 million. And while we recognize the three-win player doesn’t actually receive a one-year deal worth $27 million, when the money is spread over a multi-year deal and the presumed decline from aging is factored in, the wins paid for over the life of the contract come out in roughly that manner. For example, Hyun-Jin Ryu is projected to be roughly a three win player in 2020. But over the course of four seasons, he is likely to be worth closer to nine wins; he signed a contract for $80 million, which comes out to right around $9 million per win. Not every case fits so neatly, but Ryu is one example.

The question now is whether the above is still true. In 2017, Matt Swartz examined the seasons through 2016 and found that the cost of a win was still linear. Since then, a narrative has emerged of slightly lesser players getting squeezed. Heading into the 2017 season, Travis Sawchik discussed baseball’s embattled middle class as players appeared to be getting frozen out of free agency. He followed that up in 2018 after another slow winter provided more evidence of a market in dire straights. Providing further support, the crowdsourced contract estimates our readers provide as part of our annual Top 50 Free Agents exercise have generally overshot free agent contracts under $40 million the last few years.

To flesh out that narrative further, and test whether the cost of a win is linear, I separated free agents into two groups: players with projections of at least two wins, and players below that level. This demarcation separates the average to above-average players from players below that bar. Over the last three seasons, here are the cost of a win figures, adjusted for the qualifying offer and with a 10% adjustment due to projections overshooting playing time and production:

Cost of a Win in Free Agency: Above-Average v. Below-Average Players
Offseason Cost of a Win in Free Agency 2+ Proj WAR 0-2 Proj WAR Difference % Change
2018 $9.3 M/WAR $9.4 M/WAR $9.1 M/WAR $0.3 M/WAR -3.2%
2019 $7.8 M/WAR $7.9 M/WAR $7.6 M/WAR $0.3 M/WAR -3.8%
2020 $9.1 M/WAR $9.5 M/WAR $8.2 M/WAR $1.3 M/WAR -13.7%
TOTAL $8.7 M/WAR $8.8 M/WAR $8.3 M/WAR $0.5 M/WAR -5.7%

Based on the above, there’s a pretty good case to be made that WAR is still linear. We do have a slightly larger change this season than from years’ past, but it’s still not massive, and over the past three years, the difference is just a 6% drop from above-average players to below-average players. We could argue that the change this season shows a difference in the way players are valued, but given the seeming drop in pay for below average players the past few years, it is awfully odd to find that there’s actually almost no change in those offseasons. I moved the groupings around a bit in terms of projected WAR, and took different subsets of the groups above, but the above data held. But there is one major problem with the dataset that could be deceiving our totals: relief pitchers.

Setting the bar at two wins generally sets above-average players apart from below-average ones, but that’s not the case for relievers. A two-win season for a reliever means that pitcher is one of the best of his kind in baseball. A one-win relief season is a good, and even half a win is a little above average over 60 innings pitched. As a result, including relievers in the below-average group means there are a bunch of above-average players in the dataset. I took relievers out; this is what the resulting the table looks like:

Cost of a Win in Free Agency: Above-Average v. Below-Average Players
Offseason Cost of a Win in Free Agency 2+ Proj WAR 0-2 Proj WAR, No RP Difference % Change
2018 $9.3 M/WAR $9.4 M/WAR $8.1 M/WAR $1.3 M/WAR -13.8%
2019 $7.8 M/WAR $7.9 M/WAR $5.3 M/WAR $2.6 M/WAR -32.9%
2020 $9.1 M/WAR $9.5 M/WAR $6.7 M/WAR $2.8 M/WAR -29.5%
TOTAL $8.7 M/WAR $8.8 M/WAR $6.6 M/WAR $2.2 M/WAR -25.0%

Now, the difference is obvious. The $1.3 M/WAR change we observed in the previous table seems more easily explained by the lack of quality relievers in this free agent class. Once relievers are removed from all years, we see the disparity between above-average and below-average players. Because the above-average players are the ones making all of the money, the change is obscured in the total cost of a win in free agency, while relievers hide the change for below-average players. Once we separate relievers out, we see a 25% discount in the cost of a win for below-average players. The last two years, the change has been over 30% and it has the chance to continue to drop this season. There are 17 more free agents on our Free Agent Tracker with projections between 1.0 and 1.9 WAR. Only Nicholas Castellanos seems likely to do much better than the current $6.7 M/WAR figure, and many of those remaining players seem likely to do worse.

The numbers above also help explain some of the curious non-tender deadline activity from earlier in the offseason. While César Hernández actually has a projected WAR just above two, players like Kevin Pillar, Jonathan Villar, Travis Shaw, and C.J. Cron might have been decent values in a linear free agent market. That’s not the case when the alternatives in free agency are cheaper. Overall, it’s pretty plain to see that the cost of a win in free agency is no longer linear given the huge discount being applied to below-average players. As for why this is happening, there are multiple reasonable explanations, including the availability of more good, young players, good teams not adding once they’ve reached a certain expected win total threshold, and bad teams not trying at all. The win-curve and a team’s financial outlook are topics that will be explored in my next piece.





Craig Edwards can be found on twitter @craigjedwards.

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Thrasius
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Thrasius

One other reason, 4 wins in 1 roster spot is presumably more valuable than 4 wins in 4 roster spots.

Edit: For clarity’s sake, that is 4 one win players.

bernardgilkeyhasaposse
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bernardgilkeyhasaposse

Yep, I’ve always felt that way. Linearity would also locigally presume a “perfectly liquid” market (my terminology is probably wrong), in that any team should be able to package two 2-WAR guys and a 1-WAR for a 5-WAR player.

But that doesn’t really happen, does it? This coming from a fan of a club that grows 2-WAR players on trees (thank you Devil Magic). Surely our front office would like to turn that into a star, but only under very limited circumstances (1 year of Heyward) has that even almost happened.

mikejunt
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mikejunt

The reason that it’s always been close to linear is that while you have a benefit from concentrating value in one spot (you can fit more players around), it is naturally counterbalanced by a risk for concentrating value in one spot (you lose more value if that one player is injured).

Now in practice, a good baseball team needs some of both elements in order to be succcessful. A team with just 2 war guys at every position doesn’t win enough games. A team with an extreme stars/scrubs structure can have wildly varying outcomes depending on the fluctuations of the star player’s season and injuries.

It doesn’t make it actually linear, but what it does do is make it approximate linearity up through the 3-3.5ish war level, and that represents an enormous majority of baseball players (nearly 90% of MLBers). So when it stops being linear, you are dealing with the best of the best and the supply/demand factors are radically different, as well as the impact on budgets (eg Trout is always going to be underpaid at any $/war calculation, because Trout’s war is worth the entire payroll of some teams according to a $/war calculation).

Linearity has never been a rule, it’s just been an an assumption of reasonable approximate accuracy given the relative inaccuracy of all our tools of measurement (including WAR itself).

In a world where pitchers in particular weren’t at such a constantly high risk of injury, it might look a little different, but every pitcher on earth is one pitch away from breaking like a cheap chinese toy and in many cases not even they know it is coming.

It’s not the best.

Matthew Tobin
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Which makes the choice of grouping players between 0-2 WAR odd to me. A player worth 1.8 WAR can get a pretty nice, even multi-year contract. A player worth 0.8 WAR probably gets a minor league deal with an invite to spring training. When you look at players worth < 1 WAR, you are looking at an entirely different model. Every team seems to have a handful of minor leaguers that can be roughly 1 WAR players, but not roughly 2 WAR players. At a certain point(I am curious where), a team may prefer a minor leaguer with a slightly worse projection for various factors such as roster flexibility, lower cost, and youth/more potential upside. If we calculate $/WAR of 0-2 WAR players right now, the result is probably skewed. This is because as I mentioned above: players on the lower end of this range sign just before or during spring training.

The upper end of WAR is interesting. Say Mookie Betts really wanted to be a Tampa Bay Ray. He was willing to sign a 4Y/$140M deal(with full no-trade)with them. Would they do it? They are clearly getting a bargain on a terrific player. But I am not sure. On the other end with teams like the Yankees or Dodgers, they also face the luxury tax, which makes investing so much in one player even riskier. Say a player has 4/$128M contract while contributing nothing on a team exceeding the luxury tax for first time(by $40M). The additional cost due to the luxury tax is roughly $70M, making the entirely sunk cost 4/$197.5M. In addition to that, you need replace the roughly 4 WAR(at say $8.5M/WAR)you are paying the bad player to not produce, so the cost essentially doubles. For those 2 combined players(let’s assume you replace bad player with a 4-WAR player on the same 4/$128M deal), you are paying 4/$395M for 16 WAR or roughly $25M/WAR.

My logic or math might be severely flawed. I know owners are insanely rich and agreed to this. But even a benevolent owner would find it hard to stomaching essentially burning $35M a year in taxes because they signed a bad contract and still want to be competitive.

sadtrombone
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sadtrombone

In theory, if you had a roster made up entirely of replacement-level players, it should be the same. In practice, it’s not like that at all, which is why you’re 100% correct here. Teams usually have an array of 1 and 2-win players, meaning a 1-win player is rarely an upgrade on what a team already has and a 2-win player is only a moderate upgrade in many cases.

And it gets more pronounced the further up you go on the win curve, because you get teams that are loaded with 2-win players and the only way they can improve is by getting 4+ win players. The Cardinals of recent years are the canonical example, who seemed to have 2-win players at every spot on the diamond and couldn’t really improve unless they got a Paul Goldschmidt (which is what they did). Furthermore, these are the teams most likely to spend in free agency.

rizdak
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rizdak

Agree. And some more marketing value for the 4-win player vs. 4 1-win players. I’d be curious how that could be calculated in attendance or other metrics.

Joe Joe
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Joe Joe

If WAR calculations were perfect and the market was perfectly liquid, 4 WAR in one roster spot (roughly 6 wins*) would be the same value as 4 WAR in 4 roster spots (roughly 12 wins*) as the R component of WAR (replacement if replacement is set correctly) takes care of the additional roster spots being used up.

*very rough only used to show that 4 one-WAR players are worth more wins than 1 4-WAR.

In practice, calculations aren’t perfect and market isn’t perfectly liquid. As such, WAR has a non-linear region for lower level players. This analysis suggests the upper range of the non-linear aspect of WAR is near 2 WAR.

Shalesh
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Shalesh

Can you explain this more? How are 4 Yolmer Sanchez’s worth twice as many wins as Josh Donaldson? Are you referring to $/WAR or WAR?

Adam S
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Adam S

1WAR = 1 win so I’m not following how you get from 4 WAR to 6 or 12 wins.

Joe Joe
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Joe Joe

1 WAR = wins minus replacement. I’m roughly saying a replacement player is worth 2 wins such that a 1 WAR player is worth 2 wins of replacement plus 1 win above replacement.

mikejunt
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mikejunt

Yeah this is not how it works, my friend.

Replacement level is the zero. There is no further baseline below it. Specifically, it works like this:

Step 1: Calculate the run environment. This tells you how many runs would be scored and allowed by a .500 baseball team.

Step 2: Now that you know that, you can calculate how many runs would be scored and allowed by a team of entirely replacement players. For consistenecy, Fangraphs, Baseball Reference and Baseball Prospectus have all agreed to define a .294 winning percentage as that of a replacement level baseball team.

That is the 0. It works this way so that talent that has no cost (freely available talent, eg infinite minor league scrubs who dont really belong in MLB but can fill in for a week) align to 0 war.

There is no ‘1 win for the replacement level’. Replacement level is absolutely, catastrophically awful.

The structure is set up so that any player with an economic value will also have a value above replacement level.

Using average doesn’t work; if you try to do a $/WAA analysis, you will end up with negative numbers because there are players who are below-average but MLB teams are willing to pay them real money because their performances are valuable to them (eg many 4th and 5th starters). This makes economic analysis impossible, because you get an inversion from guys with a 95 ERA+ who, in fact, get paid millions of dollars to do that quite consistently.

Hope this helps.

dbminn
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dbminn

Teams don’t start with a blank roster. They have – at least – readily available replacement players. That’s why WAR is used for the calculation.

To use “WAR + replacement wins (RW)”, including the roughly 2 wins a replacement player provides, the calculation needs to be made to the base data, not at the end. Replacement Wins need to be based on slightly above league minimum, maybe $1M for the 2 RW.

Paying $36M for a 4-WAR player becomes $37M for a “6 RW+” player, or $6.2M/RW+. Paying $13.2M for a 2-WAR player becomes $14.2M for a “4 RW+” player, or $3.05M/RW+. Makes the difference look even greater than with WAR on a percentage basis.

Thrasius
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Thrasius

wat

RoyalsFan#14321
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RoyalsFan#14321

“In practice, calculations aren’t perfect…”

No kidding.

Adam S
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Adam S

Conceptually this is true, certainly at scale — 5 4 win players are more valuable than 20 1-win players because you run out of places to use meaning they aren’t all 1-win players for your team. But there’s also a risk in bunching your WAR in one player. Unlikely that 4 players all get hurt or all suck whereas if your star gets hurt or suddenly falls off the cliff, you’re really hurt.

So, teams have paid close to linear for wins.

I think there are two factors driving change:
1) As noted below, teams are realizing that 0-1 WAR veterans can be replaced by a replacement level player or scraping the free agent bottom of the barrel for 1/10th the cost.
2) Middle relievers are so inconsistent that they’re fungible and have almost no upside. It’s not worth paying for a 1-WAR reliever like he’s going to deliver .8-1.2 WAR with near certainty.

Shalesh
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Shalesh

Middle relievers are the ones who keep consistently getting paid though even in the last 2 years of diminished FA. They were the standout group, other than the SuperStars.

I agree more with you, but the league seems to keep signing them.

Smiling Politely
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Smiling Politely

One, 4 WAR player is also easier to trade and exchange for value than are four, 1 WAR players

Dan B
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Dan B

I think teams are getting more confident in their ability to find 1-2 war players amongst their older, more limited, less sexy prospects.
I wonder if the replacement level baseline is a tick too low?
It might be that it looks more linear if you set the replacement level to 0.x of a win?

Dan B
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Dan B

What I think I am saying is that i don’t think what we see here is nonlinearity, it is just a slightly out of whack baseline and if you were to regress wins against dollars, then it would give you an equation like
$ = a + b * war (where a is slightly negative) which would fit the data pretty well.