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This article started with the realization that I had been wrong. Believe it or not, it’s happened before; unlikely as it seems, it may happen again. I think I’ve got it right this time, but in the spirit of intellectual honesty and taking responsibility for my past arguments, I wanted to note that I’ve been down this road before.

If you’ve seen…well, just about anything I’ve published about baseball over the last several months, you may know that I am particularly interested in how markets in baseball work (and I don’t just mean for players). At some point in the months-long process of writing my senior thesis, I had what I thought was an epiphany: If MLB teams are anything close to rational, the cost of a win on the free agent market must be approximately equal to how much a marginal win is worth across the league. It seemed logical, and it was eloquent in its simplicity.

I now have a different idea about where the cost of a win comes from—it’s more convoluted, but it also makes more sense. And in order to properly explain why I think I’m right this time, I have to start with why I was wrong before.

(Note: For the purposes of this article, I define the value of a win not in terms of the concrete increase in revenue or net worth that a team would realize from improving its position in the standings by one game, but by the total value of the monetary and nonmonetary benefits a win would provide to a team and its owner(s).)

Where I Went Wrong
The logic of my previous interpretation was compellingly simple. In any given season, there is some number of wins available for purchase via free agency, and the 30 MLB teams combine to spend some number of dollars on them. If teams valued wins at more than the sum of their free agent expenditures divided by the number of wins available for purchase, they would bid up the prices of free agents until the cost of a win was commensurate with its value; if the cost of a win exceeded what it was worth, teams wouldn’t be willing to pay it, and the market price would come down. It followed that the cost of a win equals what a win is worth to an MLB team.

The first sign that I had something wrong was the implication that a win is worth the same amount to every team. If the cost of a free agent win is more or less universal (as it should be in a rational market) and that price reflects one’s willingness to pay for it, then the value of a win must be homogeneous across those who buy them. Any team considering entering the free agent market would do so only if it either were willing to pay market price for them or thought it had found an inefficiency in the market, and if the market is anywhere close to rational the latter wouldn’t hold for very long.

In practice, all 30 teams buy into the free agent market to some degree every year. By this logic, that implies that a win has basically the same worth to every team. This is possible if one does not think of the value of a win purely in terms of increased revenue, but it doesn’t pass the sniff test.

At issue here, I realized, is something Matt Swartz had first suggested to me in a totally different context: capacity constraints. Though I believe it is fair to assume that wins are generally fungible, roster limitations prevent teams from buying wins ad infinitum. No matter how much money you’re willing to spend, you can’t add 26 players to your 25-man roster. With several dozen noteworthy free agents hitting the open market every year, the market simply could not function if the few richest teams were setting the market price for everyone else.

Put it another way: Imagine if Magic Johnson instructed the Dodgers front office to build literally the best team money could buy and made it known throughout the game that he was willing to pay $100 million per marginal win. The Dodgers would surely be able to outbid any other team with whom they would be competing. But at this extreme—and to be clear, it is an extreme—there’s a limit to how many wins they can buy.

According to my research, there have been somewhere between 150 and 200 free agent wins for sale each season in recent years. Depending on the specific players available that year and the team in question’s preexisting talent, the absolute ceiling of how many free agent wins a team could buy is probably somewhere around 50. (Given how good the Dodgers’ current roster is, it’s likely quite a bit lower for them.)

So if there were, say, only 20 wins available one year, the Dodgers would presumably buy all of them, given that they were willing to shell out $100 million a pop for each (though they needn’t pay quite that much for a win unless someone else is offering $99,999,999). But if there are nearly 200 wins available, the Dodgers cannot possibly represent more than a (substantial) minority of the market. And the remaining free agents can’t demand that other teams match what the Dodgers are giving their peers if no one else values a win as highly as they do.

The Inelasticity of Supply, and Why it Matters
At issue here is the principle of price inelasticity. For those who have never had the pleasure of taking an economics class, elasticity is the measure of how responsive the supply of or demand for a product is to changes in its price. If a marginal increase in price leads suppliers to increase their production by a proportionally large amount, supply is said to be relatively elastic; if a change in price doesn’t have much impact on the quantity that producers are willing to provide, supply is said to be relatively inelastic.

In constructing the market for free agent wins we run into a special case of a supply curve: one that can be expressed as approximately perfectly inelastic. In other words, the number of wins for sale will be about the same no matter what the market price is for them. Sure, you’ll get a few guys who will go to a foreign league or retire if they don’t get satisfactory offers from MLB teams, but those who do generally don’t add much to the supply of available wins. It’s not as though Robinson Cano would have hung up his cleats this winter if the alternative had been settling for an eight-figure deal. On the other hand, Mike Trout can’t quit and offer himself up as a free agent if the cost of a win rises.

If supply is perfectly inelastic, then the equilibrium quantity Q is a given. The equilibrium price is thus defined as whatever price causes the 30 MLB teams to collectively want to purchase Q wins. So the cost of a win isn’t a reflection of how much an abstract generic win is worth—it’s derived from the value of the Qth win available.

How This Works
As an overly simplistic example, consider two teams from my hypothetical E Street League: the Jungleland Runners and the Tenth Avenue Rivers. Based on their current projected position on the win curve, the size of their media market, and the impatience of their fanbase and ownership (among other things), the Runners value their next marginal win at $10 million, but additional wins beyond that are worth only $1 million to them. Given their different team circumstances, the Rivers value every win they can add at $5 million apiece.

The two teams are competing to sign a single free agent, Sandy, to a one-year deal, and they both project her as a one-win player for the upcoming season. (Suppose there are enough other teams in the league that the win whoever signs Sandy gets from her won’t come at the other’s expense, but that only the Runners and Rivers are interested in signing free agents.) In this case there is only one win available (i.e., Q = 1), and the Runners value it more highly, so they will sign Sandy for somewhere between the $5 million she could’ve gotten from the Rivers and the $10 million they think she’s worth, depending on how the bidding process works.

At this point, let’s add another identical free agent to the mix: Rosie, who plays the same position and is considered by both teams to be exactly as valuable as Sandy. In this case there are two wins available, though the first is already spoken for. The Runners know they’re getting one win, and the next win would be worth only $1 million to them; meanwhile, the Rivers would offer at most $5 million for whomever the Runners don’t sign. Now that Q = 2, the price of the second win has to fall to at most $5 million (but possibly as low as the Runners’ best offer of $1 million, depending on how the bidding process works) in order for both Rosie and Sandy to sign. And in an open market without price discrimination, if the equilibrium price is $5 million, then both wins will cost $5 million regardless of how much Sandy would have gotten had she had the market to herself.

Back to reality: What happens when 30 different teams, each in a different place on their differently shaped demand curve, compete for Q wins on the open market? Abstract as it sounds, finding a concrete solution to such a model would require determining which team values the first win most highly, which team values the second win most highly conditional upon who would win the bidding for the first win, etc. After Q – 1 wins have been hypothetically distributed, the highest amount that anyone is willing to pay for the Qth win will be the equilibrium price.

…or something like that. In practice, the team that values the Qth win most highly needs only to beat the runner-up bidder’s best offer. If the highest-bidding team knows what its chief competitor’s maximum bid is, it can simply buy the Qth win at one dollar more than what anyone else is willing to pay—and set the market price in kind. Realistically speaking, the market isn’t neatly organized enough for the team in question to make that precise an offer. And after Q – 1 wins are already off the board, the differences between the top two bidders for the next win would likely be small enough that the minimum winning bid would serve as a good approximation for the market-clearing price anyway.

What This Means
If you’ve stuck with me through this jargon-filled journey into the nuances of baseball economics, it’s probably occurred to you that the real world isn’t nearly as neat and organized as my economic models would imply. Teams aren’t fully rational and they don’t have perfect information about their own respective circumstances, let alone full knowledge of who’s willing to pay what for the next marginal win. I don’t mean to imply that we should have blind faith in the invisible hand to move teams’ decision-makers along these lines. But on some level, when groups of smart people running billion-dollar baseball teams spend their days trying to do their jobs as efficiently as they can, it’s reasonable to think that they move toward collective rationality.

And so, whatever you think the equilibrium price of a win is, it isn’t necessarily a reflection of how much a win is worth. When I estimate that the cost of a win in 2013 was $7,032,099, that implies that it took lowering the price to $7,032,099 for all the available free agent wins to be purchased (or that the next-highest bidder for the Qth win would have paid $7,032,098 for it). The same applies to Matt Swartz’ various models, which peg last year’s price at closer to $8 million. Or Nikolai Ballevski’s $7.5 million projected cost of a win in 2014, or the various estimates that Dave Cameron came up with for how much teams paid per wins this past offseason.

That said, I still think the price of a win serves as a satisfactory estimate for how much a win is worth to the average team. On the one hand, it’s a clear underestimate of what a win is worth: If it took lowering the price to the equilibrium point to sell Q wins, presumably the first Q – 1 wins were worth more than the market price to the teams that bought them. But signing a player usually requires thinking he is more valuable than the other 29 teams do; sometimes that’s a function of team-specific circumstances, but it usually means the signing team is somewhat overrating the player it is acquiring. If teams think they’re spending less per win than they actually are, the amount they pay per win could be more than what they would willingly spend. Assuming that these two opposing biases basically cancel each other out—and admittedly, that is a big assumption—then the cost of a win would be approximately equal to the general value of a win (to the extent that such a universal figure can exist).

This doesn’t invalidate the general rule that teams shouldn’t willfully overpay for free agents (if a win is worth twice the market price to you, why offer your maximum bid when you could buy two wins instead?), and it leads to a number of questions about how the interplay between the cost and value of a win affects a team’s optimal player-signing and employee-hiring strategies (a topic I plan to explore in more detail at BP in the near future). But even on its own, I think how the market sets the price of a win is important information—especially since it’s not just a question of what a win is worth.

Thank you for reading

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I think you are missing one important point in your construct. Given the "25 roster spot" constraint, I think there is also a different marginal value per win for the highest WAR players (since their supply is especially limited, and their value to the richest teams is uniquely higher). I end up with the Canos, Ellsburys and Tanakas valued at $8 or $9 million per win because their wins are more efficiently packaged vs. roster spots, but a 0.5 WAR player only commands $4-5 million per win (since there are far more of them, and a finite supply of roster spots to employ them). You get test this hypothesis by seeing if the residuals are positively skewed vs. your "average" valuation for the most expensive players.
Excellent point. When the picture of Hanley that accompanies the article was captioned "If Hanley hits free agency, he'll hope the cost of a win continues to climb." I was hoping that this was the reason why. He's precisely the kind of guy who could get the elite player bump.
That's a good point, but as I said the idea of the roster constraints actually limiting how many wins you can be is a truly extreme case.

If the Dodgers were trying to put together literally the best team money could buy then yes, they'd be willing to pay a premium for being able to condense it into one roster spot. But that would really be an exceptional scenario.

To use the Cano example, are the Mariners really a good enough team that having Cano and his X wins is more valuable to them than having two players who would each be worth X/2 wins? I strongly doubt it that their replacement level is that high. That doesn't mean that they're not willing to pay a premium for a star, but it does mean that doing so isn't rational.

Matt Swartz has done some good work on this in the past that showed that, in general, the cost of a win is linear—i.e., a player worth X wins really does make about double what a player worth X/2 wins makes. I believe Dave Cameron found evidence of a premium for star players this past offseason, but his methodology is less similar to mine than Matt's is and he noted that this winter was the exception to the norm.
PECOTA thinks Cano is worth 5.2 WARP. In CF, they start Abraham Almonte (0.5), But at every other position, the Mariners field a player worth 1.5-2.5 WARP.

Absent Cano, they'd presumably play Nick Franklin (2.5 WARP in 2013) at 2B. Franklin's value is glove-heavy, so they can't slide him to DH, and thus lose him to AAA. So adding Cano is still +2.7 WARP, despite a worst-case situation where the backup choice looks unusually good, but yet can't move laterally to replace a different starter.

Yet adding two half-Canos looks worse unless one can play CF, since they'd be replacing MLB starters who average around 2.0 WARP each, and the Mariners would only total +1.2 WARP for the two Half-Canos (+2.6 each).

Keep in mind that high-end free agents displace starters, not the 25th man, and starters aren't usually worth 0.0 WARP. Now, the starter they displace often does displace the 25th man, but with reduced playing time, the value of that is minimal unless an injury happens.

A more thorough study (for example, constructing a 'what-if' tree for each team's WARP change upon losing each starter) could find the optimal strategy. Matt Swartz answers what teams _are_ doing (a start), but not necessarily what they _should be_ doing. As your thesis shows, those don't always intersect.
But there is another problem -- it is not entirely clear at least intuitively that each players financial contribution to the team can be measured by wins, especially in the short term. For example, how much was Mariano Rivera worth to the Yankees in attendance, souvenirs, TV audience, refreshments, etc. last year? His financial impact to the team may well have been significantly greater than his proportionate "win" value. And that is not only a Yankees phenomena -- Mets attendance when Matt Harvey pitched versus all other games; keeping Giancarlo Stanton in FLA; the Phils not blowing up the team; Miggy's contract extension all may make a lot more financial sense than they do if you just took total WAR of free agents/ total spending to come to an average. Many people pay to see the stars, even if two regulars add up to the same number of wins. My point is that the outlier players are likely to get disproportionate payment, even apart from the roster limitation/opportunity cost issue.
You're right about that, and that's something I'll be exploring in my next article. The problem with that way of thinking is that those kinds of marquee players are very much the exceptions, not the rules. Of the dozens of free agents who hit the market every year, how many of them really carry that kind of celebrity value? I'd guess at most only 10-20% of them bring any significant amount of extra revenue besides their on-field performance. Sure, it matters for a few players, but I'm not convinced that it would have a substantial impact on the market price of a win.
In the example, is the next win valued at 1 million because of budgetary constraints? If that's the case, after they sign player 1 to a 7 million dollar deal, they should now be and to offer player 2 four million, which means player 2 will still get 5 million from the second team pushing the value of the win above five million.
Not necessarily.

Imagine a team that's 90% sure it will make the playoffs. Adding one more win could maybe bump its odds up to 99%, so that could be worth a lot. But any additional wins beyond the next one don't matter very much. That's more what I had in mind.
And you wanna be my latex salesman? :) I'm no Art Vandelay but he told me you guys are awfully good at math. What I glean from this discussion is that dbiester is on to something.
I thought it would come out in the piece but the Yankees must win consistently or the edifice topples. Their winning is worth so much more money than anywhere else. This has been written about elsewhere.
Yes, but the article acknowledges that different teams have different priorities on the demand side, so a win to the Yanks may be worth more than one to the cubs, and then you get into marginal win values and play off scenarios, but there is also a value above the win contribution for certain players -- call it the "celebrity value" if you will, which makes the "average" value of a win interesting but incomplete.
Another perspective on this: I'm sure you're not advocating that teams in any way get together to reach "collective rationality", as that would have extremely serious implications for baseball's antitrust exemption. Thus, another factor in this calculus is that teams are essentially required to operate in the dark, at least at the micro level.
The market will never be flat because, first the teams have widely variable budgets (so $3M to Miami means more to them that $3M does to the Dodgers), and due to relative wins.

By this I mean that, as another poster said, the Free Agent will be displacing another starter with some value. As different teams have different valued starters, the relative wins the Free Agent adds depends on who is being displaced.

For example, if a team has no incumbent at Shortstop, then the value of the Free Agent would be its highest, or equal to their wins. However that same player going to another team with a serviceable incumbent, would have less value because there is a difference in wins. And another team might actually have a better incumbent than the Free Agent, but want to bring them on board as insurance. So its not as simple as buying wins.

Also, because new players are always graduating to the major leagues, and are all relatively free, there is always certain downward pressure on free agent costs on those teams with good player development. So 2 teams with similar payrolls in similar markets will value the same player differently if one's farm system is better than another. They aren't as desperate to fill the hole and possibly block a free player in the future.
One thing that a lot of people taking this approach don't take into account is something I like to call "Someone's Gotta Play The Games."

Assuming relative rationality on the part of teams, the management is reasonably aware of a team's chances of competing in a given year. But even if a team isn't going to compete, they are still responsible for putting a reasonable entertainment product on the field. Now the traditional response from the sabermetric community has been, "This is no problem, freely available talent, replacement-level players!"

The problem is that running out a squad of 4-A guys concedes to everyone--both the industry and, probably more importantly, local fans--that the towel is out before the first game is played. Beyond that, the 4-A guys may end up being a lot worse than advertised or may end up being embarrassments at the Major League level in conduct, bonehead mistakes, etc.

As a result, while it drives sabermetric types crazy, I'd argue it actually is rational to pay some amount of money above the Major League minimum salary to veterans who project to play at or below replacement level simply to purchase a kind of respectability--even if that purchase nets the team no wins.
Is that you, Dayton Moore?
I buy that for sure and I think it fits very well with my theory.

If you'll notice, I haven't made any claims about what any individual team's demand curve for wins actually looks like nor what it should look like. What you're basically saying (I think?) is that the 50th win is generally worth more to a team than the 80th win. That actually explains why even bad teams enter the market every year.
I was following along pretty well until you introduced the "equilibrium quantity Q" stuff. Would you mind elaborating on that concept for the uninitiated?
In any given market, there is a quantity Q and a price P of goods/services produced/provided. If you have graph with supply and demand curves (both are expressed as Q as functions of P), the equilibrium point—i.e., where the market will produce—will come where the two lines cross. At that point, the price will lead consumers to purchase exactly as much as producers are willing to supply.

In this case, we take the equilibrium quantity Q as given because the supply of free agent wins is basically static regardless of price. The question, then, is: at what price P will the 30 teams combine to want to buy exactly Q wins?
Thanks, Lewie! That clears things up for me considerably.