Wins Above Replacement (WAR): What is it good for? (not “absolutely nothing”)

A few days ago, Patrick Minton wrote an article about the Myth of the Free Replacement-Level Player (at  And after trying to write a few comments on this post, I promised I would write a post of my own outlining my thoughts on this issue.    These thoughts will essentially serve as a quick primer on the basics of labor economics. Hopefully this primer will be of some help.

For those not familiar with the idea, the concept of a “replacement player” is utilized to construct the Wins Above Replacement (WAR) measure used in baseball and the Wins Above Replacement Player (WARP) developed for basketball by Kevin Pelton.  Essentially, it is argued that to measure a player’s performance you MUST compare that player to the value of a replacement player.   In sum…

WAR = Player Productivity — Productivity of Replacement Player

I have questioned the usefulness of this measure.  Having read more and more about this, I think WAR is useful.  But I don’t think this measure is quite what some proponents seem to believe.

To explain the problems I have with this measure, let me start with a very basic research question: What question is this measure designed to answer?

In looking at the comments to Patrick’s post and other writing on the subject, it seems to me that people are using this measure for very different questions.  Unfortunately, I do not think it works for all the questions people are asking.

I am going to start with a few questions I think WAR does not answer.  Then I am going to discuss the question I think it does actually address.  Along the way, I hope to explain some of the basics of labor economics that I think you need to understand if you wish to understand how workers contribute to a firm’s success and how these workers are compensated.

1. How much did the additional worker produce? 

I think this is the primary question people think WAR is addressing.

But if you want to measure the productivity of a worker, I think you need to go back to the basics of labor economics.

Let’s begin with a simple term “marginal product”.  This is simply defined as “the output from an additional worker”.

Applying this to sports, if we assume a team is trying to maximize wins (and that is not always an assumption that matches what the team is doing), then the output we wish to measure is how many wins the additional player produces.  These additional wins are called the player’s “marginal product”.

In calculating marginal product, we simply look at what the additional worker adds to the firm’s output.  WAR, though, does not take this approach.  WAR is a player’s production of wins minus a replacement player’s production of wins.  But this is not how we measure marginal productivity.   And I would argue, this is not just economic theory.

Think of your own job.  When your employer determines your productivity, do they subtract off what they could have received from a replacement?  I used the example of someone selling cars.  If a person sells 5 cars in a month, the car lot will not say “yes, but a replacement is worth 2.  So your actual output is only 3 cars sold”.

Not only does this idea not exist in labor economics, I do not think you can find someone measuring production of a worker in this fashion (outside of WAR enthusiasts).  And that is why I do not think “how much output did the additional worker produce?” is the question WAR is really designed to answer.  Subtracting off the value of a hypothetical replacement player is simply not how you measure productivity.  A worker’s productivity is simply what that worker produces.  So in basketball, an individual player’s productivity is how many wins the player produces.

2. What is the economic value of the worker?

To answer this question, economists use the term “marginal revenue product” (MRP).  A worker’s MRP is calculated as follows:

MRP = Marginal Product of worker * Marginal Revenue of output

In simple words… MRP is the productivity of the additional worker multiplied by the value of that output in the marketplace.   For example, if a player produces 5 wins and each win generates $2 million in revenue, then the player’s MRP is $10 million.

The MRP of labor is labor’s contribution to the revenue generated by a firm.  And the MRP of capital is capital’s contribution to firm revenue.  It is important to emphasize that MRP only includes the productivity of the input (i.e. labor or capital) and the value of that production in the market where the firm sells its output.

Again, there is no mention of a replacement player.   We do not calculate MRP by subtracting off the value of a hypothetical replacement of labor or capital.

3. How much SHOULD an additional worker be paid?

You should note that so far — in addressing questions #1 and #2 — I have not used the word “pay”.  That is because a worker’s contribution to firm revenue (i.e. MRP) and what a worker is paid is not the same thing (and because this key to the story I am telling, I am going to repeat this point again below).

Now it can be the case that MRP and salary end up being the same number.   If a labor market is competitive (assuming MRP can be measured and is measured accurately) then competition over a worker’s labor will result in wages equal to MRP.  So in this particular case, wages and MRP are the same number.

And we tend to think that a competitive market is the ideal. Therefore, we tend to argue that wages should equal MRP (notice I put the word “should” in bold; that’s because in a moment I am going to discuss what a player “is” paid).

4. How much IS an additional worker paid?

We define “exploitation” as the case where the wages paid to the worker are less than MRP.  How can “exploitation” happen?  It happens when the labor market is not competitive.  The history of baseball provides a great example.  Prior to the 1970s, baseball’s labor market was dominated by the reserve clause.  Because of this clause, player could only negotiate with one team.  And when workers are not selling their labor in a competitive labor market we expect wages to be less than MRP.

Consider the case of Reggie Jackson.  In 1973 he was named MVP of the American League while playing for the Oakland A’s.   But the reserve clause was still the law of the land so his salary was only $70,000.  Four years later, though, free agency came to baseball.  And when Jackson joined the New York Yankees, he was paid $525,000.    One could imagine that Jackson’s MRP increased by more than 700% between 1973 and 1977.  However, a more plausible story (and one more consistent with published research) is that the reserve clause led wages to be less than MRP and when the reserve clause was removed, Jackson’s wages were much closer to his MRP.

Jackson’s story (and published research from economists) suggests that labor market institutions impacts what a player is paid.  But the move from the reserve clause to free agency did not impact Jackson’s marginal product or MRP.

The same story would be told about roster limitations, imposing a minimum wage, caps on salaries, caps on payroll, and luxury taxes.  These institutions could impact the bargaining power of owners and players (and therefore impact the wage the player’s are paid).  But such institutions do not impact a player’s productivity on the field (i.e. his marginal product) or what that productivity is worth in terms of ticket sales, broadcast revenue, etc.. (i.e. the marginal revenue of output).   A player still produces the same wins. And those wins are worth the same in the marketplace where the firm is selling that output.

And this the big point I am making about WAR.  When we measure MRP we do not consider such issues as the minimum wage paid to workers or hiring constraints.  Again, these can impact what a worker “is” paid.  But they do not impact the worker’s MP or the firm’s MR (and thus, the worker’s MRP).

So WAR can’t help us measure MRP.  But it can help us think about the compensation of workers.  And that topic includes the concept of “opportunity cost”.

4. What is the opportunity cost of signing a player?

In deciding whether or not to hire a player at team considers a player’s future MRP.  And they consider the wage the player is demanding (and/or the wage they think other teams might be willing to pay).  But this is not the entire story.

Consider the story of Carmelo Anthony.  The Knicks recently signed Anthony to a five-year contract worth $124 million.   Last year (according to my calculations), Anthony produced 6.9  last season.  If he could continue to produce about seven wins a season for the next five years, and each win was worth $3.5 million to the Knicks, then Anthony’s MRP (35 wins * $3.5 million) would be about equal to his $124 million salary.

Of course, it seems unlikely that a 30-year old player will maintain his production.  And although it is not clear what a win is worth to the Knicks*, it seems likely that $3.5 million is too high.

But let’s pretend for a moment that Anthony is being paid exactly his MRP over the next five years.  Is this a good deal for the Knicks?   That would depend on the team’s opportunity cost, or the value of the team’s next best alternative.   For example, the San Antonio Spurs spent $63 million this past season and won 62 games.  So the Spurs spent about $1 million per win.  That suggests that it is possible for a team to spend far less than $3.5 million per win.

In sum, there is an opportunity cost to sign Anthony.  And although we may not know all the alternatives the Knicks had to re-signing Anthony, there is some evidence that wins could have been purchased more cheaply.

I note the issue of opportunity cost since it seems to me that this is part of the WAR story.  In calculating WAR, people are comparing a player’s production to an alternative.   Of course, a “replacement player” is not the only possible alternative.  So WAR is not a complete analysis of the opportunity costs of signing a player.

So what is WAR good for?  The song says “absolutely nothing”.  But in this case, I think that would be wrong!

  1. At what level of production would I care to pay a worker more than the minimum wage?

Let’s go back to the definition of WAR:

WAR = Wins above Replacement

So what exactly is meant by “replacement”?  For an answer, let’s turn to the definition offered at FanGraphs (where WAR is calculated): “Replacement is defined very specifically for my purposes: it’s the talent level for which you would pay the minimum salary on the open market…

If you follow this conversation between TangoTiger and Rod Fort (an economist at the University of Michigan), you see the same story.  The replacement player is defined as a player who is paid the league minimum.

So why would we want to compare a specific player to someone making the league’s minimum wage?  The answer to that is related to this question:  How much does a player have to produce to be worth more than the league minimum?

This seems like an important question to ask.   A minimum wage player does something.  But how much does he do?  And is the player the team thinking of hiring really doing much more?

Hence the need for WAR.  By comparing the player a team is seeking to hire to the minimum wage player, the team can see how much it might be willing to spend on the new employee (and again, that spending choice would still be impacted by other labor market institutions).

Of course to make this work, one has to make a couple of assumptions.

First, we are assuming that the measurement of the “replacement value” actually matches the productivity of the minimum wage player.  Since different people have different estimates of “replacement”, it appears that identifying the exact productivity of the minimum wage player is difficult.

Secondly, we are assuming that pay and productivity are closely related.  Specifically, we are assuming in this calculation that the minimum wage player is the least productive player.  There is reason to think this is true in baseball (I am working on a paper with Paul Holmes and Rob Simmons on this topic). But in the NBA pay and productivity are not as closely related.  Remember, the Spurs don’t spend much but they keep winning. The Nets and Knicks spend much more and are far less successful.  Apparently someone in the NBA isn’t quite measuring productivity very well (and yes, there is plenty of research showing that productivity is not measured very well in the NBA).

So I am skeptical that WAR is a useful concept beyond baseball.  But in baseball, it seems like a concept that answers a specific question facing a decision-maker.

Summarizing the Story

I think it important to emphasize again what I am saying:

1. WAR is not a measure of MRP.  WAR is looking at how a player compares to a minimum wage player.  The existence of a league minimum wage can impact the negotiations between owners and players.  But it does not change the productivity of a player on the field or the value of that production in the market where output is sold.

2. WAR does answer a specific question decision-makers have in baseball.  But it is not clear this approach would work in a sport where the link between pay and productivity is not very strong.

Calculating MRP is difficult.  One has to link what a player does to wins.  And you have to determine how wins relates to revenue.  So it is not easy to determine what a player “should” be paid.  Likewise, determining what a player “is” paid is also difficult.  Labor markets in sports include a number of distortions that have to be considered.

Given the difficulties inherent to these questions, it is not surprising that simply subtracting a replacement player’s production of wins from a player’s productivity will provide an answer to every question.  But I think it can answer a question.  And I think it is important for people who wish to apply this concept to the analysis of player performance understand what question this metric is actually trying to answer.

So if you are wondering how much more Albert Pujols is worth beyond a minimum wage player, look at his WAR.

But if you are wondering if the Knicks got a good deal in signing Carmelo Anthony… a replacement player approach is really not the approach necessary to answer that question.  For that question you need to go back to the basics of labor economics.  And hopefully this post helps you see some of what labor economics teaches on the subject.

– DJ

* – once upon a time, calculating the marginal revenue from a win in sports was fairly easy to calculate.   When revenue was primarily about the gate, all one had to do was look at the statistical relationship between gate revenue and wins.  Today (as a paper I have written with Peter von Allmen and Michael Leeds notes) the story is more complicated.  National broadcasting revenues do not change in response to the wins of individual teams.  And so, determining the economic value of a player in sports is not nearly as easy.

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