There is not a set value for a win. It depends on how many wins a player adds, and how many wins the team has. That’s why the non-linearity is important.

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And focused on just players listed as “free agent” or “waiver.” (I understand this is no way duplicates the Fangraphs study – again, I’m just trying to work through this with some data in front of me, to help me visualize this.) If I take the sum of salary divided by the sum of WAR, I get roughly $4.9 million per win. If I run a regression, I get a constant of 3320576.267 and a coefficient on WAR of 1339016.354. (This reflects as much as anything a failure of looking at observed WAR rather than projected WAR – I suspect a large number of those 0 WAR or thereabouts players were ones who were given contracts in anticipation of far greater amounts of playing time.) If I fix the intercept of the regression at 0 (I know, I know – again, just trying to work through some of the ideas here) the coefficient on WAR rises to 2260335.473, still far less than $/WAR.

If I take and draw a scatterplot of actual salary to wins, and then draw a similar scatterplot of WAR*(sum of dollars/sum of WAR), it looks pretty much like the textbook illustration Bradbury presents.

]]>Anyway, I guess I could see an issue using the y=0 constraint here, because it ignores revenue issues and only uses the amount of money teams are currently spending on payroll. It’s the difference between “how much of MLB’s marginal revenue is a player’s production worth” and “what how much of MLB’s payroll expenditure does a player’s production deserve”.

]]>You are using the term “marginal” different from they way I’m using it. The $-per-win approach takes the aggregate number of dollars for players who are considered to be adding “marginal wins,” then divides by the aggregate number of wins produced. When this calculation is made, the intercept is then assumed to be zero. Marginal, in the economics sense, is the value added following a change in wins. As the first diagram above show, the change is not linear. The second diagram, which is theoretical and not directly applicable to the numbers here, shows the marginal impact to be linear, and with a less-steep slope with a positive y-intercept.

If you want to estimate worth from market salaries, you should use a multiple-regression approach like Krautmann, which does not constrain the intercept to zero. If it is zero, then it will be estimated to be so.

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