## On \$-per-Win Estimates of Baseball Players’ Worth

Frequently, I receive a comment or e-mail that brings up the dollar-value estimates at Fangraphs.com. Fangraphs is a fine site with lots of interesting numbers, but I don’t think the dollar-value estimates listed on the site (or any simple wins-to-dollars conversion) properly value players. Here’s why.

1) The derived estimates are based on the assumption that there is a constant linear relationship between wins and dollars. This assumption is incorrect: there are clear increasing returns to winning. This is the revenue function I estimated for my book, converted to wins instead of runs.

2) By dividing the total value of free-agent contracts (Y) by total “wins” added by the signed free agents (X), this method assumes the y-intercept (b) is 0, which biases the estimates. Y = mX + b, when you assume b is zero when it’s not, bad things happen to slope m. The graph below from the popular econometrics textbook Understanding Econometrics: A Practical Guide by A.H. Studenmund demonstrates why this assumption biases the estimates.

Due to the thinness of the free-agent market and the potential for market mistakes, I prefer a fundamental-value approach to valuing players as opposed to a market-valuation approach. However, if I want to use the free-agent market to value talent, I prefer Anthony Krautmann’s “free-market returns” approach, which can be implemented in ways to avoid the problems mentioned above.

In Chapter 4 of my book, I explain why I prefer the Gerald Scully inspired approach to the free market returns approach. This is not to say that market prices are not useful for valuing free agents. In my book explain where free market returns helped me shape my estimates. Also, here is a working paper in which I discuss the pros and cons of the Scully and Krautmann methods.

### 16 Responses “On \$-per-Win Estimates of Baseball Players’ Worth”

1. studes says:

That Studenmund graphic convinced me!

2. Colin Wyers says:

Professor Bradbury, can you explain how the Fangraphs approach (as detailed here) differs materially from the free market returns approach Krautmann outlines?

http://www.fangraphs.com/blogs/index.php/win-values-explained-part-six/

(I mean in terms of the dollar value of a win – obviously there’s disagreement in how to identify the marginal products.)

3. JC says:

There are numerous differences, but the main difference is the Krautmann approach uses multiple regression analysis to estimate the revenue contributions of players and does not constrain the y-intercept to zero. The Fangraphs approach is division of aggregates. There are links to explanations of both approaches in the post.

4. Sky says:

Regarding the Studenmund graphic: are you suggesting the axes are absolute wins and absolute dollars, or should one or both be marginal wins or dollars?

5. JC says:

Absolute dollars and wins. The slope of the line m maps the marginal change to dollars from wins.

6. Colin Wyers says:

I’ve read the Krautmann paper, although it’s been a while. I’ve read parts of your working paper, although it’s a busy morning for me and I will probably have to finish it later.

My confusion largely arises from point one, about the different marginal values for a win based upon a team’s record. Krautmann’s model doesn’t seem to address that point, either.

7. JC says:

Past estimates by Krautmann are linear. I believe this aspect of the model is mistaken. When I refer to the Krautmann approach, I am referring to using market salaries to estimate player worth, not the specific estimates.

8. Sky says:

But Fangraphs is using marginal dollars and marginal wins… That’s NOT forcing the y-intercept to be zero on the absolute axes. It’s translating the origin to a point that makes logical sense in a marginal sense. Now, one could challenge where that point is placed, but once you place it, it IS the zero point. Moving the frame of reference back to the absolute graph, the regression line is NOT constrained to a y=0 intercept.

9. JC says:

Sky,

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.

10. Sky says:

Ok, let’s remove the “free agent rate” part from Fangraphs’ model. I agree it has some problems (but also some uses). One could use their same process, but instead use the money paid to ALL players, not just free agents. Instead of about \$4.5M per win, it’s maybe \$2.25M per win. With those numbers, it all adds up to total salaries paid out, instead of a much larger number (since not all players are paid at the free agent rate).

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”.

11. Joe says:

So according to you, JC, how much is a “win?”

12. Colin Wyers says:

I took a quick look at this – not so much to pretend I did a study of the issues at hand, but to see if I could visualize it for myself. So I took this:

http://www.baseball-reference.com/leagues/MLB/2010-value-batting.shtml

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.

13. JC says:

Joe,

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.

14. Ernie King says:

I’m econ drop-out but this one seems pretty easy. You’re comparing apples to oranges here. FG is looking at the marginal COST of a win and JC is looking at the marginal REVENUE of a win. Mind you, in a perfectly competitive market, marginal cost = marginal revenue = price, BUT I don’t think MLB is anywhere close to satisfying that condition. Thus, we should expect a difference between MC and MR, with that difference being the level of inefficiency in the MLB market and I would guess the level to which wins are overpaid/underpaid for (I would assume overpaid for in the aggregate.) So wouldn’t we want to use both approaches and compare? As an owner, or a player, I would want to know how much additional revenue an additional win will produce and use that to evaluate how much to offer, or how much to ask for (as a player). Or am I missing something, or a few things, here?