# Wins and Salary

There’s a pretty neat discussion on the Internet airwaves today regarding baseball economics and the value of players on the free agent market. It began with JC Bradbury’s post about the proper way to evaluate salaries and performance (be sure to read the comments) and continued with more discussion on the Book Blog.

I used to work a lot with free agent salaries and performance, but I purposely shied away from developing a model to project future salaries. My approach was retrospective; I liked to find out which players had been the best and worst free agent values each year.

Still, I was intrigued by JC’s two graphical representations (one of which was created by an economist with a suspicious sounding last name) and I thought I would create one of my own. Here it is, a graph of the Win Shares Above Bench contributed by free agent position players in 2007 (the last year I had detailed information) compared to the salaries they were paid above the minimum. The listed salaries were actual moneys paid to the player that year, not an average of long-term contract payouts.

I didn’t include pitchers in the graph because there are many disconnects between expected and actual contributions for pitchers. Of course, there are similar issues with position players, but they’re less extreme.

I added a fitted line, which basically states that a free agent’s salary in 2007 (retrospectively) was equal to $3 million plus $0.5 million times the number of Win Shares Above Bench he contributed. In other words, the equation doesn’t begin at zero dollars (above minimum) for zero wins (above bench/replacement). This is probably due to two reasons: the bench level for WSAB may be higher than a typical replacement level and teams never negotiate a free agent contract with a player they think will be below replacement level—but it does happen in retrospect.

Remember, there are three Win Shares per win, so the equation equals $4.5 million for a player who is one win above bench/replacement.

The line isn’t a bad fit, as these things go. To one of J.C.’s points, however, the data does appear to be curved. If you look, you can see that there are more data points below the line when WSAB equals zero, and more data points above the line as WSAB increases.

As I said, this analysis is retrospective in nature, so it doesn’t really add anything meaningful to the discussion. But it gave me a chance to revisit some old data, draw a graph, and let you know that Vince Gennaro has a terrific article in this year’s Hardball Times Annual that analyzes the value of free agent contracts both BEFORE and AFTER each contract was executed. Vince has consulted with several major league teams, and I think you’ll find his comments pretty interesting.

JC Bradbury also has a book out, Hot Stove Economics: Understanding Baseball’s Second Season. I’m reading it right now and I’ll have more to say about it in a couple of weeks.

**Addition:** Here’s the same graph, with the axes reversed (see the comments below):

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holy outlier batman

^why would that make a difference? it’s isomorphic.

I am saying I think if you take Pena out you will lose a lot of attenuation in that slope that makes it appear like a nonlinear relationship. I wouldn’t be surprised if it was nonlinear, but I don’t know how to interpret all the points clustered at 0 SAM – does this mean they lack service time to have become FA? I wouldn’t include these guys, but I think I just realized I don’t know what the point of this is, so nevermind…

Dave,

The difference is that WSAB is biased as an independent variable, but salary shouldn’t be. Players with high WSAB totals probably got a little lucky, and so will have more WSAB than we might expect based on their salary; vice-versa for low WSAB players. The same does not hold true for salary. Of all players paid a given salary, the average WSAB should be exactly what was expected of them.

The top right is A-Rod. Lower right is Carlos Pena.

Okay, when you flip it around, the equation intercepts the axis at 0 WSAB (-0.1, actually) and equals 0.6 WSAB for every $1 million in salary above the minimum.

I’ll upload the graph to the article. What do you think?

That is much, much better. The intercept at 0 indicates that WSAB sets replacement level about correctly, and combined with what seems to be a good linear fit, confirms that value above replacement is how teams value players. Moreover, the slope of the line indicates that teams in 2007 were paying about $5 million per win, which sounds about right.

I think it is best to have your guys paid on how they perform. A-Rod is totally overpaid and yes I’m biased because I am a Texas Ranger fan.

Dave,

What happens if you fit it the opposite way—that is, make salary the independent variable and WSAB dependent?