Can WPA Explain How Teams Buy Relievers?

Over the last decade, much of team behavior has, sooner or later, matched up with new sabermetric discoveries. The Athletics’ “Moneyball” focus on on-base percentage is no longer a secret; teams have begun to quantify the value of defense and spend accordingly; arbitration-eligible players have become more valuable on the trade market. But there is one area in which sabermetric findings quite clearly do not mesh with team behavior, regardless of general managerial regime: spending on relief pitching. At least using Wins Above Replacement, it just doesn’t make much sense. Whereas every non-pitching position on the diamond — as well as starting pitchers — make roughly the same amount per WAR, relief pitchers are on another level, frequently making three to four times more per WAR than other players.

This only doesn’t make sense, however, if we think of teams as buyers of WAR. They aren’t. The teams with the most WAR, although typically in a very, very good spot, will not necessarily win the most games. Teams buy real wins, and the best way we have to measure real wins is with Win Probability Added.

As implemented, our version of WPA is hardly perfect, largely due to the impossibility to give credit to fielders given the data available. What we would need is a purely hypothetical version of WPA with perfect information — a true “Holy Grail” statistic which would give the proper credit to every fielder for every play (and while we’re at it, gives proper credit to David Eckstein for being so scrappy), and then adjust it for replacement level instead of average. But for the purposes of trying to solve the problem of why relievers are seemingly so overpaid, our version as implemented works fine — we’re not going to look at individual players necessarily, and the issues of fielder credit should, at least theoretically, be spread out relatively evenly across all relievers.

The first question — why do we use $/WAR in the first place? Why not $/WPA? The answer is because due to all the factors of WPA which are out of the player’s control, WAR tends to predict future WPA better than even WPA itself. Even in-season, WAR and WPA tend to correlate pretty well.

Observe, for starting pitchers and hitters:

Click to embiggen.

Pretty simple stuff — the correlations are pretty solid. The formula isn’t “1 WAR = 1 WPA” due to the differences between measuring by average and by replacement level, but there is an obvious relationship between WAR and WPA, as we would expect. But the clever reader will have noticed the formula for the relief pitchers trend line in the image, and it does not follow the same relationship as for the rest of the players at all:

Click to embiggen.

For a number of reasons — leverage chief among them — a relief pitcher with one marginal WAR is likely to have a WPA one full win higher, unlike the 0.5 extra WPA per WAR we see among the starting pitchers and position players. So it appears when, say, Ruben Amaro pays Jonathan Papelbon $50 million over four years, he is giving his new closer full credit, or nearly full credit for the situations he is placed in. Although this doesn’t completely explain the discrepancy relief pitcher $/WAR and the rest of players, it does close much of the gap — halving the $/WAR relievers get puts them at $8.4 million per WAR. Still high, but not as clear of an outlier.

The discussion of how to appropriately give credit for high leverage situations is just one of the discussion points addendums to this post can cover. A few others I plan to look at: How does Baseball-Reference’s implementation of WAR differ with relievers? Does what we’ve seen here differ when we filter just for elite relievers? Have the big, multi-year contracts for relievers actually worked out based on WPA? It seems clear that simply declaring “teams are overpaying relievers” ad nauseum does very little to advance our understanding of how the game actually works, and perhaps a look from a different angle can help us try to understand what is actually going on in the marketplace of Major League Baseball.