1 Albert Pujols > 2 Casey Blakes (Or, All Wins Are Not Created Equal)

I'm a big believer in WAR, but something about it has been nagging at my brain for a while.  The WAR leaderboard tells us that Albert Pujols was worth 8.4 WAR ($38 million) in 2009, and that Casey Blake was worth 4.3 WAR ($19.2 million).  So let's say you're John Mozeliak.  The devil comes to you and offers you the following proposal: In 2010, you can replace Albert Pujols with Casey Blake and his identically skilled twin, Blake Casey.  Under the terms of the deal, Pujols and Blake (and Blake's twin) will perform exactly as they did in 2009.  The combined salary of Casey Blake and Blake Casey will be the same as Pujols' salary.  Which do you choose?

Even without accounting for fan reaction, the answer seems obvious: Pujols. For one thing, he only takes up one position and one roster spot, allowing you more roster and lineup flexibility.  The Blake twins may provide the same total WAR as Pujols, but this value is mitigated by the loss of potential WAR from other spots.  Having that value concentrated in one player allows you to add another player to your lineup; provided you have another above-replacement-level player available, who wouldn't prefer that?  In essence, the comparison here isn't so much 1 Pujols vs. 2 Casey Blakes as it is Pujols and say, Skip Schumaker vs. the 2 Blakes.

An even larger problem is that of scarcity.  Think of a major league player as a diamond (the gem, not the infield), and WAR as the carat count of the diamond.  All else being equal, a 1-carat diamond will always be worth much more than twice a 0.5-carat diamond.  Why?  Because the relationship between carat count and rarity is non-linear.  It is exponentially harder to find a 1-carat diamond than it is to find a 0.5-carat diamond.  Thus, a 1-carat diamond is worth exponentially more than a 0.5-carat diamond.

Similarly, an 8-WAR player is much, much rarer than a 4-WAR player:


The preceding graph is compiled from FanGraphs WAR data (position players and pitchers) for the past 3 seasons.  The exponential relationship is clear.  Over the last 3 seasons, there have been an average of 4 8+ WAR players and over 68 4+ WAR players per season.  In these seasons, an 8-WAR player has been about 8.5 times harder to find than a 4-WAR player.  4-WAR players are (comparatively, anyway) a dime a dozen.

Shouldn't the 8-WAR player be much more valuable than two 4-WAR players, then?  I'm not arguing that Pujols is worth 8.5 Blakes, but certainly he's worth more than 2 Blakes, based on the scarcity of his skill set.

There are (at least) two ways of resolving this.  The first would be to make an exponentially-scaled WAR, or an alternative WAR to dollars conversion that incorporates this relationship.  I don't want to replace the current, linear model, which does a good job of measuring the market value of a given performance.  What I would like is a market-independent valuation that can be used to figure out what portion of MLB's salary resources a player's performance should command in a perfectly rational, performance-based economy (yeah, I know, that'll never happen, but still...).

The other solution would be to find a way to compare 1 player to 2 players (or 3 players; or 2 players to 3 players, etc.) that accounts for the added value of performance being concentrated in fewer players.  In other words, we'd need a scale that could convert the combined performance of a number of players into an equivalent, 1-player value.  This would be a particularly useful tool for evaluating multi-player trades.

Has anyone worked on anything like this already?  Does anyone have ideas for how to do so?  Am I crazy, or missing some key element?  If you disagree, please enlighten me; I'd be interested in hearing any arguments for the Blake twins.  Thanks for reading.

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