Linear Dollars Per Win, Again

For those of you who have been reading sabermetrically inclined blogs for the past five years or so, you’ve probably come across a discussion of whether a player’s value is linear or exponential – in other words, is each extra win added more valuable than the one that came before it? I wrote about this here about nine months ago, in fact, but given the discussion (uproar?) that has arisen over yesterday’s post on the potential rewards of trading Tim Lincecum, it’s probably time to talk about it again.

The goal of every team is obviously to get as much production as possible out of their roster. Each team is essentially working towards this goal with two constraints – a budget constraint and a roster constraint. Teams can only spend up to a certain figure on their total payroll, and they can only have 25 players on their team at one time. Within those limits, each team – one that is trying to contend, anyway – tries to maximize production from the roster as a whole.

The discussion of whether a player’s value is linear (all wins are worth about the same amount) or exponential (each additional win added by a player is worth more than than the one before it) essentially is born out of the roster constraint. Since a team has a finite number of spots to hand out, getting consolidated value creates a higher potential maximum than if everyone on the team was of equal value. If you have a +5 win player and a +0 win player, you can theoretically replace the +0 win guy and end up in a better position than if you have two +2.5 win players who you don’t really want to replace.

This idea certainly has merit, and given that premium players give teams higher potential production from the roster as a whole, it’s natural to think that the market would account for the value of the scarcity of superstars in this way. However, in reality, it doesn’t really work that way.

So far this week, we’ve seen a plethora of guys who could be described as about +1 win players (Chien-Ming Wang, Juan Rivera, Jeremy Affeldt, Javier Lopez) get valued at right around $4-$5 million apiece. They’re each role players who have some value above a generic replacement level guy, but clearly, they’re on the lower end of production from what you’d want taking up a precious roster spot. And yet, for these guys, the going rate has been pretty close to $5 million per win.

This week, we also saw CC Sabathia re-sign with the Yankees. He is, of course, one of the game’s elite pitchers, as 2005 was the last time he finished with a WAR of below 5.0, and he’s been over +7.0 in three of the last five seasons. Even as a 31-year-old pitcher with a decent amount of mileage on his arm, he’s clearly a +5 to +6 win pitcher for 2012. If wins were exponentially valuable, we’d expect Sabathia to be one of the recipients of this premium. Instead, he signed for essentially $142 million over six years, or just a shade under $24 million per season, and had to give the Yankees a health-related out clause on the final year as a concession.

That deal puts Sabathia in very close proximity to the salary that Cliff Lee – last year’s premium free agent starter – received a year ago, and is slightly above the expectation produced by our contract crowdsourcing project here. In other words, Sabathia’s deal should be interpreted as market value, and there was no “home town” discount that needs to be adjusted for.

Now, because Sabathia’s deal is a multi-year contract instead of a single year deal, the dollar-per-win calculation is a little more complex. A basic model that has proven to be fairly accurate is to essentially assume a decline of 0.5 WAR per season and 5% inflation in the dollar per win rate each year. If we use that basic model to project Sabathia’s value, and we believe he’s a +5.5 win pitcher for 2012, then a linear dollar-per-win model would give us the following totals from 2012-2017:

Year	WAR	$/Win	Value
2012	5.5	5.00	27.50
2013	5	5.25	26.25
2014	4.5	5.51	24.81
2015	4	5.79	23.15
2016	3.5	6.08	21.27
2017	3	6.38	19.14
	25.5		142.12

Six years, $142 million. Also known as the exact dollar figure that Sabathia signed for. Using a model of linear dollars-per-win, a very basic aging curve, and best-guess inflation assumptions projected Sabathia’s salary to the tee. You can go back through history and see that these sorts of basic models have done pretty well at projecting free agent pricing over the years. In fact, The Book Blog has the archives and most of the data right there for you if you want to look it over.

The reality of the MLB free agent market is that premium players do not get a substantial bump that reflects that teams substantially value a high WAR concentration from a single roster spot. In fact, most of the evidence (including the Kenny Williams quote from the piece I wrote in February) suggests the opposite, that most teams are more interested in risk avoidance by spreading their production out over multiple roster spots. Based on the actions of the teams, we can safely state that there’s a preference in the sport for multiple good players instead of a single great player.

The roster constraint means that teams will not choose quantity over quality in every situation, as they aren’t going to just keep doubling down until they have 25 +1 win players, but the evidence simply doesn’t support the notion that the market dictates exponential dollar-per-win valuations for premium players.

You can make a case that teams are currently being too risk-averse, and that this is an inefficiency that could be exploited, but the people currently in charge would probably argue that they have a significantly better handle on the actual risks of having all of your eggs in one basket than us outsiders do. The fact that not even the well capitalized and extremely well run franchises of the northeast have begun to pay those kinds of premiums should suggest to us that there are legitimate reasons why the market isn’t supporting the theory, and that the focus on consolidated value is missing key elements that affect actual roster construction.

I understand why people think that premium players are worth more than a linear dollar-per-win conversion might suggest, but the evidence simply isn’t there to support that kind of market valuation in Major League Baseball right now. Teams pay for wins (or what they perceive will lead to wins, anyway), but they don’t pay extra to get more of those wins in one package.