The Assumptions and Linearity of the Cost of a Win

This morning, I presented some data on the off-season price of a projected win in free agency, noting that one could come to a reasonable conclusion ranging between $5 and $7 million per win, depending on preference for average versus median or how significantly to discount future spending. That was mostly just an explanation of the assumptions and a data dump, but there’s plenty more to say about the market price of a win, so let’s dig into the data a bit more.

One of the very first discussions that happens any time we bring up this subject is the question of whether or not teams are paying on a linear scale. In other words, does a team pay roughly the same — in terms of dollars per projected WAR — to acquire a +5 WAR player as it does to acquire a pair of +2.5 WAR players. On one side, there’s an argument to be made that the consolidation of value within a single roster spot is highly valuable, and that elite players deserve higher pay for freeing up playing time that could go to another player, who can also add value and create a higher total overall.

On the other hand, consolidation of value also brings increased risk, since one injury can negatively affect a team’s success more dramatically than if the roster is balanced. Additionally, many teams simply can’t afford to pay non-linear values to star players, leaving just a handful of rich teams to compete at the highest end of the pay scale, dragging down demand due to a reduced pool of potential buyers.

So, what does the 2014 data show about the linearity of pricing over the off-season? Well, without trying to skirt the issue, the reality is the answer isn’t entirely cut and dried.

For reference, here’s the chart of the $/WAR values for each of the 83 players listed in this morning’s table plotted against their projected 2014 WAR value. I’ll list all three calculations, ranging from just straight dollars and wins to both sets of the NPV calculations that discount future spending.




As you can see, the line trends down for all three calculations, but as I noted this morning, there are a few outlier $/WAR calculations that are driven by very low projected WAR numbers for moderate price role players. Essentially, the small denominator in the calculation is making for absurdly large valuations on guys who are being signed for things like willingness to serve as a reserve, pinch-running skills, defensive flexibility, and overall good guyness. In reality, I think we’re probably better off looking at the market for players who are expected to be regulars separately than the market for part-time role players, since the low denominator of the projected WARs for those players can skew the data for the rest.

So, let’s go ahead and shrink the pool of players we’re talking about to some degree. Instead of focusing on the entire pool of free agents, let’s just look at players forecast for +1 WAR or higher in 2014. This leaves us with just 47 free agents, but they combined to sign for $1.64 billion in total commitments, or 93% of the $1.76 billion that our entire pool of 83 players signed for. So, while we’re tossing 36 players out of the sample, we’re left with the ones who teams are spending real money on, and the players that we generally care about when analyzing contracts to begin with.

What does the data look like for the remaining 47 free agents forecast for at least +1 WAR? Here are the same graphs as above, just with all the 0-1 WAR forecast players excluded.




Now, with the focus solely on players with forecasts between +1 and +5 WAR for next season, there absolutely is a non-linear escalation in the price of a win. If we’re asking if teams paid the same for a +5 WAR player as they would have for a pair of +2.5 WAR players this off-season, the answer would pretty clearly be no. The average $/WAR for the six players forecast for +3 to +5 WAR in 2014 was $7.5 million per win; the average $/WAR for the 14 players between +2 and +3 WAR was $5.5 million per win. The contracts for Cano, Ellsbury, Choo, McCann, Tanaka, and Kuroda represent a significant premium over the rest of the market.

However, you might notice that four of the six premium players happened to sign with the same team this off-season. The Yankees, by themselves, represented the buyer for 67% of the upper class end of the market, and they were reported to have made significant offers to both Cano and Choo, even though they signed elsewhere. The Yankees bid on every single free agent projected to produce at least +3 WAR in 2014, and signed most of them in the process. While there’s often one extremely aggressive team setting the market each winter, it isn’t always a team with the Yankees ability to sign anyone they really want, and I think it’s probably fair to say that they won’t do this every winter. This off-season may not be all that representative of future off-seasons if the Yankees decide to pull back on the spending spree next winter.

Caveats aside, however, the data essentially forces us to conclude that the 2014 free agent market price for wins was non-linear, at least at for the range of players that we generally care about. Whether this continues in the future or not remains to be seen; perhaps MLB clubs will begin to put larger premiums on higher end players than they have before. However, with many of the game’s best players no longer opting to reach free agency, we could also seeing a shift in the utility of evaluating the market price of a win solely based on free agent signings. Perhaps we need to expand the model to include players signing long term contracts before they reach free agency, opening up the pool of contracts to evaluate to a much larger sample. After all, free agency isn’t the only place to spend money, and increasingly, more and more teams are allocating real dollars to non-free agent extensions.

None of this is designed to be the final word on the issue. As we talked about this morning, there are enough assumptions required that reasonable people can reach differing conclusions based on the same data, and even the range of figures can be drastically altered by whether or not a discount rate is applied to future spending. If we use NPV calculations to bring the total dollars committed into present day valuations, the range of values shrinks pretty dramatically. On a straight $/WAR conversion, the +1 to +5 WAR group has a spread of $5 to $9 million per win, while the 10% discount rate model suggests a spread of only $5 to $6 million per win. The conclusions drawn are heavily influenced by variables that don’t have a clear cut right or wrong answer.

And, of course, it’s also quite possible that we’re simply evaluating some groups of players incorrectly. Note, for instance, that four of the six lowest $/WAR values in the second group belong to part-time catchers: J.P. Arencibia, Kurt Suzuki, Geovany Soto, and Dioner Navarro. Our calculations have those four signing for between $1.7 million and $2.6 million per win, which would make them fantastic bargains relative to other players of similar value. However, instead of the right answer being that there’s a market inefficiency related to acquiring cheap part-time catchers, it may very well be that WAR isn’t capturing the deficiencies of these types of players particularly well, and is systematically overrating mediocre backstops. Maybe they aren’t so much a bargain as our valuation of them is wrong.

Additionally, there’s also the issue of incentives and how they can change the calculations. Our salary estimates are essentially only including the amount guaranteed to a player, and none of the additional funds that get paid if a player reaches certain benchmarks. By ignoring incentives and vesting options entirely, we’re almost certainly understating the costs of acquiring players on one year deals, where a great majority of the incentive-heavy contracts are located. For instance, Dan Haren looks like a nice bargain at +2 WAR for $10 million, but if he actually pitches enough to reach +2 WAR, he’ll almost certainly trigger his player option for 2015 and get paid an additional few million in salary. Dan Haren’s actual cost to the Dodgers, if he reaches the projected value that we’re giving him, is probably something closer to $12 million for one year, plus whatever value we’d assign to Haren’s right to opt into a guaranteed 2015 contract at a similar price.

All of this is a long way of saying that we certainly don’t have this thing nailed down to an exact science. There are a lot of moving parts, and even with the additional adjustments made, there are still many assumptions that may or may not be valid. As you saw when we restricted the pool of players down to those with just +1 WAR to +5 WAR forecasts, the entire direction of the non-linear trendline shifted. These small adjustments can make big differences, and depending on which decisions are made when building the model, the results can come out very differently.

So, I’m not here to make any strong proclamations. I will suggest that the market for free agents this winter behaved more along the lines of what the non-linear crowd argues in favor of, which is different from what we have seen in the past. Whether that’s a new trend or simply an artifact of either the Yankees spending or flaws in these calculations remains to be seen.

Dave is the Managing Editor of FanGraphs.

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8 years ago

Great read.

One quick side note, in the second paragraph you added an extra e. Hate to be that guy but yea.

One one side

On one side

Thanks for the great piece

8 years ago
Reply to  Teddy

And you’re missing an ‘h’, so I guess you two are even.