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.

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Dave is a co-founder of and contributes to the Wall Street Journal.

50 Responses to “The Assumptions and Linearity of the Cost of a Win”

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  1. Teddy says:

    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

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  2. Mike says:

    I do’ recall you (or anybody else frankly) only looking at 1+ WAR players in the past. I think it would be worthwhile comparison to go thru the last couple of winters worth of FAs with this same restriction in place.

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  3. GilaMonster says:

    I feel like this is something that should be examined AFTER the 2014 season because teams should be paying for future performance in $/WAR, not past WAR.

    This would negate some of the factors of under performers in 2013 seemingly being underpaid and those who overperformed in 2013 seemingly be underpaid.

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  4. John W. says:

    If you hate to be that guy, don’t be that guy.

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  5. tz says:

    Dave – could somebody at Rotographs do a similar analysis on auction-style leagues (auction prices actually paid vs. formula-based prices using projected 2014 stats?)

    Seriously, I think that both real and fantasy baseball are subject to the law that “dollars remaining fill most of the available budget space”. The Tanaka contract, and to a lesser extent the Choo contract, come to mind as late-in-the-game bids similar to what happens in fantasy auctions.

    It wouldn’t surprise me to see fantasy auction values have similar scatterplots to what you have for the 2013-14 offseason FA class.

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  6. RC says:

    “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”

    Which is what I’ve been saying all along, and you’ve been fighting me on it. The universe of 0-2WAR players is an absolute mess, and there are teams that seem to do really well sorting through it, and teams that do really poorly.

    For the teams that do well sorting it out, an 8WAR player is drastically more valuable than two 4 WAR players. For a team that does poorly, its not.

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    • tz says:

      RC, I totally agree with this point on a historical level, when a fair number of players kept everyday jobs despite showing themselves to be below replacement level. These teams were counterbalanced by teams like the Orioles, who got a lot of value out of well-constructed platoons giving close to +2 WAR for the weakest positions.

      Going forward though, I wonder how many teams will employ the “proven veteran like Yuni” strategy to fill their roster gaps. I bet that most of the mess in the sub-2 WAR zone will end up being performance-related noise rather than a big variance in how well teams value back-of-roster players.

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    • Jason B says:

      You seem to think this is a personal vendetta against you and your hallowed idea. Deep breaths man. You’re just not that important in the Dave-verse.

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  7. RC says:

    Also, when your trendline is as far off as it is in the first few graphs, you’re usually looking at a problem with outliers, or you’re curve fitting with the wrong order polynomial.

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    • tz says:

      I think the first graph might become more informative if it plotted $ vs WAR instead of $/WAR vs WAR. The denominator wreaks havoc where the WAR is low.

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      • TanGeng says:

        Taking that approach, you’d have to find a parabolic function for best-fit.

        A parabolic best-fit handles the error much better numerically, but for the audience it’s much harder to visualize and understand.

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        • jessef says:

          Calculating the mean, you could also weight the value of each sample based on the expected WAR because a player who is expected to provide 0.5 war but overperforms by 0.5 war ends up looking like a much better bargain than a player who is expected to provide 4 war and overperforms by 0.5 war.

          Basically, we don’t really know what teams are expecting from players. With all-stars, it’s not that big of a deal but with back-of-roster players small differences in expected war produce much, much greater $/WAR values (certainly no team is deliberately spending 30 M per war so datapoints like that should probably be, at the very least, capped in calculations of the mean

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        • tz says:

          Great point jessef. For example, even if you had Willlie Bloomquist and Delmon Young at 0.5 WAR, you probably would give Bloomquist a higher valuation because he can give you that 0.5 WAR at several different positions, making him more valuable as bench insurance.

          I also think that WAR may be off +/- 1 win or so for various types of “part-time” roles. A guy with major platoon splits is probably worth less than the given WAR if he has had the benefit of facing the preferred handed opponent, while someone who gets called a lot to pinch-hit against top-tier closers might be worth more than their WAR due to the better quality of pitcher faced and/or the higher leverage of when they bat.

          And that last point, I believe, is why reliever WAR values seem to miss the mark as well. WAR is a great framework, but there is plenty of fine tuning that could be done.

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    • pinch says:

      no kidding, outliers? really? in that graph? you think the $40 mil/WAR for Jeff Baker is an outlier? gee whiz. some hot sports-math takes over here. no wonder dave hates you personally so much when you see through his cunning illusions so easily.

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  8. ImmanuelKant says:

    What about the linearity (or not) of the specific wins that teams take themselves to be buying. I.e., you’ve often mentioned that the 90th win (or something) is more valuable to a club than the 75th. So, I wonder if teams pay more for the wins that they expect a player to get them relative to the relevant playoff thresholds, etc.

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    • Matthew Murphy says:

      On one hand, this makes a lot of sense, a team is willing to spend more if they place a higher value on a win in 2014 because they’re projected to be in the thick of the playoff race. However, the market is set by the entire league, so if the market rate of a win is fixed (or fixed relative to 2014 WAR), it doesn’t make sense to spend more than the market dictates just because you place a higher value. This would simply encourage more spending at the market rate.
      The argument for consolidation of WAR is that the teams that are looking to make the playoffs and are in higher leverage positions may already have very good rosters with fewer positions to upgrade at. You can argue that a team like the Rangers would have been better off spreading out the money they gave to Choo, but they didn’t have a lot of places where it would be easy to upgrade (at least before Holland got injured), so they were willing to pay a higher price for a big upgrade at one position.

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      • RMD says:

        The market rate of a win is NOT fixed across the board.

        When teams are negotiating with a player, his dollar value solely comes from his specific value from the specific team he’s negotiating with. As Mr. Cameron stated, the Yankees threw his $/WAR values off. It’s because they’re right at the threshold for making the playoffs. Tanaka’s really not worth as much as he’s paid… but he certainly is to the Yankees.

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  9. threespeedbike says:

    One quibble on an otherwise excellent article; the relationship between $/WAR and 2014 projected WAR appears to be linear (it’s represented by a line running through all the plots), but it’s slope is not 0. Would it be more correct to say something like the relationship between $/WAR and projected WAR is not constant across all talent levels? Again, that’s a small technical point, but does that make sense to anyone else?

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    • Eric R says:

      Linear or not isn’t about the slope, it is about a straight line being the best fit to the data.

      If you were to graph X vs x^2, it wouldn’t be linear not because the slope is positive, but because a non-linear formula would be a better fit than a linear formula.

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      • threespeedbike says:

        I agree. That’s why I think it’s more correct to state that the relationship between $/WAR and projected WAR is actually linear. A straight line is the best fit for the data as seen in the graph above.

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        • Jon L. says:

          Hi threespeedbike,

          I think the confusion is that the straight-line relationship in the graph is not the same as the linear relationship to which Cameron is referring.

          The graph represents the relationship of $:WAR as WAR increases, and it appears dollars per WAR increase steadily (~linearly) as WAR increases. However, the linear relationship Dave Cameron is testing is between $ and WAR. If that linear relationship existed, $/WAR would remain constant. Instead, it appears to increase steadily as WAR increases.

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        • threespeedbike says:

          Sorry to beat a dead horse here, but I still but there is still something in the “non-linear” language that I don’t think is technically correct, but if I’m still not getting something obvious please let me know. I get stuck on things sometimes.

          Isn’t is right that the relationship being tested is in fact linear, meaning it is not represented by a non-linear function (such as an exponent or log)? The relationship is a straight line. What is being tested is whether the line is positively sloped, meaning $/WAR increases as projected WAR increases, or has a slope of zero, meaning $/WAR remains the same even as projected WAR increases. The conclusion at the end of the article is that the relationship is linear and positively sloped. This means that the $/WAR is not constant across expected production levels. That conclusion does not support the claim that the relationship is non-linear.

          This is a small point, so I will let is go after this. Do let me know if I’m still not getting something or if this is merely a semantic point that I haven’t caught.

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  10. garett68 says:

    I am starting an Ottoneu League. It is an auction draft, fangraph points, $99, full keeper league on march 23 at 8pm. the league details are here:

    Email me @ for any questions.

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  11. Mike Green says:

    Thanks, Dave. The two highest star contracts for $/WAR were Granderson and Pence; I don’t think it is a Yankee thing. When I looked at Pence’s BBRef age comps, it might be that 10 WAR is a better projection than 9 WAR for him. When I looked at Granderson’s age comps, the figure of 6 WAR looks reasonable.

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  12. Gabriel says:

    I like the direction you’re going, but the sample size seems way too small to me. The left half of the graph has a fairly large sample, but the right half is composed of 5 guys, the majority of whom were paid by the richest team. A multi-year analysis would help out a lot.

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    • ian says:

      I think part of the idea is to do a 2014-specific $/WAR analysis, though. Bringing in other years of data might be interesting, but it’s answering a different question.

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  13. Avery says:

    Another potential issue with this data is the confounding role of the qualifying offer. Generally speaking, the “stars” that you’ve identified are signing very large contracts, thus spreading the cost of the QO out and lessening its impact on the player’s $/WAR value. Marginal QO players aren’t signing as large of contracts in dollars or years, and therefore their $/WAR value appears much lower than they actually are. Since this phenomenon is predominantly impacting the lower end of the projected WAR spectrum, it could help introduce the observed non-linear trend in $/WAR.

    I doubt this effect is strong enough to account for the majority of what we’re seeing here, but it likely lessens the degree of it.

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    • ian says:

      It’s funny – it’s not like this is the first year of the QO or draft picks being attached to FAs. Teams may just be valuing draft picks more, or FA players are getting dumber about rejecting QOs. Or it’s just kind of random.

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  14. pft says:

    Its unfortunate that the article limited its look to 2014. I feel the market, especially this year is being influenced by collusion at the non-elite end of the free agent market. This season will have a knock off effect for arb eligible players and extensions, and down the road even at the elite end.

    Collusion today is smarter since teams/industry agree on a single model which they create or influence and argue that their valuations are based on the science behind the model, and we all know models don’t always get it right (weather, financial, climate, etc).

    A crude estimate of the value of a win is MLB’s total income divided by the number of replacement wins. Today that number is 8 million per win. The average player is getting 47% of total revenues, which includes arb and pre-arb players. I don’t see how a free agent market value can be argued to be 5 million, not by a long shot. If its still 5 million it would mean free agent market prices have been stagnant for several years while revenue is booming, which would be clear evidence of collusion if true.

    From the viewpoint of supply and demand, and in a competitive market, free agent prices should increase with MLB revenue. It seems to be working at the top end of the market, although as Dave pointed out 1 team influenced that segment of the market this season. Remove the Yankees and Mariners and the rest of the league seemed to have lost their wallets, which does not make sense in a competitive market where the commodity is in relatively short supply.

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  15. BenRevereDoesSteroids says:

    For the record, either Soto or Arencibia won’t be a part time catcher, as they are now on the same team. In all likelihood, Soto will be getting a good majority of the playing time.

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  16. DrFarmer says:

    I downloaded the original data to play around with it, and I thinking you’re making an error by comparing 2014 projected WAR to the average $/WAR over the life of the contract. You should compare the average annual projected WAR to the average annual $/WAR. When you make that comparison, you get a slightly different relationship. When removing annual projected WAR below 1, you get a regression that yields an r2 of 0.055 and p-value of 0.12 on the slope. In other words, there isn’t a lot of evidence that the cost of single player WAR SIGNIFICANTLY increases with increasing WAR.

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  17. Paul says:

    As far as payrolls…the top 5 teams are ‘outliers’ when it comes to what most MLB teams operate as far as revenues and salary spent. Additionally, 1 team was lower by 2 SD’s below that ‘confidence interval’ of teams. That said, I wonder what the average cost/win would be for those teams (Excludes NYY, NYM, BOS, LAD, PHI, HOU) for players.

    So if one were to exclude those teams, as well as your below 1 WAR group, if the numbers would normalize better. And perhaps have a very true value/WAR ratio. Perhaps put a R^2 to it that may approach close to 1.

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  18. cs3 says:

    I wonder how many people you confused by repeatedly mentioning “non-linear” and then drawing straight lines on all your graphs.

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  19. x says:

    Those slopes don’t mean anything without the error bars on them- is it a slope of, say, 0.1+/-0.001? Or more like 0.2+/-0.25?

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  20. jiveballer says:

    Generalities (non-exhaustive) of multi-year FA deals:
    1. The early years are presumed to be better than the later years.
    2. Guaranteed long-term contracts place higher the risk on the team than the player.
    3. It’s more difficult to predict far-future performance than near-future performance.
    4. Inflation of the dollar, payroll allocation & salaries, and $/WAR at the back end of contracts can mitigate items 1-3.

    Basically, I think long-term deals skew the 2014 $/WAR/WAR pretty severely. If these were all single season contracts the prime superstar would probably be getting paid, like, a lot more per season than what the AAV represents. The actual, on the field value would thus be more strongly represented in the $/WAR/WAR and the apparent linearity would evaporate.

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  21. rusty says:

    What if instead of throwing out all players projected for <1 WAR, you just weight each data point by projected PAs? That covers your part-time player issue while still allowing, say, Michael Morse's contract to be included in the data set.

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  22. Joebrady says:

    Besides taking off any players with < 1.0 WAR, I might also take off any players with 1-year contracts. My position is that you can always overpay for one year. If, for example, the choice is paying $36M/3 or $14M/1, I'd usually opt for the $14M/1. That will reflect on the WAR/$ because the risk is lower.

    The one-year contracts are also harder to value for these purposes. A lot of contracts have minor buyout provisions, but it seems to me that they would relatively more impactful when allocated over 1 base year. Some, like Beltre's 2nd year guarantee with the RS, probably defy valuation.

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  23. Big Daddy V says:

    The entire “trend” in the $/WAR graph is a giant blob of guys on the left, and then 5 outliers on the right. Remove the 5 highest-paid guys, and there’s no correlation at all.

    This just seems like an example of trying to make the data fit the theory.

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  24. jdm says:

    There are a few reasons why this works out this way. The first obvious reasons have already been articulated as the combination of saving a roster spot and the ability to marginally upgrade one position more easily. But beyond that I think there are other factors at play.

    When signing a star to a deal – obviously there is a scarcity of them but the reason they are a star is because they have had consistent production over a number of years (free agents showing their resume for more than six years obviously) which demonstrates the a posteriori distribution of the player and implies a lower probability assigned to the downside risk of the player. No one expects Miguel Cabrera’s production to drop off whereas when signing a Cuddyer or a Marlon Byrd the chance they tank is much higher and this is factored in. This is not always the case (see Albert Pujols and Josh Hamilton) but not many expected Pujols to drop from superstar status to above average.

    Additionally, elite players usually have a more diverse and complete skillset. For instance if you take a look at a player like McCutchen (not a free agent but just an example) or Robinson Cano, they are five-tool players. Even if their defensive capabilities diminish they still have four other tools where he provides value which limits his downside risk. This non-linearity (or non-constant) $/WAR shows GMs risk aversion.

    Additionally, when signing these superstars, the buying teams are much more subject to the winners curse. When every team is vying for Robinson Cano you have all 30 teams competing in a bidding war and the team that pays the premium usually pays above market value for his services, whereas signing Cuddyer or Byrd there are fewer teams in the market so their services should be closer to market value. The Mariners presumably had a good option for the future at 2B with Nick Franklin but still decided to pay Cano all the same. By signing this free agent a team signals that this is the most expensive price any team is willing to pay and therefore they are least likely to receive market value for the player. As I mentioned above, the Angels signed Albert Pujols expecting him to maintain his value, but were the top team bidding for his services and have seemingly overpaid.

    I also think the free agent market might be a market for lemons that other GMs don’t realize. The Cardinals obviously wanted to maintain a hometown hero like Albert Pujols, but their reluctance to pay Pujols despite being offered a hometown discount for his services shows this information asymmetry.

    Additionally, the qualifying offer confounds the true value paid for a player’s service. If a team has to give up a draft pick to sign a player they are not going to want to sign the player for a short term deal, they will want to lock up the player so that the marginal impact of the qualifying offer only slightly impedes the super stars but acts as a big road block for the average player. If a +5WAR player is modeled by a half-WAR decline each year he is still serviceable several years into the deal but if the same model is applied to a +2 or +3WAR player that model limits his value to the team to only a couple of years which makes teams a lot less willing to pay and give up a draft pick.

    Additionally, I think the model discounting future wins is most appropriate. Even if you don’t factor inflation, recently we have seen booming revenues for teams, especially teams signing massive television deals. The Angels signed one right before they signed Pujols giving them the financial leverage to sign stars. If this boom continues we could see players portion of the revenues to stay constant but exploding $/WAR valuations proving that the dead years of the contract aren’t as costly as they were perceived to be when originally inked.

    And finally, we must also factor in the marginal value of WAR for each team. In addition to the Winners Curse, for teams that are on the cusp and want to make a run and boost revenue, it might be worth it to overpay for the stars to display the fan interest in this newly competitive club that enables the club to ink big TV deals.

    All in all, I think these reasons of risk-aversion/a priori belief in a superstar limited downside, winner’s curse, information asymmetry, the confounding qualifying offer, and increasing revenue streams all contribute to this non-linear (constant) phenomena. But who knows this year might be an aberration and these reasons might not be appropriate to model the FA market.

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