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  1. tl:dr

    seriously, be thinner skinned. your votto trade proposal was silly. it wasn’t dumb, but it wasn’t good. people raised legitimate objections and you just “pshawed” them. when you say something wrong, get indignant about people questioning you, and arrogantly dismiss them, people are going to come after you.

    #6org will never die

    Comment by fjmanuel — November 4, 2011 @ 2:23 pm

  2. Fantastic explanation. Even I understood it. Thanks.

    Comment by WilliaminMaine — November 4, 2011 @ 2:26 pm

  3. So what you’re saying is nobody is going to surrender $30 million/year and one of the top prospects in baseball for Tim Lincecum? Just checking.

    Comment by Kevin S. — November 4, 2011 @ 2:29 pm

  4. Dave got pretty butthurt over this one.

    Comment by Guy — November 4, 2011 @ 2:30 pm

  5. What about the fact that at these higher contract values, the free market is thinned down to 5-6 teams, instead of the standard 20+. So, salaries are kept lower than they would otherwise be based on non-linearities in value?

    Comment by mettle — November 4, 2011 @ 2:30 pm

  6. Haters gonna hate. At least he did something productive and wrote a well-reasoned and quantitatively sound article. What have you done to help further sabermetrics?

    Comment by futurecfo — November 4, 2011 @ 2:31 pm

  7. I actually think the $$$/Win calculation is a little bit more complicated than being linear or exponential. This is purely from my personal observations, but it seems that teams overpay for mediocre 1-win players (the Wang/Rivera/Affeldt/Lopez deals), but also overpay on big-name long-term contracts (probably because they don’t properly account for injury risk and decline).

    Also, I know that the 0.5 WAR decline is a pretty standard figure, but where does that number come from? Would it be more accurate to use a % decline? And where does the 5% inflation figure come from? That seems high to me.

    Comment by Yirmiyahu — November 4, 2011 @ 2:38 pm

  8. Could the reason that clubs don’t pay a premium for elite talent have to do with other constraints? Specifically payroll. If you pay $5 mil for 1 win and that guy gets hurt, most teams can replace that no problem. If you pay say $28 mil for 5.5 wins and that guy gets hurt, not only will it be equally or more costly to replace that guy with a 5 win player, but even bringing in a 2.5 win player to replace him simply might not be budget-able. So, the various risks involved with relying on a pricey player suppress his top end earning power such that it appears in line with a linear $/win estimate.

    That’s one idea anyway.

    Comment by Brad Johnson — November 4, 2011 @ 2:40 pm

  9. i saved latin, if that counts.

    if that’s not relevant enough, i refused to be a sycophantic lapdog.

    Comment by fjmanuel — November 4, 2011 @ 2:44 pm

  10. Eh… sounds like the analysis might suffer a bit from sample size. Sabathia had a good thing going in New York, and I doubt the man wanted to strain relationships if he intended to keep pitching there. There’s more to be accounted for in that situation than WAR.

    Also, I doubt that simple salary-rate analysis makes this argument fly. The original analysis of 5.0 and 0.0 vs 2.5 and 2.5 was not one from which we should move on simply because one possible counter-example is presented. The Mariners can stick Carp, Wells, Robinson, or a ‘cheap’ FA in LF next year… having 3-4 upside options to pair with Fielder as DH/1B means the 0.0 in the 5.0+0.0 is VERY likely to be more easily upgradable than the two 2.5 players.

    Dave’s not out to left field or anything (no pun intended), I just disagree with his analysis here. (I also think 5 years of cheap Pineda-plus-not-much is close to a fair swap for 2 mid-priced years of Votto, so there. Discredit me all you want to, fellow posters. ^_^)

    Comment by KingCorran — November 4, 2011 @ 2:47 pm

  11. Isn’t it common knowledge that Lee preferred Philadephia and took less than the Yankees were offering to go there? Sabathia didn’t, in fact, opt out to become a free agent available to all thirty teams. To say that wasn’t a hometown deal is, well, wrong, even though it is the Yankees and it is a lot of money. Those two arguments of the point aren’t correct, so it doesn’t stand up.

    Comment by Derek in The Rock — November 4, 2011 @ 2:53 pm

  12. Yankee exception. They would have a roster of 25 6+ win guys at market value if the talent was available. It just isn’t though.

    Comment by AK707 — November 4, 2011 @ 2:59 pm

  13. Try google searches adding in, site:fangraphs.com to your search.

    Comment by Barkey Walker — November 4, 2011 @ 2:59 pm

  14. Yankees offered Lee 6/140 – source. Lee signed for 6/134 with the Phillies. No, he didn’t take substantially less from Philadelphia to sign there.

    Sabathia’s deal is exactly in line with what similar prior free agent pitchers have received. The collective FG readership guessed that he’d sign for 6 years and $136 million, $6 million below what he actually got. If there’s evidence that he left money on the table, it’s not obvious.

    Comment by Dave Cameron — November 4, 2011 @ 3:02 pm

  15. I guess I am confused how you are going to call out someone who at least put some facts and comparisons to support his point and yet your support is: “This is purely from my personal observations…” and “That seems high to me”

    Basically you want to disagree with him and then also require him to provide all the data…this doens’t help your case at all.

    Comment by Myk — November 4, 2011 @ 3:03 pm

  16. I used the Sabathia example because it’s current. This isn’t an isolated incident, though – players have been getting linear dollar per win valuations for years, regardless of how good they are.

    Comment by Dave Cameron — November 4, 2011 @ 3:03 pm

  17. Dave presents a reason they might be super-linear, you have identified a reason that they might sub-linear.

    It is interesting because you are positing that the teams want to buy insurance by diversifying their assets. However, teams sign longer term deals with this type of player and so actually *sell* insurance when you think they should be *buying* it.

    The answer to this might, however, be marketing–bigger deal players make more sense to plug to the local market and the brand is easier to sell for multiple years straight than year by year.

    Comment by Barkey Walker — November 4, 2011 @ 3:03 pm

  18. Sorry to respond to my own post, but I wanted to add that the FG interpretation is that wins are wins and that fans respond to that, so my third paragraph explanation is antithetical to that presumption.

    But then again, so is the Rays low numbers, so, go figure.

    Comment by Barkey Walker — November 4, 2011 @ 3:05 pm

  19. Good column. Real baseball is not roto; there is no meaningful “position scarcity” that articifically raises the value of players at specific positions.

    Comment by Blue — November 4, 2011 @ 3:06 pm

  20. Just want to get both sides of the argument straight:

    Dave: Provides his side of the arugment and the facts that he feel supports his argument.

    The Others: Disagree with Dave’s argument because it doesn’t “feel right” and that is sufficient evidence.

    Seems a bit confusing why one side would consider this a legit response.

    Comment by Myk — November 4, 2011 @ 3:07 pm

  21. Where, exactly, did he call Dave out? He asked where Dave derived his figures from.

    Comment by TomG — November 4, 2011 @ 3:07 pm

  22. If I’m not mistaken, taxes are also considerably higher on income in NYC. Ostensibly, Lee would have bought NYC real estate for ease of access (or at least a property in a nice neighborhood outside the city) where it is more costly than near Philly. So those offers are even closer than they appear at first glance.

    Comment by Brad Johnson — November 4, 2011 @ 3:09 pm

  23. Wasn’t disagreeing with anything Dave said. Just saying that it could be more complicated than purely linear vs exponential, and then was asking for where his numbers came from on the other two numbers. I’m sure there’s some analysis behind them that I don’t know about.

    Comment by Yirmiyahu — November 4, 2011 @ 3:10 pm

  24. The wins are worth increasingly more, but risk increases on a similar path.

    The risk is a huge factor for teams that can’t sustain a top payroll. One bad contract on the team can devastate the budget.

    Comment by Anon — November 4, 2011 @ 3:11 pm

  25. I suspect 5% inflation is historically low, though may be a touch high going forward.

    Comment by Bookbook — November 4, 2011 @ 3:14 pm

  26. As you note, there would seem to be benefit in having more wins in consolidating wins in one player, yet the market doesn’t seem to reflect that. Perhaps that added premium is offset by the added risks associated with consolidating wins that way. The largest is probably the added financial risk – teams can’t secure those players in the open market without offering a long-term contract. The second is the performance risk associated with that player suddenly losing effectiveness. In one member of a 2.5 win tandem goes down, the team still has 2.5 wins. But when the 5 win single player alternative goes on the DL, the team loses all of the production. Those differences should be feeding into the calculus in some way.

    Comment by Basil Ganglia — November 4, 2011 @ 3:15 pm

  27. It’s not an issue of “feel.” For a team to pay a 5 win player the same as two 2.5 win players is theoretically nonsense, at least in a simple model that considers only talent and roster constraints.

    As I commented above, what I suspect is actually happen is that the 5 win player is more likely to cause a variety of other constraints not being discussed in that simple model. The result of which is that either roster constraints and “other” constraints virtually cancel out, or the clubs are simply opting to throw out all the constraints as noise that is close enough to zero sum to ignore.

    We could also be seeing a sort of winner’s curse effect where the “other” constraints are actually negatively stronger than the positive roster constraint. The team that wins the player is the one which is most willing to ignore all constraints in that scenario. Then, in such an environment, if a team is set on hiring a free agent, then they are incentivized to ignore constraints.

    Comment by Brad Johnson — November 4, 2011 @ 3:19 pm

  28. I’m glad to see so many commenters instantly recognize that this is a debate about two overly-simplistic models, one which fits the data but doesn’t make sense and one which makes more sense but doesn’t fit the data.

    One should never settle for a model that fits but doesn’t make sense. We can certainly use it as a proxy, we just need to identify it as such.

    I could write you an article about the theoretical elements of a robust equation, but I could not give you the coefficients for the variables involved. That is why proxies are good, reality is usually too complex to model.

    Comment by Brad Johnson — November 4, 2011 @ 3:26 pm

  29. Yes, the Yankees would run a $750 million payroll. Seriously people, are you actually assuming the Federal Reserve opened up a branch in the bowels of the Death Star?

    Comment by Kevin S. — November 4, 2011 @ 3:28 pm

  30. Really good article Dave. Always a pleasure to find new posts from you.

    Comment by Caveman Jones — November 4, 2011 @ 3:29 pm

  31. The .5 WAR decline comes from a simple rule of thumb that Tom Tango uses in his Marcels. Like linear wins, it fits the data well but isn’t necessarily theoretically robust. Consider it a useful proxy for back-of-the-envelope analysis.

    Comment by Brad Johnson — November 4, 2011 @ 3:30 pm

  32. Linear vs. exponential growth is a false dichotomy. I think Dave has fallen into the common trap of using “exponential” (2^n) when he really means “polynomial with degree greater than one” (e.g. quadratic: n^2). For reference:

    http://en.wikipedia.org/wiki/Asymptotic_analysis

    This makes a big difference: linear growth values an 8-win player 2x as much as a 2-win player, quadratic (n^2) growth values the 8-win guy 4x as much, and exponential growth (2^n) gives him 16x the value.

    There are good arguments that a linear model might not be the right one, and there’s clearly room to argue about the degree of the polynomial: the right equation might be something like:

    VALUE = $5M * WAR^1.2
    instead of
    VALUE = $5M * WAR,

    but to say value grows exponentially with wins is absurd.

    Comment by Bryce — November 4, 2011 @ 3:32 pm

  33. There is a non-linear value to addition wins built into the equation already because it uses Wins Above Replacement, not Wins above zero. Even a replacement team will win some games, and by these equations, these wins are “free” (but each replacement player still does use up a roster spot). For example, let’s say a team full of replacement players would go 50-112 (and thus each “replacement” player is worth 2 wins, but 0 WAR)… so a +1 WAR player actually leads to 3 wins, and +5 WAR player is worth 7 wins. Therefore, paying a +1 player $5M really costs $1.67M per win, and a +5 player $25M costs $3.57M per win. One of the beauties of the WAR statistic though.

    Comment by Erik — November 4, 2011 @ 3:36 pm

  34. Agreed. Great article. Thanks

    Comment by tim elas — November 4, 2011 @ 3:40 pm

  35. You could complicate things further by getting more granular and valuing different win totals differently. For example, $5M * (WAR4^1.2) = Value

    I might have expressed that wrong or confusingly. Basically fewer than 2 wins are raised to the 1st power, wins 2-4 are raised to the 1.1th power, and wins above 4 are raised to the 1.2th power.

    Ultimately, your suggestion is just the simplified version of what I just struggled to express.

    Comment by Brad Johnson — November 4, 2011 @ 3:42 pm

  36. The reason fans get in such a tizzy about this is that fans love individual outliers. For whatever reason, people love quality over quantity (something very exploitable if you’re a video game nerd like me btw). Rabid fans cannot imagine their team without Mauer, Pujols, or as proposed recently, Lincecum or Hernandez. So much so that logic is totally tossed out the window. They bring up theories that are clearly bunk like the classic “but think of how much merch and how many tickets this guys sells!”

    What I find so strange about this concept is that if a “Franchise” player is traded or hurt or whatever, fans are extemely quick to forget their love affair and move on. They still buy jerseys and tickets and posters, despite claims to quite fandom forever.

    People are fickle, illogical creatures. Obviously I’m smarter than the masses though, just like how I’m an above-average driver! :)

    Comment by The Difference — November 4, 2011 @ 3:43 pm

  37. If your 20+ million 7War guy gets hurt it might sink your season, but if he doesn’t you will survive several injuries to everybody else.

    If you spread the money around, maybe now 2 injuries sink your season. If you focus on one big guy 2 maybe 3 injuries is no big deal (as long as it isn’t the big guy).

    Comment by joeiq — November 4, 2011 @ 3:44 pm

  38. Let’s say league minimum is $500k.

    Really what you’re doing is paying $250k for two wins and $4.5 million for one win due to the assumptions of the replacement level theory. You ALWAYS pay $500k for 2 wins and then the rest of the contract pays for the remaining expected value.

    Comment by Brad Johnson — November 4, 2011 @ 3:44 pm

  39. Funny, all people think they are above average, but only half are right. 1/3 if you say average is the middle third, and 1/3 are above and 1/3 below.

    Comment by joe — November 4, 2011 @ 3:45 pm

  40. should read: “paying $250k for (the first) two wins”

    Comment by Brad Johnson — November 4, 2011 @ 3:46 pm

  41. There are 2 separate uses of WAR that might be separated:

    1. What a win goes for on the free agent market. This seems to be linear.
    2. What a win should go for on the free agent market. Theory suggests it should be non-linear.

    I see no reason why a model can’t be built for 2, even while knowing that 1 is out there. For example, fangraphs can report that Pujols will likely be worth $25M on the open market assuming X year, Y WAR projections, and Z cost per WAR. Definition 1. But that wouldnt prevent fangraphs from also reporting that a player such as Pujols should be worth $30M if team’s were maximizing wins under Alpha budget and Beta roster constraints. Definition 2.

    Comment by Abreutime — November 4, 2011 @ 3:50 pm

  42. by the way, I’d go for the big guys, and then fill the holes with bad players, you have more chance of finding breakout guys.

    You might not now who the best guys are at the beginning of the season, but you will by the end.

    Winning might be the most important thing that sells tickets, but Stars sell more tickets than good players. Also more merchandise.

    Comment by Joeiq — November 4, 2011 @ 3:50 pm

  43. There’s no doubt this is a simplistic model. It is not comprehensive, nor is it intended to be. Yet, it still works pretty well. That tells us the other factors are not all that important. That doesn’t mean we shouldn’t try to figure out what they are, but it does tell us with some reasonable amont of certainty that elite players aren’t getting some massive bump due to consolidated value that makes $/WAR analysis useless.

    Comment by Dave Cameron — November 4, 2011 @ 3:52 pm

  44. Yeah, I had a few paragraphs on that in the first draft but they hit the cutting room floor. I think part of what we’re seeing is that there’s basically no way to have a cheap +6 win player (Evan Longoria excluded), but if you have two +3 win guys, you can potentially exploit inefficiencies in the market or in arbitration and pay them less than what they’re actually worth.

    You’re not going to be able to get an underpaid star, but you might be able to get a couple of underpaid good players, and then you’re getting similar production at a lower total salary.

    Comment by Dave Cameron — November 4, 2011 @ 3:54 pm

  45. Not so much that I fell into a trap as I was trying to write for a decent-sized audience. Exponential is a much less scary word than polynomial for most people, and since the point was the same, I went with the less intimidating mathematical term.

    Comment by Dave Cameron — November 4, 2011 @ 3:56 pm

  46. If you live in NYC, yes, taxes are high. A lot of the players – especially the ones with families – live in North Jersey or in Westchester county. Taxes are much lower for those guys.

    The tax issue is also blown out of proportion – you get taxed by the state you play a game in. You pay tax on half your income to the state you play your home games in, and the other half gets taxed by all the states you play in while on the road.

    Comment by Ed — November 4, 2011 @ 3:56 pm

  47. “That tells us the other factors are not all that important.”

    Just a little nitpick, what it tells us is that other factors don’t have a noticeable effect on the end result. They could very well be important but offsetting.

    Comment by Brad Johnson — November 4, 2011 @ 4:01 pm

  48. Good point: not truly “free”, but the basic logic still holds.

    Comment by Erik — November 4, 2011 @ 4:02 pm

  49. Case in point, Placido Polanco. Undervalued skill set with a lot of value relative to contract.

    Comment by Brad Johnson — November 4, 2011 @ 4:04 pm

  50. That’s working under the assumption that the second theory is correct and teams are being overly risk averse. That’s possible, but it’s also possible that outsiders simply are underestimating risk due a lack of first-hand knowledge of the circumstances.

    The market isn’t perfectly efficient, but I’m not comfortable saying that it’s clearly wrong when there aren’t any teams exploiting that inefficiency.

    Comment by Dave Cameron — November 4, 2011 @ 4:12 pm

  51. “here’s basically no way to have a cheap +6 win player (Evan Longoria excluded)”

    So there is no way to… say, get a player with a 2.27 ERA who pitches 250 innings and wins the CYA for $3.8 million?

    Comment by Barkey Walker — November 4, 2011 @ 4:29 pm

  52. I disagree. I think the chances of a 2-3 win player breaking out and becoming 5 win player are higher than a 0-1 win player suddenly becoming a 3 win player. I admit I base this on nothing but my first reaction.

    Comment by Dissenting Opinion — November 4, 2011 @ 4:48 pm

  53. I think there’s another factor that’s missing in this – contract length tradeoffs. The longer the contract, the riskier it is for the team. You don’t just have to worry about normal age related decline, you also have to worry about injuries making the decline faster or even ending a career.

    I think the actual behavior we’re seeing is elite players are trading dollars in their prime years in exchange for more years on the contract. Most players aren’t hitting free agency until their late 20s or sometimes even early 30s, meaning they’ll be past their prime years when their first free agent contract ends. Their best bet is usually to go for as many years as they can get, even if they have to give up some money per year. In the case of elite players that hit free agency early, they’ll try to mitigate this by getting an opt-out clause. A-Rod, CC, AJ Burnett, JD Drew, and others used this to their advantage.

    Comment by Ed — November 4, 2011 @ 4:58 pm

  54. Aren’t the high WAR players receiving multiple year deals though? Isn’t that the compensation?

    It doesn’t effect the model in anyway, though.

    Comment by josh — November 4, 2011 @ 5:00 pm

  55. Really good article, makes things seem a little more logical. A couple of questions, though:

    -Does the same model apply to trades? Do teams bid on the same scale when dealing with players as money? My intuition would say that there is less “bidding” when it comes to free agents and more meeting a price with the player’s demands. Consider though that there are theoretically more teams with the means to acquire a star via trade than free agency. This is because players ultimately make the decision where they end up. Any team may have a chance to trade for a star player, but perhaps he never would choose to go there on a free agent contract. If other teams have that same opportunity, does that change the model?

    On the same note, does the model change when the opportunity for a star player is condensed into .5-3 years? For example, did the Braves not vastly overpay for 1.5 years of Teixiera going by the linear model and is this so out of the ordinary? Does it change when teams are going for a postseason birth in any given year?

    -Also, do players/representatives sacrifice salary for stability? If Albert Pujols asked for a three year contract and had teams bid on that, would he get a higher salary?

    Thanks, and again, this was a good simplification.

    Comment by dan — November 4, 2011 @ 5:01 pm

  56. You can’t make this analysis when a team resigns a player with no other bidders. You can only make this analysis if some team besides NY/BOS/PHI signs a free-agent, because that implies they were not willing to pay any more than what the player signed for. In this case, no one knows how much NY or BOS would have been willing to pay for Sabathia.

    Comment by Cliff — November 4, 2011 @ 5:02 pm

  57. oops! twice that, looking at the wrong year. Still, points stands.

    Comment by Barkey Walker — November 4, 2011 @ 5:07 pm

  58. 0-1 win players are often raw prospects who have a lot of development left and could end up as stars.

    Comment by Cliff — November 4, 2011 @ 5:12 pm

  59. another thought I had: Does anything change when you consider that the roster constraint of “starters” is usually more like 15 players and less when you account for cost-controlled players that teams are more likely to get surplus from?

    Comment by dan — November 4, 2011 @ 5:16 pm

  60. Well, as someone who has touted importance of non-linear factors (though not in this most recent round of articles), I think there is at least mounds of data that teams are indeed paying for WAR at an nearly linear rate. So for descriptive analysis, this definitely has legs.

    For normative analysis, it’s kind of silly to just say “Well, teams probably know better than us though.” I’ve got a PhD with serious background in modeling, including probabilistic models. I’m not saying I’m better than their guys, but if I was willing to put in the time with the data- I’d definitely be competitive. And yourself, the rest of the Fangraphs crew, and the Fangraphs community aren’t exactly shufflers- you’re all quite capable and have done a lot of analysis on player valuation.

    There is a very good possibility that teams are indeed being risk averse. As near as I can tell, teams tend to be risk averse for big contracts but also indicate time-discounting (e.g. wasting money on later years is less bad than wasting them on current years). The two factors work in opposite directions, so it’s very possible they nearly cancel out. There’s also issues of uncertainty at play, which are entirely different than risk. So those are your big psychological impacts on descriptive explanations.

    With that said, I’m not sure why we should throw up our hands about enumerating the structural factors impacting normative player value. I already took a stab at this during the comments in the Pujols article in Feb. An except of factors that I enumerated:
    “””
    Increases value (Better to have more value in one player)
    ———————————
    – Lineup/Fielding position constraints
    – Roster slot constraints

    Decreases value (Better to diversify)
    ——————————
    – Lineup sequencing (e.g. HR hitter worth more with high OBP ahead)
    – Defensive synergy (e.g. double play combinations)

    Unclear Effect on Expectancy (But impacts distributions)
    ——————————
    – Salary constraints (provided people pay based on correct value)
    – Injury risk (expected loss of WAR should be the same, only variance changes)
    “””

    So… I mean, we’re already talking about at least 6 factors. I’m sure we could think of more- but probably not THAT many more. I am also not convinced that teams explicitly use a coherent analysis to integrate these factors, and may very well just throw up their hands and use a linear analysis because it’s close enough.

    But either way, I do not accept that we should just give up on pushing the bounds of normative analysis on these matters. The market might be right, it might be wrong. Why give up before we’ve even bothered to frame the problem?

    Comment by B N — November 4, 2011 @ 5:18 pm

  61. Well, normative models don’t have to fit data. That’s why they’re normative. ;) Descriptive models have to fit data. So linear is a good approximation of what happens. I’m not quite sure why we’re throwing up our hands at framing the function of what a normative assessment of value would look like, however. People model systems more complicated than this all the time.

    Comment by B N — November 4, 2011 @ 5:24 pm

  62. No.

    Dave: Provides examples of the current state of contracts; states that because no one currently does it, implies it isn’t a viable strategy.

    Which is basically equivalent to saying that all theories have been empirically tested in MLB front offices and what we currently see must be the most efficient model.

    While the article is useful in its re-enforcement, by example, that the market doesn’t believe the theory to be true, it is by no means a rebuttal of the theory.

    Comment by Marver — November 4, 2011 @ 5:30 pm

  63. This. You’re talking about the difference between a descriptive versus a normative model, FYI. Glad somebody else is… With that said, there are other structural factors around too and some go the opposite direction: Defensive synergy and lineup sequencing could both favor spreading out your talent. E.g. double play combinations, lineup protection, etc.

    It’s easy to see these when you look at the extremes: what if all your WAR was in one player? If he’s a batter, they’d walk him and K everyone else. If he’s a pitcher, you’ll win every 5th day and lose otherwise typically. Basically, you might not get maximum value because you lack the supporting parts to facilitate it.

    My gut still says that the roster spots are important, but monetary restrictions are the only negative factor with enough significance to outweigh them (e.g. nobody can afford to play the top players what they’re actually worth and still field a team).

    Comment by B N — November 4, 2011 @ 5:35 pm

  64. And, for the record, I would believe that such a theory probably won’t be tested anytime soon, since it exposes the decision maker to considerable job security risk.

    Besides, if the theory leads to the conclusion that ‘high WAR players should be paid more’, then of course looking at market rates won’t be supporting evidence, since the conclusion implies that high WAR players are currently underpaid.

    It’s like me saying “you should add salt to this soup” only to have you test the salt levels of other soups to state that you shouldn’t add salt. Until it is empirically tried, it certainly isn’t proven either way.

    Comment by Marver — November 4, 2011 @ 5:36 pm

  65. It is possible to draft a cheap 6-WAR player. Unfortunately, to do so you also have to draft a lot of negative-WAR players, and those signing bonuses add up.

    Comment by Jeff — November 4, 2011 @ 5:37 pm

  66. If teams are risk averse to long contracts for big talent, why would some of these team not offer Pujols a 3 year deal at 35 or 40 mil per?

    Comment by zack — November 4, 2011 @ 5:37 pm

  67. I never implied that it wasn’t a viable strategy. I said in the piece that this could be evidence of an inefficiency. I’m not sure I believe it is, but I certainly didn’t say or imply that it isn’t.

    Comment by Dave Cameron — November 4, 2011 @ 5:40 pm

  68. Why do 1 win players always get 1 year deals (at most two years) while 5+ win players get 6-8 year deals?

    Comment by Matt — November 4, 2011 @ 5:44 pm

  69. Was anyone actually saying that nonlinear $/W is valued in the market? Appeared to me that the yelling was more along the lines of ‘it should be a part of the market’, not that it is.

    I suppose I either believe that you implied it isn’t a viable strategy — second to last paragraph regarding northeast teams — or that the article doesn’t really quell the gyst I get from the majority of the nonlinear crowd: that this is a theory that is potentially undervalued.

    If anything, showing it isn’t part of the market only fuels those who believe it is an efficiency.

    Comment by Marver — November 4, 2011 @ 5:54 pm

  70. Nice article.

    A team of league-average everyday players* is actually a reasonably decent team (I calculate 36.5 WAR from 8 position players and 5 starting pitchers) so perhaps roster constraints, and the even smaller number of everyday players on a roster, isn’t that much of an issue.

    *Quick and dirty look at the 2011 stats:
    Median WAR for hitters (batting-title qualified) seems to be about 3.
    Median WAR for starting pitchers (minimum 100 IP) is about 2.5

    Of course, if you want to run Aaron Hill and his -0.8 WAR out there every day, well, that’s when you have to have a Jose Bautista on your roster to average things out.

    Comment by Jeff — November 4, 2011 @ 6:06 pm

  71. @BarkeyWalker point stands if and only if that player was signed on the free market and his performance was something expected. Otherwise the comparison fails, and your point collapses.

    Comment by Travis L — November 4, 2011 @ 6:09 pm

  72. How much would Sabathia make if he were will to take one year contract like the 1-win guys are? Seems like some risk might be priced into long term contracts.

    Comment by Chad DeShon — November 4, 2011 @ 6:13 pm

  73. exactly…just becuase the labor market doesnt compensate the player for the exponential value they are adding doesnt mean it doesnt exist.

    Comment by Jason — November 4, 2011 @ 7:14 pm

  74. To be clear, I don’t think sabermetricians should be throwing their hands up. If they have the skills, I much rather prefer a theoretical normative model that pretty much works to a descriptive model that works but is overly simplified. That’s my personal preference.

    I however am throwing my hands up since my modelling skills don’t really extend beyond panel data in Stata.

    Comment by Brad Johnson — November 4, 2011 @ 8:16 pm

  75. I don’t think anyone has the balls to set that precedent.

    Comment by Brad Johnson — November 4, 2011 @ 8:18 pm

  76. For normative analysis, it’s kind of silly to just say “Well, teams probably know better than us though.”

    B N, he didn’t actually say that; what he said is that teams would argue that.

    Comment by The Ancient Mariner — November 4, 2011 @ 8:39 pm

  77. I don’t think that using CC Sabathia and the Yankees is the best way to make this point because any time the Yankees get involved, the economics gets messed up [thanks to their almost half a billion in yearly revenues]. You do address what a lot of the complaints of the linearly extrapolating from the 1 WAR player range where you have more data points to the 5+WAR player range where there is much less data. The initial reaction to saying the 8 WAR MVP candidates are worth $40M that year is generally going to be strongly against.

    A few points here. First, it should be noted that players often make the money in the long run that they get underpaid for up front in free agent deals, especially if you start to factor in the risk that baseball teams take on in long term deals.

    Secondly, I feel like this is confuses two ways that we measure value. Is a true talent 8WAR player worth 2 4WAR players, all else being equal? No, I’d argue they are worth more due to their rarity.

    Part of the way that can be proven is if you start at the 1-2 WAR players for whom the $5M/WAR figure is seemingly based off of and then go the opposite direction from the elite players (go towards the replacement level players). I wrote something that got picked up by Tango at the Book Blog regarding this (link below), but there are many more below average and replacement level players than average players in the majors. Smart teams should spend less than $5M/WAR on these players because there are so many of them. Taking this logic the other way, elite players are worth more than the average $/WAR amount…the issue is that you’re not likely to see it in real AAV because there aren’t enough teams with the financial means to do it. I’d argue that it ends up coming out in the extra years given, but that’s an analysis that I’ve been saving for this offseason.

    http://www.insidethebook.com/ee/index.php/site/comments/talent_distribution_in_mlb_bell_shaped_or_right_tail/

    Comment by Prashanth Francis — November 4, 2011 @ 8:40 pm

  78. I don’t think there is a major puzzle. With the exception, perhaps, of the Yankees no team has so much money that it makes sense to pay non-linear amounts for wins at a specific position.

    Comment by Blue — November 4, 2011 @ 9:10 pm

  79. “Taking this logic the other way, elite players are worth more than the average $/WAR amount”

    This is true if and only if roster size/full time roster slots are a significant constraint on the number of wins a team generates. The data suggests that this is not the case.

    Comment by Blue — November 4, 2011 @ 9:13 pm

  80. I also think that fan good will and the effect of marketing have a breaking point. Any player making over 30 mil will turn off many many fans and undermine all marketing efforts. This I would consider a reason players are paid(to outsiders like us) in a linear fashion.

    Comment by ofMontreal — November 4, 2011 @ 10:04 pm

  81. @TravisL, not sure what you are talking about. Evan Longoria was not signed as a FA.

    Comment by Barkey Walker — November 4, 2011 @ 10:28 pm

  82. I haven’t thought about this carefully, so this could be stupid. But if you pay an elite player more $/WAR, then it seems to me that the stars are the players I don’t want, because their cost is disproportionately large compared to their on-field impact.

    For example: if a 4-win player is paid as much as two 2.5-win players, then for the same cost in both salary and roster spots, I can either buy a 4-win player and a replacement level player, or I can buy two 2.5-win players. Clearly, I want to do the latter, and this is true regardless of the details of the $/WAR curve as long as it’s an increasing function of WAR.

    Comment by Brendan — November 5, 2011 @ 12:14 am

  83. If it were your tens (hundreds?) of MILLIONS of dollars on the line, would you not be extremely risk averse as well?

    Comment by Logic (again) — November 5, 2011 @ 1:04 am

  84. Yes, but only because teams currently select for raw prospects in need of development.

    If you need a bunch of 0-1 WAR players to fill out your roster, you’re not going to be able to be as selective as most teams are.

    Comment by Bhaakon — November 5, 2011 @ 2:48 am

  85. Are we doing the 5mil per win narrative AGAIN this offseason?

    Seems like the whole salary inflation meme was done enough last winter when Dave continued to beat that drum and throw out 5mil per win to support it (with no data whatsoever).

    Looking at the actual valuations on the 2011 players, it appears the 5mil/win was just a wee bit off and the whole “we’re in for salary inflation” story was just a bit overdone.

    Rather than just continue to throw out 5mil per win arbitrarily, can’t a stat based site use the ACTUAL data from last year and apply an inflation rate to it? If that # turns out to be less than 5mil, the CC Sabathia example may suddenly not look dead on linear.

    Comment by Joe — November 5, 2011 @ 4:26 am

  86. Dave you are using a model/projection to validate another model. You are assuming teams are using similar valuation models, and more importantly similar assumptions that you are using.

    There are several problems with this
    1) Last year you were tossing about the 5mil/WAR figure as well and that turned out to be considerably off… why not use a data driven baseline (use last year or last 3 years of $/WAR and use an inflation rate)

    2) The inflation rate itself is a guess (even if it is based on historical data), it’s hard to apply it to any specific 5 year period with certainty

    3) your WAR starting point is yet another model with it’s own built in error bars.

    I’m guessing you took the last 3 years (a little over 18 WAR) and divided by 3 for the Sabathia example? Even if we believe FIP based WAR is accurate (I’d think you’d want to use a xFIP based model for something like this), shouldn’t you also correct for an aging curve in that period? In other words if he posted 18 WAR over 3 years that means he’s at 6 WAR in the MIDDLE of that period not at the end of it (basically it would be something like 6.5, 6, 5.5)… this would start your model at 5 WAR (not 5.5)

    The bottom line is the Yankees could very well be paying a premium for higher WAR…. there’s no way of knowing this unless you understand what their specific assessment of $/WAR, inflation, current true talent and expecting aging curve is.

    Since you deem that it fits (despite some of the issues I pointed out with your assumptions), you conclude that this is evidence that teams must be valuing wins linearly… all it means is YOU are valuing wins linearly.

    Comment by Hank — November 5, 2011 @ 5:57 am

  87. I think you to consider that the market for <1 WAR players is not driven by market forces. The market for these players is completely fungible, but teams have to pay the veteran minimum, arbitration value, etc. This skews the analysis. I'd argue that the first win is essentially free. Any team with any degree of resourcefulness will be able to scrape up a 1 WAR player. After that, I don't doubt that the $/win realtionship is pretty linear.

    You can see this if you consider that there is no number of fully priced 1 WAR players a team would trade for a fully priced 6 WAR player. They would be indifferent from a "value" perspective, but their team would be significantly worse off. It's a replacement level issue, basically. No team actually ever plays Willie Bloomquist. They sift through the dregs until they up with someone who can play a bit.

    Comment by pierre — November 5, 2011 @ 8:50 am

  88. I don’t know about that. Sure there is real exponential value from consolidation but the reason you don’t see it is because of market constraints. If team A pays exponential value for one player they are usually but not always limited in their ability to continue paying exponential value for other players or even linear values for other players. Saying exponential value exists is probably correct in a salary cap free world, but whether or not there is a cap in rule, there is a cap in fact as teams simply cannot afford to do that. So if the labor market could never really exploit exponential value then does it really exist at all?

    Comment by Colin — November 5, 2011 @ 10:49 am

  89. This thread is useless without graphs. I mean, 2 data points = line!

    You could even do a prediction (future contract value vs. expected WAR) graph and postdiction (paid contract value vs. actual WAR).

    Comment by zenbitz — November 5, 2011 @ 12:03 pm

  90. @Logic: Quite the opposite, actually. People end up with hundreds of millions of dollars EXACTLY because they are willing to take major risks. There’s a ton of studies of business moguls and elite stock traders that demonstrate that the best ones take on more risk, not less risk.

    Comment by B N — November 5, 2011 @ 1:36 pm

  91. @Mariner: Yes, he did say that, but then he followed it up with the fairly unsupported comment of “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…” Which basically implies that we should just trust that the market is efficient because smart people aren’t making those moves.

    I mean, I’m a Red Sox fan, and I greatly respect our front office. But I’d be the first to admit they’re not infallible. Likewise for the Yankees and also the Rays (though the Rays are about as close as you’re going to come to infallible, they play a bad hand better than any team in baseball).

    Comment by B N — November 5, 2011 @ 1:54 pm

  92. Collin… I’m not sure I follow, but the labor market does “exploit” exponential value in the sense that the players add it but aren’t compensated for it.

    Either way – my point was just that if your hypothesis is that exponential value of concentrated war doesnt exist, you can’t test that hypothesis using player salaries.

    Comment by Jason — November 5, 2011 @ 7:04 pm

  93. Jason,

    I think now that I’m reading it again my point was simply to question whether a player added value in exponential fashion to a team if said team cannot capitalize on it.

    Comment by Colin — November 5, 2011 @ 9:09 pm

  94. if you take the value placed on risk aversion the long term deals (5-7) are depressed because of the risk of losing players to injuries is MUCH higher than a one-three year deal. It probably plays out more with pitchers than position players. Some of that discount may be seen in the WAR decline, but there is also the value (negative) of having that percentage of your overall payroll invested in 1 player that if he goes down for a significant part of season (or multiple years) it creates an incredibly difficult hole to fill. The only way to fairly compare is if the players ALL had the same number of years, like a perpectual

    PS dave please stop spouting off the implication that people who equate Yu Darvish to other Japanese pitchers are being racist. They use other japanese pitchers as a reference point because they played in the same league. Most compare him to pitchers who have had similar hype from Japan. You’re assuming incorrectly that everyone has a knowledge of Darvish’s scouting skillset. Very little information was available for him prior to the last few weeks in terms of scouting report in mainstream arenas. The statistics from the NPL are very similar between DiceK and darvish. In terms of NPL stats, development path and HYPE they are very similar. That is the reason people don’t compare him colby lewis. Im not sure about the stats, but neither the developmental path or HYPE are anywhere similar with Yu. The truth about NPL pitchers coming over is that they have been VERY unpredictable. That’s what people are indicating when they compare to NPL players and are skeptical. NOT racial profiling or racism

    Comment by gimpcom — November 5, 2011 @ 9:49 pm

  95. meant to add in the first paragraph: the exponential value that are putting into the contract is the RISK. there are other things involved with risk beyond years like previous health, but the years are much lower because they don’t have to take that risk on for players that are fairly replaceable.

    Comment by gimpcom — November 5, 2011 @ 9:59 pm

  96. Did I reference my college career yet?

    Comment by CircleChange11 — November 6, 2011 @ 4:33 am

  97. Wins are a fungible commodity. There is no reason they should be treated as anything but linear.

    People want to believe that the 5+ WAR player has some unique value just because of the scarcity of that player’s particular individual talent. But a win is a win is a win. There is no advantage there over two +2.5 WAR players.

    The argument that it is easier to upgrade the +0 player than the +2.5 player is simply not accurate, unless you have set replacement level incorrectly. You are never really dealing with a decision about only 2 roster spots; you always have 25 spots to play with, which leaves enough options overall that there are always upgrade alternatives somewhere in the marketplace, provided the only interest is wins.

    Keep in mind, the average team only produces about +33.0 WAR. With 25 roster spots, that still only comes to +1.3 WAR per player. Or figure +2.0 WAR per regular (8 pos + 5 SP) and +0.6 per bench or bullpen player.

    Suppose you now replace one regular with a +5.6 win player. Now, to still win 81 games, you might now need your other 12 regulars to average +1.7 WAR each, rather than +2.0 WAR. Or if you consider that bench and pen upgrades are also an option, you now need +1.14 WAR from each of the other 24 players. These small changes in WAR needed simply aren’t going to increase the available supply of talent enough to measurably lower your costs per win for the rest of the roster.

    Maybe you could invent a scenario where some non-linearity existed, by putting significant restrictions on nearly all other roster spots. But real world teams don’t have that many restrictions. Teams just aren’t locked into enough multi-year contracts for this to be the case. Even the Yankees or Red Sox don’t have more than 10 players who they are locked into for 2012 for more than $5M. And many of the players who do have contracts are tradeable, so the team still has alternatives.

    To create the kind of non-linearity some are theorizing about would require significant restriction or market failure. There really has to be no other options. Even suppose you were to create a situation where the Yankees are trying to create a 110 win talent team, and have locked themselves into contracts for every other roster spot, all with no-trade clauses, and so must have one particular player. That might cause non-linearity; but non-linearity would then only apply to that one player. The rest of the market would be unaffected.

    Comment by acerimusdux — November 6, 2011 @ 11:41 am

  98. all you have to do to convince yourself that $/win is not linear is to look at the list of players with .5 to 1.5 WAR in 2011. Except for the rookies and the odd Justin Morneau, these are players that either aren’t regulars because they aren’t good enough or players whose teams are/should be desperate to replace them. Chris Getz and friends. The first win above WAR=0 can be had for, say, a low-level C- pitching prospect. The A’s make this trade every year. They fill voids with 1-2 WAR players like Raj Davis, Ryan Sweeney, Jack Cust, or Scott Sizemore.

    Comment by pierre — November 6, 2011 @ 12:16 pm

  99. Of course, in February you also said that Kenny Williams believed that the White Sox were better off with Dunn and Rios over Pujols and I think you implied that you agreed with that sentiment. I realize it’s a retrospective analysis of a move that looked good prospectively but I still gotta ask…how’d that work out?

    Comment by MikeS — November 6, 2011 @ 6:00 pm

  100. This is an extremely good point that’s completely being missed here – how many players do you see getting 6 year, 3 mil/year deals?

    The contract size/WAR curve isn’t linear – it may be linear in $/WAR/year, but it’s non-linear in guaranteed years (and thus becomes completely non-linear in total guaranteed $$). You can come up with all sorts of reasons for this – the lack of a need to lock up mediocre players, or even having something like a luxury tax (and even more generally having teams that want to spend approximately what they take in) forces this to happen, as overpaying per year can create a situation where you can’t field a full competitive team, whereas overpaying in # of years allows you to go through boom & bust cycles.

    This is a non-trivial effect, too – if you’re (in true value) a 5 mil/ year player, you’re probably constantly signing 1 or 2 year deals… and having one bad year can drop your next contract down to 2 or 1 mil/year, and over a 5 year period you may wind up making closer to 10 or 15 mil instead of the 25 mil that your value would indicate. On the other hand, if you’re a 20mil/year player, you’re never going to sign a <4 year deal barring major injury concerns, and so there's no risk – even if you have one random crappy year in the middle there, you're still going to make 100 mil in 5 years.

    Comment by Eric — November 7, 2011 @ 2:58 am

  101. One big factor that may drive down non-linear salary structure is what it would mean for future contracts. Remember A-Rod’s first contract? If he signed for 8/$200, which was, reportedly, close to second-highest bid, Manny doesn’t get 8/$200, Jeter doesn’t get 10/$171, many others have to take a bit less on their contracts. Teams may want to pay above linear for the best players, but mid-level guys would then start asking for linear value compared to the best, and MLB GMs (or any group of GMs in the world) don’t have enough discipline to say no. And MLB is back in the salary spiral from the start of the century.

    Comment by Davor — November 7, 2011 @ 4:15 am

  102. Also, if a team has 9 hitters and 5 starters producing 3 WAR each, bench producing 3 WAR and bullpen producing 5 WAR, that’s 100-win team. They don’t need to pay premium for 7 WAR player, specially in the offseason. Before season starts, everybody is healthy, and there are plenty of options on the market. Why would anyone set the offseason plan as “I’ll overpay for 6WAR player and leave my black hole, so that I can upgrade him during the season”, instead of filling every hole possible? Without researching trades in more detail, it seems to me that non-linear premium exists in trades.

    Comment by Davor — November 7, 2011 @ 5:12 am

  103. (Excuse three posts in the row)

    Most of those who believe in non-linear scale don’t think that 7 WAR player should be paid for 10 WAR, but that 6 WAR player should be paid for around 6.5 WAR. So, it’s pretty hard to actually see that effect. The example here is CC’s contract. Most people(both fans and professional analysts I read) believe that he left at least several million on the table by not opting out. If he could have gotten $10M more, that’s some 0.2 WAR more. If Yankees consider him to be 5.4 WAR pitcher, or win to be worth $4.9M, there’s premium over linear model, but not big enough for us to see.

    Comment by Davor — November 7, 2011 @ 8:55 am

  104. My guess is that the reason is b/c Pujols has no interest in $110 M when he can get $200 M. Not being sarcastic, I’m just saying that the player tends to prefer a long-term deal b/c it provides more long-term security. He’s trading off the possibility of a bigger payoff for the security of having more $ guaranteed.

    It’s quite possible that teams have mentioned to Sabathia or the Cards maybe even to Pujols something along those lines and he and his agent have just dismissed it immediately. The report then to the media goes something like “Sabathia won’t sign for less than 6 years” or something like that. We assume that means he won’t take a 5 year deal w/o knowing the particulars — that he doesn’t even want to talk about a 3 year $105 M deal.

    Comment by chuckb — November 7, 2011 @ 9:01 am

  105. I understand that for sure. My point is just that in a free market, the market should be able to define a short term threshold as well as a long term threshold. There has to be a point, such as at 3yr 140 mil, that Pujols is okay with the risk that he’s going to get hurt or decline, for the reward of another giant contract after the 3 years.

    You don’t see this kind of situation very often. Is it the players union… or just agents wanting to make as much money as possible? I wonder.

    Comment by zack — November 8, 2011 @ 1:04 pm

  106. Per fangraphs own value calculations:

    2002: 2.6
    2003: 2.8
    2004: 3.1
    2005: 3.4
    2006: 3.7
    2007: 4.1
    2008: 4.5
    2009: 4.5
    2010: 4
    2011: 4.5
    2012: ???

    When should I expect WAR/$ to start inflating again, and how much? Everything was oh neat and fitting until 2008. But if we just naively extrapolate from the entire data set:

    2012: 4.9
    2013: 5.1
    2014: 5.4
    2015: 5.6
    2016: 5.8
    2017: 6.0

    This rate projects Sabathia’s value at $137.45 M, and Pabelbon at $44-48 MM depending on how you model his value and his decline (I used 0.25, because .5 just feels too steep for a relief pitcher who is maxing out at 2-3 fWAR/year at his peak).

    Comment by Phils_Goodman — November 12, 2011 @ 2:11 am

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