THE BOOK--Playing The Percentages In Baseball

OK, read the whole thing.  First, you and Dr. Fort should be congratulated for undertaking this extended discussion and trying to understand each other’s reasoning. 

To me it seems where the discussion gets hung up is with the concept of marginal product.  I think the difficult thing for Dr. Fort to conceptualize is the idea that a MLB team can only employee 25 workers at a time.  For most Economics discussions concerning compensation an employer has no limitation on the number of employees that he can have producing for him at any one time.  A law firm, for example can decide to hire an additional lawyer even if that lawyer can only bill 2000 hours a year and the average associate in the firm bills 2500 because an additional 2000 dollars still adds value to the firm.

But that same law firm would not hire that lawyer if they were limited to 100 lawyers and the worst lawyer already in the firm was already billing 2001 hours because the new lawyer would actually be lowering the total product of the firm since they would have to fire a more productive lawyer to hire the new one.  You are claiming that would mean that the new lawyer would have a negative MP and Dr. Fort seems to be arguing that since the lawyer can bill 2000 hours that that would be his MP.  The closest he came to accepting your point of view was when he talked about “normalizing” MP.  But it really isn’t normalization, it is the limitation on the number of employees that creates a need for a different definition of MP.

If you can get him over this hurdle I think that you will be close to agreement.  But I would drop the use of “replacement player” for a while as I think that language is not familiar to Dr. Fort and is making the discussion more difficult than it needs to be.  Put the empahasis on the limitation on the number of employees.  I know you mentioned it a couple of times, but I don’t think Dr. Fort completely understood the ramifications to the limitation to the team’s MP and how that could decrease even when they were hiring a player that had some ability to contribute, i.e a positive personal MP.

Fascinating discussion.  Thanks for sharing.

I agree that my use of the term “replacement level” from the outset made the discussion more difficult, because “replacement” means something different to him and me.  Specifically for Rodney, you are always replacing someone else, who himself provides some MP, and so you are always paid for your production above whatever that’s guy’s salary is.  It’s like a huge chain effect.  And I was trying to force the chain all the way down to one baseline.  And to Rodney, it seems the baseline is zero / no-worker.  And to me, it was the best player not in MLB.

Once we got to the point of Mendoza/Jacque-Ponson, I think the discussion picked up.

This probably explains why JC thinks that Francoeur was worth 12MM$: he’s a worker who is on the field, and so, he must provide production.

And I agree with Peter that the constraint of (exactly) 9 players on the field (and 25 players on the team) is a critical requirement in the model.

Tangotiger -
Interesting discussion.  While conceptually I believe in a general league-wide valuation based on WAR and a common replacement player base, I feel there’s a variable in play based on past performance.  A ‘security/risk aversion’ variable by which a team will offer someone like Jermaine Dye more money than an AAAA OF.  It could be that certain teams (say the Royals with Ankiel/Podsednik) just overproject player values but I think some of it has to do with the security of a ‘proven commodity’/’someone who’s been there’ vs. an unproven commodity like Mitch Maier.  How else does Garret Anderson still have a job after posting a -0.9 WAR? 

Another comment on the tail end of the conversation.  The definition of a ‘replacement player’ in fantasy baseball is made more complex when the rostered players in a fantasy league is less than the number of rostered players in MLB. 

The shallower the league, the more replacement value varies by position.  In a 12-team league with one catcher and 5 OFs per team, the replacement value of an OF is much higher than that of a Catcher.  In a 12-team AL-only league using 2 catchers + 5 OFs per team, the replacement value for both positions converge to 0 as the rostered teams contain all players getting any playing time.

The limit on the # of players is crucial, but it’s also important that these players’s skills are not transferable to another industry.  In Peter’s example, a firm the could only hire 100 lawyers would in fact still have to pay the 100th lawyer something close to his MRP, or else their 100th lawyer would always be hired away by another firm.  More similar would be if only 100 people in the United States were allowed to practice law (not a bad idea), but even then the 100th lawyer would be pretty well compensated because someone with that skill set would have lots of other suitors (lots of people with law degrees make good money working outside the legal profession).  Baseball players, however, have no other market for their baseball skills.  Most of them would have trouble earning more than 1/4 of the MLB minimum salary (unless they’be first become a star playing baseball).  So they really are like a coal miner in a company town, with only 1 employer and zero leverage.  Without the union, the MLB minimum salary would probably be only slightly more than AAA salaries.

Guy, actually we know exactly what happens when the union is not involved.  Babe Ruth for example earned more than the President. 

I don’t know what the MLB salary was relative to the workforce in the old days.

Guy brings up a good point about the lack of options that a skilled baseball player has.  It reminded me of another point that I want to make.  Dr. Fort may be having trouble with another concept that makes the MLB employment model unusual.  The lawyer that gets fired from the limited to 100 lawyer firm when that firm finds a more productive lawyer still has full control of his productivity and can sell it to another firm for his own benefit.  In the MLB employment model if a team finds a more productive player and hires him to replace a lower producing player the team still retains an interest in the lower player’s productivity.  Tom mentioned this fact to Dr. Fort during the email exchange but I am not sure that Dr. Fort completely understood how this “remainder value” in a player that is being fired affects what the team is willing to pay a better producing player.  The MLB employment model is just very different than that which an economist usually encounters.

"Guy, actually we know exactly what happens when the union is not involved.  Babe Ruth for example earned more than the President.”

And thanks to the union, last season Robb Quinlan made more than the president.

The Babe’s 80,000 contract in 1930 (when he had a better year than the president), in inflation adjust terms only comes out to about 1 million in 2010, using the BLS inflation calculator.

Another way to look at it is the Babe was paid 4000 ounces of gold.  Today that would be worth about 4.5 million.

The average player of his day was paid like an upper middle class professional, and the very top stars were rich.  Now anybody who lasts more than 3 years in the bigs is set for life.

"actually we know exactly what happens when the union is not involved.  Babe Ruth for example earned more than the President.”

Which is why I specified the MINIMUM salary, which would be much less than $400K sans union.  I’m not saying star players would make nothing.  On the other hand, I feel pretty confident that players did not receive anything close to 60% of total revenues before they had a union.

Guy: my bad.

What Dr. Fort didn’t understand was

1.) the limit of 27 outs in a game.
2.) The concept that a player had the same value no matter what the team.

What you didn’t get to was is multiplier - will the Yankees pay more for a player than the Royals, since wins return for revenue for the

Yankees than the Royals

He didn’t understand that a player below replacement level would actually have a negative value. That is true because he is taking opportunities from more talented players. If I could pitch in the majors and have a runs against average (RAA) of 9.00, I would still project as a .170 win% or so. The problem is, I would be taking innings/outs from someone who could be 5.00, and be worth more wins. I would be hurting a team and in actuality be worth negative runs.

The same holds true for Mendoza; I am using more outs, and limiting the attempts by better teammates get. If he makes 500 outs and gets on 100 times,he may contribute runs; but if player750 played there would be more runs with the same number of outs.

I believe that would be a scarcity of resources.

For 2) I think he wasn’t familar with the concepts. At the extreme end you may get a player worth more to a bad team than a good team. Additionally, some wins are worth more than others, which might be the 2nd part of his equation.
Going from 75 wins to 83 wins makes a team a playoff contender, and would be worth more than going from a 55 to 63 win team, or a 108 to 116 win team. That might be included in the 2nd part of his equation.

I don’t think it’s a given that there’s a greater multiplier for the Yankes than Royals (other than the playoff-potential).

For those discussing comparisons of MLB compensation to compensation in the broader employment market, keep in mind that most professional athletes invest many years of intensive training and in return experience a very short and high-risk career in which a return on that investment must occur over a very short time period, compared to lifelong professions such as doctor, lawyer, economist, etc. It is not a coincidence that profesional athletics for over a century and even today has tended to attract a disproportionate number of entrants from economically disadvantaged and immigrant communities, or underdeveloped nations, where other career options are limited.

For position players, is it possible that player A could have more MP than player B for a given team? Maybe the team in question does not have anyone with a high OBP and player A has a high OBP and player B has a very low OBP. And maybe it is reversed for SLG (but, say, the each have an identical wOBA-I know this may not be possible). Maybe the team would pick player A since they can stick him at leadoff where OBP has a high MP relative to other stuff.

It is also possible that the kind of guys you want to add to a low scoring team might be different than for a high scoring team. See

http://www.beyondtheboxscore.com/story/2006/9/26/93215/5836

I know that is based on my OBP/SLG values for each lineup slot, which may not be a good way to set a lineup. But that is not what I was trying to do. Anyway, maybe the same thing can be done usiing simulations or that Markov lineup model.

The results might only show a small difference. But a team with a low OBP and a bunch of sluggers might benefit more if they add a low SLG guy with a high OBP. And that guy might not add as much to a high OBP team with a low SLG.

Tango

Forgot to say thanks for putting all of this together.

Cy

The other point about different values for different teams would be the value of a closer to the Yankees, or a catcher to the Twins, etc.

The 15th best closer as much more value to others.

A lefthanded slugger may have more value to some than others.

Yes, everyone everywhere is worth different to someone in all facets of life on Earth.

But, we’re not talking about a big deal here.  If Mauer is a 5.2 WAR catcher for the Twins, he’s 5.0 to 5.4 for every team.

(Off-field value not considered.)

Tango #18, exactly right.  Park effects can also make one player more or less valuable to one team than another, but as Tango says, the effect is minimal.

If Dr. Fort does understand that players have the same MP (I hate using these economic terms - can we just say WAR?) regardless of the team, then the discussion ends there.

Another sticking point with a non-saberist is the use of the words “zero value.” A replacement player has more than zero value of course. He has at least the value of the major league minimum.  As well, players worse than replacement value have some positive value too.  They are needed to replace replacement players when someone gets injured or retires.  Or they are young and might get better, such that they have future value.  Etc.  You can keep using the term “zero value” as long as you explain that you don’t mean “zero value” and even then, that can be confusing for a non-saberist.

The other sticking I have is MR and MRP.  MR in baseball is the value of a marginal win, right?  That varies greatly among teams, right?  To the Royals it might be 1.5mm and to the Yankees it might be 10mm, right?  Let’s forget about the whole playoff sweet spot issue for now.

So according to the standard formula salary=MR*MP, the Yankee players would get many more times the salary of the Royals players, even for the same quality (MP) of player. Why is that not the case?  Is it only because teams can only hire 25 players?  Is it because there is some kind of implicit collusion among teams?  It is because the Yankees only have to pay slightly more than the team with the next highest MR (and what if that team has only a slightly smaller MR and the next team only slightly smaller than that?).

Inquiring minds want to know…

That was great, and I was incredibly happy to have the saber view represented so competently. Tango is not only one of our top researchers, he’s also our Carl Sagan.

One of Dr. Fort’s best and most reasonable statements is that the opportunity cost of using a slightly better measure of MP might outweigh the benefits of using it. But that apply equally well to the pursuit of analyzing baseball with a strict academic MP*MR = MRP model.

If holding MP baseline constant at zero is empirically defensible, and the MP delta of a player on different teams so small such that WAR / Player salaries is stable for free agents among all teams, is it worthwhile to work so hard to identify those instances where MR makes a replacement level player’s MRP positive? If yes, I think an exploration of why would be very interesting to read.

Took two classes from Dr. Fort.  Interesting guy.

MGL/19 - I would say that value = MR*MP rather than salary = MR*MP.  Salary is going to be determined by a number of different factors, many of which you alluded to in your post.  Roster limits are set at 25.  The number of players a team can use on any given night is set at 9 or 10 at a time.  The number of teams is limited to 30.  The presence of a limited number of non-homogenous teams means that the market works like an auction in that, to get a player, the winner must only pay just more than the team whose valuation is second.  Long-term contracts create a certain stickiness in the market.  Players under team control provide excellent value and effectively reduce the number of bidders for players in free agency.  These are all substantial violations of the perfectly competitive market assumptions, which is where the salary = MR*MP result comes from.  Salary = MR*MP is only true at the margin, anyway.

That does not mean we can not think of a player’s marginal value as being equal to MR*MP, which should vary widely among the teams, primarily due to differences in MR.  They won’t pay that salary, but that is their value.  The difference between MRP and salary would represent a marginal profit for the owner.

I haven’t finished reading the discussion yet, and really felt like I should be taking notes, but here are a few comments.

Dr. Fort was sticking to calculating MP (WAR) relative to the team the player is on, while Tango represented the community view of calculating it in a neutral context, vs a league baseline. I understand that in the example of Greinke on the Royals or the Yankees, the Yankee being replaced (5th starter) is likely of higher quality than the 5th starter of the Royals. The Yankees can then trade their 5th starter, causing a ripple effect as rosters around the league rearrange, but it’s still player 751 who’s out of a job.

Dr. Fort also seemed to assume that any additional revenue (MR) added to the teams by winning more games is automatically passed on the player payroll. What about profit? Just because the Yankees have a larger pot of money to spend on players does mean they will. Find the best available players for the lowest possible cost and pocket the difference (see Marlins).

The idea of different teams paying substantially different amounts for the same player, theory, is simple:

Let’s say that Boston, the Yankees and a few other teams are interested in a good SS.  Obviously the Yankees and Boston can pay the guy a lot more than the other teams and still make money, because their MRP (WAR times dollars per win in revenue/profit) for that player is a lot higher than for the other teams.  So what happens?  Well, Boston and the Yankees only bid for this player, and presumably the Yankees get him for a little more than Boston can afford to pay.  But that number is a lot higher than the other teams could afford to pay.

Now, when Boston or the Yankees no longer need a SS, then the other teams can get one. And of course they get one for a lot less $ per WAR than the last SS.

So why don’t teams like the Yankees and Boston pay substantially more per marginal win than the other teams?  More generally speaking, why is it that teams do not pay for marginal wins in some proportion to the profit/revenue per marginal win?

The ONLY reason that I can think of is that there is some kind of unspoken rule (among the teams) that no team will pay a lot more than any other team for a player (per WAR) or that teams with less revenue/profit will overpay for players, mostly by paying the same rate per win as all the other teams, including the Yankees and Red Sox, but buying lesser players so that their total salary (and of course their expected wins) is less.

Our $ per WAR numbers come from players actually signed, but the decision of which team a player signs with has to do with offers.

Let’s say the Yankees rotation has WAR of 6,5,4,3,2 - while the Royals rotation is 4,3,2,1,0.

If the Yankees add a 3 WAR (vs league replacement) starter (say Gil Meche), 2 replaces 2 for a net of 1, while the Royals can net 3 wins with the same player.

The Yankees have a better team, and more revenue, and thus can offer more money than the Royals, but if the player only adds 1 win to the Yankees, they don’t need him as much as the Royals do, and thus will likely offer a smaller contract. The player, acting rationally in his own best interest, takes the largest offer and signs with the Royals.

Even though the Yankees have the largest resources, they improve themselves the most by only signing the players who offer the most marginal benefit - where the player being replaced is the closest to being replacement level.

Which leans back towards Tango’s argument that in the end, the team specific MP of players actually signed will be closer to their league normalized MP than you might expect.

Brian, that is wrong, because you are assuming that the Yankees drop their 2 WAR pitcher, he doesn’t pitch, and they still pay him!  No!  They net 1 win in their rotation, but they trade the 2 win player that Meche replaces and get another 2 win player to replace someone else that they need, or they trade him for prospects or they simply trade him and don’t pay him anymore.  I am afraid your logic is completely wrong.

Eight years ago, Voros had this great article:

http://www.baseballthinkfactory.org/files/primate_studies/discussion/mccracken_2002-08-19_0/

The first question is the big one. I tried a series of various functions trying to see if the amount of revenue per win varied across different markets (IE, is a win more valuable in New York than Kansas City). I tried as hard as I could, but I kept coming back to the same conclusion: if there is any change in the value of a win across markets, it’s that a win is worth more in the smaller market but to an extent that it is irrelevant and also not statistically significant. In layman’s terms, using the same revenue per win figure as a constant across all teams will give you as good an estimate as any other method I could come up with.

You know me, I’m a simple guy, and I love simple rules.  Why put somethign to some exponential function, if a linear function works just as well, or almost?

Basically, there is no multiplier effect in wins by market.  A win is just as valuable in a small market as a big market.

This is just like park factors: it’s not a given that there is a multiplier effect, as opposed to an additive effect.

And Voros’ research says that it’s linear.  Since linear makes our life 100 times easier, why try to make it more complicated? Other than a playoff sweetspot parameter, there’s no extra multiplier effect.

Now, that research was done 8 years ago.  If someone wants to reproduce what Voros did, by all means, I’d love to see that research.

A win is worth just as much in a small market as a big market?  Really?  Then why don’t the Royals and Pirates and Marlins spend as much on players as the Yankees?

I always thought that a win was worth the same *up to a certain threshold*.  If the Royals paid league minimum and won 40 games, the fans would desert them and they’d lose money.  Going from 40 to 70, each win is worth the same to the Royals as (say) going from 70 to 100 is to the Yankees.

Putting it another way: a win is worth the same to all teams *at the margin*.  And each team has its own margin, because each team as its own revenue-to-wins curve (which includes playoff potential, of course).

Am I wrong on this?  Is it really true that (a) the average teams are spending the right amount, (b) the Yankees would make more money if they spent a lot less, and (c) the Marlins would make more money if they spent a lot more?

Nice, civil discussion between Rodney and Tango. But it seems clear at the end that Fort still does not accept either replacement value nor the idea that players’ MP is basically the same on all teams.  So that leaves a pretty wide gulf.

*

“So according to the standard formula salary=MR*MP, the Yankee players would get many more times the salary of the Royals players, even for the same quality (MP) of player. Why is that not the case?”

That is a good question, and exactly the kind of question that the sports economists should be able to help us understand (but haven’t).  I assume the answer is a competitive labor market forces a constant price for WAR that all teams must pay.  The Yankees could afford to pay $5M/WAR for a 5-WAR SS.  But then let’s say a 4-WAR SS goes for just $4M/WAR to St Louis.  Why should the Yankees do that?  They can hire the Cards’ SS, and for the cost of 1 win they pocket $9 million!  They can then use that to buy a lot more than 1 win elsewhere.  So in the end, the cost of a win will be the same for all teams.

Does that mean Pitt and KC are actually taking a loss when they hire FAs?  Do they pay more than MRP?  It’s possible, I don’t know.  Again, this is exactly where I’d like a smart economist to help us out. 

*

“One of Dr. Fort’s best and most reasonable statements is that the opportunity cost of using a slightly better measure of MP might outweigh the benefits of using it.”

Yes and no.  Sure, marginal improvements in the model may not be worth the trouble. But the standard model in his field is the Scully model, which measures the value of players against not a replacement player, but NO ONE PLAYING AT ALL.  This is how Scully, and Bradbury, came up with some wacked player valuations.  So if that’s the standard, then WAR offers a quantum leap forward.

Right (#27), if Voros is correct, then either teams like KC and FLO are spending WAY too little or teams like Boston and the Yankees are spending WAY too much.  That makes little sense to me.

I seem to vaguely remember that in either one of the THT Annuals, the Wages of Wins book, or the BP book (I forgot the name - not the annual), or some other publication, that they got the value of a win being very different among teams, exactly the opposite of Voros, but I could be misremebering.

I don’t understand this:

“The Yankees could afford to pay $5M/WAR for a 5-WAR SS.  But then let’s say a 4-WAR SS goes for just $4M/WAR to St Louis.  Why should the Yankees do that?  They can hire the Cards’ SS, and for the cost of 1 win they pocket $9 million!”

If the Yankees can afford 5mm per win and the Cardinals only 4mm per win, then if a player is available to both teams, and both teams want him, then the Yankees will get him at something more than 5mm per win.  As I said before, eventually the teams who can afford higher $ per in will fill up their rosters (they will get ANY player they want) and then, and only then, other teams will get players at less $ per win.  Teams at the bottom of the heap, who can only afford, say, 1 or 2 mm per in, will get the last group of players, after all the other teams’ rosters are filled.  In reality that is not how it happens (because baseball teams are not the epitome of the efficient business and they are not all in it to maximize profits), but IF teams each had their own $ per win value, above which they lose money, then that is exactly how it SHOULD happen.

Hi Guy,

I agree with what you’re saying. I didn’t mention WAR by name because I was simply agreeing with the principal. And the principal suggests, to me at least, that if WAR and its assumptions work and hew closely to what we observe happening in the MLB labor market, isn’t it equally valid to ask what can be gained by using a more rigorous and time consuming model like he proposes?

Suppose there are shortstops X, Y, and Z, with WARs of 4, 2, and 0, respectively.  Teams A, B, and C value wins at $15, $10, and $5, respectively.  Every team needs exactly one shortstop at that rate.

What happens?  The bidding limits for the three guys are:

Player X: A $60, B $40, C $20
Player Y: A $30, B $20, C $10
Player Z: A $00, B $00, C $00

Suppose bidding is simultaneous.  You’d think,

Team C will bid $20, $10, and $0.
Team B will bid $40, $20, and $0.
Team A will bid $60, $40, and $0.

But team A realizes that it needs only to bid $1 more than the next highest possible bid.  So

Team C will bid $20, $10, and $0.
Team B will bid $40, $20, and $0.
Team A will bid $41, $21, and $0.

Now, B knows it can’t beat A on the player A wants.  So it needs only outbid C.  So it bids $1 more than C:

Team C will bid $20, $10, and $0.
Team B will bid $21, $11, and $0.
Team A will bid $41, $21, and $0.

A knows that B is thinking this, and now A knows that it need only bid $1 more than B’s updated bid:

Team C will bid $20, $10, and $0.
Team B will bid $21, $11, and $0.
Team A will bid $22, $12, and $0.

So A gets the best player for $22, B gets the next best for $11, and C gets the worst for $0.

Now, the increments are really a penny rather than a dollar, so

A gets the best for $20.02
B gets the next best for $10.01
C gets the worst for $0

All three teams pay $5 per WAR!

That is: every team in MLB pays the same price, not because WAR are worth the same to them, but because it’s the worst team’s value of WAR that sets everybody’s price.

It doesn’t really work that way in real life—our conditions are pretty special (one SS per team, last SS worth zero).  But, as an approximation,

1.  All teams pay roughly the same $ per win
2.  That $ per win is probably closer to that of the bottom team than the top team
3.  The teams with the higher $ per WAR tend to get the better players.

I suppose you can come up with an example where there might be a time where $/win gets bid up: only one SS, with two teams needing him badly.  If you only have team A and B arguing over the best SS, B will bid $30 and A will bid $30.01, getting him for $7.50 per win instead of $5. 

However: A has options!  It might be able to sign a 2B at $5/win and make a trade.  Even if it has to throw in $1 in cash, it still gets its SS for $6.

So, I think, on balance, all teams pay the same $/WAR, and it’s below what the average win is worth to the average team.

Another way to look at it:

Each of us has a different value we put on a new Toyota Camry that costs $30,000.  Mine is low, because I don’t like them—call it $10,000.  Others who actually buy one must have a figure in mind more than $30,000, or they wouldn’t get one at all.  There are some who probably like the car so much they’d have bought one even at $50,000.

Think of us all as teams, and a Camry as a win.  Even though each of us has a different $/win, everyone pays the same price for a Camry.

Same with wins.  The Yankees are more desperate for wins than anyone else, but they wind up paying the same as everyone else, just like Bill Gates pays about the same for the same Mercedes as you or I would.

The logic in the two cases isn’t exactly the same, because the supply of WAR is fixed (inelastic), but I think the result works out similarly.

Yep. If my econ knowledge hasn’t completely deserted me, what you’re talking about is consumer surplus and your example above at first assumes first degree price discrimination (or that a player can successfully charge a different price to a different buyer based on their willingness to pay). But, as someone pointed out earlier (Tango?) GMs have WAR now so MLBPA can’t rely on price discrimination any longer. The market is much more efficient and the Yankees get the benefit of the consumer surplus.

One more way to look at it:

Say everyone is a free agent at the same time, and so there are about 1,200 WAR available in the league any one season.

Isn’t that like 30 consumers facing 1,200 Cabbage Patch Dolls?  The Yankees earn $3 of happiness for each of the first 40 dolls, then $2 for the next 20, then $1 for the next 10.  The Royals earn 50 cents worth of happiness per doll up to, say, 20 dolls, then 40 cents for the next 20.  And so on.

Add up all those “quantity of dolls demanded at price X” curves, intersect it with the inelastic supply curve, and you get the equilibrium price per doll.  That’s the price all teams will pay.

That is: even though the Yankees may be willing to pay $10MM per win instead of (say) $4MM, they don’t have to.  There will always be a supply of players willing to sign for $4MM, just because the other teams aren’t willing to buy them all at that price.

Right. There are (roughly) three inputs into a player’s free agent “price”:

A) The value that other teams are willing to pay;
B) The player’s reserve price (that is, the minimum amount he is willing to accept); and
C) The amount that player’s production is worth to the team in question.

Hypothetically, every team’s free agent offer should be:

(A|B)

Of course, there is imperfect information about all three, and so sometimes teams offer more than they “should” (or at least, more than they would if they had perfect information about one, two or all three of those factors).

Let’s try that formula again, shall we?

(A|B)<Offer<Worth

Colin/35: Agreed, even though your “less than” signs don’t come out in the post. 

I’d go one step further and say that a free agent’s price should be exactly:

(His projected wins) * (the 30 teams’ common equilibrium price per win)

When the actual result varies from that, it’s because of one or more of:

1.  Teams’ or players’ projections vary
2.  Teams or players are not acting rationally
3.  Temporary imbalances in supply/demand for wins among the positions
4.  Idiosyncratic preferences (Mariners and Junior).

Two things to add:

1) The Yankees can only sign 3 Class A free agents a year. That means that even if they wanted to pick up 4 because it would improve their team, they can’t.

2) They only have 25 players on their team. That means that if they have a nice cost controlled player doing well, they won’t necessarily pay for a marginal improvement at market rates since the benefit may not outweigh their actual cost per win (even if they would if both could contribute).

Think of it in terms of a bench player. For the Yankees, the playing time is going to be much lower than the same player would get as a starter elsewhere. Let’s say it’s 25% of the playing time. So for the Yankees, who can afford $10 Million/win (let’s say), they’d value a 2 WAR bench player at $20 million x 25%, or $5 million. On the Royals, who can only afford $4 million/win, he would be a starter and worth $8 million, so they will still win.

The biggest limitations are in regard to the amount of talent that the Yankees can obtain in a year, and further limited by how much talent they can put on the field—they are limited by how many innings/outs each player can play.

This is totally different from how a business would normally work.

I think mfan’s post #21 describes the difference between salary and value perfectly and in terms understandable to economists because he studied under Dr. Fort. And then Phil uses those concepts in his wonderful example in post #31 of how they would work to create a constant value for WAR in the baseball free agency model.

OK, I believe you guys.  I learned something new today, which pleases me…

So for the Yankees, who can afford $10 Million/win (let’s say), they’d value a 2 WAR bench player at $20 million x 25%, or $5 million. On the Royals, who can only afford $4 million/win, he would be a starter and worth $8 million, so they will still win.

If that player were a free agent and the Yankees wanted to retain him presumably they would have to pay him 8 million a year or else he would sign with the Royals.  And they might be willing to do that as a form of insurance to be able to use that player full time as a replacement if one of their 5 WAR starting players got injured.

"as an approximation,
1.  All teams pay roughly the same $ per win
2.  That $ per win is probably closer to that of the bottom team than the top team
3.  The teams with the higher $ per WAR tend to get the better players.”

I think we all now agree on point #1.

I’m not sure that #2 is right.  In a normal market, it sounds right:  a firm should never pay more than MR.  But in a normal market, firms that couldn’t generate enough revenue to cover the market cost of labor would go out of business.  Here, small market teams might be willing to pay more than MR up to a point.  Even if FAs cost more than the immediate benefit, Pitt or KC might feel they have to make some effort to field a competitive team in order to prevent the league from taking away their revenue sharing of forcing them to sell the team.  Owning a franchise in this monopoly business may be valuable enough that it makes economic sense to take a loss on a few mid-range FAs.  So the market price for WAR could exceed MR for some franchises. (I think)

#3 is only true indirectly.  In a normal market, the higher MR teams should should end up with the most TOTAL value-- not necessarily with the best individual workers.  But in baseball, given constraints on PT and rosters, having the most total value means having the best individual players.

Guy/42:

Right, #2 doesn’t make sense ... all teams have a marginal win value of $4MM or whatever it is, so saying that it’s closer to the bottom than top doesn’t make sense. 

However: the value of the win *includes* the value of MLB not taking the team away.  If you want, you can argue that the first 20 wins above replacement (say, to 62-100) are very, very valuable—they keep MLB from taking action.  But that doesn’t matter.  It’s like how the first gallon of water a day keeps you alive, and is worth hundreds of dollars to you, but that doesn’t matter: you still pay only 1/10 of a cent, or whatever it costs.

Agreed on #3.  Since there are only 25 roster spots, the big-market teams wind up with the most value, and to do that, they have to sign the best players.  I wouldn’t call that “indirectly,” though, but I agree with what you mean: that it’s *possible* the Yankees will get their 60 WAR by buying 25 2.2 WAR guys instead of getting their 60 WAR by buying an assortment of 7- and 6- and 5- and 4- and ... 1-WAR guys.  That’s unlikely, though.

So for the Yankees, who can afford $10 Million/win (let’s say), they’d value a 2 WAR bench player at $20 million x 25%, or $5 million. On the Royals, who can only afford $4 million/win, he would be a starter and worth $8 million, so they will still win.

If that player were a free agent and the Yankees wanted to retain him presumably they would have to pay him 8 million a year or else he would sign with the Royals.  And they might be willing to do that as a form of insurance to be able to use that player full time as a replacement if one of their 5 WAR starting players got injured.

Sorry, my poor use of subjects (living in Japan too long?).

The Yankees would not be able to get that player, because he would be worth less to them (even at an increased value per win) than he would be to the Royals because of the limited opportunities that can be given to a team.

I realize there’s the insurance possibility, and it could be a consideration, but the point I’m trying to make is that the innings they can give players is limited. Therefore, once they hit the innings limit with superstars, they get decreasing gains from further investment, which is why they have a smaller incentive to spend additional money per win.

Not to mention the fact that because of trading, teams don’t want to have massive contracts that are hard to offload.

Phil/31,

I think your example would make an economist cringe here (though, when reading into it a little more, holds some insight if these things are not completely simultaneous):

“Team C will bid $20, $10, and $0.
Team B will bid $21, $11, and $0.
Team A will bid $22, $12, and $0.”

Under a standard competitive bidding situation, we would have Team B bid $23 for that top player, and so on. Why would they let him go for $22 when they value him at $40?  Because then Team A is out of the market for the second player, and bidding on Player 1 would reduce the surplus from just buying Player 2.  BUT if these things are simultaneous, no one is out of the market, and the best strategy is to bid the MRP (or total WAR $ value as you define for each team).  Or, if you know the next best bid, to bid $1 higher than their value, so that you do end up at $41, $21, $0 (or, the minimum I guess).

Otherwise, Team B can still get a $17 surplus on Player 1 (by bidding $23), rather than a $9 surplus on Player 2 at $11.  Simultaneously, Player 2 is also being bid on by Team 1 beyond $11 for the same reason, given the surplus for each team (they have a $28 surplus on Player 2 at $12 and $28 on Player 2 at $32, making them indifferent).  Now, Team B has a choice, they can bid on Player 2 at $13 for a $7 surplus, or Player 1 at $33 for a $7 surplus.  Team A will always outbid them for ONE of the two because they value them higher.  The same goes for Team A, but with larger profits. 

If either one holds off their bid as to keep the price from going up on both players, then we’re getting into an area of collusion (not impossible, but I think it’s fairly doubtful that’s happening, otherwise MLBPA has a strong case).

If things are not simultaneous, I think we’d still end up at $32, $11, and $0, if teams are forward looking.  Team B would drop out of the running for Player 1 when they know they can get at least the amount of surplus on Player 2 as they would competing for Player 1.  Since once Team A already has a SS, they’re no longer bidding, they can get Player 2 at $11, as you say.  But that means PLayer 1 goes for $32 (since at $31 they’d get a $9 surplus, but at $33 the $7 surplus is less than they’d get for Player 2).

I realize that using the word ‘cringe’ above might give a strong negative undertone.  Not meant in that way, Phil.  Just want to make it clear (and I’m not cringing). 

But I am curious as to how the progression happens with bidders in MLB.  Maybe it does happen the way you speak of it, as having 2 bids out for 2 SS may not be a good strategy (the ultimate winner’s curse).

I think Phil is right here, and it’s a fantastic example.

First off, before someone asserts that the Yanks/Sox are paying at a higher marginal $ per win compared to the Royals and Pirates, I’d like to see the evidence.

As it stands, the status quo is that bigmarket and smallmarket teams ARE paying at the same marginal $ per win, even if (a particular) economic theory says it shouldn’t happen.

Phil has shown an example of the mechanism that could take place, where there’s no collusion, and everyone prices wins the same.

***

I’m reminded of an experiment I just saw, whereby you had 10 men and 10 women.  Each man had a number on his forehead, as did each woman.  And the numbers were unique from 1 to 10.

The object was to pair up with the highest number you can find.  But, you don’t know what your own number is.  You infer it based on how much the opposite gender courts you.

In the end, every man paired up with a woman that was within 1 of his forehead number.

I see the same thing here, whereby not every team is going to try to bid up Figgins so that the Redsox will end up with him because their $ per win is higher.  There’s a courtship going on, and every team is not going to bid on every player such that you have one winner and 29 losers for each player. 

I’d say it’s more like a dutch auction, whereby you want to pay the same as everyone else for the $ per win, even if for you, you will value those wins more.  It’s not necessary that you get Figgins if you can get Scutaro or Beltre or Polanco.  You do that because that leaves you with more money in your pocket that you can spend on more players.

Anyway, the point is that we see the results of the market, and so, you need to have an economic theory that models the reality.

Barry Zito went to SF, not NY.  Hampton went to Rox, not the Mets.  Joe Mauer could have waited, but he signs with the Twins.  Greg Maddux stays with the Braves, instead of Yanks.  Cabrera with the Tigers.  ARod with the Rangers, not Mets.

The only reason that the Yanks end up with a disproportionate number of big contracts is because they have more money, not because they value those players more.

And teams are not in the business of “investing” on players that are going to give them more money than they are used to.  For example, Carlos Beltran was fairly paid at 7/119.  Any team could have paid that, and they would have gotten their money’s worth.  But, to most teams, that’s an investment that’s disproportionate to their CURRENT revenue stream. 

The Mets had a high (past) revenue stream, have a high (current) revenue stream, and so, Beltran is a portion of that stream.  Even if they are going to get a higher revenue stream that matches his performance.

To the Astros, they have a lower revenue stream, and even if they wanted him, they’d compare his future salary obligations to their past revenue stream, and say that it’s out-of-whack.  They are risk averse.

The weird part is say the Royals that give up on Beltran.  They don’t factor in that they will lose revenue when Beltran is gone.  They hope/plan that they can replace his wins with cheap pre-freeagent players.

Sorry, not dutch, but reverse-dutch (I think it’s called).  The way Google did their IPO.  They have one million shares outstanding, and everyone bids from say 1$ to 300$ for a total of 10 million shares, and the price is the highest price at which all shares are bought.  If that price is 100$, then everyone gets it at 100$, even if there were people who wanted it at 300$.

Tango,

I’m glad you bring up the total revenue argument, as I think I tried to go through an example with it before.

In each city, it’s likely that there is a win total at which profits are maximized, rather than spending at a per-win basis by player.  Teams go about buying talent to reach this threshold, but further spending isn’t useful, since at the margin, teams at different places on their revenue/profit function could end up valuing the talent higher. 

For example, with Barry Zito, the amount of revenue he would have added to the Yankees isn’t necessarily a linear function of the wins he would provide for them.  It could be that where the Giants were at the time, their marginal value to a win, given their current talent level, was higher than the Yankees (steeper revenue slope for Wins at that point of the function).  Given this alternating demand for wins between large and small market teams, we don’t see a difference in the $ per win as a whole.  Just a thought, not sure if that’s right.

While not a perfect model, this paper comes to mind:

http://findarticles.com/p/articles/mi_hb5823/is_2_25/ai_n29418493/

I’m still trying to decipher everything the authors are trying to say.  The Burger/Walters paper they discuss has some estimates as to the ‘value’ of a win to each team, though I think the relative values are a bit much.  That’s just a surface view of everything though, I can’t defend it.

Millsy/45:

I think you’re right, the argument needs another qualification for it to work. 

Your argument is this: If team B waits for team A to get the best player, it can get the second best player and a surplus of $10.  But if it bids higher for the best player—say, $23—it can get a surplus of $17.  So why wouldn’t it bid for the best player? 

If there’s only one player, then, fair enough.  But what if there is an unlimited number of auctions?  That is, suppose after shortstops, the same thing happens for second basemen, and then catchers?

The team has two choices for the shortstops:

-- Bid $23 for the best player and earn a surplus of $17
-- Bid $10 for the second-best player and earn a surplus of $10

The second player is a better investment: it earns 100%.  The best player earns only 74%.

If a team has $23 to invest, it’s better off getting two $20 players for $10 (with $3 left over) than a $40 player for $23.  The first way, it has a surplus of 20.  The second way, it has a surplus of $17.

So it’ll concentrate on bidding for second-best players, which are the best investments under its circumstances.

But Why won’t it spend $43 for all three players?  Isn’t the best shortstop a good investment too?  Eventually not.  You have to assume, don’t you, that the marginal value of talent is declining as the team acquires more and more talent?  While it is still at the point where the surplus exceeds the cost, it will choose to buy the players who are the best investment—which are the second-best players.

So the team will try to buy a bunch of $20 players at $10, rather than a bunch of $40 players at $23.

Does that fix the problem? 

P.S. Cringe away!  If an argument is wrong, it’s OK to cringe. 

Right, I think that’s what Phil, or someone, was getting at.  That given the current situation, every team’s $ per win is the same.  But if say the Yanks were to decide to dump all their stars, and have a team that wins 60 games a year, then their fans may leave in droves.  And so, to get to 85 wins, they have to pay say at double the $ per win we are used to.  And then, after that point, they pay the standard rate.

However, I would bet that teams really have little idea what it means to add Carlos Beltran to their team that would be different from the league.  First off, if Beltran is a 5 WAR player, he might add 3 to 7 wins.  Furthermore, even if the Mets are a 90-win team, they can win by luck 80 to 100. 

There’s so much noise all over the place that I doubt teams are thinking that they can spend more efficiently (different from the standard).  What if the Pirates believe they are an 80-win team as opposed to say a 70-win team?

The only place it’s different is the sweetspot of the playoffs, and you see the cost there based on the quality of prospects that teams in contention give up at the trading deadline.

Phil/50,

Hmmm.  You might be right that the 2 players are a better return in some way(assuming they have the same marginal value if on the same team...say we’re just moving one to 2B or something). 

But I think your point applies to both teams, right?  The margin for the crappier player is also larger for the richer team, which means they might have the same logic, and go for the two players at the cheaper price.  In the case of buying two players, the margin to buy the 2 (now $12) guys that the rich team values at $40 (but middle team values at $20) would be $28 + $28 = $56 surplus, and an even larger percentage return than the 100% that the middle-level team is getting.

Ultimately, I think both get bid up, though I don’t think it’s necessarily MRP=Pay, depending on how the bidding process takes place.  It seems that it could be some variant of who’s involved and how many players are available with significant value.

Tango/51,

Definitely was throwing out uncertainty there.  When talking about baseball, you’re correct about the significant uncertainty of what would happen.

But in general, that’s my hypothesis as to why teams mostly pay the same rate (one that could be very wrong).  They’re all right at their margin in terms of overall revenue/profits, and rather than pay addition $$ for wins, they buy players that come packaged with more wins...or less wins.  So the Yankees don’t bother with Barry Zito and his 5 wins (making that up, given he was good at the time of signing), they just pay that same going rate for a 1-win player.  In terms of accuracy, uncertainty, and efficiency...well I leave that up to someone who knows a lot more about measuring talent than I do.

Millsy/52:

The rich team values the second best guy at $30, not $40, so it’s indifferent to whether it gets $60 guys at $20 or $30 guys at $10.

I think Tango/Phil are pretty much wrong here, and Millsy is correct. The ROI argument Phil/50 makes is silly: If the signing has surplus value (in corporate finance, we’d say it has a positive NPV), it’s a good deal for the team regardless of its ROI. Unless the team is cash strapped and therefore must stick to a set budget, they’re going to want to make any signing that will result in a profit (i.e. surplus value, i.e. positive NPV). So ROI is not important. In fact, I’m almost positive I’ve seen Tango make this argument a hundred times, which is why I’m really surprised by Tango/51. The point about the Yankees in Tango/51 is also off: The Yankees will spend the same on players regardless of their win level, except that they may spend LESS (not more) if they have a really crappy team, since those teams will be worth less to them.

What actually happens is that teams do end up paying about the same $/win, but that number tends to correspond to the value a win presents for the teams that derive the most (actually second-most) value from an additional win, not the least (as in Phil’s example). Let’s say there’s a free agent who would miraculously be a great fit for every team (maybe we’ve made all catchers first basemen and so every team is going after Joe Mauer). The Yankees derive the most value from that player, and they will end up getting him. The amount they end up paying will be slightly more than the highest offer that player could have received from any other team (probably the Red Sox). So in this case, the $/win paid in the free agent market will be slightly higher than the $/win number for the Red Sox. If the Yankees were not in the market for that player, the Red Sox would get him and the $/win would be slightly higher than what, say, the Cubs would offer. And so forth.

This is why top-flight free agents tend to go to the large market teams, which tend to be the teams that derive the most value from them (according to my research). And since the $/win number is heavily influenced by the biggest contracts, this is why the overall $/win we find is close to what the biggest teams get from adding an extra win (again, based on my research). That number is around $4-$6 million.

David/50:

One-pound bags of apples: $1 each.
Two-pound bag of apples: $2.25.
You value apples at $2 a pound. 

The one-pound bag gives you a surplus of $1.  The two-pound bag gives you a surplus of $1.75.

Do you buy the two-pound bag because it has the bigger surplus?  Of course not!  You buy the two one-pound bags.  That way you get a $2 surplus instead of only $1.75

Now, as you say, since both bags offer a surplus, why not buy both bags?  Sure!  Except that you only value apples at $2 a pound for the first two pounds.  The third pound makes you throw up and is worth 50 cents to you to freeze for later.  Since both bags of apples cost more than 50 cents a pound, you stop at two pounds.

So, under those circumstances, you wind up caring about your ROI, you buy the two one-pound bags instead of the two-pound bag.

David/50:

As I said in #43, I take back my guess that the price per win is calibrated more to the smaller teams than the bigger teams.

What I should have said (and I think this still disagrees with you) is that, at the margin, all teams value a win at the same price.  If the Yankees buy 60 WAR (for a 100 win projection) and the Marlins buy 20 (for a 60 win projection), the Yankees value the 101st win at the same price that the Marlins value the 61st win.  If that wasn’t the case, one team would buy a win from the other (or would have let the other grab the last 1 WAR free agent).

I now think it makes no sense to say that the value of a win corresponds more to the Yankees’ value or the Marlins’ value without further clarification of what that means.

This is why top-flight free agents tend to go to the large market teams, which tend to be the teams that derive the most value from them (according to my research).

Are you saying that the dollars per win is being paid more by the large market teams?

***

In my case, I am saying the free agents distribution is based purely on the budget amount minus salary obligations. AND, the budget amount is preset based on current revenue streams. 

So, if Joe Mauer were to present himself to the Royals or Yankees or Redsox or Reds, he’d have a similar effect in terms of extra revenue.  But, the Royals and Reds won’t budget for that.  They won’t increase their revenue estimate from 120MM to 150MM and thereby justify the 25MM salary for him.  What they might do is bump up the revenue estimate by 5MM or 10MM.  And so, they don’t have enough money in the budget.

It’s simply impossible for a team to keep going back to the owner and say: “Listen, Player A is worth 1 win more than Player B, and we can get him for only 2MM$ more.  You need to increase the budget by 2MM.” It’s just not going to happen.  Maybe once, or twice, but they can’t keep going back to the owner saying that.

This is how corporate america works for the most part.  There is very little in terms of huge investments (which is what a Mauer signing would be).  Teams and companies much prefer something predictable, and investments based on what they are used to investing.

For the Yankees, it’s different.  They lose a free agent, they replace him with another free agent.  They’ve got a revenue stream pattern that is predictable.  The owner is not interested in seeing what might theoretically happen if you add Joe Mauer to your team.

Some teams though are happy to go against this pattern.  Say like the Cardinals. 

But in any case, the $ per win is constant.  I think we all agree to this.

The question seems to be what would happen if the status quo was shaken for a loop, if the Rays were all of a sudden granted a 50MM$ increase in their budget, or the Redsox had to cut their budget by 60MM$.  What would happen to the marketplace if that were to happen, and how would that affect the $ per win.

Phil/55,

The issue is that you care about is not ROI but NPV (or surplus). So in your case, the more expensive player had a higher NPV, and so that would be who you acquire. The issue with your analysis was that you assumed they only had $23 to spend, but unless they are cash strapped, that is not the case. Think about it this way: What if each player was paid exactly half of what he was worth to you. Who would you want in your lineup? Nine $10 million guys being paid $5 million, or nine $20 million guys being paid $10 million? The ROI is the same, but obviously you would prefer the latter. The number to pay attention to is NPV, not ROI.

As for Phil/56, yes at the margin every team values a win at the same price (with some constraints imposed by the fact that there are only so many plate appearances and innings to go around). However, remember that at the beginning of free agency, we are not at the margin. This is why all the large market teams go out and buy top free agents—because those free agents are worth more to them than anyone else. And so when we add up the number of dollars spent in the free agent market and the number of wins purchased, because those top free agents make up such a large share of all the dollars and wins handed out, we end up with an overall $/win number that is pretty close to the value of a win for a large market team. To put it a different way, I bet if you looked at expensive free agents who signed early and cheap free agents who signed late, you would find that the $/win paid to the first group would be much greater than the $/win paid to the latter. This is a great study for someone to undertake, by the way.

Tango, the problem is thinking about this in terms of budgets, which I don’t think is a very good framework at all (except for maybe a few teams like the Dodgers at the moment). Team payrolls can and do evolve—when the Brewers went from a perennial loser to a potential contender, they increased their payroll to get a few much-needed wins, while the Padres have on other hand slashed payroll as the quality of their team declined. So forget about budgets. If Joe Mauer is worth $25 million to the Royals, they’ll spend $25 million on him—in fact, that’s what the Twins just did! The Twins, by the way, are another great example. For years, they had a poor team and low payroll. But as their team improved, their payroll expanded and it’s now well above-average. So stop thinking about budgets—teams will invest in positive-NPV projects (i.e. players who cost less than they bring in), while declining to pay players who cost more than they can bring in. The reason the Rays don’t add $50 million to their payroll, for example, is because their fans don’t appear to be very responsive to winning, and therefore it makes little sense to pay much more than $1 million/win. The reason the Red Sox don’t cut payroll by $50 million is because their fans ARE responsive to winning, and if the Sox win less they’ll lose a lot of money.

I agree, David, that the budgets certainly evolve.  Take a look at any franchise, Seattle, Cleveland, etc. and you’ll see this happen.  I think there’s a causation problem, as the article I cited above talks about with a simultaneous equations model.  Untangling the direction of causation with wins and demand for baseball is a mess, though the causation definitely goes both ways (wins bring more people to the park, while more people in a market provide more revenue for more wins).

However, I think the overall payroll is a long run choice, based on the elasticity of the market to winning.  I think you’re kind of saying this as well.  I’m thinking that you and Tango and I are saying almost the same thing here, outside of the ROI argument.  I’m not sure on that one, to be honest.  I suspect it’s simple positive NPV, but there is definitely an opportunity cost to not spending the money elsewhere (perhaps even in non-baseball investments).

Given the price of talent, the Yankees maximize their profits/revenue at, say, 93 wins, while at that same price the Royals maximize their profits at only 65 wins.  This long run choice makes the budgeting idea feasible, at least within a 3 year span.  Seeing the payroll costs as fixed, with short term adjustments (i.e. buying talent at the margin to get to 93, or 65, wins).

As for the ROI mattering, I think I’d agree with you in general.

Budgets: well, I think you are wrong about that as much you think I am wrong.  That’s simply not the way it works.

***

This is why all the large market teams go out and buy top free agents—because those free agents are worth more to them than anyone else. And so when we add up the number of dollars spent in the free agent market and the number of wins purchased, because those top free agents make up such a large share of all the dollars and wins handed out, we end up with an overall $/win number that is pretty close to the value of a win for a large market team.

I dispute this assertion altogether.  The large market teams pay as much per win as the smaller ones.  And if you were to split up the $ per win by market size, you’ll still get the same $ per win.  And by date as well (up to a certain point, say a week or two before spring training).

I can say this because every time I look at each offseason signing, they are almost all so unsurprising.

The only split I have seen is the one-year v multi-year deals.  No other split (bias).

To put it a different way, I bet if you looked at expensive free agents who signed early and cheap free agents who signed late, you would find that the $/win paid to the first group would be much greater than the $/win paid to the latter. This is a great study for someone to undertake, by the way.

Other than the bias I noted, I say you are wrong here.

Bay and Holliday signed late as a fer instance.

Yes, budgets evolve.  Evolve, as opposed to just jump up.  And they are much more likely to evolve to retain existing players (like Mauer) than they are to bring in new players.  That’s because the owners have a comfort knowing that they have Mauer as opposed to Chase Utley, even if they are equivalent.  That is, if Mauer were to be a free agent and Utley is a free agent, an owner is much more likely to increase his budget to maintain Mauer on the team, than to increase his budget by the same amount to lose Mauer and bring in Utley.

Tango/62,

I’d be interested in seeing the data on that.  Not because I don’t believe it, but because it might indicate the hometown value of a superstar.  Have you looked at the claim empirically?

Millsy: just anectodal.

Tango/62,

Two issues: (1) These two things are inextricably linked. Yes, the Twins paid Mauer and Morneau a lot of money, but these are also the players that made them contenders and moved them up the win curve, therefore making it worthwhile for the Twins to make bigger investments. It’s like the chicken and the egg—you can’t separate the two. And (2), it’s not just that teams spend big to keep their superstars while keeping spending on everyone else constant. Again, look at the Twins. Their payroll this year is $97 million (IIRC), and that’s without the Mauer extension! Your argument about budgets makes no sense in terms of economic theory (which does not make it wrong, by the way), and it also is contradicted by empirical examples (which in conjunction with what theoretically makes sense does mean that the burden of proof lies with you, not me).

This is the Twins payroll relative to league average from 2003-09:

yearID payrollIndex
2009 74%
2008 64%
2007 87%
2006 82%
2005 77%
2004 78%
2003 78%

In 2010, there was a jump.  According to USA Today Salary DB, the jump was 32MM$.

15MM$ of that was on returning players. 

On top of which, their budget likely increased because of their new stadium.  This is precisely my point!  If they were still at the Metrodome, their budget would not have increased.  I haven’t kept up, but can I guess that their season ticket sales have jumped up quite a bit?  Did it go up by, I dunno, 10,000 extra tickets per game?  If a season ticket package is 2000$ or 3000$, well, there you go.  That’s 30MM$.

And, they spend the extra 17MM on Pavano, Hardy, and Hudson.

I don’t get what your point is supposed to be regarding the Twins.

sports.espn.go.com/mlb/news/story?id=5041531
http://www.startribune.com/sports/twins/78928167.html

They sold 24,000 this year, up from 11,000 last year.  And I presume they are more expensive this year.

So, if you make them 2500$ a seat this year and 2000$ a seat last year, the difference is 38MM$.

I don’t see anyone giving me any reason to go against the budget idea.

If, as I am arguing, winning leads teams to boost payroll, we should find a relationship between winning more games and increasing payroll. Now unfortunately, we cannot simply ask whether winning in one season is correlated with increasing payroll in the next because we essentially want to remove team quality from the equation. Tango/62 agrees that good teams will see their payroll go up, but he argues that this is due to teams retaining their players rather than bumping their budgets to acquire outside players.

What we need is an instrument, and I think I have quite a clever one. Specifically, we can remove team quality from the equation if, instead of looking at the correlation between a team’s record in one year and the change in its payroll the next, we focus on the difference between its actual and Pythagorean records, with the Pyth standing in as an indicator of the team’s true talent, and the change in its payroll the next year. If Tango is right, there should be no correlation; however, if I am correct, then teams who outperform their Pythagorean will tend to boost their payroll, and that boost will have everything to do with their record and nothing to do with the talent that is already there.

So what are the results? Well, I took every team since 1998 and found its payroll index (i.e. a team with a payroll 10% above the league average will have a payroll index of 110), as well as its actual and Pythagorean record. Then, I looked at how their payroll index changed in the next year and regressed that against their Pythagorean variance. The results? For each “extra win” (i.e. one more win than the team’s Pythagorean record it should have had), teams boost their payroll index by 0.3 points the next season (p = .20). That’s not quite at the usual thresholds of statistical significance, but I think it’s significant enough.

After all, economic theory tells us teams should spend whatever amount maximizes their profits, rather than sticking to any sort of strict budget. So if teams think they are higher up on their “win curve” than is actually the case, they will tend to spend more money to try to make it to the playoffs, and if they think they are lower down they will spend less. My results seem to confirm that. I think a more rigorous study with proper adjustments would show a stronger result than what I was able to find in 30 minutes.

David/68,

See the one I link to above.  They get at the idea that you are getting at as well with a simultaneous equations model.  I wouldn’t doubt there are short term adjustments made to payroll, but I think in general there is a fixed payroll budget for a team in a given year, or couple year span.

Perhaps the team was seeing revenues increase to begin with, and they therefore increased their budget to go toward payroll.  It could be that there’s just more of a taste (more people, more money, more baseball fans) for winning each year for whatever reason, that would give you what you find.  Though, I’m not averse to the idea that a few good years in a row might indicate further investment into payroll.

I am arguing that if the teams make money, they will then turn around and spend it on players.  That is, they have cash in hand.

What teams will not do is spend beyond their budgets (investment) on extra wins that they aren’t used to, on the idea the the dollars will follow these new wins.

The process is:
1. spend to your budget
2. if you win more than your payroll deserved, you make more money
3. that money is spent on players

What teams won’t do is spend beyond their budget, on the theory that the extra wins are so theoretical that they can’t really be counted on as “real” wins.

Teams are happy to keep spending on existing players, because those wins seem more real to them.

Has anyone here calculated $/win for A-Rod’s new contract? That one might indicate if Yankees have higher valuation of win (even though they usually pay the same as other teams).

Do you think it happens that a team has an 90MM$ payroll, on the expectation that they will win 81 games, they instead win 71 games, they actually now have a new true talent level of 73 games, the payroll is still 90MM$, they expect revenues to go down because the true talent level has gone down by 8 wins, and then they will increase their payroll by 32MM$ to purchase 8 new wins to put them back at 81 wins?  No.

Do you think it happens that a team has an 90MM$ payroll, on the expectation that they will win 81 games, they instead win 91 games (and attendance goes up, with big profit), they actually now have a new true talent level of 89 games, the payroll is still 90MM$, they expect revenues to continue go up because the true talent level has gone up by 8 wins, and then they will increase their payroll by 12MM$ to purchase 3 new wins to put them at 92 wins? Yes.

The money spent on players is based on the planned budget.  And the planned budget will not give 4MM$ per marginal win, over and above what the team already is set at.  Maybe the planned budget will be 1MM$ per marginal win, over and above what they are used to.

It’s all budget-based, in a risk-averse environment.

"However, remember that at the beginning of free agency, we are not at the margin. This is why all the large market teams go out and buy top free agents—because those free agents are worth more to them than anyone else....I bet if you looked at expensive free agents who signed early and cheap free agents who signed late, you would find that the $/win paid to the first group would be much greater than the $/win paid to the latter.”

I would be shocked if this were true, David.  Your model only works in a world of almost zero transparency, and zero historical knowledge.  If the Yankees know nothing expect their own MR/WAR, then they might bid players up close to that level.  But that’s not how the world works.  The Yankees know what the market price of a WAR is.  They know that if other teams want to pay an above-market rate for top FAs in November, they can wait and get bargains later in the winter.  And if it’s a weak FA crop that year, they can fill their needs in the trade market, which reflect the same market price for WAR.  Moreover, every other team also knows what the market price is.  So they won’t make the mistake of overpaying in the first place. 

This notion that the market somehow starts from scratch each October, with rich teams blindly overpaying for talent, doesn’t make theoretical sense nor does it reflect what we actually see.

*

“And since the $/win number is heavily influenced by the biggest contracts, this is why the overall $/win we find is close to what the biggest teams get from adding an extra win (again, based on my research).”

I’m not sure how the market price relates to the range of team MRs.  Perhaps it’s the average, maybe it’s lower than that.  But I can’t see how it can be true that it’s set by the Red Sox.  That would mean that 28 teams are losing money every single time they buy a free agent.  Why would they do that?

I’m not sure how the market price relates to the range of team MRs.  Perhaps it’s the average, maybe it’s lower than that.  But I can’t see how it can be true that it’s set by the Red Sox.  That would mean that 28 teams are losing money every single time they buy a free agent.  Why would they do that?

Actually, that isn’t as stupid as it sounds at first. Let’s not forget that teams make great profit from non-FA players, revenue sharing and centralized revenue. But they have to have certain quality not to alienate fans, as others have said. If another couple of signings can increase win value from $2.5/win (fans think team ha no chance and isn’t even trying) to $3/win (team is at least trying), and your existing lineup is paid $2/win, it is worth paying another 20% of your payroll to players who cost $4/win, because that increases value of cost-controlled part of the team and increases profit.

"Actually, that isn’t as stupid as it sounds at first. Let’s not forget that teams make great profit from non-FA players, revenue sharing and centralized revenue. But they have to have certain quality not to alienate fans, as others have said.”

That was me (#42).  So I agree, to a point.  But overpaying for minimal respectability, to maintain revenue sharing, is one thing.  David’s argument is that 28 teams are taking a loss on every single dollar they spend on FAs.  That strikes me as unlikely. 

And anyway, the Red Sox and Yankees can only buy so many FAs.  If they really were massively overpaying (compared to what other teams could pay), then poorer teams would just wait until Sox were done buying and prices would then fall.

“If another couple of signings can increase win value from $2.5/win...to $3/win...and your existing lineup is paid $2/win, it is worth paying another 20% of your payroll to players who cost $4/win, because that increases value of cost-controlled part of the team and increases profit.”

As wins increase, it’s the revenue on each extra win that rises.  The prior wins don’t then generate more revenue.  If it worked this way, it would always make sense to spend more.

My guess is that teams do in fact generally spend up to point where adding payroll produces no additional profit (or less), with the proviso that some may be excessively conservative as Tango suggests.  For the Yankees, that’s $200M; for small market teams, far less.

Guy,

It’s not 28 teams—probably more like 20. Based on my research, here is the list of teams for which the most valuable win is worth at least $3.5 million (this is just about the least strict way to look at things):

Yankees
Red Sox
Cubs
Dodgers
Phillies

That’s five teams. Now my research is not finished, so I absolutely do not regard it as definitive in any way. Importantly, I have not attempted modeling TV revenues (though I do include concessions, merchandising, etc.). But at current free agent prices (around $4.5 million/win), the cost/benefit analysis only makes sense for the Yankees, Red Sox, and Cubs by my numbers. I have a hard time believing that adding in TV revenues as well as improving my model would add more than 7 other teams (since, remember, TV contracts are negotiated fairly rarely—obviously teams with RSNs will be more sensitive, and indeed those are the teams which I would expect to be added to this list). Most teams spend money on free agents because (a) It keeps the current system in place, where teams can drastically underpay young players, and (b) Owners surely derive non-monetary utility from having a winning team. But purely financially, it is almost certainly not worth it for most teams to spend the going rate for free agents.

And by the way, though my work is unpublished, you can look at Vince Gennaro and Nate Silver’s work to get a similar picture.

David:  If you’re saying FAs are a break-even or better proposition for 12 teams, then I agree that’s plausible.  That essentially means the market price = average MRP.  I just have a hard time believing market = Bosox. 

But I remain skeptical that the price of a win is significantly higher:
1) for rich teams than for poor teams;
2) in November than in January;
3) for great players than for other players.
If I understand your argument, all three should be true to some extent.  Right?

76

Cubs - how are their wins worth the most(tied) at the margins? The belief in Chicago is that it is the opposite, and that adding wins adds no revenue just expenses, and the team has been happy to sit at the level they are at, since it is the most profitable.

Have a ballpack that attracts fans, a good tv deal, a team that contends for the playoffs, spend enough for your fans to get excited and not give you grief.

But dont spend Yankee money, because winning 95 games instead of 85 doesnt generate the revenue to justify it.

Guy/77,

I think #2 may be wrong; I’d have to think about it a little more. #1 and #3, yes, I stand by. As for market price, I am convinced that it is a losing proposition for all but a few teams (maybe its better to draw the line at, say, the Phillies than the Red Sox).

dq/77,

Cubs fans are not insensitive to their team’s record--actually, based on my research, they are about average in that regard. However, because ticket (and concessions, merchandise, etc.) prices at Wrigley are sky-high, the Cubs gain a lot more revenue per fan than most other teams. Add the two factors together, and the Cubs are right behind the Red Sox and Yankees in terms of the value of a marginal win.

Actually, I’m not sure about (1) and (3), either. I have to think about this a little harder.

The Cubs won 89/79, and 66 games 2004-2006 and averaged 38,660 38,272 and 38,558
fans

THey expanded the capacity and won 85/97/83 games 2007-2009 and averaged 40,154 40,743 and 39,611
fans.

They won 90 games in 98 and drew 31,990, The next 2 years they won 67 and drew 34,739 and then won 65 and drew 34,438

If the Cubs win 76 games a year they will draw 39,000 fans. If they win 96 they will draw 40,000

As wins increase, it’s the revenue on each extra win that rises.  The prior wins don’t then generate more revenue.  If it worked this way, it would always make sense to spend more.

My guess is that teams do in fact generally spend up to point where adding payroll produces no additional profit (or less), with the proviso that some may be excessively conservative as Tango suggests.  For the Yankees, that’s $200M; for small market teams, far less.

I’m saying that there is a point where additional 5 - 6 wins mean difference between hope and complete apathy or anger from fans. As I said, for some, it may be evading 100 losses, for others being in the hunt for playoffs. I gave the wrong example, way overestimated the effect, the real one would be something like a team that has average value of $3M/win, but 5-6 wins which may bring them into hunt for 70 wins may be worth $5M/win. So, even though $4M/win player is too expensive, on average, it is worth buying him if team is just under that sweet spot.
Most teams have two such spots: one where they convince their fans that they are serious/moving into right direction, and the other when those victories mean serious run at playoff spot.

In a weird way I think Tango and David are saying pretty much the same thing. Tango is saying:

am arguing that if the teams make money, they will then turn around and spend it on players.  That is, they have cash in hand.

What teams will not do is spend beyond their budgets (investment) on extra wins that they aren’t used to, on the idea the the dollars will follow these new wins.

The process is:
1. spend to your budget
2. if you win more than your payroll deserved, you make more money
3. that money is spent on players

David on the other hand is saying:

Team payrolls can and do evolve—when the Brewers went from a perennial loser to a potential contender, they increased their payroll to get a few much-needed wins, while the Padres have on other hand slashed payroll as the quality of their team declined. So forget about budgets. If Joe Mauer is worth $25 million to the Royals, they’ll spend $25 million on him—in fact, that’s what the Twins just did! The Twins, by the way, are another great example. For years, they had a poor team and low payroll. But as their team improved, their payroll expanded and it’s now well above-average. So stop thinking about budgets—teams will invest in positive-NPV projects (i.e. players who cost less than they bring in), while declining to pay players who cost more than they can bring in.

But aren’t these two viewpoints essentially the same?

David is saying that as teams do better they expand payroll because they are close to the 90 win point and the next marginal win is worth a lot more in revenue than it is to say a 70 win team. And as I see it Tango is saying exactly the same - if teams have extra cash because of doing well they’ll spend it on players. Same argument.

David’s little experiment by looking at pythag records confirms BOTH points of view. Teams who outperform believe they are better than they are so they invest in more players. However, because they outperformed their pythag they’ll also get more revenue that year (this becomes more pronounced if they are close to the playoffs), which (as per Tango) they’ll (more likely to) spend on players the following year.

I don’t particularly want to rehash the whole debate again but I think what is incontestable is that the $/win is set by the higher revenue teams because they can afford to spend more. I think the debate is at what level is this set.

On this point I suspect David is right. I think it is driven by the top 5-8 teams. That makes sense right?

I wonder if one way to think about this is as two separate markets. There are the elite free agents who are looking at long term deals and then there are the older FA who will probably sign for a year or two.

For the first market there will only be a select few clubs bidding for these contracts as they are the only ones that can really afford them. Sure, every so often a Texas Rangers signs A-Rod but the only way they can do that is by offering more $$$ than the big boys. Therefore it stands to reason it is the top teams (ie those with higher MR) that are setting the $/win price. The smaller teams haven’t really got the payroll capacity to lock into really long term contracts. And typically when they do the outcome is bad (Helton, Hampton, A-Rod etc.)

In the second market it is a bit different as all 30 teams are in the market for these players. Intuitively you’d expect the $/win here to be a little lower as the big market teams soak up wins with expensive free agent contracts. However there are a few reasons why these $/win numbers should converge:

1/ There aren’t that many FA contracts to go around so even the high MR teams need to dip into this pool frequently, which sets a market price

2/ Wins are more or less fungible so you’d expect $/win to converge

3/ The ‘risk premium’ in market 1 may be larger than we think which means that if these guys signed a 2-3 year contract we’d see a much higher $/win that we’d expect to (after adjusting for aging) when they sign a 6 year deal.

Sure there are other reasons too ... Anyway, it all suggests that the $/win is set by the larger teams, which makes sense.

I don’t think John’s characterization of how close we are is accurate.

David is suggesting that the Royals would invest 25MM$ on Mauer, even if they have no cash in hand, on the expectation that dollars follows wins.  I’m saying they won’t do that.  That they can only do that if they already have cash in hand.

He used Twins as an example of investing, but I said that’s not a good example, because the Twins have some 30-40MM$ of cash in hand from the new stadium.

David said forget about budgets, and I said MLB operates mostly on the basis of budgets, and will give great deference to resigning their own players to go over budget rather than dropping their star players and signing other star players to go over budget.

Okay - yup I understand the difference. I’ll let David respond.

However, one thing I will say is that David’s pythag experiment also supports budget Tango’s point because if a team outperforms its expected true talent and as a result gets more revenue, that extra revenue will be spent on players who (should) contribute more wins - it other words revenues were better so budgets can be increased for players ...

Putting economic theory to the side for a second there is no question in my mind that the ‘budget’ approach is what teams use - that’s what every company does. 

The NPV lens isn’t necessarily the right way to look at this. Sure it is academically and economically sound and is the way that companies value ‘projects’, but scarcity of resources means that only a few NPV positive projects get up - a whole ton will never make it. And it isn’t necessarily the best NPV projects that get up. Strategic alignment is often a lot more important. Also companies that are cash limited can’t afford to invest in the best NPV projects if that puts a constraint on their finances.

In real life what happens is that companies (teams) are looking to spend a fixed pot of money as optimally as possible - the budget approach. If they reap additional profits then they may choose to invest that in the business next year. It is very unlikely they’ll spend ahead of revenue unless they are certain of the additional revenue ala the Twins’ additional season ticket sales.

Thanks for your comments, John. I think the debate is pretty much settled—Tango and I agree on certain points and disagree on others. I did specify that my point about spending on positive NPV projects (players) only applied to teams which are not cash-constrained, so perhaps there is some reconciling Tango and my viewpoints. Certainly, it is likely that a team near the peak of its win curve (i.e. would benefit most in terms of playoff probability added from adding a couple of wins) also already had a pretty good performance the previous year, and thus would, under Tango’s definition, have some cash on-hand to spend.

There are very few teams, if any, that would operate like Google or Amazon in the early days (or any company that invests heavily in R&D), where they’d be happy to invest in their product on the expectation of future revenue that comes from seeing a quality product better than what people are used to.

I’d like the hear when the last time that happened in MLB?  The Angels with their new owner signing Vlad?  And what of the other three sports? 

Anyway, I agree, it seems that we’ve exhausted expressing our point of views in as many ways as possible, without really moving the discussion forward.