Tango on Baseball Archives

Evaluating A-Rod (December 8, 2003)

We've all known that Alex Rodriguez is an incredible player. But maybe we haven't realized how much: He's so good that even after a drastic market correction, the $252 million man still isn't overpaid.

I prefer the neat clean way out. PECOTA projects ARod to produce +7.4 wins above replacement per year, over the next 4 years. A player "generates" about 2.5 million$ of revenue / marginal win. That's about 1.8 million$ of salary generated per marginal win. ARod is worth 15 million$ / year.

If we remember what my Win Advancement list showed, the top start from 1999-2002 produced an average of 6 to 7 wins above average, or about 8 to 9 wins above replacement. So, if you knew after the fact, you can pay the top player about 16 to 18 million $ / year.

Sorry, but I would essentially cap anybody's salary at 15 million$ / year (with anything extra being marketing driven).

Of course, you can argue that wins in certain markets or conditions, like playoffs, might generate more money.
--posted by TangoTiger at 02:36 PM EDT

Posted 6:14 p.m., December 8, 2003 (#1) - Nate
I'll put this here, since I need to solicit TangoTiger's help and since there is a somewhat loose connection to A-ROD.

My question for you sir is in regards to players with poor discipline (ala Nomar, and the very loose connection) and their aging patterns. Have there been any studies to see if players with good discipline age better or worse than players with poor discipline?

Over on Sox Therapy quite a few posters have said that Nomar doesn't figure to age well. They argue that since he swings at everything he has to be in peak phsyical shape. And they argue that when he starts to lose a slight step, he'll have more trouble than the normal batter because of his swing at everything approach (because balls are harder to hit than strikes, and Nomar swings at and hits a ton of balls). This may make sense logically, although not the most sound arguement ever. I wanted to use a study to evaluate this statement.

I'm not sure if BBs, BB/K or even pitchers per plate appearance would be best. I might think that PPE would be best, since in this case it is prolly a better measure of someone's "hacktasticness" (a new word!).

Thanks in advance Tango.

Posted 6:14 p.m., December 8, 2003 (#2) - J Cross
I'm not following this line:

"A player "generates" about 2.5 million$ of revenue / marginal win. That's about 1.8 million$ of salary generated per marginal win.

Shouldn't marginal revenue = marginal cost for a player? How did you get 1.8 from 2.5?

Posted 9:42 p.m., December 8, 2003 (#3) - Tangotiger
I was just taking a look at aging patterns a few weeks ago. I really didn't see anything with K/BB ratios, or batters who are hackers. I did notice something with fast players (they age slightly better), and fast players with no power (they also age well, presumably because they have a bit more potential).

You might want to ask Nate Silver, as I'm sure he has taken a look at this. I can present some of my preliminary research, but I didn't really want to,because I really didn't do any thorough work on it.

****
I may generate 200,000$ of marginal revenue for my company, but my salary won't be that. If a company is going to spend say 10 million$ on a hitter, they don't want him to generate 10 million$ of revenue (that's not a good ROI), but more like 12 to 14 million$. And that 12 to 14 million will really be more like 6 to 20 million$ (lots of variability in player performance).

I would consider a player like a junk bond (or maybe the coupon rate of a bond), though perhaps others here have a better security to compare them too.

Posted 11:07 p.m., December 8, 2003 (#4) - Nate
Thanks Tango.

How could I contact Nate?

Posted 7:41 a.m., December 9, 2003 (#5) - studes (homepage)
In a recent issue of the "Journal of Sports Economics", a couple of guys ran a study in which they found that the average revenue per marginal win increases as total wins increase. That is, the 80th incremental win is not worth as much as the 85th, which is not worth as much as the 90th, etc. (No, I don't subscribe, but I did backorder the issue)

I think the market approaches ballplayers in the same manner. That is, I think the market pays more for a single player's seventh win than the sixth win. I think that's rational, but only within the context of the total team's position (something that wasn't well-considered with ARod). Over time, I'm exploring this thought on my site.

There are several other important aspects of baseball economics that drive up the salary paid for an incremental win. One is the relatively high fixed cost of running a baseball team. I've worked and consulted in several industries with high fixed costs, and it's interesting to see what the market does. Sooner or later, each market (if it's competitive enough) squeezes the incremental profit margin out of the product.

That is, the temptation to offer a player virtually all of your incremental revenue is too hard to resist, because ANY contribution to your bottom line helps. I believe baseball owners often look at baseball players this way. The players are the mechanism whereby owners realize their return on their large fixed investment.

The other issue is that the "win market" is a zero-sum game. There is a cap on the number of wins that a "market" can produce. An individual team can't invest more money and create more wins, beyond some reasonable number, because the total can't grow.

So game theory comes into play here, and I'm no expert in game theory. But I believe the zero-sum nature of the win market turns the entire affair into a bit of a gambling casino.

The junk bond thought is a good one. Think of your roster as a diversified portfolio of investments. In order to increase your total wins, you will decide to take on some riskier investments. That is, you'll price a junk bond like a AAA bond, in order to reach a certain level of wins, as long as the overall risk of your portfolio is under control. This is also rational, within the appropriate team context.

I'll stop here -- too many thoughts are making me ramble. But my basic point is that there are reasons to pay a player more than $15M. Some of them are rational, some are built into the economics of the business, others are irrational. But I believe the case can be made.

Posted 10:48 a.m., December 9, 2003 (#6) - tangotiger
You can contact Nate from the "contact us" link at BP.

I was just taking a look at my sim scores (great way to waste half an hour). Nomar is not just a hacker, but a smart, strong, and fast hacker, which makes him different from most. Right now, I have my program set up to look at players aged 23 to 26 only. (Nomar was born in 1973, and so, his age 23-26 would be 1996 to 1999). And I haven't lg/park adjusted yet.

His 4 best comps among players from the same age group are:
Billy Williams, Tony Oliva, Ruben Sierra, Reggie Smith

Hope this helps.

And I spoke to soon regarding the profiles of players that don't age differently, so I'll take that statement back.

Posted 10:50 a.m., December 9, 2003 (#7) - tangotiger
Studes, I don't think your last statement is any different than mine:

Sorry, but I would essentially cap anybody's salary at 15 million$ / year (with anything extra being marketing driven).

Of course, you can argue that wins in certain markets or conditions, like playoffs, might generate more money.

Posted 10:55 a.m., December 9, 2003 (#8) - studes (homepage)
Right. I decided to argue. :)

Posted 11:39 a.m., December 9, 2003 (#9) - J Cross
I may generate 200,000$ of marginal revenue for my company, but my salary won't be that. If a company is going to spend say 10 million$ on a hitter, they don't want him to generate 10 million$ of revenue (that's not a good ROI), but more like 12 to 14 million$. And that 12 to 14 million will really be more like 6 to 20 million$ (lots of variability in player performance).

Well, geez, my Microeconomics teacher told me that your salary WILL be your marginal revenue. Now you're going to come and tell me that's not true? Next thing I know you'll be telling me that there isn't perfect information and that people don't always act rationally.

Seriously though, if a player's marginal revenue is $12-$14M I'm sure teams would love to pay him $10M but some other team would offer $11M. I can't see any reason for teams to be risk averse with respect to player's salaries and I'd think that teams would bid up a free agent until his salary equaled his expected marginal revenue. I don't think the investment analogy is a good one since (for the most part) you are receiving the revenue at the same time as you're paying the salary. In fact, with a small pool of free agents and a situation with, say, one star shortstop but several bidders for that shortstop with varying expectations we might expect a little winner's curse and a salary greater than the expected marginal revenue.

Posted 11:51 a.m., December 9, 2003 (#10) - tangotiger
Seriously though, if a player's marginal revenue is $12-$14M I'm sure teams would love to pay him $10M but some other team would offer $11M.

I'm sure that's how Kevin Brown got his 15 million/year.

I can't see any reason for teams to be risk averse with respect to player's salaries

Uncertainty to their actual true talent level + uncertainty as to their expected true talent level = RISK! So, ARod may have produced over the last 4 years at say +9 wins above average / year, that's not what his true expectation is today, nor would it be for the next 4 years.

and I'd think that teams would bid up a free agent until his salary equaled his expected marginal revenue.

I think they used to, and now they don't.

I don't think the investment analogy is a good one since (for the most part) you are receiving the revenue at the same time as you're paying the salary.

I don't think it's that close, but it might be. A win today will add some money tomorrow, but a bit more the weeks after, and even some next year. So, I think there's probably a 6-month lag between revenue and performance. Plus, the playoffs too add more risk, as this is an all or nothing revenue source.

In fact, with a small pool of free agents and a situation with, say, one star shortstop but several bidders for that shortstop with varying expectations we might expect a little winner's curse and a salary greater than the expected marginal revenue.

Because we can't "short" a player's stock, the market price is not the average market price, but the highest price a market will bear. So, there is inefficiency here.

You can have 5 bidders for the ARod stock, and it could be $21, $22, $23, $24, $25. It'll sell at 25$, but that's only because you don't have any shorts to keep them honest (and of course, you only have 1 share outstanding).

Posted 12:16 p.m., December 9, 2003 (#11) - J Cross
Uncertainty to their actual true talent level + uncertainty as to their expected true talent level = RISK! So, ARod may have produced over the last 4 years at say +9 wins above average / year, that's not what his true expectation is today, nor would it be for the next 4 years.

Okay, but that's not what I meant by risk aversion. I'm just saying that if after you've calculated your expected return to be $10M next year I don't see why a 50% chance of $4M value and a 50% chance of $16M value is worse. In fact, if a team is more likely to miss the playoffs than make them (and most teams are) the situation with higher variance likely increases "penants added."

Posted 12:18 p.m., December 9, 2003 (#12) - studes (homepage)
Well, geez, my Microeconomics teacher told me that your salary WILL be your marginal revenue.

Well, that's a weird statement for an economics professor to make. It only makes sense to me if it applies to a situation in which the owner and labor are the same person.

If business management does not get to keep any of the marginal revenue generated by a player, then there is no incentive to hire that player. None. In every deal that has been made, the GM/owner made some sort of assumption about incremental revenue and the portion of that revenue he (or she) would get to keep.

Now, their assumptions may have been completely wrong. But that's a different matter.

Posted 12:59 p.m., December 9, 2003 (#13) - J Cross
studes, maybe I'm not explaining it well but that's what's happens in the economic model of perfect competition. Zero economic profit. Marginal revenue equals marginal cost. The owner gets the revenue that's due to capital. The worker gets the price/product*marginal product of laber ie the marginal revenue his labor.

Posted 1:06 p.m., December 9, 2003 (#14) - J Cross
"If business management does not get to keep any of the marginal revenue generated by a player, then there is no incentive to hire that player."

Ah, but there's also no incentive NOT to hire that player. That's why it the margin.

Posted 1:45 p.m., December 9, 2003 (#15) - tangotiger
Yes, they can take that 10 million$ and buy a government bond.

Posted 2:10 p.m., December 9, 2003 (#16) - J Cross
Yes, they can take that 10 million$ and buy a government bond.

or they could take the $10M in revenue the player produces and buy a government bond. same difference.

To make it clearer I should say that the marginal revenue of the player is the present day value of revenue he will bring in and the marginal cost of the player is the present day value of the salary he will be payed.

Posted 2:17 p.m., December 9, 2003 (#17) - tangotiger
Well, present day value makes a big difference. 10 million$ invested in a government bond is still 10 million$ in present-day dollars.

However, the performance of a player won't be discounted at the T-bill rate. It will be much more likely to be discounted at a junk bond rate (15 to 20% or whatever).

Even after you account for the best-guess true talent level of the player today, the expected future earnings of that player will have such a high variability, that a team would be crazy to pay a player equal to the marginal revenue he's expected to generate.

Maybe if they had 500 players, they would do it. But, they've only got a handful of players.

Posted 2:32 p.m., December 9, 2003 (#18) - J Cross
Even after you account for the best-guess true talent level of the player today, the expected future earnings of that player will have such a high variability, that a team would be crazy to pay a player equal to the marginal revenue he's expected to generate.

Why? It's the variability of your assets as a whole that matters. As far as I understand it people hedge Beta and buy stocks from difference sectors to make sure that the risk across their assets isn't correlated. When you consider than any one player is just part of a team whose injury risks are completely unrelated and that each team is only part of an owner's portfolio I just don't see why an owner would be averse to variability of outcomes. Again, a player who has a 50% chance of being worth 10 runs and a 50% chance of being worth 30 runs is, for most teams, more likely to add a pennant than a player (if there was such a player) sure to be worth 20 runs.

If BP's data shows that A-Rod is a good investment compared to other FA's and also shows that he's being payed at or higher than his marginal revenue then aren't most free agents being payed at or higher than their marginal revenue? Are these teams all crazy? I'd think you'd have a tough time convincing and economist of that.

Posted 2:38 p.m., December 9, 2003 (#19) - studes (homepage)
J., I agree with most of your last post. I went to business school, and have been a hardcore businessman for twenty years, so economic theory language often doesn't make sense to me.

I believe that when Hicks cut the deal with A-Rod, he truly believed that the marginal revenue generated by A-Rod for his team would be greater than his marginal cost/salary. Turned out, he was wrong. So now he wants to trade him.

To the Red Sox, who also believe that the marginal revenue he will generate will be greater than his marginal cost/salary -- particularly in contrast to the Manny situation.

The notion of increasing marginal revenue as total player wins increase is important to the discussion, too.

Posted 2:52 p.m., December 9, 2003 (#20) - J Cross
Ofcourse, a players real marginal revenue goes down when you're sharing part of the marginal revenue with other teams. Tango, was that $2.5M/win figure from before or after the new CBA? I guess Hicks didn't see that one coming. He must also have imagined that the Rangers would contend which would (according to that article in the new issue "Journal of Sports Economics") have made A-Rod's marginal revenue higher.

studes, I'm not an economist or even an econ major so my economic theory language might not be so good.

Posted 3:06 p.m., December 9, 2003 (#21) - tangotiger
That was based on a study by Voros, from I think 1999-2001. A good rule of thumb is that 1 marginal win = 2% marginal increase in revenue. A MLB team would have around 120 million$ in revenue.

*******

Perhaps you can enlighten me on stocks.

You have a basket of 30 stocks, where the correlation in price movement among the 30 stocks is the least. So, while the beta for every individual stock might be 2 or 3 or 10, as a basket, the beta is 0.5.

Now, the market doesn't know that you've figured out how to reduce the variability to this extent, and so, each individual stock's price has a high discount rate (good for you!).

However, in baseball, player performance are naturally not tied to others on the team, and so, the team beta will generally be the same for every team. While the discount rate for any one player might be 20%, you are saying that the discount rate, when looking it from the team's perspective, should be closer to 5% or something. Is this what you are saying?

I'm not sure I'm buying into this, yet.

Posted 3:45 p.m., December 9, 2003 (#22) - J Cross
I think that's what I'm saying.

So, if your star player is going to be worth $15M with a standard dev of $7.5M and all you players have the same variance/value then a team worth $60M might have a SD of something like $15M, right? So, let's take a typical 81 win team with $75M value (is this a contradiction?) that would have a $16.8M (6.7 win) variance in expected performance. My question is (and I realize that this is a ridiculous idealized situation), wouldn't you rather have a team good for 81 wins with a SD of 6.7 wins than a team that was going to win 81 games for sure? I'm suggesting that the variance could be considered a good thing.

Posted 3:48 p.m., December 9, 2003 (#23) - J Cross
okay, the numbers in that last example don't make sense ($15M with a SD of $7.5M???) but I think you get the idea.

Posted 10:09 p.m., December 11, 2003 (#24) - Guy
I think there's a fundamental problem with the logic that says A)wins are worth X ($1.85M), B) A-Rod is good for about 7.5 net wins, therefore C) it only makes sense to A-Rod around $14M. The unstated premise is that all wins are worth the same, that 3 players producing 2, 2.5, and 3 wins are cumulatively worth exactly the same as A-Rod, that if I don't "overpay" A-Rod I can always buy more wins with my $$ elsewhere. In many economic markets that would be true, but I don't think it applies here, for two reasons. First, a team faces artificial constraints within which it tries to accumulate wins: a roster limit and, more importantly, PA and IP limits. Second, the market does not provide at all times a wide array of players of different values at each position. In that context, a player who can generate 7.5 wins while consuming about 1/10 of your team's offensive playing time, especially at SS, may be worth much more than 3 guys in the OF contributing 7.5 wins over 30% of your PAs.

I look at it this way. Tango says a team of replacement players will win about 49 games (.300). Suppose my goal as owner is to win 100 games and advance to the postseason, so I need 51 marginal wins. Realistically, I need to generate most of those wins from 14 players -- 9 position players, 4 starters, and a closer. I don't have the option of buying 30 1.5-win players for $2.8M each -- because I can't play them all! So those 14 players have to generate an average of 3.6 net wins to reach my goal; assuming my 10 backup players produce something, let's call it 3.2. That's a very high average level of performance (PECOTA projects Thome at 4.1/season). Can I get there without buying one or two 5-7 win players? It's possible, but very hard.

It's hard because of the second factor: a limited and "illiquid" market for baseball talent, especially when you need to fill 8 discrete defensive positions. How many 3-win or better catchers, SSs and 2B are available? Not many. If I don't sign one or two very elite players, putting together a top team is like trying to fill an inside straight -- I need to find very good players everywhere else. In contrast, if I have A-Rod, I have more flexibility to gamble on 1 or 2 cheaper players with the potential to overperform. And I may figure that I can find some good-hitting corner IFs or a LF at below-market prices (see 2003 Red Sox), which you can never do up the middle.

Given this constrained market, and the scarcity of very high "win density" players like A-Rod, the price naturally goes up.

That leaves the question of whether it makes economic sense to pay more than $1.8M for a marginal win. Certainly for an owner who wants to win (they do exist!), it's easy to see them paying at least the full marginal revenue ($2.5M)for a player who can contribute so much. There's no profit, but also no loss, and your team is better. And if Studes is right that wins 95 and 100 generate more revenue than wins 70 and 75 -- which certainly sounds right --then a lot of that higher revenue will go to the very best players. For it is they, not the 1-2 win players, who make it possible for a team to win at that level. So it's not hard to see how a 7.5-win player could be worth considerably more than $14M to a good team.

Posted 11:06 p.m., December 11, 2003 (#25) - Tangotiger
Guy, I will concede that there is an extra variable here, and that is the cap on the number of roster spots. I still haven't spent any time trying to figure out the cost of that.

***

Let me also clear up a little something on the distribution of those marginal wins above replacement. A team of all replacement players will win around 49 games, and realistically, what you want is a team to get you to 95 wins, or +46 wins.

Your 8 hitters, 4 starters and 1 closer, if they were all average, would contribue about +26 wins. Make them all above average a little (+1 win above average, or +3 wins above replacement), and you are at +39. Your backups would generate about +4 wins or so. That makes it +43, or 91 wins.

So, I'm not sure the real constraint with the roster size.

Posted 10:14 a.m., December 12, 2003 (#26) - Guy
Doesn't sound like we're far apart, Tango. To get to your benchmark of 95W, the 14 key players need to average 3.4 wins above replacement (I guessed 3.2). The question is, how hard is that to do without buying one or more elite 5+ win players? You describe +3W as "a little above average." But how many players consistently perform at that level? And specifically, how many catchers, ss, and 2b? My guess is it's fewer than one would expect. In the PECOTA table in Bloom's article, the players projected to perform in that range are I-Rod and Cliff Floyd. That strikes me as a very high level of performance to average over 9 position players and 5 pitchers. And assuming you will inevitably fall short at 2 or 3 positions, you then need to exceed that level elsewhere.

This illustrates the larger problem with using the arithmetic mean to define "average," which is common to almost all baseball analysis. Mathematically, you have to do it. But analytically, you always have to remember that the mean performance is much higher than the median, or what a typical MLB player delivers. In part, this is because good players get most of the playing time, boosting the mean performance. Also, as Bill James pointed out long ago, the distribution of talent in baseball is a pyramid, not a normal curve. Consequently, there are far fewer "above-average" players than "below-average players," and the number who are one full game above average is much smaller still.

You're in a better position than I to evaluate this, but it seems to me that assembling a core of players that average +3.4 wins is no mean feat. And if the owner wants a high degree of confidence that the team will win 95+ games (as the Red Sox clearly do today), having an A-Rod/Vlad/Bonds type player is almost essential.

Posted 11:12 a.m., December 12, 2003 (#27) - Tangotiger (homepage)
mean performance is much higher than the median

This is not true. I direct you to the above link. If the true talent level of the average player is 100, and the replacement level is 80, the median player will be around 95 if I remember right.

While James is right that the talent is kinda of pyramidial (i.e., the tail-end of the right-hand of a bell curve) he is wrong when he said something about 20 players below 10% of average to 1 player of 10% above average. If I remember right, it's more like 4 or 5 to 1.

As well, because of the playing time component, you end up with an almost normal distribution of "talent times playing time".

Posted 12:22 p.m., December 12, 2003 (#28) - Guy
In defining median, are you including all players who play during a season? Just those who play with some regularity? If the former, median can't possibly be 95% of mean; if latter, I remain skeptical, but maybe so.

In any case, this is a sidebar (if interesting). How common are 3.4 wins over replacement players?