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Wednesday, June 04, 2003

Super Linear Weights, 2000-2002

Revised methodology with complete 3-year results.

Several years ago I introduced a new series of metrics for evaluating the complete performance of a player, called Super Linear Weights.  I have once again revised some of the methodologies, most notably defensive linear weights, or Ultimate Zone Rating (UZR).  For a complete discussion of UZR, see my two-part series.

Like traditional offensive linear weights (lwts), each of the categories is expressed as a number of runs above (plus) or below (minus) the league average, which is naturally defined as zero. For example, a player who had a defensive lwts of -5 was a below average fielder. In fact, he allowed 5 more theoretical runs to score than an average defender at that position. By convention, regardless of the category, minus is always bad and plus is always good. Because each category uses the same "currency" (runs above or below average), all of the categories can be summed to produce each player?s total linear weights, or Super Linear Weights (SLWTS). A player?s SLWTS tells us exactly how many runs above or below average he is "worth" to an average team under average conditions.

Super Linear Weights is only "better" than traditional lwts, BaseRuns, Runs Created, or other such offensive metrics, because it includes defense, baserunning, and a few other odds and ends. This makes it a much better metric or series of metrics for evaluating total player value. There are many players whose good or great offensive performance are mitigated or in some cases completely negated by their poor defensive and baserunning skills. As well, great defensive value can turn a good offensive player into a great overall player, poor defense can turn an otherwise marginal player into a terrible player, etc. The permutations are unlimited and are in fact critical in terms of evaluating and measuring total player value.

2000-2002 Super Linear Weights Results

Here are the Super Linear Weights categories and their brief explanations.  For a more detailed explanation of each category, see the links to previous Superlwts articles below:

     

  1. Player Name
  2. 2002 Team (if more than one team, the team with the most PA?s)
  3. Primary position
  4. Number of PA?s in 2002 (PA=AB+SF+BB+HBP)
  5. Defense
  6.  

    • UZR - Ultimate Zone Rating or defensive lwts - runs saved or cost on defense, "everything" adjusted, based on PBP hit type, speed, and location data.  Again, see the UZR articles for a more detailed explanation.
    • GDP defense (DP) - for infielders only. Number of DP?s turned per DP opportunity, compared to the league average at each position, adjusted for league-average number of opportunities, and converted into runs saved or runs cost. When a DP is turned, the fielder and the pivot man are both credited. If the fielder and the pivot man are the same, only one credit is given.
    • Outfield arms (ARM) - This category includes an outfielder?s outs (assists), "holds", and extra bases allowed, compared to league averages, adjusted for league-average opportunities, and converted into runs saved or cost. They are park adjusted.
    • Catching (CAT) - for catchers only, the combined run value of their SB and CS totals, errors, and passed balls, as compared to a league-average catcher.

     

  7. Baserunning
  8.  

    • "Taking the extra base" (BR) - this is the exact reverse of the "OF arms" category - it is extra bases, outs (OOB), and "holds" by a baserunner, adjusted for baserunning opportunities.
    • GIDP (GDP) - this is a batter?s number of GDP?s per opportunity, compared to the league average, adjusted for league-average number of opportunities, and, as always, converted into runs saved or cost.

     

  9. Hitting
  10.  

    • Moving runners over (MR) - a batter moving a runner from 2nd to 3rd on a fly or ground out ("giving oneself up"), per opportunity, compared to league averages, and converted into runs saved or cost.
    • Batting (BAT) - traditional offensive linear weights, including SB and CS. They are park and opponent adjusted, using regressed, 10-year (if possible) component park factors.

     

  11. Super Linear Weights
  12.  

    • Superlwts (SLWTS) ? the total run value of all categories combined.
    • Superlwts per 162 games (/162) ? same as above, only prorated to the PA equivalent of 162 games for an average player.
    • Position-adjusted Superlwts (POADJ) ? adds in a positional adjustment. For example, if an average SS in 2002 had a total Superlwts value of -15 per 500 PA, and SS A had a Superlwts value of -10 in 250 PA, he gets a plus 8 run adjustment, to give him a position-adjusted Superlwts value of -2. (In other words, in 250 PA, he was 2 runs worse than an average SS in 2002.)
    • Position-adjusted Superlwts per 162 games (/162) - the above prorated to the PA equivalent of 162 games for an average player.

     

    In order to fairly compare players across the defensive spectrum, it is important to use position-adjusted Superlwts for two reasons. One, it is assumed that a player at a more demanding defensive position has better defensive skills than a player at a less demanding defensive position, therefore less offense is "tolerated" from that player. Two, a player?s run values in all categories other than the defensive ones are relative to a league average player at all positions. A player?s run values in the defensive categories are relative to an average player at that position, so it would be adding apples and oranges to combine all categories without doing a positional adjustment.

    Keep in mind that a player?s Superlwts, like any metric or stat, is only a sample of a his performance, and thus represents only the first step in estimating a player?s theoretical value to a hypothetical team (his "talent") or projecting his future performance. The other step is regression. Regression is a function of, among other things, and most importantly, the size of the performance sample, which in this case, is the number of a player?s PA?s or "games played". If one wanted to compare the theoretical value of two players using their respective Superlwts results, either both players would have to have approximately the same number of games played (or PA?s), or if not, one would have to regress one or both player?s Superlwts, and then compare them. For example, if Player A had a Superlwts value of +8 "per 162" in 900 PA and Player B had a Superlwts of +10 "per 162" in 830 PA, one could say that player B was likely "better" (of more value) than player A, all other things being equal or unknown.  However, if Player B was +10 per 162 in only 300 PA, then one would have to either first regress Player B?s Superlwts to make it "equivalent to" Player A?s 900 PA sample, or regress both Player A and Player B?s Superlwts in order to make them equivalent to each other.  In either case, it is now likely that Player A is the better player (Player A?s +8 in 900 PA might get regressed to +6 and player B?s +10 might get regressed to +4). 

    The reason for this is that in order to compare players with different numbers of games played or PA?s, in terms of deciding which one is likely "better", we might first estimate their true Superlwts (true "talent") from their sample Superlwts.  In order to do this, as with any other sample stat, we must regress the sample result towards the mean.  If we know nothing else about a player but his position-adjusted Superlwts results, that mean is usually going to be zero.

    The charts at the links below contain the results of the 2000-2002 seasons, both separate and combined.

    SLWTS, 2000-2002 Totals

    SLWTS, 2002 Totals

    SLWTS, 2001 Totals

    SLWTS, 2000 Totals

    Related Links

    SLWTS 2000, part 1


    SLWTS 2000, part 2

    Mitchel Lichtman Posted: June 04, 2003 at 06:00 AM | 46 comment(s) Login to Bookmark
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Reader Comments and Retorts

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   1. Silver King Posted: June 04, 2003 at 02:11 AM (#611227)
A terrific vantage point on performance information - thanks!
   2. GGC don't think it can get longer than a novella Posted: June 04, 2003 at 02:11 AM (#611232)
Wow, it looks like Super LWTS catches just about anything that happens on the ballfield. Is this what Tango means by granular?

Thanx, MGL
   3. Dave Studenmund Posted: June 04, 2003 at 02:11 AM (#611233)
Yes, Erstad is the ARod of the field. That is, his runs saved in the field equal ARod's runs created at bat. This was covered in the UZR discussion; I believe outfielder UZR's are suspect, because of the "ball hog" phenomenon (my phrase) in which teammates allow the "better" or more aggressive outfielder to "take it." Interesting that Andruw doesn't have the "ball hog" phenomenom go his way.

Seems slwts is still a work-in-progress, but it does represent an important conceptual move forward. Good work, mgl. Keep it up.
   4. Dave Studenmund Posted: June 04, 2003 at 02:11 AM (#611237)
Pachoo, some of that difference is time. I think even the Win Shares methodology would recognize that Andruw is not the fielder he was a few years ago. Other, more readily available data, such as adjusted range factors, indicate that his skills have slipped.

Also, I have a feeling mgl will point out that there is a sample size issue here. Making absolute fielding prowess statements based on one year's worth of data is stretching the methodology.
   5. bob mong Posted: June 04, 2003 at 02:11 AM (#611239)
UZR must be drastically different from the defensive rating system in Win Shares that Bill James came up with.

In his system Andruw Jones ranks as easily the best defensive outfielder to ever play the game (in his words). Erstad doesn't register anywhere close to Andruw in that system.


Something to keep in mind is that the Win Shares book was published in March 2002 - and at that point, Darin Erstad had only one season under his belt as a CF - in 2000 he played 112 games at LF (30 at CF), in 1999 he played 78 games at 1B (2 in CF, 67 in LF), in 1998 he played 70 games in LF (70 at 1B, 2 in CF), in 1997 he played 126 games at 1B, in 1996 he played 36 games at CF (out of ~57 games).

In other words, when Bill James wrote his book, Darin Erstad was a LF/1B who was converted to CF for one year. It makes sense that he won't rate as high, defensively, as someone who had been playing excellent CF, fulltime, for four years at that point (Andruw Jones was fulltime CF from 1998-2001).

MGL's ratings, on the other hand, are from 2000-2002 - and during that time Darin Erstad has played:
2000: 112 LF, 30 CF
2001: 146 CF
2002: 143 CF

It's weird to think that Erstad wasn't the Angel's CF until 2001. At least, it is weird to me.
   6. tangotiger Posted: June 04, 2003 at 02:11 AM (#611240)
I HIGHLY recommend you look at the 2000-2002 data, and not any single year, so that the sample size is not as much an issue.

Andruw Jones was very high from 1999-2001, and below average in 2002. Whether this is just luck, or a change in his talent, we don't know.

Your perceptions of Jones must have built up since you've seen him play, and not since you've seen him play in 2002. Therefore, try to compare any system on the same timeline basis.
   7. MGL Posted: June 04, 2003 at 02:11 AM (#611244)
Pachoo,

You must have overlooked their (Piazza, et al) their catcher ratings. Piazza is -27 for the last 3 years, Posada is -7, and Pudge is +14. Piazza is a good example of how peripherals can reallt mitigate a player's overall value. Of course, Piazza is such a good hitter compared to the average catcher, that he is still one of the all-time best (43 overall runs per 162 games better than an average catcher for the last 3 years). However, if you include all of the peripherals, it is clear that Pudge has been quite a bit more valuable (+55), and is probably the better the player overall (not counting "game calling" of course, of which it is said, right or wrong, that Piazza is good at).

As far as Erstad's, Andruw's, or anyone else's UZR, much has already been said on the UZR thread. If anyone is interested or "troubled" by any of the UZR ratings, I suggest they look at that thread. A few comments:

I ditto Tango's above post. Also, there seems to be a steady decline in UZR as a player ages, almost from day one. The aging pattern for UZR appears to be similar to the aging pattern for triples, especially for CF, SS, and 2B. In fact, in a premilinary study I did with a limited number of data points, players appear to lose around 2 runs per year in UZR at CF, SS, and 2B, and 1 run per year at all other positions except for 1B. At first base, players appear to get better after age 30 and I forgot what happens before age 30. I don't know if that's a statistical blip or if it has some merit. I think a a little bit of both. Anyway, that was an aside.

Anytime you see a player with a bery high (like Erstad) or bery low UZR rating for ANY period of time, you can automatically assume that part of that very high or very low rating is good or bad luck. That is the essense of regressing a player's rating, UZR or any other, towards a mean of zero. That means that the further a player is from zero, the more you are going to regress his sample rating in order to estimate his true rating or true talent. So it is unlikely, for example, that Ersatd will have a 2003 UZR rating of 50 (per 162). Our best estimate (projection) of his 2003 UZR rating is more like 30 or 35, including an age adjustment. The same is true of offensive ratings (lwts or whatever metric), but because offensive metrics are more reliable than , say UZR ratings, you would regress them less in order to project a player's offensive performance or estimate their true offensive talent. So, for example, if Player A had a 3-year Superlwts of +40, whereby most of that +40 is offense, and player B had the same 3-year Superlwts and most of that was defense, Player A would have a higher Superlwts projection and most likley be the "better" player.

Keep in mind that UZR is far from a "perfect" metric - IOW, it does not reflect a player's defensive contribution to his team perfectly reliably and accurately. This goes without saying and is true for all metrics. However, for various reasons, I (and other people) am confident that it is a very good metric and is probably the best defensive metric out there, by far and away (Diamond Mind and maybe some others have a similar metric, but they do not share their exact methodology, so I am excepting them from this statement - I am mainly referring to things like ZR, RF, etc.). That doesn't mean that it doesn't make lots of small mistakes, a few big mistakes, and 1 or 2 gigantic mistakes. All metrics will do that by virtue of random fluctuation alone. Can we identify those mistakes? Probbaly not. Can we assume that what it sugests about a player's defensive value is true ands accurate? Yes! That's the whole point of a good (not perfect) metric! To augment or even supplant (that's the dirty word in scouting/sabermetric debate) what we think we know about a player. To borrow (and paraphrase) an axiom from Bill James (I think), if we are not shocked by some percentage of a certain metric, the metric has no use! That should be intuitively obvious! If we look at a metric and go through it and say "Nah that doesn't look right (A. Jones in 2002, Torii Hunter, J.T. Snow), I'll reject that one (or even, worse, I'll reject the whole metric)," or "Yeah that's what I thought, I'll accept that one," what would be the point of the metric? You would need only go with whatever you thought in the first place. A rule of thumb is that if a metric is good, you should be surprised at maybe 10 or 20% of the results and very satisfied with the rest. I think that UZR falls nicely into that framework, although I've never gone through any of the files and calculated the percetnage of "surprises". Now, that 10 or 20% will vary depending upon the sanmple size of the ratings you are looking at of course. The smaller the sample size, the more there should be surprises. Also, even if it is a good metric, that doesn't mean that for all of those surprises the metric is correct and your presumption was not. Burt that's another topic altogether (using Bayesian probability), which I won't get into right now.

As far as "out stealing" which someone mentioned (an OF'er taking more than their shar of easy fly balls, thus inflating their UZR and deflating another fielder's), although it is an inherently problematic area in UZR, I don't think it is THAT much of a problem, and I wouldn't automatically attribute unusally high or low UZR ratings to that.

Thanks for the comments thus far.

More later...
   8. MGL Posted: June 04, 2003 at 02:11 AM (#611246)
Rally,

That's right, around .50 from season to season (min 120 games or so), which is about the same as BA. Of course, correlation coefficients are about reliability and not accuracy (some metric might be terrible at measuring defensive talent but if whatever it measures it is good at that, it still could have high year to year corr. For example, FA and RF may have great year to year correlations but they are terrible at measuring defensive talent), so you never know.

I don't think you would have nearly a large enough sample size to do what you suggest doing (year to year corr. of OF'ers who swtich teams), and give you any reliable or meaningful results...
   9. MGL Posted: June 04, 2003 at 02:11 AM (#611247)
One way perhaps to look at the problem of out stealing is to look at the entire outfield UZR of teams who have a partiucalr OF'er who has an unusally high UZR (you suspect him of out stealing). If part of the reason for that high UZR is out stealing, the entire OF's UZR would be smaller than that of the "suspect" OF'er (the other OF'ers that he is presumably stealing from would have negative UZR's on the average). I did this and did not find that was the case. As I said, I suspect that out stealing is not much of a problem...
   10. User unknown in local recipient table (Craig B) Posted: June 04, 2003 at 02:11 AM (#611248)
Vinay: Interestingly, Garrett Anderson, who is considered a terrific defensive LF, only rates at +6. So it's certainly possible that he's letting Erstad catch all the balls in the gap that they both reach.

MGL: As far as "out stealing" which someone mentioned (an OF'er taking more than their shar of easy fly balls, thus inflating their UZR and deflating another fielder's), although it is an inherently problematic area in UZR, I don't think it is THAT much of a problem, and I wouldn't automatically attribute unusally high or low UZR ratings to that.

I think, though, that there's more to "out stealing" than the discretionary outs (though those are huge; I've been doing an anecdotal count recently of how many flyouts wind up as "discretionary outs" and the figure is over 50% for infielders and easily 25% for outfielders).

Positioning is a major factor in the spread of flyouts, and the presence of an Erstad is going to affect the positioning of the other outfielders to a significant extent... having Erstad out there to run down everything in the gaps lets the LF and RF position closer to the foul lines, taking more hits away overall but resulting in less perceived value for the LF and RF... more balls are going to be hit in the gap than down the line.
   11. Chris Dial Posted: June 04, 2003 at 02:11 AM (#611249)
MGL,
pardon my Swahili, but wow, this is some m*****f***ing fantastic work.

*This* is what you submit to SABR. I *greatly* appreciate it, regardless of what flaws you see in your own method. Keep up the great work.
   12. MGL Posted: June 04, 2003 at 02:12 AM (#611252)
the notion that eg A Jones may in fact be peforming at an average defensive level right now doesnt strike me as horribly scandalous, especially when sampling issues are taken into account.

Now that is a reasonable thought!

JK and Chris, thanks! As far as SABR is concerned, personally I prefer whether 3-legged Sam would have been in the HOF had he hit in the old "Groley down by the Pit" stadium rather than in Sportsman Parks, but that's another story...

As far as the talk about regression, it is the same concept as if one used a "regression formula". The regression formula would be derived from, say, regressing a player's 3-year UZR (minimum of 300 games perhaps) on his next year's UZR. The resulting regression formula would yield regression coefficients. Using a regression "coefficient", like the 25% or 50% that I am talking about above in ordert to project a player's UZR, is essentially a crude way of guesstimating what the real regression coefficients would look like if you did a regression analysis. OF course, the crude method assumes that the regression equation would read something like y=mx+b, where b equals zero, x is a player's sample (3-year in this case) UZR per some unit number of games or chances (say 162) and m is the "coefficient" (50%, 75%, or whatever).

It would be interesting to match up all the 3-year Superlwts with each player's salary during that period or their 2003 salaries, to see which ones are under or overpaid. Whenever I have done that in the past, you get lots of amazingly ridiculous results, which gives you some idea as to the state of player evaluation by major league teams...
   13. tangotiger Posted: June 04, 2003 at 02:12 AM (#611254)
that an SS must be a good fielder that every MLB manager puts someone there who is competent, thus creating an artificially high "average" level compared to other positions like LF - where the average is driven down by the habit of many managers of sticking defensive monstronsities out there to get bats in the lineup

Exactly my thoughts, though you articulated it better than I could have.

If you assume that, historically (say over the last 20 years), that each position has been balanced to the point that they are all overall equals, and you find that SS are -10 runs below the overall average player offensively, this MUST mean (based on my assumption), that the average SS is +10 runs above the overall average player defensively.

If you've got Ozzie Smith being +25 runs above the avg SS defensively, and the avg SS is +10 runs above the overall average player defensively, then Ozzie is +35 runs above the overall average player defensively.

My suggestion is that MGL should introduce a adjUZR that takes the player's UZR (relative to position) and adjusts it to relative to all players. While he makes this adjustment at the end, I think it would make it clearer in this way. (You still get the same overall answer). From this standpoint, you'd get say Helton 1B UZR value to go from +15 (relative to position) to +2 or something (relative to all), and makes comparisons fielding-wise, if not very good, at least better.

I second David's point that in order for the community to buy into UZR, intermediate results must be presented. This is up to MGL as to how much time/effort he wants to put into it, and whether selling UZR is important to him as an analyst.

We are of course all thankful with the generosity of what he's presented so far.
   14. Depot Posted: June 05, 2003 at 02:12 AM (#611258)
I still confused about where these numbers are coming from (as I think JK is as well). How are the offensive nmumbers determined? I assume from the term "linear weights" that we're talking about a regression which tells you what the average single, double, etc. is worth. Is this correct? What level of data is used?
   15. washerdreyer Posted: June 05, 2003 at 02:12 AM (#611260)
Has anyone who has access to 2002 Win Shares done a comparison or correlation between three year rankings by Win Shares and three year rankings by SLWTS? I'm not sure what the point of this would be other than checking for the degree of agreement or lack thereof, but I for one would find it really interesting. Actually, looking at the correlation of just their defensive rankings could be great, since it could be surprising to see how much or how little accuracy Win Shares loses by not using Play by Play data.
   16. MGL Posted: June 05, 2003 at 02:12 AM (#611262)
MKT,

The numbers in the Superlwts charts are not regressed. They are absolute numbers. When I talk about regression, I am talking about translating a player's absolute Superlwts into a projection or into an estimate of their "true" performance level (what would their Superlwts' numbers look like if they played an infinite number of games, hence there was no random fluctuation?). The easiest way to get a handle on this if you are not familiar with the difference between a sample performance result in baseball (or in anyhting else) and an estimate of a true result, is to think in terms of BA. If a player hits .488 in 50 AB's (like Matos of Baltimore), and you know nothing else about him, what do you think his "true" BA is (which is basicaly the same thing as asking, "What is the best estimate of what he will hit during any period int he future, not considering age progression and things like that?")? Intuitievley you know it's not .488, and that the answer is somewhere between an average major league BA of .260 (or whatever it is) and the .488. That is "regressing to the mean", the mean being the .260. In the case of any of the Syperlwts categories, the mean is zero (since an avreage player has a run value in any category of 0, by defintion (I make it that way)). You also know intuitively that the fewer AB's a player has, the more we ignore his actual BA and stick with something close to the league average. If a player is 0 for 1 and that's all we know, what is the best estimate if his true BA? Again, intuitively you know that it is right around league average (slightly less, due to the fact that the 0 for 1 tells us SOMETHING). So that is why and how we regress a player's sample performance towards the "mean", the mean being the league average for that type of player, and the fewer the number of AB's (or chances for defense or GDP's, etc.), the more we regress. This is always true, and always must be done for any sample measure of performance, whether it be UZR, BA, or ERA, if you want to "translate" that sample performance (like the .488 in 50 AB's) into a projection or an estimate of that player's true performance. As the article says, if a player has a Superlwts of +40 per 162 in 1000 PA, that is NOT the best estimate of his "true" Superlwts. The +40 must be regressed towards zero. +30 might be a better estimate. Ditto for a -20. -15 might be a better estimate. If a player has a +40 Superlwts (per 162) in only 100 AB's, then like the player with a .488 BA in 50 AB's, if you want to estimate his true Superlwts value, you must regress the +40 A LOT. SOmething like +10 might be a reasonable estimate of his true value or his projection (these are pretty much always the same thing - a player's "true" talent and their projection, although technically "true talent" is context neutral and a "projection" may be for a particular team or in a particular park, or in a particular lineup, etc. But if by "projection" you do not specify any particular context, then "true talent" and projection are pretty much the same thing. But remember, all this talk about regressing the Superlwts numbers as a function of the sample playing time (the sample size) is only applicable in terms of comparing one player to another to see who is likely "better", or a player's "true talent", or for a player's "projection". The numbers in the Superlwts charts are absolute numbers and do not include any regression. That is left for the reader or analyst to do if he wishes (and must be done if you want to "value" a player for salary purposes, trade evaluation, player acquisition, etc).

BTW, Tango meant "sold" in terms of convincing readers of the efficacy of the various Superlwts methodologies, particularily UZR. I'm not interested in selling any of my stuff to anyone. Heck, I'd give it away if a team were interested, and I'm POSITIVE that I could (as well as many regulars on Primer and Fanhome) save and earn a team millions of dollars in proper player acquistion, salary, optimal lineup formulation, (that's another story) etc.

I also agree with Tango and the guy who originally suggested it, that the reason there is more disparity among the clasicallly non-defensive positions, is that in order to play SS, you have to have a decent glove. It is a given that all the SS's who can't hit (which is most of them) have a decent glove (although if you look at S-lwts, you will see that meny of these so-called "glove-men" are actually "schlub-men" - Womack, Cruz, Guzman, etc.). Even if you put a good hitter at SS who is not that good defensively (Jeter) or gets too old for that postiion, you can always move him to a less demanding position (Ernie Banks). On the other hand, there are lots of terrible outfielders (and third baseman) who are only there because they can hit, and there are only so many DH spots available (and some players refuse to DH, especially at a young age). This brings down the average defensive talent in the outfield (left and right), whereras the average defensive talent of CF and SS (and to a lesser extent 2B) stays high because there are plenty of good glove, no-hit players to go around, and, as Bill James says, you can alwyas move someone down in the defensive spectrum, but not up...
   17. Jason Posted: June 05, 2003 at 02:12 AM (#611265)
Well I think UZR is a good MVP tool, but I'm not convinced as currently constructed that it would help you build a team any better than other methodologies. Mostly what you want for a team building device is an interactive player description. A single number is dare I say meaningless because for comparison everything is neutralized. Stickign with the Anaheim example though having a neutralized CF whose better with the stick and worse with the glove yet overall equal to Erstad would seriously hurt the club because their context is not neutral they rely strongly on CF defense to catch all those fly balls.
But the interactions can get more nit-picky though not neccessarily less important. Let's say you build your pitching staff to crank out ground balls to SS and 3rd base where you find the biggest range guys you can get. Problem is their arms tend to be wild. Well Having Doug Meincevick isn't going to be a big help, but John "scoop shovel" Olerud is going to shave runs off the board like no tommorrow. In essence you want something more like Diamond mind and less like Grand Unified Stat theory.
But those examples are extremes and generally teams fall close to the median like a well behaved normal world should, so they aren't really important you might argue? I'm going to disagree because this is all about finding little advantages. And as we move to an era of more mix and match players and less overpaid bloat proper team construction is a huge edge. OK guys we'd don't have the budget for a good RF, do we want the Matt Stairs all hit no defense model, the Alex Ochoa no hit all field model, or just a guy with a cannon who can hit OK? You make a lot of these interchangeable parts decisions already and more will be needed in the future. And I'd wager that gaining an advantage in optimizing your team on those margins will be a huge advantage.
   18. tangotiger Posted: June 05, 2003 at 02:12 AM (#611267)
I agree with Jason and Colin. (Colin: I think MGL talked about this issue regarding confidence in subcomponents.)

Jason: there's no question that you are right. If you view this in baby steps, you can view ZR as step 1, basic UZR as step 2, and MGL's UZR as step 3. If a team would stop there, they'd be better off than not using anything of course, though, we don't know if they'd be better off than discarding their scout evaluation.

While you could probably say that the "true talent" of a hitter is recognized almost entirely in his performance numbers of the last 3 years (with maybe a some scouting to supplement that), we have not established at all the balance between UZR/scouting. Maybe hitting/scouting is 80/20 and maybe UZR/scouting is 60/40. I don't know.

In any case, what you are describing is really creating a true model that will use these true talents, within the expected context of a game for that team. I agree with this 100%. Step 4 or Step 5 or Step 6 would be able to use that information.

If we let's say go to hitter's as an example, say we find that Larry Walker is +50/600 PA in a context-neutral setting, but against lefties groundballers at Coors, he's +30/600 PA. And if Walker is expected to face alot of those (more than average), then he's not as valuable as you have him.

Same deal with fielding, and anything else. While taking him into a context-neutral setting is a "fair" way to rate a player, you could conceivably say "hey, rate him relative to the average player, but in the most optimal setting for this hitter [or fielder or pitcher]". All of a sudden, Hurst/Ojeda/Tudor look horrible with respect to fenway, but look sensational with respect to their optimal park.

By the way, as MGL does UZR, he is NOT putting everyone in the same context-neutral setting. What he is doing is taking a league average player, and putting him into Erstad's or Roken's context setting. If Erstad is showing +50, you can read this as "Erstad is saving 50 runs more than a league average CF would, if this league average CF would play in the same context that Erstad faces". (This is what I call value.)

If you switch it around and ask "how would Erstad's performance numbers translate if he faced the same context as the league average player, and how would Erstad compare to the league average player", maybe Erstad (who let's say derives alot of his value from getting alot of opps that he can maximize more than others) will come in at +35. This figure, after you've regressed it somewhat, is what I call ability.
   19. Jason Posted: June 05, 2003 at 02:12 AM (#611268)
Thanks for the clarification I had how he was doing context backward, and it does answer the team perspective issue relatively well, but it makes it somewhat imprecise for making player to player comparisons, since based on opportuity their average maybe different. I think I'll go work on some simpler problems like detecting planets oribiting stars hundreds of light years from us.
   20. Buford Sharkley Posted: June 05, 2003 at 02:12 AM (#611271)
Excuse me if I'm missing something, and I don't mean to be critical, but.....

....is this saying that in 2000, the MVPs, Kent and Giambi, were less valuable than Geoff Jenkins and Bobby Higginson?
   21. tangotiger Posted: June 05, 2003 at 02:12 AM (#611272)
but it makes it somewhat imprecise for making player to player comparisons

Yes, that's correct. The comparison baseline (for UZR anyway) is the average player in player x's context. The comparison baseline (for batting runs) is player x's in a league-average context. (I think I got that right.)

As for Higginson/Jenkins, yes, you are reading it right. They are not necessarily better players, but according to sLWTS, they did perform better. However, confidence levels are not attached to the metric (when they really should be). How certain are you that his fielding was +30 runs? With all the manipulation of the various context and subcontexts, maybe there's a 15 run swing in there.

Giambi might have performed very well with men on base, and so he doesn't get the extra credit for doing well (that he should get, if you talk MVP). So again, you might have a 10 run swing in there.

(And this is true for ALL metrics, and not just sLWTS.)

So, when you look at this lists, if they don't provide it, you have to try to figure out a confidence interval, and maybe it'll come out that "I'm 40% sure that Higginson performed better than Giambi did".
   22. Kevin Harlow Posted: June 05, 2003 at 02:12 AM (#611273)
The positional adjustment is not correct. The players are categorized by primary fielding position and then the positional adjustment is made based on that primary fielding position. The adjustment should be made for each position played. This doesn't affect most regulars since they either almost always play one position or they play two positions close in the defensive spectrum. However, it is not accurate for players like Melvin Mora. In 2002 he was categorized as a LFer since he played more innings there than any other single position. The LFer adjustment was -11. However, Mora played about 619 innings in LF+RF and 648 innings at 2b+ss+cf. Mora's positional adjustment should have been close to 0.
   23. tangotiger Posted: June 05, 2003 at 02:12 AM (#611274)
Kevin, another good reason to make the pos adj to UZR at the beginning.
   24. Boileryard Posted: June 05, 2003 at 02:12 AM (#611275)
Higginson being so highly rated in 2000 is far from out of the realm of reality. He had 19 outfield assists, which is a lot of defensive runs. He also had a nice 44-double, 30-HR, 74-walk season. Keep in mind, the Tigers challenged for a wild card spot that year. That had to come from somewhere.
   25. MGL Posted: June 05, 2003 at 02:12 AM (#611277)
I'm a bit confused. First, I was under the impression that MGL did prorate the positional adjustment according to time spent at each position. If not, that would be a worthwhile refinement. Second, as far as Tango's explanation of UZR, I thought MGL was doing the opposite--assigning opportunities based on league avg context, not the context of the player's (Erstad's) own team.

No, I use a player's primary position for the positional adjustment, which is not correct of course for player's with multiple positions (I will refine that in the future).

And yes, a player's rating in each category is prorated to the league average number of opps, so I'm not ecactly sure what Tango means. If Erstad got 300 opps in 100 games and had an absolute UZR of +20, but the league average number of opps is only 200 opps per 100 games (say because ANA was a flyball staff), then we use 150 games for Ersatd's +20 and NOT 100 games, so his UZR per 162 is NOT +32.4, but +21.6. So yes, David you have that right. I think what Tango means is that aside from the "chances" (opps), Erstad's UZR is based on the fact that he may have had 20% of his chances hit in tough zones and 80$ in easy zones, whereas an average CF'er may get 25% hit in tough zones and 75% in easy zones. I don't think this makes much of a difference.

Jason, while you have a good point, I think you are WAY overstating the issue! If teams used nothing else but S-lwts to construct their team (with no adjustments for conext) they would be WAY ahead of the game. Look at the salaries of players next to their S-lwts and you will see how many grossly under and overrrated players there are. Changing contexts is NOT going to make much of a difference in a player's overall S-lwts rating! It's not going to make a +20 player a +5 or a -10 player a +3. Some of the components are barely susceptible to a change in context, and none of them are going to VERY susceptible. If you can give specific examples of how MEANINGLESS the context neutral S-lwts numbers are, I would love to hear it! What makes you think that if you replaced Ersad with a player who was average defensively, but as good offensively as he is defensively, that it wouldn't be a wash (give or take a few runs)? Please explain that to me in concrete terms. (Besides the fact that 1 run saved is always a little better than 1 run gained - i.e. defensive runs are worth slightly more than offensive runs because a run in a low scoring environment is worth more "wins" a run in a high scoring environment.)

Also, you can easily take the S-lwts components and adjust for context! I do it in my sim. For example, an OF'er UZR projection gets "prorated" according to the number of fly balls that a particular pitcher allows (based on his BIP percentage and his G/F ratio).

One thing I found when I was adjusting UZR for the G/F ratios of each team's pitching staff was that is unusual for an entire pitching staff to be an extreme GB or FB staff. The pitchers tend to cancel one another out. That should be obvious.

Sure, every single one of the Super-lwts ratings can and should be adjusted for context in constructing a team, evaluating trades, etc., and like I said, that's a very good point and I'm glad you mentioned it, but if you were to to do the work (say take 9 random players and put them on the ANA team and then refigure all of their S-lwts according to the context of the ANA pitchers, the intereaction of players, etc.), I will GUARANTEE that the differences you will find in their total "adjusted" S-lwts compared to their context neutral S-lwts will be MINISCULE. Try it if you don't believe it!

Your argument is like saying that A-Rod, Rey Rey's, Sosa's, L. Walker's, etc., OPS or EQA is meaningless becuase all of these things are context neutral. They would be different if you redid them in the context of their respective teams and lineups (using custom linear weights or whatever). For example, when Rey Rey played for the Mets, his true lwts values would be completely different from L. Walker's becuase of the home parks they play in. How much do you think of a difference using these custom lwts values would make in terms of characterizing these players?
   26. MGL Posted: June 05, 2003 at 02:12 AM (#611279)
Concrete examples huh? Sounds like work :) I wasn't trying to slam your system and meaningless was tossed out in a good natured way.

No prob, and as I said, a very good point. If I were advising a team, S-lwts would be a starting point. I would definitiely adapt (adjust) each player's rating in each category to that particular team (home park, pitching staff, other players, proposed lineup slot, etc.), and THEN estimate that player's value as compared to another or in terms of salary valuation...
   27. Kevin Harlow Posted: June 05, 2003 at 02:12 AM (#611280)
I know that beggars can't be choosers, but on the other hand beggars, by definition, beg, so I'd like to beg for Super Linear Weights to be presented with separate lines for each player based on team and position. I would assume that this would require modifying your code that processes the data, so I realize this is a large item to ask for. However, in preparing this detailed data you would end up being forced to calculate the correct positional adjustments based on playing time at each position. Furthermore, this data could be used to take a more team-oriented view without letting trades and such complicate matters. Also, I'd like to see how players skill at different positions compare(selection bias noted).

Some misc. comments: 1) even though it probably doesn't make too much of a difference, using not only league average opportunities but also league average opportunities in each zone probably does make more sense for UZR. 2) Is using league average opportunities for ARM and CAT strictly correct since the %success is a function of # of opportunities, or alternatively that the # of opps is a function of success rate? 3) A separate value metric where everything adds up on the team level would be nice, especially for comparison and validation of the use of Win Shares for historical purposes. 4) Without going back thru the posts I believe that someone else suggested the inclusion of pitcher stats, which I second. It would be very nice to have PBP-based "win shares" (I'm not suggesting that your ability-based metric isn't incredible though!). 5) Do I understand correctly that the positional adjustments are based on what is needed to make each position's slwts equal to zero for that season?
   28. MGL Posted: June 05, 2003 at 02:12 AM (#611282)
Rally, good post!

If I had time I'd look at 3 year SLWTS and compare them to 2003 salary for all of this year's offseason free agents.

Come on, make the time - I'd love to see that!

Kevin, lots of good points. I can and will redo the charts so that each player has a separate line for each position. If you look at my UZR files (Tango, where are they?), you will see the UZR rating of every player at every position (2000-2002 and combined).

Yes, the positonal adjustment is such that all players at that position add to zero, or alternativel;y, if you add the Superlwts of all the players at that position, not including the positional adjustment, the sum (per 162) IS the positional adjustment.

Your point about the value of a category, like arm, or the SB/CS part of the catcher being a function of the number of opps is of course true. How I should handle that I'll have to think about.

Yes, most "statheads" thought that Erstad was a terrible signing (at that price). I think Neyer wrote a column as such. I'm not sure I'm willing to give ANA credit for "knowing" that Ersatd was worth so much defensively. It may have partly been a fluke (what about Rey Ordonez - for years the Mets thought that his defense was worth the money and the playing time, plus teams' think that certain players are great defenders when in fact UZR does not suport that notion (T. Hunter, Snow)). Also, again, if I am evaluating a team, I have to give less weight to any defensive evaluation for 2 reasons. And that is why you have to regress the sample UZR numbers more than the other ones. One, any measure of defense is less accurate and reliable than other measures. Two, defense is very sensitive to age, injury, and size (bulk and weight). If you sign a player to anm expensive long-term contract becuase of his defense, you are asking for trouble. You are much better off giving out long-term contracts for offense...
   29. tangotiger Posted: June 05, 2003 at 02:12 AM (#611285)
If you go to the above link, you will see MGL's csv file. If I remember right, it has one line per player/team/pos/year.

If you go back up one level, there are a couple of UZR files. One definitely has player/team/pos/year for 1999-2002. Another is an HTML file that I created based on that.

There's tons of data in there to keep many sabermetricians happy for months to come.
   30. Silver King Posted: June 05, 2003 at 02:12 AM (#611286)
Kevin wrote:
"I know that beggars can't be choosers, but on the other hand beggars, by definition, beg..."

hee hee

"...so I'd like to beg for Super Linear Weights to be presented with separate lines for each player based on team and position."

I was gonna beg for that too. But here's my other beg. In the past, you've indicated that it'd be possible to go back further years with your SLWTS revelations, back to the edge of the eighties, even?!? If that's still even semi-possible, hey it'd be great to see that eventually.

The discussion is being really neat.
   31. Charles Saeger Posted: June 06, 2003 at 02:12 AM (#611287)
You know, if you're going to use PBP fielding data, why not use PBP hitting data, reflecting changes in base/out states? You've done so much work with PBP fielding that the hitting pales next to it.
   32. tangotiger Posted: June 06, 2003 at 02:12 AM (#611291)
Charlie, check out the above link for something that I did when Ichiro went MVP. On my site I also have LWTS by the 24 base-out states. I also last week published LWTS by the Game State (inning/score/base/out) on Clutch hits.
   33. MGL Posted: June 06, 2003 at 02:12 AM (#611303)
UZR uses average values for all hits (and outs and errors). It doesn't "care" about the actual outs or baserunners.

Actually for hitting or for fielding, I find using bases/outs states uninteresting and not particularly useful. Even if there were evidence that clutch hitting or fielding "exists", there would be absolutely no way of separating "real" clutch performance from phantom or random clutch performance.

The only thing you can really do (and it is not easy or particularly useful for various reasons) with PBP data to "enhance" a player's offensive evaluation is to look at things like "Did a player have an inordinately high or low number of bloop hits, line drives caught, infield hits, etc. Even then, you would have to be careful, as players hit line drives at different average speeds (the better hitters hit harder line drives), so that while it might look like Player A got "lucky" because few of his line drives were caught, it might be that his line drives were hit a lot harder, or it might look like a player got an inordinately high nubmer of infield hits, and you might attribute that to luck, but it might be because he is very fast, or it might look like Player B was lucky in that he got he an inordinately high number of bloop hits, but it might be because the OF plays him very deep (he has lots of power), etc...
   34. tangotiger Posted: June 06, 2003 at 02:12 AM (#611306)
MGL, since players change their hitting approach by the 24 base-out states, and since leadoff hitters especially have a very skewed distribution PAs in those states, then you would have to at least account for this.

The line-drive thing is interesting, though I'm not sure how reliable it would be.
   35. MGL Posted: June 06, 2003 at 02:12 AM (#611309)
MGL, since players change their hitting approach by the 24 base-out states, and since leadoff hitters especially have a very skewed distribution PAs in those states, then you would have to at least account for this.

Yes, I agree. A hitter's stats should at least be "adjusted" for his average hitting state, like I do for fielders in UZR. It's probably even more important for hitters, since, as you say, hitters will probably vary more than fielders as far as their average hitting state (bases/outs)...
   36. tangotiger Posted: June 07, 2003 at 02:13 AM (#611317)
how well a first baseman can catch bad throws and strech out to make an out

How about relay throws, blocking the plate, and calling a game? Of course we should have all that. I just don't think there's enough information in the pbp to figure that out.
   37. MGL Posted: June 07, 2003 at 02:13 AM (#611323)
Tenkev,

Yes, for example, supposedly Snow is great at receiving throws and perhaps saves many errors by the infielders per season (the data supports this suggestion). It is difficult to measure, however, but not impossible to measure it to some degree. One, I could automatically assign some of the value, negative or positive, of each infielder's throwing errors, relative to league average, to the first baseman. Two, I could look at throwing errors of all infielders with first baseman A and compare that to the throwing errors of those same infielders with first baseman B. I might take a look at this in future versions.

Relay throws and blocking the plate? Hmmm...
   38. Kurt Posted: June 09, 2003 at 02:13 AM (#611332)
Hey there,

First off, MGL, very cool stuff. Thank you.

The idea of comparing Super Linear Weights to 2003 Free-Agents Signings sounded very interesting to me, so I collected all the data I could. I'm still in the midst of running some analysys on these, but I've gotta run to work, and thought I'd post at least the data. Correlation between SLTWTs and annual salary seems to be about .64; that's more than I (and probably some on this board) would have guessed. Marquis Grissom is the biggest standout of this whole set (ugh). Some things to keep in mind when doing the SLTWTs to Salary comparison: (1) I only included anual salary (so far) and contract length should be factored in somehow (as a 3 year $9 mil contract is "better" than a 1 year $4 mil) (2) the SLTWTs MGL posted are only a description of the last three years, and I would assume contracts are based on what a player is expected to do, not what they have done (at least I would hope).

Anyway, about 1 in 5 of the 2003 FA signings I couldn't find dollar information in their contracts, many were pitchers, and of the rest there were only 47 guys. Not the biggest population, but a starting point. Here they are:

Name $/yr(mil) 3yr SLWTS(PO/162)
Alfonzo, Edgardo 6.5 29
Alomar, Sandy 0.7 -13
Bell, David 4.25 22
Bell, Jay 0.55 -11
Bordick, Mike 1 19
Catalanotto, Frank 2.2 -5
Colbrunn, Greg 1.75 1
Coomer, Ron 0.65 -36
Cordero, Wil 0.6 -44
Cruz, Deivi 1.2 -26
Cruz, Jose 2.8 -13
Durham, Ray 6.7 32
Finley, Steve 5.625 1
Floyd, Cliff 6.5 17
Galarraga, Andres 0.5 -35
Glanville, Doug 1 -46
Goodwin, Tom 0.635 4
Grace, Mark 2 -1
Grissom, Marquis 4.25 -39
Harris, Lenny 0.8 -18
Haselman, Bill 0.55 6
Hernandez, Jose 1 18
Hollandsworth, Todd 1.5 8
Houston, Tyler 1 -10
Kent, Jeff 9.1 41
Kreuter, Chad 0.75 19
Lofton, Kenny 1.025 18
Loretta, Mark 0.625 -1
Mabry, John 0.85 -32
McGriff, Fred 3.75 -9
Mueller, Bill 2.25 -2
O'Leary, Troy 0.75 -10
Olerud, John 7.7 13
Ortiz, David 1.25 2
Perez, Neifi 2.375 -13
Pratt, Todd 0.85 9
Reboulet, Jeff 0.6 -11
Rodriguez, Ivan 10 55
Sanchez, Rey 1.3 12
Sanders, Reggie 1 7
Sierra, Ruben 0.6 -19
Stairs, Matt 0.9 -13
Stevens, Lee 0.4 -19
Thomas, Frank 5.625 18
Thome, Jim 14.16666667 39
Ventura, Robin 5 25
Zeile, Todd 1.5 -5
   39. tangotiger Posted: June 09, 2003 at 02:13 AM (#611337)
To turn sLWTS into $, do the following:

1 - take the 3-yr sLWTS / 162, divide by 10, and multiply by 2 million$ (maybe 1.5 these days?).
2 - add 4 million $
3 - figure out the % of expected playing time (say 145/162 GP)
4 - multiply #2 to #3

1 - Let's take Fonzie. Supposing he's +27 runs / 162. That works out to about 5.4 million marginal dollars above average.
2 - Now he's at 9.4 million total dollars
3 - Say 90%
4 - Now at 8.5 million $

I understand the length of contracts, the discount rate is an issue, as well that the sLWTS value should be regressed to better match his true talent level. His age should be considered as well to make for a better projection.
   40. tangotiger Posted: June 09, 2003 at 02:13 AM (#611347)
Rally: it works out to the same thing.

Let's suppose you have -20 runs / 162 GP. That's -2 wins / 162 GP, or -4 million $ / 162 GP compared to average. Add 4 million $ / 162 GP that an average player gets, and this player is worth 0 dollars.

In essence, what I have calculated is dollars over replacement! The step-by-step shows how you don't have to initially think about replacement level, that you can make the process based on comparison to the average level, plus the playing time.

To summarize:
wins over repl per 162 = wins over average per 162 + 2

salary = (wins over average per 162 * 2 + 4 ) * GP/162
salary = ((wins over repl per 162 - 2) * 2 + 4) * GP/162
salary = 2 * (wins over repl per 162) * GP/162
salary = 2 * wins over repl

Take the player's total wins over repl (not /162), double it, and that's his salary.
   41. tangotiger Posted: June 09, 2003 at 02:13 AM (#611348)
Rally: I posted before I saw your followup post.
   42. tangotiger Posted: June 09, 2003 at 02:13 AM (#611351)
ARod and Bonds, for example, are about 300 runs over replacement according to sLWTS, or +30 wins, which translates to 60 million$ of "salary earned on performance" over the last 3 years.
   43. dlf Posted: June 09, 2003 at 02:13 AM (#611357)
I join Chris Dial's poetic Swahili of SLWTS. Good stuff.

On June 4th, Vinay Kumar wrote "...Win Shares uses basic traditional data (PO, A, estimated innings, etc.)..." Generally correct, but Bill James expressly cautioned against estimating defensive innings and, from what I could tell, didn't use them in his metric.

On June 5th, Jason wrote " Well I think UZR is a good MVP tool ..." No, no, a million times, no! SLTWS and its components expressly attempt to elimate value. A player who is +10 per 162 games will always show up in SLWTS as "better" than one who is +1/162; however, in every sense of the word, the later is more valuable if he plays significantly more than the former. SLWTS attempts to eliminate context from the equation and is, as such, an attempt to distill ability, not value. SLWTS could easily be used for MVP-type discussions by simply presenting the material per actual opportunities instead of per normalized opportunities.

I wonder how much of the .5 correlation between UZRs of adjacent years is due to the adjustments? Does the correlation improve or worsten if park adjustments are taken out. Back on baseballboards or fanhome or whatever, it was reported that Torii Hunter's road UZR was significantly positive while his home UZR, after adjustment, was significantly negative. That makes me wonder whether we are measuring performance or adjustments.

Does the PBP data indicate whether a runner was going on the pitch if the ball is put into play? I find the hit and run to be an incredibly exciting and enjoyable play, but have never seen any analysis supporting or debunking it. As a first approximation, I think the SB/CS data on swinging strikes versus all other pitches would be interesting, but obviously it leaves out all successful efforts.

Nitpicky details aside, I think the concept behind SLWTS is fabulous and thank MGL for his willingness to present it publicly.
   44. Kurt Posted: June 09, 2003 at 02:13 AM (#611360)
OK, when I posted the data in text it looked like crap, so here's a graph. Each blue point represents a free-agent signed this year, and his SLWT and his annual salary. The pink line is the "best fit" for this distribution. An interesting thing about the best fit line is that a hypothetical free agent with zero SLTWTs would be payed $2.7 million in 2003, which (if memory serrves corect) is league average salary. Kinda neat that. Also, the line croses $400,000 (league minimum salary) between -12 and -13 SLWTs, meaning (based on prices payed for 2003 free agents) you shouldn't have signed a FA with SLWTs less than -12 in 2003 at all (if he's expected to play full time, that is).

Tango, how did you come up with that formula for salary from SLWTs? I know I used a line to approximate also, but it strikes me that we should be using a curve; the marginal utility of each additional SLWT will increase the beter the player is. That is, if going from 0 SLWTs to 1 SLWTs in one player worth $200,000 wouldn't it make sense that going from 40 to 41 SLWTs would be worth more than $200,000, as its significantly harder to improve on? This would be a result of having only 9 men on the field at a time, and 25 men on the team at at time, and 40 men on the roster at a time. Since you have limited number of players, improvements in the good ones will be more "valuable" than improvements in the marginal ones.

Have you ever considered using an exponential equasion to fit SLWTs (or something similar) to expected salary?
   45. tangotiger Posted: June 10, 2003 at 02:14 AM (#611370)
There's several independent ways to calculate the marginal $ / marginal win:

1 - I figure that 1 win adds 2% attendance to a team. Assuming that 2% carries to all team revenue (including tickets and TV), and the average team has 100 million$ of revenue, 1 marginal win = 2 million $ of revenue. Maybe 1.5 should go to the player?

2 - The average team has a 60million$ payroll. The minumum payroll is 10 million$, or +50 million $ of marginal payroll for an average team over a replacement team. A replacement team should win 40 to 60 games, which makes an average team +20 to +40 wins above replacement. That gives you about 1.5 million $ / win.

3 - I did a a study at fanhome a couple of years back that shows that the average team paid 3 million $ / win.

2 million is a nice round number. If you want to use 1.5, that's ok, and probably what we should use.

As for an exponential, that makes it too complicated for something that is not yet proven. I understand the roster constraints, but I really don't see it right now. Maybe teams pay a premium for "known" quantities, but again, too complicated for my tastes. I just find it easier to explain and justify "1.5 million marginal $ per marginal win", and leave it at that.
   46. Jeff K. Posted: September 23, 2004 at 07:28 PM (#872291)
Just a random post. Never mind me.

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