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Last year, I presented a theoretical model as to the talent distribution in MLB (actually, the world, too).

Can we figure out what it actually is, and can we use that to figure out the replacement level?

Here are my quick thoughts on the matter...

What is the spread in talent among MLB players?

Just doing a quick analysis, and making some key assumptions, let's say we have:
- distribution among nonpitchers = 18 runs per 680 PAs (includes off and def)
- distribution among pitchers = 12 runs per 162 IP
- we have 9 nonpitchers and 9 pitchers, each playing equally

If the hitters are randomly distributed, the team nonpitchers will be distributed at:
sqrt(18^2 * 9) = 54
and the pitchers at
sqrt(12^2 * 9) = 36

Anyway, assuming that pitchers and nonpitchers are also randomly distributed, that gives us:
sqrt(54 + 36) = 65 runs

That is, a set of random teams with random players will produce 68% of the teams of true talent within 65 runs of the league average.

Using a 10.5 runs per win converter, that gives us 6.18 wins per 162 games, or 1 sd .038 wins per game.

We would have expected .5/sqrt(162)= .039

So, my quick model here is decent enough.

Note: I am aware that this does not model MLB exactly, and my theoretical model that I linked would make more sense. But, this is as far as I can go in 15 minutes and my current limited statistical abilities.

What does this tell us? Well, in this stylized example, we have established exactly what the distribution of baseball talent is! Furthermore, if you decide that the "replacement level" is say the bottom 16% of all players, then we also know exactly what the replacement level is: 18 runs for nonpitchers, and 12 runs for pitchers. Or, if you want the replacement level to be the average of all players below 1 SD, then.... well, I'm sure someone has a handy function to tell us that. I'm guessing about 24 runs (per 162 GP) for hitters, and 18 runs (per 162 IP) for pitchers.


The true distribution among teams is not random, as evidenced by the actual team win% being distributed at .080, meaning that the true talent team win% is .070.


Given the priors for player distibution, we should be able to come up with a distribution so that we end up with a true team distribution of .070. My question: how?

Talent Levels and Replacement Level | 24 comments | Create New Account
The following comments are owned by whomever posted them. This site is not responsible for what they say.
Craig B - Friday, September 24 2004 @ 02:38 PM EDT (#20669) #
Tango, this is an excellent method for describing a theoretical "replacement level", from which we could presume the observed variation is a result of individual players out- or under-performing their true talent level.

One real-world problem among many : Players are not in fact chosen rationally according to their talent level. In fact, they tend to be chosen according to their observed level of talent, as well as for other factors thoguh these are probably minor but not insignificant. So the "replacement level" players, those not initially preferred, will in fact include some players whose true talent level is higher, in some cases considerably higher, than the regulars. Calvin Pickering is one of many examples immediately to hand.

What I usually do to gauge a "replacement level" for a league, just generally, is to look at who the replacements actually were... i.e. sum together all the actual performances by players with few appearances, like all starters with 8 starts or less or all players with fewer than 100 at-bats. That sort of thing. It's quick and very dirty, but it does a pretty good job! Obviously, that sort of thing is not really close to what you're doing to set a theoretical level, but it's an interesting counterpoint.

My own method is prey to selection errors... David Newhan, for example, was a replacement player by definition (not good enough to make a major league roster until there was nobody left to fill a role), but he doesn't fall in my hypothetical study because he performed too well!

The problem I've always had with the notion of "replacement player" is that is makes a great deal of counterfactual presuppositions about the structure of major league baseball. In fact, when applying the notion of "replacement level" to real-world situations, the actual "replacement" is quite identifiable... he's the guy on the bench, with the AAA guy getting the remainder of the time that used to belong to the bench player. Comparing players to a "replacement level" is bogus in my view, a sort of fake-GMery; if you want to make a counterfactual comparison, make it to the player who was actually going to be the replacement. I know I'm in the distinct minority here, and it's not important - I've lost the battle (hence the development of stuff like Win Shares Above Replacement, which takes "missing the point" to new levels of incredulity).
_tangotiger - Friday, September 24 2004 @ 03:02 PM EDT (#20670) #
"Replacement level" as a term is frought with ambiguities, like "valuable". I usually try to use "baseline" instead, so I can define my own meaning.

Replacement could mean: (1) immediate replacement from within, or it could mean (2) immediate replacement from outside the organization (scrap heap), or it could be (3) mid-term replacement, or (4) long-term replacement.

If Ty Wiggington went down with an injury in May, replacement could be:
1 - David Wright
2 - some guy cut from a 40-man roster in spring training
3 - whatever you can get in a trade for a guy who is costing you more than he's putting out (Kaz Matsui), but that some team is willing to put up with that
4 - a draft pick, or a rebalancing of your team (like getting Beltre for Reyes or something)

I try to view it monetarily. The minimum wage is 300K, so, what player, or what shifting of my players around, will get me an extra player on my roster for 300K. I don't see the point of replacement-level, from a practical standpoint, beyond that thinking.

All I want to do is create a function that says:
salary = true talent level above baseline * PAs above baseline + 300K * % of days on the 40-man roster

Other than that, I don't have much use for replacement level talk, in a practical standpoint.


The shorthand though is that we never know the true talent level, so that needs to be regressed. So, if you employ a particular baseline level, you kill 2 birds with one stone by using a replacement level. I think that hides what we are really after.
_Jim - Friday, September 24 2004 @ 03:04 PM EDT (#20671) #
Craig’s post is exactly why I like things like Lee’s RCAA/RSAA in his ATM reports. It could be much more useful if there was a positional adjustment, but average is much easier to work off of (for me at least). Throw in the fact that I think most of the ‘replacement level metrics’ show replacement level as higher then I feel it truly is. I believe that SNVA uses a .425 pitcher as replacement level. IRL the Yankees have a hard time finding replacement pitchers who would work out at .425 (Halsey, Loaiza, ect.)
_studes - Friday, September 24 2004 @ 03:08 PM EDT (#20672) #
hence the development of stuff like Win Shares Above Replacement, which takes "missing the point" to new levels of incredulity

Oh, good grief. I've stated repeatedly that true WSAR is yet to be established. I've just used thresholds once or twice when I thought that not doing so would be worse. But I've always admitted there's more work to do.

This offseason, I was thinking of taking exactly your approach, Craig. Using Win Shares, Expected Win Shares and Win Shares Percent, it should be relatively easy.
Craig B - Friday, September 24 2004 @ 03:27 PM EDT (#20673) #
studes, I don't mean to poke fun, sorry. WSAR is actually a very useful stat to have, and I look at it all the time.

I just mean that WS was sort of set up as a way to get "replacement-talk" off the front burner (I think WS if flawed, though, because it then proceeds to import it all back in via the back door).
_studes - Friday, September 24 2004 @ 04:04 PM EDT (#20674) #
Thanks, Craig. I understand, and I agree about replacement level coming in the back door of Win Shares.

Seriously, if you have thoughts about ways to properly attack WSAR, I'd love to work with you on it during the offseason. Replacement level is an extremely tricky question but, as Tango says, it is important in contract evaluations, if not other places.
_tangotiger - Friday, September 24 2004 @ 04:38 PM EDT (#20675) #
That should really be:

= true talent level above baseline * Leveraged Index * PAs above baseline
+ 300K * % of days on the 40-man roster

True talent level is expressed as a runs or wins rate per PA or per IP. I like to use lwtsOBA, which is
= .72*bb+.9*1b+1.24*2b+1.56*3b+1.95*hr all divided by PA (excluding sac bunts and IBB).

Leveraged PAs are all PAs, weighted by how much they expect to occur in game situations. For hitters and starters, the weight would be "1". For relievers, it would be between 0.5 and 2.5, depending on how they are used.

The only point of contention would be the "baseline" levels.
robertdudek - Friday, September 24 2004 @ 04:55 PM EDT (#20676) #
I think replacement level is an extremely important concept in baseball analysis.

Every year a certain number of players lose their jobs, retire or are injured to the point they can no longer play. And in each case, another player who did not have a job gets to play. I speak here on a major league wide basis.

So, it would seem to me that replacement level should be the average performance of the players who have landed in a major league job for the first time. I don't think this is difficult to compute - take the first season over 50 PA and 10 IP of every player in the major leagues.
_tangotiger - Friday, September 24 2004 @ 05:14 PM EDT (#20677) #
Robert: you didn't say why it was important.

As for taking all first year performance:
1 - those are performances, and not true talent levels. You still need to regress those performances to get to true talent levels.

2 - those players will not be weighted the same. Guys with 600 PA will count for more than 60 PAs. If you have a pool of available players, they should all count the same.

3 - why have any cutoff? A guy who went 5-45 probably will have been sent down, but he was a viable candidate (or so they thought, and maybe still think).

4 - those guys are probably going to average 24 years. Not necessarily what people think of as readily available players.

A replacement level player is more likely a guy who had less than 200 PAs last year, and less than 300 PAs last 2 years, and over 25. Just a definition I'm throwing out, and of course, using arbitrary cutoffs.


In any case, my model here would be a better way to do this, since I'm trying to create a model that would explain the performances. Once we have that, it's just a matter at setting your line.
_Magpie - Friday, September 24 2004 @ 05:16 PM EDT (#20678) #
I think replacement level is an extremely important concept in baseball analysis.

Yes, and replacement level should be where the "free" (comparatively, anyway) talent is available. Marginal players on the fringes of a job... (This is how Toronto got Rance Mulliniks, for example.)

Now the levels are different for different positions, right? A replacement level 1B is probably going to produce more offense than a replacement level SS.
robertdudek - Friday, September 24 2004 @ 05:26 PM EDT (#20679) #
It's important because baseball value needs to be evaluated in comparison to a baseline. Replacement level is the most sensible baseline because it corresponds to something real in the baseball world: the guys you can bring in from the reserve pool of players when needed.
robertdudek - Friday, September 24 2004 @ 06:04 PM EDT (#20680) #
I don't think so - a young prospect is in the replacement pool. He's one of the players a GM can choose to fill in for an injured player or replace a retired or poorly performing player. Maybe people don't think of these guys as in the reserve pool, but I think in a real sense, they are.

But I should probably add players who have had less than 300 PA the last two years - they are in the replacemnet pool in that they don't have regular jobs.
robertdudek - Friday, September 24 2004 @ 06:09 PM EDT (#20681) #
Choosing the baseline to regress to is going to be tricky isn't it? You're likely to end up with a result that is close to whatever baseline you choose. And on what basis to you choose the baseline to regress to?

Why not just have a threshold and weight everyone equally regardless of PA above - then take the average (or bettert yet, the median). Some guys will be performing above true talent and some below - so overall those tendencies should cancel each other out.
_tangotiger - Friday, September 24 2004 @ 09:21 PM EDT (#20682) #
Other than for salary or "posterior" (MVP, HOF) purposes, I see no application to needing replacement level.

I agree that how you regress will determine the talent level of your player. For example, you could legitimately regress a player to the MLB average, even if he had only 1 PA. But, since you know he had 1 PA, you would use that information as well.

That is, say you have 5 rookies, and they had 4 PAs (0-4, 1-4, 2-4, 3-4, 4-4). The question becomes: what is each of their true talent levels?

Typically, you would simply regress to the league mean OBA of .333, and say that they are all going to be around .330 to .336 or so.

However, you and I both know that if you get only 4 PAs in a season, you are probably not that good to begin with. So, you scour the historical records for rookies with 4 PAs, and you find that their performance in year x+1 was .280 OBA. Now, all of a sudden, we have a new piece of information: the population mean of these players is the 4PA mean of .280.

You can probably construct a function based on PAs, age, and years of service that would establish what the population mean you would regress a player towards.
_tangotiger - Friday, September 24 2004 @ 09:33 PM EDT (#20683) #
Btw, even for salary purposes, you don't really need replacement level.

Say for example that the repl level is -2.5 wins below average, per 162 GP. And that the marginal $ / marginal win is 2 million $.

Say that the average payroll for nonpitchers is 45 million$. So, a full-time average hitter would get 45/9=5 million$.

How much would the repl-level hitter get? -2.5 * 2 = -5 million$ relative to average. Since average is 5 million$, then a repl hitter get 5 - 5 = 0 million$.

How about a guy who is 5 wins above average? Well, that's +10 million$ above average, or 15 million$ total.

5 wins above average is also 7.5 wins above repl, right? 7.5*2=15 million$.


How about we look at a part-time player. An average hitter in 81 games should get 5 million$ / 162 * 81 = 2.5 million$.

How about a guy who is +5 wins per 162 games, but only played 81 games? He's +2.5 wins above average x 2 million$ = 5 million$ above average, which is 7.5 million$.

If we first calculated repl-level, then he'd be +7.5 wins per 162 above repl, or +3.75 wins over the 81 games above repl. Multiply by 2 million$, and you get 7.5 million$.


See? Even without the need to determine a baseline level, I can still get you the player's earned salary. So, all that leaves us with is MVP/HOF talk.
robertdudek - Friday, September 24 2004 @ 09:44 PM EDT (#20684) #
What if you want to know if a certain below-average player should be replaced?
robertdudek - Friday, September 24 2004 @ 09:45 PM EDT (#20685) #
Say for example that the repl level is -2.5 wins below average, per 162 GP. And that the marginal $ / marginal win is 2 million $.

Yes, but how do you know that replacement level is -2.5 wins?
_tangotiger - Saturday, September 25 2004 @ 08:15 AM EDT (#20686) #
It would have to be, if the marginal $/win is 2 million$.

My definition of replacement level is to set his earned salary to 0$. If the average salary for a full-time player is 5 million$, then the only way to meet my definition is to set the repl level to -2.5 wins per 162 GP.


As for whether he should be replaced: why do you need to know the repl level for that?

If the avg player is 0 runs per 162 GP, and your player is -10 run per 162, he is worth 3 million$ to you, over 162 GP. If you are paying him 6 million$, replace him! (Yes, that's you Roger Cedeno, when with teh Mets). If you are paying him the league minimum, keep him!
_studes - Saturday, September 25 2004 @ 10:19 AM EDT (#20687) #
Sorry, to be slow, but how do you know what the worth of a marginal win is? By looking at the total wins/total salary for all regular players?
_tangotiger - Saturday, September 25 2004 @ 03:19 PM EDT (#20688) #
No. Marginal implies marginal utility, so you can't look at total utility, as you are asking.

You measure the impact in terms of change in revenue and change in wins. That this figure (2 million$/win) and the other figure (-2.5 wins per 162) can be measured independently, and come out to exactly the average salary (5 million$) is good enough for me.
robertdudek - Saturday, September 25 2004 @ 10:42 PM EDT (#20689) #
I'm still not convinced there is a straight line relationship such that you can come up with a marginal dollars per marginal wins figure that is applicable to both very good and very bad teams. But I'll drop the argument about that at this point.

I think that while you are starting at the "average" baseline you are bringing in the CONCEPT of replacemnet level in through the back door (i.e. by defining zero value).
robertdudek - Saturday, September 25 2004 @ 10:55 PM EDT (#20690) #
I know I said I'd stop, but let's consider two hypothetical teams - one with 25 players making the minimum and the other with an average salary of 700,000 per ballplayer. The second club's payroll will be 10 million dollars higher than the first. The first club will have a marginal wins total of 0 (and whatever number of actual wins - which I'm not sure how you derive).

So your formula suggests that the club with an extra 10 million in payroll will expect to win just 5 more games over the course of the season.

I have my doubts. I don't think that's the way the market for ballplayers works. Rather, as you get into the high salaries, you get diminishing returns in wins (because there is more competition between teams for these good players, the price is bid up).

If I recall correctly, you derived your marginal dollars per marginal wins by analyzing teams. But most teams are made up of high and low priced players so the effects of these two groups (if any) would counterbalance each other to a large degree. So, I'm not at all sure what applies on the team level applies to individual players.
_tangotiger - Sunday, September 26 2004 @ 12:30 PM EDT (#20691) #
My concept is about salary earned.

Since teams are way overpaying for topflight starters, my model doesn't worry about that. If I worked for a team, I would create a model for empirical practices as well, so that you could take advantage of those inefficiences. The bidding war for Pettite between Yanks and Sox was one of those incredibly ridiculous things, and I said so at the time.

Back to your scenario, if the payroll is 10 million$ higher, then, you should expect 5 more wins.
_studes - Monday, September 27 2004 @ 05:38 PM EDT (#20692) #
I know this is simplistic, and not in line with the most recent comments. But here's my basic idea for ascertaining replacement level from the Win Shares data.

I took all second basemen and shortstops from both leagues (just for a sample) and ranked them by Expected Win Shares (which is a measure of playing time). Taking the top 60 players (30 teams times 2 positions), I found that they averaged a Win Shares Percentage of .505. (WSP is a proxy for winning percentage).

I then took the next 60 guys and found they had a WSP of .365, or 72% of the regulars. This is one way to judge replacement level. I think it's consistent with Craig's ideas.

The question I have is the sample size of the replacements. Should it be smaller?

The smaller the sample, the higher the replacement level. For instance, if you include 50 players, the replacement level rises to 78%. At 40 players, the replacement level is 82%.

Er, I mean baseline level.
Talent Levels and Replacement Level | 24 comments | Create New Account
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