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Relative and Absolute Scales

A Primer

The following was an e-mail I wrote a few months ago. I didn't intend to write this as an article, but I suppose that the information here might act as a good reference point for some readers. Consider it to be a primer on relative and absolute scales.

Terminology

First of all, it would do well to define some terminology, so that the ambiguity will be removed, or at least lessened.

skillset - a player's set of skills that is not determined by any context. For example, a player's strength, bat speed, strike zone judgement, intelligence, foot speed, acceleration, baserunning intelligence, throwing arm, release, etc, etc, etc. All these can be qualitative or quantitative.

performance level - the manifestation of a player's skillset within the context of an actual game. For example, an actual game could be Fenway, afternoon, Mussina, Jeter at SS, 78F, bases loaded, 2 outs, 7th inning, down by 1, etc, etc. Essentially, any variable outside the player's control is a context. How a player's skillset responds to that context (whether by luck or design) will result in his actual performance. Sosa going 3-3, 3 HR, 9 RBIs, for example.

true talent level - the expected manifestation of a player's skillset within a context-neutral environment of an expected game. So, this is to put all the players in the same expected environment. What expected environment? Well, if you are considering trading Boggs for Brett, then you would have to look at Fenway in particular. However, in general, we are trying to get a context-neutral environment for everyone, and therefore, it would make sense to use 1/30th Fenway, 1/30th Wrigley, etc, etc. You would use 1100 parts Randy, 800 parts Pedro, 300 parts Mariano, etc, etc. You'd use 45% bases empty, etc, etc. Essentially, you come up with an environment that you would expect to face, and that would be the league average environment. That league average environment will be our context-neutral environment, for the purposes of establishing a player's true talent level.

marginal utility - the additional worth of something, given a particular context. For example, a single when Pedro is pitching has less run impact than a single when a bad pitcher is pitching. A run scored at the Astrodome has more win impact than a run scored at Coors. All this is expected to be true, because of the expected behaviour of the context.

valueless - non-value-added, or that the marginal utility is zero. This player still has absolute value and still contributes wins. He just does it at the same rate as the guy who is not in the league. Unless a batter never gets a hit, or a pitcher or fielder never records an out, then every player must have absolute win value that is greater than zero.

True Talent Level

Ok, let's talk.

A player's true talent level can be described in many ways. For example, as a number from 0 to 1, where the average is .500. Mike Mussina would be a .620 pitcher. Mariano Rivera would be a .670 pitcher. These numbers have a certain implication, in that these players have these values assigned, assuming that the rest of the team (hitters and fielders) will be league-average, and that their opponents will be league-average. Again, we are establishing a context-neutral environment here in order to fairly assess a player's theoretical contributions to it, and establish his true talent level. And the best way is to assume that the environment around them is average.

Now, we have to make a distinction between actual contributions (past-tense) and expected contributions (future). In terms of actual contributions, we look at a player's performance level in the context of the environment in which it was played in, and compare that performance to some baseline player in that same environment. The question therefore is "which baseline player".

Baseline

Now, we could say that Mussina is +.120 wins / 9 IP better than average, and that Rivera is +.170 wins / 9 IP better than average. These are true statements. However, the value that these players have derived (past-tense) is not to be better than average, but to be better than someone that is paid the league-average minimum. Their marginal utility to the Yankees is what they produce, over and above what the Yankees would have produced without one of these guys around. So, we can better describe their value (past-tense) as +.220 wins / 9 IP x 230 IP for Mussina, or +.270 wins / 9 IP x 80 IP for Rivera (assuming a .400 baseline player). In this fictitious example, Mussina is worth +5.6 wins over our baseline value-less player, and Rivera is worth +2.4 wins. And if the marginal $ / marginal win is 2 million$/win, then Mussina should be paid 5.6 x 2 + .3 (the .3 for the value-less player who still gets paid), or 11.5 million$.

In establishing their trade value, again, playing time is an important component, and again some below-average baseline level would be required.

However, if playing time was not a component in the discussion, then you don't need to know about any baseline level. .670 is better than .620 and therefore, .670 has the higher true talent level. The .670 (in this case) just won't manifest itself as much.

(Relievers of course have higher-leverage impact to their games, and Rivera's +2.4 wins would be worth +4 wins, but that's a separate discussion.)

Marginal Utility

Let's look at hitters. The walk has a certain run value only because of the theoretical expected behaviour of the environment in which that walk will occur. A walk has a marginal utility of say +.20 runs with Pedro on the mound, and +.40 runs with Rojas on the mound. The walk, by itself, really has no value. Baseball run-scoring is interdependent and non-linear, and therefore, we cannot analyze the events that occur as if they occurred in a vaccum.

Therefore, every event has a certain run impact, given the expected behaviour of the environment. The value of the average out, in a league average setting, is about -.270 runs. That is its marginal utility, within the context of the average game.

If you were to apply all these marginal values to all the events, the total value will equal to zero. Therefore, you could have, over 300 PAs, Tatis being -10 runs. And you could have, over 600 PAs, Ventura being -15 runs.

Negative(?) Value

Does this mean that these guys have negative value? Negative is a loaded word, and therefore, best not to use it. All this implies is that these guys reduced the amount of runs that an average team would have scored by 10 or 15 runs. An average player is worth 4 million$. Therefore, even a slightly below average player still has alot of worth. A .490 player, or a -2 runs / 600 PA player is essentially the same thing, but expressed with a different scale.

You could choose to describe that -2 runs / 600 PA player as a 78 RC / 600 PA player, given that the league is 80 RC / 600 PA. It's really irrelevant, since you are describing the same thing. 0 Celsius is 270 Kelvin. They both describe the same thing, but in different scales. Celsius cannot be used in multiplication or division. +5 celsius is not 5 times hotter than +1 celsius. However, the Kelvin scale is an absolute scale, and can be used in this manner. Linear Weights is a relative scale.

Playing Time

Let's go back to Tatis and Ventura. Tatis is -10 runs below average ....GIVEN that an average player received 300 PAs. This is an important, in fact, critical statement to make, one that is never made. The problem is that there is still another 300 PAs to give away.

Now, a value-less player (one worth the league-average minimum) would produce at a rate of -15 runs / 300 PA (let's say). Therefore, Tatis' actual (past-tense) value is that he is +5 runs above the value-less player (over 300 PAs).

Therefore, whether we describe Tatis as being -10 runs / 300 PAs, or -20 runs / 600 PAs (relative to average) or as being +5 runs / 300 PAs, or +10 runs / 600 PAs (relative to the value-less player) it is the same thing.

In establishing a player's (past-tense) value, playing time is an important component.

So, the league average context is important to know so that you can establish the various run value impact that every event has. The baseline value-less player is important to know the level of contribution that a player had.

My point is not that one is better than the other, but rather that their uses have certain merit, depending on the question being answered.

Replacement Level

I'll make this point as well, regarding replacement level. If a hitter's replacement level is 20 runs below league average, and a fielder's replacement level is 20 runs below league average, does this mean that the average player's replacement level is 40 runs below league average? No! In fact, if you take the top 240 players by PA, and determined their run values relative to league average in hitting and fielding, and look at the rest of the players and did the same, you will find that the overall replacement level will *also* be 20 runs below league average.

Afterthought

It is important that a player's contributions (past or future) be understood within the league average context, and the marginal utility it has. While replacement level comparison (or some other baseline level) is usually the desired output, the comparison against league average is a necessary middle-step, and sometimes, the final step.