# Fantasy values by parallax

*parallax* n. *: The apparent displacement of an object caused by a change in the position from which it is viewed.*

Generating dollar values for fantasy players can be tedious. A common approach is to sum the stats above replacement level in a category and then divvy up those stats among a portion of the total budget and add up the contributions for each player. That’s doable, but there are challenges. For one thing, there are wrinkles to handling rate stats like BA and ERA and “clumpy” stats like saves and steals. Also, there is something unrealistic about treating categories as freely floating when there are obvious dependencies, such as between home runs and RBIs, or ERA and wins.

There is another approach. This one has its own challenges, including a longer time to derive the values, but it sidesteps the bumps with the usual method, and it’s easily tailored to many formats.

The key is to look at fantasy value from a different angle. Suppose that Roy Halladay is valued at $30 in your league. It’s true this says that Halladay’s stats are “worth” $30. But you could re-state this to say that **paying $30 for Halladay neither helps nor hurts your odds of winning**. If you get Halladay for less than $30, then your odds of winning go up, and if you pay more than $30, then they fall. But paying $30 neither raises nor reduces your odds; if it did, then $30 would be the wrong price.

So we have turned a statement of value (“Halladay is worth $X”) into a statement of probability (“Drafting Halladay at $X neither raises nor lowers your odds of winning your league”). Why is this good? Because now, to find the value of a player, **we need only to find the price at which ownership of the player doesn’t alter your odds of winning**. There are no other calculations—no defining of the spread of player stats, no breakdowns of categorical value.

Note that this method works in fantasy because we have a fixed budget. In the real world, things are looser—there is no price at which owning C.C. Sabathia “hurts” your odds of winning. However, real businesses are in the business of maximizing profits, and C.C.’s salary can surely hurt those.

So we have the bare bones of an approach. Let’s create a two-team league. (In this exercise, we’ll stick with pitchers, so that we don’t have to worry about accommodating multiple positions.) On one roster, we’ll put our player of interest—in this case, Roy Halladay. Halladay always appears on this roster. The other eight slots on Roy’s roster, and all nine slots on the other one, are open:

Roster #1 Roster #2 ============ ========= ROY HALLADAY Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher Pitcher

The open slots will be randomly filled with 17 distinct pitchers (no duplication within or across rosters.) After populating the rosters, we will determine the side that “won,” based on whatever categories we have in our league, and behaving as if these were the only two teams in our league. For example, in standard 5×5 roto league, there would be five categories—wins, saves, ERA, WHIP, and strikeouts. Finishing first in a category in our two-team league is worth two points, and finishing last is worth one. We’ll repeat this exercise 1,000 times for various roster configurations and track the winners.

(Why do we need to track only two rosters, even if our real league has more teams? Because each Halladay-less roster is identical. Suppose that there are 10 other rosters like Roster No. 2. Each is indistinguishable from Roster No. 2, because all rosters draw from the same pool. If we can balance Halladay’s roster with Roster No. 2, then we’ll also have balanced Halladay’s roster with the other rosters. A one-in-two chance of beating Roster No. 2 equates to a 1-in-12 chance of beating the league.)

Our ultimate aim is to make Halladay expensive enough that his team wins exactly half the time. “That’s swell, but you have no dollar figures. So you can’t turn your probabilities into prices.” And that’s true. We need points of reference.

How many points? Perhaps as few as two. If we have two points of reference, we might be able to adapt the method of parallax, which is used by astronomers to determine the distance to stars. But that’s getting ahead of ourselves, because we don’t have two points of reference.

But we do. For any fantasy league, there are two statements that we can say with certainty (both statements require us to identify the draft-worthy pool of pitchers—we’ll tackle that later):

**1. The last drafted player is worth $1.**

**2. The worth of a slot that freely floats among all draft-worthy players is the average price spent on that slot.** If owners in a 12-team league historically spend $99 on nine pitchers, then a pitching slot that freely floats among all 108 draft-worthy pitchers is worth $11.

Now, in a real auction, you can’t draft a “freely floating” slot. However, in our simulation, we can—in fact, in our diagram, each slot labeled “Pitcher” is exactly that. In a particular run of the simulation, the slot could be worth $1, or it could be worth $50. But the expected value of the slot is $11. (Actually, it is slightly less, since one pitcher—Halladay—is not available. But $11 works for our purposes.)

Armed with our two points of reference, we can employ parallax. Here’s the approach: Roster No. 2 will never change—it will always contain nine freely floating pitching slots. For our first 1,000 runs, Roster No. 1 will also be the same. Over time, though, we’ll swap free-floating slots (worth $11) for the last drafted player (worth $1). Each switch means a drop in value of $10 for Halladay’s team.

Eventually, we’ll reach a point at which Halladay’s roster wins only half the time. Since the odds are the same, the total value of each team must also be the same. We know the value of Roster No. 2 ($99), and of the non-Halladay slots on Roster No. 1 (either $1 or $11), so it’s easy enough to solve for Roy’s value.

If we replace all eight floating pitchers, we could end up with a graph like this (not real numbers):

Here, when Halladay is paired with eight freely floating pitchers, his team wins more than 75 percent of the time. However, when he’s stuck with eight $1 pitchers, he wins only about 15 percent of the time.

To find Halladay’s value, just read off the point at which the trend line crosses 50 percent. In this case, that’s around 3.5. So Roster No. 1 would be balanced with Roster No. 2 if 3-1/2 slots worth $11 were replaced with the same number of slots worth $1. Ergo, Halladay is worth $35.

That’s the idea, anyway. Will it work?

NEXT WEEK: Will it work?

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That’s a compliment!

This is one of the lamest posts I’ve ever read…you must have been on drugs

guy and real real high…..I guess I’m pretty lame for taking the time to post a comment on this post…

How is this lame? Because you’re not bright enough to understand what he’s trying to do?

Good work, John: I’ve been thinking about this question for a little while and am wondering what kind of results you’ll get.

Yeah i’m quite impressed with the creative, ingenious method you came up with. Kudos for some great work John.

Quick Q: when you’re running this simulation pre-season, are you simply taking your 108th best pitcher projection to plug in for the 1$ player, and filling the rest of the slots with a sample from the top 108 projections?

The main attractions of this method are that it is interesting and more versatile. Is it “more accurate”? Remains to be seen, though I can’t believe that the traditional piecemeal method of apportioning value can’t be improved on. Still, any difference is probably not large, and a drawback of my approach is that you won’t get a definite result: In one cycle of simulations, Halladay might be worth $31, and in another, $29. There may be value is showing player values as a small spread rather than as a single number, but it’s bound to frustrate some people.

KY, I did not ask where your projections come from. I asked “How do you determine the ‘best projections’”?

You seem 100% devoted to the current system. Presumably you have had success with it. Me, I do not believe that our methods are as sound as they can be. What I am trying to do here is explore an alternate and untried means.

We’re starting to talk past each other. I’m concerned not with the best projections but with the truest dollar values for a given set of projections. How did McFly come up with $30?

You said that the simple way is “You take the best projections, assume those will be the stats for the end of the year, and divide them by the dollars available to spend.” I don’t think that way is quite so simple. But this isn’t the forum for hashing that out.

Anyone who believes that the old system produces dollar values with “0 error margin” and “no debate as to whether that is true” will have no need of the ideas in my article.

If your inputs to it are the end of year numbers. how can it not? Its a calculation. xHR = x$

If you think the way many people and draft sites calculate xHR = x$ is wrong you should publish that article because it would be very important information.

John – Good start here, I like that you approached this from a business model POV rather than a projection POV.

KY – “…if you take the stats at the end of the season and create dollar values from them using the old system you will get exactly how much each player was worth that season with 0 error margin.” – Sadly, we all draft at the BEGINNING of the season where our error margin can be quite large.

KY, There is nothing simple about turning “xAVG + xHR + xRBI + xR + xSB” into “x$.” That is as true of the old system as of this one.

There is a popular approach that gives (at a minimum) reasonably accurate prices. But it is not simple, and it is certainly not elegant.

I don’t have a ready link, but do some googling on methods for turning projections into prices.

Just gonna jump in here quickly. Even if we know exactly what a player will do in all 5 fantasy categories with 100% accuracy, I’m not sure we are able to turn that into a dollar value with 100% efficiency, at least not yet. I’ve yet to see a perfect system to do it. KY, I’d be curious to know what method you use. I know that I, personally, have advocated Standings Gain Points in the past, although I will be the first to say that the method has some flaws.

I’m curios what a “I can imagine curves where that doesn’t suffice.” player pool in a league with no add drops using end of year stats would look like that would cause the z-sore method to generate dollar values that can be beat?

If there are no add drops and you use end of year stats all of those above problems of position and slotting are removed I believe.

KY, As I said a while ago, you have complete faith in z-scores, which means that you have complete faith that player values can be no better described than as the sum of the number of standard deviations from the mean. Whether my lack of a similar conviction is a strength or weakness remains to be seen, and won’t be answered here. Adieu.

And it would be really nice for this conversation if you had provided a single reason z-scores would not give you perfect dollar values using end of year stats with no substitutions. Since that what I spent a day asking for.

But instead you just keep saying, “I don’t think they do that.”

Adieu indeed.

KY, Distributions of numbers are characterized by more than their mean and standard deviation. Do you grant that? You must, because it’s true (for example, skewness). If you do grant it, then why should we assume that a calculation that considers only mean and standard deviation is perfect?

The nice thing about the simulation is that, in theory, it should pick up mean and standard deviation as part of its laborings, along with anything else of import. So it shouldn’t have any LESS information than z-score, and it may have more.

I should have stated this plainly 20 posts ago.

Yes, thank you very much!