# The Hardball Times Fantasy

## Fantasy values by parallax

by John Burnson
September 18, 2009

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
============           =========
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 5x5 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?

Compiled by THT Staff.