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Friday, September 18, 2009Fantasy values by parallaxPosted by John Burnson at 4:00amparallax 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 PitcherThe 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.
Dave Studeman said...
That’s a compliment! Posted 09/18 at 10:39 AM
Ed D. said...
Hi John. Interesting stuff, as always. I’m curious, if and when you run this type of valuation for the entire player population, for your “certainty statement #2” wouldn’t you just use $260/23=$11.3 as your free-floating value for a standard league, or perhaps adopt a “normal” 65/35 hitting/pitching split and use $260*.65/14=$12.1 for free-floating hitters and $260*.35/9=$10.1 for free-floating pitchers? Second question, remind me, how is this different than your WOW simulations from a few years ago? (I get that the simulation mechanic is different, but I’m wondering if you’d expect the ranked order of players to be different between the two systems or if the ranked orders would be the same, just with WOW giving you a % likelihood of being on a winning roster and this giving you an actual dollar value). Posted 09/18 at 02:10 PM
John Burnson said...
Brian, Thank you. I had to go deep into my inner nerd to pull this one out. Ed: (2) This builds on WOW, but I was never able to turn WOW into dollar values (or, rather, dollar values that I could accept). Whereas this approach, I think, is sound. I do expect the ranked order to be the same as with regular WOW. Posted 09/18 at 03:02 PM
ourcellardoor said...
This is one of the lamest posts I’ve ever read…you must have been on drugs Posted 09/18 at 04:19 PM
MrLarryDavid said...
How is this lame? Because you’re not bright enough to understand what he’s trying to do? Good work, John: I’ve been thinking about this question for a little while and am wondering what kind of results you’ll get. Posted 09/18 at 06:33 PM
archilochusColubris said...
Yeah i’m quite impressed with the creative, ingenious method you came up with. Kudos for some great work John. Quick Q: when you’re running this simulation pre-season, are you simply taking your 108th best pitcher projection to plug in for the 1$ player, and filling the rest of the slots with a sample from the top 108 projections? Posted 09/18 at 08:00 PM
John Burnson said...
I’ll cover this more next week, but the answer is essentially yes. The one tweak is that, instead of singling out one player as “last drafted,” I use a pool of the last 6 drafted plus the first 6 non-drafted. These guys are all bordeline, and in a given run, any of them could be “last drafted.” Having a pool of last draftees keeps the program from becoming biased if, say, the original “last drafted” contributed only to Saves and nothing to the other categories. Also, when you go to stack Halladay’s roster with multiple $1 players, they aren’t all the same guy. Posted 09/18 at 08:38 PM
KY said...
I’m not convinced of some assertions here. I guess I’m lost on how this system will create a more accurate measurement for Halladay that the system in the first paragraph. They both generate a dollar value, why is this way more accurate? Posted 09/19 at 10:49 PM
John Burnson said...
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. Posted 09/19 at 11:31 PM
KY said...
“but it’s bound to frustrate some people.” ...especially if it hasn’t been proven to return a more accurate dollar value yet. I don’t think the normal method is piece mail at all. Its 100% calculated in a simple way. 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. Each player gets their portion. If you used the actual end of year stats and a 50/50 split for dollars you would get exact dollar values for every player that perfectly reflect their value. If someone spent a penny more then your dollar value using end of year stats and the other guy in your two team league spend a penny less, he would lose 100% of the time. There’s nothing incorrect about the regular method, its the fact that the 65/35 split is up for debate, players enter the pool mid season, that many league allow keepers and that nobody performs to projection that cause the dollar values to not be 100% accurate. Now if you want to debate how to quantify those, that would add value. I’d love to explore a better way but it just feels like here you are misrepresenting things to say the old way “seems wrong” and that this way, as of yet, has value. Posted 09/20 at 12:36 AM
John Burnson said...
KY, So your first step is “You take the best projections”? And how do you determine the “best projections,” when you have yet to assign dollar values? You’ve had two posts now in which you could have attacked the logic in my article. Instead, you’d like to turn this into a contest of “accuracy,” which, you’ll note, is not a word that I used in my article, not because I don’t think the method is accurate, but because I don’t think that’s the highlight of it. I didn’t set out to write 1,000 words on my problems with valuation methods, and I don’t plan to do so here. That’s not my target. It’s possible there’s a flaw in my logic. But if the logic is sound, then the values are accurate. At that point, the challenge will be to explain differing prices between this method and others. Posted 09/20 at 10:06 AM
KY said...
“The Best Projections” Projections here refers to stat projections. Pujols will hit .325 34HR 111RBI and 9SB next year for example. Projections, as I was referring to them have nothing to do with dollar values. They are the stats you use to create dollar values. The original method is 100% accurate in creating dollar values if you were able to use end of year stats to generate dollar values. If someone were to use a different system and apply it to the end of year stats, unless it too was a system that translated the stats into dollar values 100% accurately, it would lose to the original system. The best it could do is tie it. There’s nothing wrong with the logic of how your system works so why would I attack it? My posts were not about how your system worked internally to itself, they refer to how it would match up against the old system. I’m saying two things; 1) I think you were not accurate when you said the original system had flaws. The flaws in the system come from the data inputs, not the way it calculates dollar values. Posted 09/20 at 12:16 PM
John Burnson said...
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. Posted 09/20 at 12:27 PM
KY said...
The best projections are the ones that come the closest to the eventual end of year numbers. The only thing you can do is look at historically who has produced accurate projections in the past and hope they do so again. I’m not devoted to anything. What I’m saying is, if you give the original method the end of the year’s results, it will beat or tie any system, because it is simply doing a calculation from what we already know has happened. There is no way to improve upon a calculation that is saying 1 = 1. At the end of a given season .287 in 50AB, 4HR, 8RBIs and 1SB, given a certain player pool, correspond exactly to some dollar value. 1 to 1. If you used Marty McFly’s sportsbook and were able to know the end of the year’s results at the time of an auction, an auction that allow no keepers or pickups, you would walk into that action knowing exactly what dollar value each player available will end up being worth. All that would remain would be for you to bid less than those dollar values while still winning the players and accumulating value. If you know a guy is going to end up being worth $30 (because you have McFly’s book) and you buy them for $15, your team just moved up from the middle of the standings. Towards the top. Some other team will have moved down as well because you just took more value out of the pool then you paid in in dollars. As long as you walk out of the auction with the most savings of any owner, you will win the league. What I’m saying is, there is no belief in the above. Everything above can be mathematically proven correct. I didn’t understand that you were claiming the above was up for debate. Now I do, but I do not agree that it should be. Also, if I am wrong and there is something that is incorrect about the above, I’d love to learn about it. Posted 09/20 at 12:44 PM
John Burnson said...
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. Posted 09/20 at 02:00 PM
KY said...
Ignore projections. What I’m saying is, 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. At the end of the season each player is worth a particular amount based on what they produced. Given a particular league, at the end of the season you can say “Albert Pujols was worth $64 this season in this league.” There will be no debate as to whether that is true, it will be a calculated value and be mathematically correct. If you said, “no he was worth $62” you would be lying. To the best of my knowledge. Posted 09/20 at 04:23 PM
John Burnson said...
Anyone who believes that the old system produces dollar values with “0 error margin” and “no debate as to whether that is true” will have no need of the ideas in my article. Posted 09/20 at 04:46 PM
KY said...
If your inputs to it are the end of year numbers. how can it not? Its a calculation. xHR = x$ 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. Posted 09/20 at 06:19 PM
thumble said...
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. Posted 09/21 at 01:42 PM
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This might be the nerdiest fantasy article I’ve seen here yet. Interesting stuff, John. I might have to play around with this sort of simulation (or a variation of it) if I can figure out how to program it. Look forward to the next article.