Fantasy Math

Fantasy Math is my active site for weekly start-sit Fantasy Football advice. Subscribers enter in their matchup info and the players they're deciding between, and get back optimal player and the probability they win.

Many sites (Yahoo, ESPN, Fleaflicker) give team win probabilities, but — as far as I know — Fantasy Math is the only one modeling player projections as explicit probability distributions.

Projections as Probability Distributions

The fundamental problem in fantasy football isn't maximizing your expected points, but maximizing the probability you score more than whoever you're playing that week.

Projecting distributions for a player's score (as opposed to point estimates or rankings) helps in two ways. It lets you take into account:

  1. Variance, how boom-or-bust a player is (1).
  2. Correlation, the tendency of players' scores to move together(2).

Both can account for scenarios where starting the guy who's ranked higher (i.e. gets the most points on average) isn't necessarily the guy who maximizes the probability of winning.

I'm obviously biased, but because it takes correlations and player variance into account, I think Fantasy Math is the best start-sit advice you can get.

The Second Best "Who Do I Start Advice" You Can Get

It's easier to understand why Fantasy Math is the best start-sit advice you can get by considering the second best start-sit advice you can get.

I have an economics background, and I've always been a big believer in efficient markets and the wisdom of crowds. I draft by selecting the biggest bargain according to ADP, and my investments are in index funds.

ECR is clearly the fantasy football equivalent of all that. And so if the goal is maximizing point totals, efficient markets would suggest — just as index funds beat stock picking — that's the best most of us (even experts) can expect to do.

Fantasy Math is essentially ECR with correlations and player variance added on top. Because the goal in fantasy is beat your opponent rather than maximizing your total points, this gives Fantasy Math a slight edge.

The Fantasy Math Edge

While real, I think the Fantasy Math edge is small.

The main reason is that optimal start-sit decisions are mainly driven by the scale parameter (i.e. start the guy whose distribution is furthest to the right), which traditional rankings and point estimates get at fine(3). Correlations and shape (aka variance aka boom or bust) are secondary.

I'm not even sure — compared to just using ECR or Boris Chen (also based off of ECR) — the impact on expected wins/share of league winnings is worth what I charge, just as a pure expected value calculation. Probably depends on your entry fee.

But some people won't rest until they've uncovered every possible edge, and Fantasy Math is for them.

How It Works

The are two parts to making the model: (1) fitting the distributions, and (2) figuring out the historical correlations between same and opposing QB, RB1, RB2, WR1-WR3, TE, K, DST.

I did (1) by fitting a Gamma distribution (both the scale and shape parameters) to historical Expert Consensus Rankings (via the weekly mean and standard deviation). For (2) it's just a giant correlation matrix.

Then I can use (2) to take generate random, uniformly distributed sets of numbers with the appropriate correlation. Then I feed those correlated 0-1 pairs through the Gamma distribution and end up with correlated simulations.

Then when someone puts in their matchup info on Fantasy Math, it queries all those simulations, figures out the percentage of time they win, and returns it.

Technology and Stack

The modeling is done using the Python data stack, and the API is in flask.

I did the first version of the front end in React using HP's Grommet framework. This worked OK, but doing this on the side, I wasn't constantly doing front end programming and didn't look at it much in the offseason.

When I went to pick it back up and wanted to make a few tweaks, I found that a bunch of the React API and frameworks I had been using were out of date. Around the same time I saw an article on Elm, and started playing round with that.

I liked it enough that — when I had hernia surgery and had to lay around for a week — I took the opportunity to recode the entire site in Elm, which is what it's been the past two season.

I like Elm a lot and might do a separate writeup on it at some point. I do sometimes wonder whether it's not boring enough though.



Fantasy Math hasn't seen wide adoption and remains fairly unknown. I think there are a couple reasons:

  • The fantasy space is very noisy and competitive, with plenty of quality free material.
  • I'm a lot better at building models than I am marketing them (working on this).
  • It's a challenge to concisely and effectively convey how the model works and why it's good.
  • People are interested in buying WDIS fantasy advice for a few weeks out of the year, which makes it difficult to design and test marketing strategies.


One of big drawbacks to Bayesian Fantasy Football was fixed shape parameter didn't allow for boom/bust guys. That's more flexible in Fantasy Math, but not to the extent I would like. This is one of the reasons I'm redoing the distribution fitting using more advanced techniques.


I want the site to work better on mobile. Ideally people would be able to import roster and matchup information too.


I'm not sure on pricing. While I'd love for it to be reasonably priced and widely adopted, that's not happening currently. I like running the site and have learned a lot doing it, but now that I'm out on my own I may need to focus on things that are profitable.

It's possible the only way that'll happen with Fantasy Math is if I jack up the price and focus on the much smaller set of users who recognize the value. They're out there (I've had people write saying they'd pay hundreds of dollars for it), and one of my goals this offseason is to figure out whether it makes sense to focus solely on them.


1 — Say you're down by 45 points going into a Monday night CLE-HOU game, would you rather start Will Fuller or Jarvis Landry? What if you were down 10 points?

2 — Imagine you have a close call — say Tyler Lockett vs Stefon Diggs — and your opponent is starting Russel Wilson. The fact Wilson and Lockett's points are positively correlated (about 0.44) might affect your optimal decision.

Precisely HOW it affects your decision depends on the rest of your matchup:

  • if you're heavily favored, the correlation might mean you should start Lockett as a hedge against Wilson blowing up
  • if you're the underdog, maybe you need Wilson to underperform AND Diggs do really well, and so Diggs maximizes your probably of winning

By modeling performance as correlated distributions, FM takes this into account. And not only this, but every pair of simultaneous correlations (e.g. the Lockett and Wilson are correlated, but they're also both correlated with Chris Carson and the QB Seattle is playing against) — between same and opposing QB, RB, WR, TE, K, DST.

3 — No matter what the correlations are, Fantasy Math is never going to tell you to bench Christian McCaffrey.