tl;dr
- We developed an iterative integer programming model for generating lineups in daily fantasy football
- We experienced limited success due to the NFL being a highly unpredictable league
- This model is generalizable enough to apply to other fantasy sports and can easily be expanded on
Daily Fantasy Sports (DFS) have exploded since platforms like Draftkings (2012) and Fantasy Duel (2009) were created. These markets pay out millions of dollars and are exploding with growth. A majority of this money is payed out to a small percent of players who have developed strategies that are robust to market dynamics.
The goal of any fantasy sports league is to select a group of players that will maximize the number of points that you score. Fantasy points are calculated based off of real sporting events. Traditionally, decisions were made on lineups at the beginng of a season with minor changes made on a weekly basis. Fantasy points were then accumulated by a user on a weekly basis and totaled at the end of the season. With Daily Fantasy Sports, lineups are chosen on a more frequent basis (i.e. weekly in the NFL) and thus the number of decisions to be made (and $$$) has increased dramatically. If you're still confused, check out Draft Kings tutorial.
We have decided to focus on the NFL for now due to the highly unpredictable nature leading to larger market inefficiencies, as well as our team's interest in the sport.
- Our goal is to maximize the total number of points scored (p_i) by all of our selected players
- Assign a binary decision variable (Xi) to each player
- Xi = 1 if player i is in the lineup
- Xi = 0 otherwise
- The total sum of all salaries (S_i) for chosen players is constrained to a budget (B)
- The lineup must adhere to DraftKings position constraints
- It is common in fantasy football to stack quarter backs and wide receivers that are on the same team. This has to do with their point production correlation. We can acheive through adding a constraint for each team such that the number of WR's >= QB's.
In mathematical notation, what we have so far looks like:
All of our data so far come right from DraftKings except the expected number of points for each player (our f_i’s). Expected points could come from any projections website such as Yahoo, FantasySharks, Rotogrinders, etc. A quick sanity check by plotting projected points scored versus actual points scored shows us just how unpredictable this game is:
So by solving the problem above, we could get one (most likely) sub-optimal solution and enter that weekly hoping we get lucky. Instead of waiting for our big break, a smarter thing to do would be to re-solve this problem a bunch of times (let's call this M), adding a constraint to the model each time that says the new solution must have an objective function lower than the previous solution. This is of course a pretty cheap way of forcing our lineup to change, but for now it's easy to integrate:
By solving this problem 50 times using a solver, we get lineups that look a lot like:
Unfortunately, our lineups don’t show much variation. Check out how many times Victor Cruz and Antonio Brown get into our portfolio... By relying on the same players over and over we really aren’t diversifying our ‘portfolio’ of lineups. Rather than putting all my money on one or two players, let’s consider some more constraints that might help with that.
- Player Frequency: set a maximum number of times any player can show up in X% of lineups
- Lineup Overlap: set a maximum number of players between any two lineups that can overlap
Since we really don't know what the values of these constraints should be, we'll have to test a bunch of different options. We develop a full factorial design of experiments so that we can easily loop through all of the different combinations of constraints to find the optimal combination. This factorial gives us 54 (3x3x3x2) scenarios to iterate across:
- Player Frequency: 5%, 10%, 25%
- Lineup Overlap: 2, 3, 4
- # of Lineups: 25, 50, 75
- Stacking: QB-WR stacking, None
We developed this model in Python using the PuLP package, which can be foudn in optimize.py. Running the model on the 6 weeks worth of data on all 54 scenarios produces 16.2k lineups in around 90min. Fast run time as well as PuLP's flexibility will allow us to play around with different parameters of the model quickly.
Before we talk about how many points we scored, it would be nice to know what we should want to break even on our investment. Draft Kings has a bunch of different contests and names them to confuse players (or so we think...). These contest types are 50-50's, multipliers, and GPP's. They differ in payout structure which is extremely important because it impacts our risk profile as well as the metrics we'll use to estimate our profitability.
- 50-50's payout 2X their buy-in to the top 43% of players
- Triple-Ups pay out 3X their buy-in to the top 27% of players
- GPP's have a very high payout structure, often giving 75% of their payout to the top 5%
Graphically, this looks like:
With 50-50's any lineup that scores above the 43rd percentile will payout 2X buy-in, so we really want to measure what percent of our lineups are above that threshold. Triple-Ups are similar, except we see how many lineups fall above the 27th percentile score and consider our earnings 2X buy-in. With GPP's we want to focus on our max score or our 95th percentile because getting just a few lineups in the money will be huge $$$ for us.
We collected data on the first 6 weeks of the 2017 season across these contest types and computed profitability using R (/rscripts/contest analysis.R). Unfortunately, none of the scenarios appear to be profitable in multiplier type tournaments. This is most likely due to the fact that we are essentially feeding noise into it because predictions are so poor.
We can see below that there were actually a few lineups above 195, which tends to be the number of points needed to score big in GPP tournaments.
For example, Scenario 6 (QB-WR stacking, 5% player frequency, 4 player overlap, 25 lineups) would have scored 3rd place out of 2408 entries in a $25k Hail Mary contest. Scenario 14 (QB-WR stacking, 25% player frequency, 4 player overlap, 25 lineups) generated 4 lineups over 195 points across multiple weeks, which is quite remarkable considering this would have payed out $750 on a $12 buy in.
Another interesting way to look at this data is to consider the lineup iteration with respect to the number of lineups that scored big (let's say 150 fpts). We can visualize this as a heat map in Tableau very easily, in order to determine the optimal cutoff for the number of iterations to use.
Some concepts we haven't looked into yet but would like to test integration with:
- Improving predictions: Models are only as good as the data we feed into them and right now we are feeding pretty terrible predictions into ours. We have tested our model on different publicized projection sites and none seem to be very good. We actually attempted our own predictions using random forests with little success. Check out the Predictions folder) if you're curious to see what we did.
- Player ownership: You can't win GPP's with a lineup of players that everyone else owns! It wouldn't be hard to find a simple proxy for ownership percentages and set constraints on the model to set thresholds on maximum player ownership (i.e. at most 2 highly owned players).
- Floor/ceiling predictions: some players may be consistent producers while others may be boom or bust. At first glance this looks like it has potential considering that there appear to be certain players with high average production value and assorted levels of variance (standard deviation).
Matt Brondum
Matt loves everything data and nothing about professional sports. He holds a M.Eng. from Cornell University in Operations Research as well as a B.S. in Industrial Engineering from SUNY University at Buffalo.
www.linkedin.com/in/matthewbrondum
Kyle Cunningham
Aside from being the Buffalo Bills #1 Fan, Kyle is a Junior Analytics Consultant at GE Healthcare. Kyle holds a B.S. in Industrial Engineering from SUNY University at Buffalo and an M.S. in Operations Research from Northeastern University.
www.linkedin.com/in/kyle-cunningham-62012383
Sakib Alam
Python guru and NBA fanboy, Sakib is currently completing his M.S. in Applied Mathematics at Rice University.
https://github.com/sa000
Matt Wood Matt is a Research Psychologist at the US Army Corps of Engineers Research & Development Center. He holds a PhD in Cognitive Psychology from Carnegie Mellon University and an M.S. in Psychology from Villanova. https://www.linkedin.com/in/matthewdwooderdc