Note: Although the title says “Fantasy Football”, the model I describe below can, in principle, be modified to fit any fantasy league for any sport.
I’ve been recently approached by several people (some students, some friends) regarding the creation of optimal teams for fantasy football leagues. With the recent surge of betting sites like Fan Duel and Draft Kings, this has become a multi-million (or should I say, billion?) dollar industry. So I figured I’d write down a simple recipe to help everybody out. We’re about to use Prescriptive Analytics to bet on sports. Are you ready? Let’s do this! I’ll start with the math model and then show you how to make it all work using a spreadsheet.
The fantasy football team rules state that a team must consist of:
- 1 quarterback (QB)
- 2 running backs (RB)
- 3 wide receivers (WR)
- 1 tight end (TE)
- 1 kicker
- 1 defense
Some leagues also have what’s called a “flex player”, which could be either a RB, WR, or TE. I’ll explain how to handle the flex player below. In addition, players have a cost and the person creating the team has a budget, call it , to abide by (usually is $50,000 or $60,000).
For each player , we are given the cost mentioned above, call it , and a point projection . The latter is an estimate of how many points we expect that player to score in a given week or game. When it comes to the defense, although it doesn’t always score, there’s also a way to calculate points for it (e.g. points prevented). How do these point projections get calculated, you may ask? This is where Predictive Analytics come into play. It’s essentially forecasting. You look at past/recent performance, you look at the upcoming opponent, you look at players’ health, etc. There are web sites that provide you with these projections, or you can calculate your own. The more accurate you are at these predictions, the more likely you are to cash in on the bets. Here, we’ll take these numbers as given.
The Optimization Model
The main decisions to be made are simple: which players should be on our team? This can be modeled as a yes/no decision variable for each player. So let’s create a binary variable called which can only take two values: it’s equal to the value 1 when player is on our team, and it’s equal to the value zero when player is not on our team. The value of (the player ID) ranges from 1 to the total number of players available to us.
Our objective is to create a team with the largest possible aggregate value of projected points. That is, we want to maximize the sum of point projections of all players we include on the team. This formula looks like this:
The formula above works because when a player is on the team (), its gets multiplied by one and is added to the sum, and when a player isn’t on the team () its gets multiplied by zero and doesn’t get added to the final sum. The mechanism I just described is the main idea behind what makes all formulas in this model work. For example, if the point predictions for the first 3 players are 12, 20, and 10, the maximization function start as:
The budget constraint can be written by saying that the sum of the costs of all players on our team has to be less than or equal to our budget , like this:
For example, if the first 3 players cost 9000, 8500, and 11000, and our budget is 60,000, the above formula would look like this: .
To enforce that the team has the right number of players in each position, we do it position by position. For example, to require that the team have one quarterback, we write:
To require that the team have two running backs and three wide receivers, we write:
The constraints for the remaining positions would be:
The Curious Case of the Flex Player
The flex player adds an interesting twist to this model. It’s a player that, if I understand correctly, takes the place of the kicker (meaning we would not have the kicker constraint above) and can be either a RB, WR, or TE. Therefore, right away, we have a new decision to make: what kind of player should the flex be? Let’s create three new yes/no variables to represent this decision: , , and . These variables mean, respectively: is the flex RB?, is the flex WR?, and is the flex TE? To indicate that only one of these things can be true, we write the constraint below:
In addition, having a flex player is equivalent to increasing the right-hand side of the constraints that count the number of RB, WR, and TE by one, but only for a single one of those constraints. We achieve this by changing these constraints from the format they had above to the following:
Note that because only one of the variables can be equal to 1, only one of the three constraints above will have its right-hand side increased from its original value of 2, 3, or 1.
Other Potential Requirements
Due to personal preference, inside information, or other esoteric considerations, one might want to include other requirements in this model. For example, if I want the best team that includes player number 8 and excludes player number 22, I simply have to force the x variable of player 8 to be 1, and the x variable of player 22 to be zero. Another constraint that may come in handy is to say that if player 9 is on the team, then player 10 also has to be on the team. This is achieved by:
If you wanted the opposite, that is if player 9 is on the team then player 10 is NOT on the team, you’d write:
Other conditions along these lines are also possible.
Putting It All Together
If you were patient enough to stick with me all the way through here, you’re eager to put this math to work. Let’s do it using Microsoft Excel. Start by downloading this spreadsheet and opening it on your computer. Here’s what it contains:
- Column A: list of player names.
- Column B: yes/no decisions for whether a player is on the team (these are the x variables that Excel Solver will compute for us).
- Columns C through H: flags indicating whether or not a player is of a given type (0 = no, 1 = yes).
- Columns I and J: the cost and point projections for each player.
Now scroll down so that you can see rows 144 through 150. The cells in column B are currently empty because we haven’t chosen which players to add to the team yet. But if those choices had been made (that is, if we had filled column B with 0’s and 1’s), multiplying column B with column C in a cell-wise fashion and adding it all up would tell you how many quarterbacks you have. I have included this multiplication in cell C144 using the SUMPRODUCT formula. In a similar fashion, cells D144:H144 calculate how many players of each kind we’d have once the cells in column B receive values. The calculations of total team cost and total projected points for the team are analogous to the previous calculations and also use the SUMPRODUCT formula (see cells I144 and J144). You can try picking some players by hand (putting 1’s in some cells of column B) to see how the values of the cells in row 144 will change.
If you now open the Excel Solver window (under the Data tab, if your Solver add-in is active), you’ll see that I already have the entire model set up for you. If you’ve never used Excel Solver before, the following two-part video will get you started with it: part 1 and part 2.
The objective cell is J144, and that’s what we want to maximize. The variables (a.k.a. changing cells) are the player selections in column B, plus the flex-player type decisions (cells D147:F147). The constraints say that: (1) the actual number of players of each type (C144:H144) are equal to the desired number of each type (C146:H146), (2) the total cost of the team (I144) doesn’t exceed the budget (I146), (3) the three flex-player binary variables add up to 1 (D150 = F150), and, (4) all variables in the problem are binary. (I set the required number of kickers in cell G146 to zero because we are using the flex-player option. If you can have both a flex player and a kicker, just type a 1 in cell G146.) If you click on the “Solve” button, you’ll see that the best answer is a team that costs exactly $50,000 and has a total projected point value of 78.3. Its flex player ended up being an RB.
This model is small enough that I can solve it with the free student version of Excel Solver (which comes by default with any Office installation). If you happen to have more players and your total variable count exceeds 200, the free solver won’t work. But don’t despair! There exists a great Solver add-in for Excel that is also free and has no size limit. It’s called OpenSolver, and it will work with the exact same setup I have here.
That’s it! If you have any questions or remarks, feel free to leave me a note in the comments below.
UPDATE: In a follow-up post, I explain how to model a few additional fantasy-league requirements that are not included in the model above.