The Case for Regression (Over Binary Classification) in Win Total Prediction
By: Jon-Ross Presta
Tags: Football, Machine Learning, Data Science, Forecasting, AI

To normal folks, today is just another day of sheltering in place. To football fans like myself, today is the glorious day when the schedule comes out.

For real, we take Schedule Day that seriously. Schedule Day is a time for planning vacations. “Oh honey, let’s go visit your parents in San Francisco this Christmas. What’s that? My Cardinals are playing at Levi’s on December 26th? Oh what a coincidence.”

Above all else, Schedule Day is the day when every hopeful fan can disillusion themselves into thinking their schedule sets them up nicely to steal a couple wins that they probably shouldn’t. “We get to play the Bucs coming off our bye? Nice!!”

In the next few days every fan of a team will look through the schedule of their team and say these words out loud: “Win, win, win, loss, win, ooh that’s a tough game but we should be we well-rested, win, win…” Then they will add up all those wins and say: “Oh 12–4?? We should be serious contenders to win the division!” This blog will be a quick explanation of why everybody does win totals incorrectly.

Whether these fans realize it or not, they are applying a (biased) binary classification model to each game. They include major factors, such as perceived skill level of each team, whether the game is home or away, is it coming off a short week or a bye, might the team be exhausted from a tough game the prior week, etc. All of these factors are pooled together, the fan calculates some implicit likelihood, and then a decision is made.

For all my loyal Data Scientists reading, the win total that we’re after can modeled as so, where I is the indicator variable of whether game G_i is a win, and N is the total number of games in a season.

Over a 16 game season, each team will almost certainly win a game in which they were underdogs, and lose a game that was an easy win on paper. The mental model described above allows no room for this error, and is prone to many shortcomings. Let’s show a simplified example:

Let’s say that by next fall, every state in America looks like The Walking Dead, except California due to our early shelter-in-place policy. Also, the Chargers accidentally moved to the wrong Carson, and are in Nevada. Because the show must go on, the season now consists of the Los Angeles Rams and the San Francisco 49ers playing each other 16 times. Because home field advantage no longer exists, let’s say that for each game the Rams have a 55% chance of winning. If I were to rely on the model described above, assuming I use a 50% threshold to make my decision, then I would go through the schedule and say “win, win, win, …” and count out a 16–0 season. This should sound wrong, because the teams are relatively evenly matched, and it is. If you model the total number of wins as a binomially distributed variable (which isn’t perfect but it close enough), you would find that the expected value of total wins is 8.8. So where did this model go wrong?

If you simplify the above equation, and make the incorrect (but close enough) assumption that all games are independent from one another, you get the following:

Now, the expected win total is equal to the sum of the expected value of all the indicator variables. And here is where every fan makes a mistake. When you are applying a binary classification (win or loss) to an indicator variable, you’re not finding the expected value, you’re finding the mode! So when we say that beautiful Jared Goff has a 55% chance of winning, we are saying that between a win and a loss, we expect to see more wins, BUT that doesn’t make it the expected value. The expected value of an indicator variable is the probability of that event happening, which is in this case 55%. When we do add up the expected value of 0.55 wins over 16 games, we get 8.8 wins.

Okay we’re almost there, here’s a quick dancing Marshawn break:

Obviously I’m not saying anything ground-breaking here. When the probability of all games is the same, the win total is literally a binomial variable. When the probability of each game is different, we lose the awesome properties of the binomial distribution, but we still can sum up the probabilities to get our expected win total.