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## Win probability in football

Though it was introduced as early as forty years ago, baseball analysis had something of a win probability revolution in the late 90s. That's how long it took for the availability of play-by-play data to increase and the price of computers to decrease sufficiently for the everyday sabermetricians to be able to really have some fun with it. As with most things in this area, football is a little behind. But it seems that we are now to a point where win probability is becoming a fairly standard part of the toolbox.

- Carroll, Palmer, and Thorn tantalizingly mentioned it in
*The Hidden Game of Football*, but didn't go very far with the idea (I'm guessing because of a lack of good data). - The site footballcommentary.com has been around for at least a few years now and has a well-developed and well-explained (and open-source!) win probability computing model.
- Brian Burke at Advanced NFL stats has recently built a model that differs from the footballcommentary model in some key ways, and has taken the extra step of building a real-time win probability scoreboard that you can use to use to follow the games each Sunday.
- I have no idea how long this site has been around, but I recently stumbled across Gridironmine.com, which has a ton of nifty win probability-related discussion, charts, and tools.

I aim to make a very modest contribution with this post.

In particular, I ran a quick logistic regression to turn the basic idea of Friday's post into a win probability equation. My model differs from those listed above in many ways. It's not nearly as precise or as detailed (and hence generally not as useful), and it probably should be totally thrown out the window once you get into the fourth quarter, but it does include one factor that the others don't: the relative quality of the two teams.

All the models above have built in the assumption that two average teams are facing off (and there are good reasons to do that). In the week four game between the Bills and the Rams, for instance, the Rams held a 14-6 lead early in the second quarter. Immediately following the Rams' second TD, the the Gridironmine model said the Rams had a 75% chance of winning. But I doubt you'd be able to find too many people willing to give you even 2-to-1 odds on the Bills at that point, much less 3-to-1. My model says the game was roughly a toss-up at that point.

Here it is:

**M** = the current point margin in the game.

**Q** = quality difference between the two teams, in points.

**H** = 1 if at home, 0 if on the road.

**In the first quarter**

WinProb =~ 1 / (1 + exp(-(-.415 + .107*M + .140*Q + .710*H)))

**In the second quarter**

WinProb =~ 1 / (1 + exp(-(-.355 + .117*M + .127*Q + .635*H)))

**In the third quarter**

WinProb =~ 1 / (1 + exp(-(-.347 + .159*M + .104*Q + .569*H)))

**Notes:**

1. This model does indicate different break-even points for different kinds of teams in the same situation. For example, let's say you're trailing 7-0 in the first quarter and you have a 4th-and-goal at the two yard line. Assuming the field goal is a guaranteed make, I calculate a break-even point --- that is the probability of scoring a TD that would make going for it a good gamble --- at about 46% if you are seven points better than your opponent and playing at home, compared to a break-even point of 37% if you are seven points worse than your opponent and playing on the road. In other words, if you think you have a 40% chance of scoring the TD, you should go for it if you're the underdog, but take the sure field goal if you're a strong favorite. This makes sense. High variance strategies, like going for TDs instead of field goals, generally are a better play for underdogs than for favorites. This is just another example of that. [NOTE: calculating the partial derivatives of the break-even probabilities with respect to M and Q might be kind of fun.]

2. The regression was based on all scoring plays in all NFL games since 1978. Therefore, it should be interpreted as the win probability of a team that *just* scored. For example, if a home team which is three points better than its opponent takes the opening kickoff and marches down the field to take a 7-0 lead, this model would give them an 81% chance of winning the game. If they force a punt on the opponent's ensuing possession, that obviously changes the win probability, but it doesn't change the model. So again, this model lacks a lot of detail that the other models include, such as down, distance, field position, and exact clock time. [NOTE: I had to take some liberties with the above interpretation in order to calculate the break-even probabilities in the bullet above.]

3. I didn't include the fourth-quarter equation because, as I mentioned, the endgame strategical decisions probably render it moot. More precisely, I guess I think it's fair to assume that the win probability for a given score and a given pair of teams wouldn't be too much different at the beginning of the second quarter versus the end of the second quarter. In the fourth, that's not true. So I wouldn't take it too seriously, but just for completeness, here it is: WinProb =~ 1 / (1 + exp(-(-.174 + .257*M + .073*Q + .348*H)))

4. Have I mentioned that this model is NOT better than any of the three I mentioned above? Well it's not. Just wanted to make sure I was clear about that.

5. For the **Q** variable, I used the teams' full-season SRS ratings for the given season. This is somewhat problematic because the SRS includes the results of the game being played. It's also problematic if we want to compute a win probability in the Bills/Rams game of week 4, because we don't know the final 2008 SRS ratings for the Bills and Rams. If the goal is at-the-time win probabilities (frankly, I'm not exactly sure what the goal is at this point, except having some fun with data), I probably should rebuild the model with at-the-time SRS ratings instead of end-of-season SRS ratings.

6. I should put this disclaimer on every post I make that involves a regression: unless you have years and years of postgraduate work specific to regression, it's very hard to be sure that the data you just ran a regression on satisfies the conditions it needs to satisfy for you to draw the conclusions you'd like to draw from it. There are all sorts of rare diseases from which data can suffer. In terms of diagnosing those diseases, I'm probably the equivalent of a first-year med student.

This entry was posted on Monday, October 13th, 2008 at 3:38 am and is filed under General, Statgeekery. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.

The relationship between the coefficients for Q & H is almost linear over the 4 models that you give (one for each quarter). The best fit line through those four points is: C(H) = -.015 + 5.229 C(Q). This has an R^2 of .97 (and no obvious pattern in the residuals, which are -.007, -.014, .040, and -.019 for quarters 1-4, respectively). That means that switching which team is at home is worth about 5.23 quality points (so HFA is about the same as 2.6 SRS points), and this advantage is basically constant throughout the game.

Since there's basically a linear relationship passing through the origin, you could simplify your model by using home-adjusted SRS as your Q (adding 5.23 to the Q of home teams) and omitting H as a separate variable. That would give you basically the same predictions, with one less parameter to estimate, and it would also allow you to use what you already know about the variability of HFA.

Doug-I just noticed your coefficients for HFA by quarter. They are go from strongest to weakest from 1st through 3rd quarters. That confirms the 'HFA diminishes through the game' observation.