## Thursday, January 13, 2011

### Why Does WP Start at Zero? (And what can we do about it?)

by Andrew Foland
All of my recent articles came from looking at the WP charts and realizing, “Hey, these start at 50/50! But Brian just told me at the start of the game, my team has a 65% chance of winning! How can this be right?!”

First I should say that I think it is actually correct to write WP in terms of average teams. It’s the only way to track WPA, which in turn, is the only way to provide an accounting of how, exactly, a team came to be the winners. That a team has a 65% chance of winning is due to the fact that it has a higher-than-average likelihood to produce positive-WPA plays. But if you start it at 65%, you won’t see that effect.

Nonetheless, it seems like one ought to be able incorporate one’s previous knowledge about the team in some systematic way. The generalized MWP(t) described previously is the main tool to do so.

Combining MWP(t) with WP(down, distance, score, possession, time etc) is fairly easy; we established how to do so in an earlier article. If we call the combination TWP (Total Win Probability, including our prior knowledge that one team or the other may be better than average) then

Logit(TWP(t))=logit(MWP(t))+logit(WP(down, distance, score, possession, time etc))

Aka

Logit(TWP(t))=Pnorm([(60-t)*Delta(MWP(0))/60] / [Sigma * Sqrt(60-t)]))+logit(WP(down, distance, score, possession, etc))

where, as before, Pnorm(Z) is the integral from –infinity to Z of the Gaussian normal PDF.

In case it wasn’t clear already, this was the motivation to generalize MWP(t) as the probability that a team would win, given that in-game information indicated a 50/50 chance of winning. That matches up with our desired quality for the probability combination formula; which we found was satisfied by the logit formula. This allows us to combine the information using the logits, even when the in-game probability is not 50%.

As described along the way, the major assumptions that go into treating MWP this way are that
1. Having an ability to outscore by P points on average over the course of a game is the same as being an average team with a P point head start
2. Scoring advantage is on average linear in time
3. Random deviations in the scoring advantage are infinitesimally normally distributed
4. Points are a “universal currency” in which advantages may be translated
5. Logit is the best way to incorporate knowledge from multiple sources about the outcome of a binary test
6. It is OK to extrapolate to time t=60 and find MWP(60) is always 0.5.

The various articles over the past week are to address the plausibility of these assumptions, or at least the plausibility of the numerical results these assumptions lead to. Although there was a lot of work writing up the assumptions and demonstrating their applicability / plausibility, the final result is relatively simple (from a coding point of view). There are numerous points at which the assumptions may be imperfect or weak, and could be improved. However, as they say, “All models are wrong—but some are useful.” I believe the various weaknesses will likely have relatively inconsequential numerical impact. I hope Brian is motivated to consider how to incorporate MWP information into the WP graphs (for instance, as a checkbox option), and that he finds this work useful in deciding how to do so.

#### 1 comment:

Andrew Foland said...

The title of this post shows how completely my brain has been living in logit space. (A statement my wife would agree with, if you remove "logit" from the sentence :) ). The WP model, of course, starts at 50% win probability.