## Thursday, January 13, 2011

### Win Probability and Point Differential

by Andrew Foland

As noted in my earlier writeup, an average team with a 13 point lead after five minutes would have a win percentage of 80%. (This, of course, is nothing more than inputting some numbers into the front end of Brian’s WP calculator—all the real work is Brian’s in creating that WP and the calculator front end.) Being only 5 minutes in, one can approximately take this result as a proxy for the result of, “If an average team had started the game with a 13 point lead, it would win 80% of the time.” (We’ll do something a mite more sophisticated in a moment, but this is the basic idea.)

Of course, no team begins the game with a 13 point lead. However, some teams do have a 80% generic win percentage. Is there a useful relationship between these two points?

A betting man might have an answer to this question: namely, the point spread. However (and this is the pedantic part), the point spread is a prediction of the final score differential of a game, while the aforementioned WP calculation is at the beginning of the game. So the real question is: is a team with generic win percentage of 80% and no point advantage at the beginning of the game, likely to end the game with a score differential equal to the score differential that an average team needs at the beginning of the game to have an 80% chance of winning?

To answer this question, one compares the point differential at the beginning of the game needed for an average team to win with a given percentage, to the average winning percentage of teams with that same expected final point differential (as expressed by the point spread.) This is shown in the table.

 WP Model BODOG WP 80 13 12 75 9.5 8.5 70 6.5 6 65 4 4 60 3 3.5 55 1 2

The table has three columns. The first column is simply win percentage. The second column comes from the WP model, and expresses the point differential needed at the beginning of the game needed to achieve this win percentage. Since the model is based on data, and there is no data on games with nonzero point differentials, to find this I used the first three columns of my previous article (score differentials at 5, 15, and 30 minutes into the game) and linearly extrapolated backwards to 0 minutes into the game.

I used the bodog conversion table of point spreads to average winning percentages to make the third column. This expresses the expected final point spread for a given winning percentage. One can find various different conversions from point spread to winning percentages, and they do not all agree. I’ve not done the relevant study to determine who is right, so I simply took bodog’s.

As you can see, in general, the answers are within a point of each other. For my upcoming purposes, this is more than sufficient agreement. So to answer the question, is a team with generic win percentage of 80% and no point advantage at the beginning of the game, likely to end the game with a score differential equal to the score differential that an average team needs at the beginning of the game to have an 80% chance of winning? The answer is: yes.

Do note, though, that agreement is not perfect, and that for some purposes, coming within a point of agreement may not be sufficient. This is especially the case insofar as the difference is systematic—the WP point differential at the beginning of the game is consistently larger than the point spread expectation of the final score. Caveat emptor.

And if you’re wondering, why anyone would mess with all this pedantry, I’ll leave you with this observation: we’ve just established a point of contact between the WP model and GWP.