by Andrew Foland
I have been asking myself an amusing question about the WP graphs the last few weeks, and it will take a couple of little pieces of work to set up to answer it. This writeup is one of these pieces.
Given a probability to win at time T0, how must the score differential change as time passes to a later time T to maintain the same probability of winning? Note that the score differential to maintain a constant winning percentage goes down as the game progresses—a ten point lead early in the game is often less likely to win than a one point lead late in the game.
I decided to set a fixed situation, namely 1st and 10 from own 20—and see how the score differential evolved at constant win probability. I did this using the win probability calculator and lots of trial and error. I evaluated this at 6 difference values of WP: 55, 60, 65, 70, 75, and 80. I evaluated the WP at 6 different time points: five minutes into the game, at the end of the first quarter, at half, at the end of the third quarter, with 7 minutes in the 4th, and with 3:30 in the 4th. At around 3:30, point differential starts behaving very idiosyncratically. It’s also worth noting that Brian probably has little data for some point differentials early in the game, so there will be some systematic dependence on the form of his functional extrapolations into that region.
Where there were gaps or skips in the probabilities, I used the smallest score differential which yielded a win probability equal to or larger than the one under consideration for that line of the table. One case deserves a little explanation. With seven minutes to go, a one point lead gives over a 60% chance of winning, but a zero-point lead gives less than 55%. In this case, based on the values for 55% at 15 minutes left and 3.5 minutes left, I entered “0.5” points into the table by hand. Obviously, one cannot have a 0.5 point lead in an NFL game, but under the circumstances it seems to best represent the quantity we are interested in.
Presumably the model is symmetric under simultaneous exchange of (1-WP) and sign of the lead.
Without further ado, the numbers and the plot: