Win Probability

Desperate for football, I'm watching the Hall of Fame Game tonight. There's four minutes left in the fourth quarter and the Redskins are up by 7 over the Colts. Can Jared Lorenzen lead his 4th-string squad to a comeback? I must be the only person in the country who cares. And I only care because I'm investigating Win Probability (WP) in NFL football.

WP is simply an in-game estimate of who's going to win based on the current score and other game variables. This post will examine the potential application of WP and will illustrate a first cut at actual WP for various scores and time remaining.

Two of my recent posts discussed measures of utility in football. I looked at first down probability and at point expectancy. First down probability analyzes how likely an offense is to convert their present down and distance situation to a first down. The success of a play can be judged based on how it changes the probability of a first down. Point expectancy measures how many points a team scores on average based on its field position. This technique not only measures success, but can provide coaches with a decision-making tool. We saw, however, that both of these techniques had their limitations.

Win Probability (WP) has been a facet of baseball sabermetrics for many years. In baseball it basically measures the probability one team will win based on score, inning, outs, and runners on base. I suppose the batter's count could be included in the calculations too.

The usefulness of WP goes beyond fan curiousity about whether the home team has a chance to win (but that's interesting in itself). Take a situation in baseball with a team down by 1 run in the 9th inning. There's a runner on first base and no outs. Should the manager call for the steal? WP should instruct his decision. We could calculate the WP of the steal decision by totaling the WPs of the potential outcomes. The WP of the steal decision would be:

WP(steal) = Pr(successful steal) * WP(runner on 2nd, no outs) + Pr(caught stealing) * WP (no runners, 1 out)

This would be compared to the various outcomes of the current at bat without stealing to decide which decision gives the team the best chance to win. There would be analagous applications of WP in decision-making in football.

Baseball is a sport well-suited to WP because it has a limited number of discrete states. There are 27 outs for each team, 3 bases, and 3 outs per inning, and there is enough historical data to accurately calculate the historical WP for each state. Football is far more complex. The states are continuous and non-discrete. For example, compare field position to runners on base. There are only eight (I think) combinations of base runners, but there are 99 yard lines. Or compare each baseball team's 27 outs (54 total) to the 3600 minute and second combinations in a 60 minute football game. There is literally over a billion potential combinations of score, field position, down and distance, and time remaining.

WP in football can be simplified, thankfully. For example, time remaining can be grouped into minute or 30 second increments. Field position could be grouped in chunks too. Even so, there are still so many combinations of states in a football game that a good WP model would need a lot of data. And I'm not talking about an entire season of play-by-play information. We'd need years of data, and even then we'd need a lot of mathematical smoothing and "best fit" estimation.

Others have looked at WP before. The site ProTrade.com built a model and it was used on ESPN.com for a time, but it seems like they used 5 minute intervals for time, something that doesn't help understand what's going on in the ever-critical final minutes. This site (which I highly recommend) also has a model, but it assumes a continuous effect of score differences. This is a fatal flaw because football scoring is in chunks of 3 and (almost always) 7. A 4 point lead is very different than a 3 point lead, and a 7 point lead is not just 1/6 better than a 6 point lead. The amount of the lead affects the strategies of the teams, which results in unique WP curves for each point of score differential.

The graph below is my own first cut at WP for the NFL. It is based on all regular season games from the 2000 through 2007 seasons. For the most common score differentials, it plots the WP of the team with possession of the ball. For example, the top curve, labeled +7, is the WP for a team that is winning by 7 points, and has the ball, at each minute remaining in a game. The -7 curve, at the bottom, is the WP for a team trailing by 7 points and has the ball.

It does not factor in field position or down and distance situations. So, this should be considered a baseline and not a finished product. But already we can see some interesting things.



A few things stand out to me right away. Notice the sudden drop off of the -7 curve. A team trailing by a touchdown with 20 minutes left in the game (5 min left in the 3rd quarter) sees its chance of winning fall dramatically. There are similar drop offs for teams trailing by 1 and 3 points at the beginning of the 4th quarter.

Also notice how the WP for a team down by 3 has a slight uptick, from .20 to .30, in the last few minutes of the game. I think this is because if they are able to score and at least force overtime, there is not much time left for the opponents to mount a scoring drive themselves.

One particular surprise is that teams down by 1 point early in the 4th quarter, who have the ball, are actually favored to win. By 7 minutes remaining, the team trailing by a point falls below even odds.

The application of WP could have profound consequences. Take a fairly common scenario. Your team is down by 3 points with 4 minutes left in the 4th quarter. You're facing a 4th down and goal from the 2 yard line. Conventional wisdom screams field goal! Get some points on the board and at least force OT. But WP might say something else. (Keep in mind field position is not considered yet, so this is rough).

If you kick the (virtually automatic) field goal, that ties the score and gives the opponent the ball with just less than 4 minutes remaining. Your opponent has a 70% chance of winning in this situation according to the graph. You've got a 30% chance.

Let's see what happens if you go for it. If a 4th down try for 2 yards would be successful about 30% of the time (which it is, at least for 2007), your PW is the total probability of the two outcomes:

WP(go for it) = 0.30 * WP(+4 point lead) + 0.70 * WP(-3 point deficit)
= 0.30 * 0.92 + 0.70 * 0.22
= 0.43

That's a 43% chance of winning by going for the TD vs. a 30% chance by going for the FG. I think this actually understates the chance of winning by going for it because if you fail, the opponent gets the ball very near his own goal line. So if you get the ball back, chances are you won't have far to go to get back into field goal range.

A lot of work still needs to be done, but the potential for WP in football is enormous.