NFL Overtime

The Colts-Chargers game yesterday went into overtime, which in the NFL is still sudden death. Sudden death, of course, gives a huge advantage to the team that wins the coin toss, and sure enough, the Chargers won the toss and scored on the first possession to win the game.

This led me to wondering, of course, just how much of an advantage winning the toss gives you. I did some research and found out that since regular season overtime was implemented in 1974, teams winning the toss had won approximately 52% of the games and teams losing won approximately 44%, with the rest ending in ties (if there is no score during the 15 minute overtime period, the game ends in a tie). However, the percentage of OT games won by the team winning the toss has grown closer to 60% since 1994, when the kickoff was moved back from the 35-yard line to the 30-yard line. This change has increased the chances of scoring on a possession.

One suggestion for improving NFL overtime is to move the kickoff back to the 35-yard line for overtime. I decided to model overtime games to see how much this would help.

It is fairly easy to construct a model based around the probability of scoring on a possession. Assume for now that the teams are evenly matched and have an equal chance of scoring on any given possession, and call their chance of scoring a probability P. Scoring on a possession wins the game, so their probability of winning the game is the sum of the probability they'll score on any of their possessions. The probability they'll score on a given possession is P times the probability that no one has scored and ended the game yet, which is (1-P)^2n, where n is the number of possessions the team has had. This creates an easily solvable geometric sum, which can be seen in this chart:



As you can see, reducing the probability that the team will score on a given possession (which is what moving the kickoff line back up would accomplish) does reduce the advantage of winning the toss, but no matter how low you reduce it, the team that wins the toss is still favored.

Proponents of the sudden death overtime say that losing the toss is no problem because your defense should be able to stop the drive. This should be obviously fallacious, but as a different way of looking at it, I modeled the case where the team that wins the loss is worse than the other team, and by this I mean, is less likely to score in a given possession. Here's a chart for the odds of winning, given their probability of scoring in a given possession, for a team 20% less likely to score on a possession (NB. When I say 20% less likely, I mean 20 percent, not 20 percentage points, i.e. if the team that won the toss has a 35% chance of scoring on a possession, the other team has a 42% chance, not a 55% chance.)



As you can see, even if the team winning the toss is significantly worse than the other team, their odds of scoring on a possession need to be brought down to 15% or less for them not to be favored to win!

Another overtime method suggested is the "two possession rule"; that would involve each team getting at least one possession in overtime, and then it would revert to sudden death. While this still favors the team winning the toss, it is not nearly as much as an advantage. Here's the chart for evenly matched teams (NB. that I was unable to account for the differences in scoring field goals and touchdowns, so I just modeled the case where both teams have to score a touchdown--which has been suggested as a variant of the two possession rule):



The blue line represents sudden death, and the red line represents the two possession rule. As you can see, the team winning the coin toss has a fairly low advantage for a wide range of scoring probabilities. What's particularly interesting is that for cases where the team losing the toss is better, there's a range where the probability of the team winning the toss winning the game goes down even as their probability of scoring on a possession goes up. To make it really obvious, take the case for a team that is 50% better losing the toss:



As you can see, the red line has a clear dip in it. This is because as the probability of scoring on a given possession goes up, it produces two countervailing forces. On one hand, it increases the chances of winning, for obvious reasons. But because of the way I've set the model up, it also increases the chances of the other team scoring even more, which reduces the first team's chance of winning. In the sudden death case, the first force dominates over the entire range. However, in the second case, the latter force dominates for much of the range. Again, a lot of this is wrapped up in the arcana of how I set up the model, but it makes one point clear: under the two possession rule, how good the team that loses the toss is becomes a lot more relevant.

Here's a link to the spreadsheet I made: NFL Overtime