The NFL announced it is considering new overtime rules. The new rules will be considered by the competition committee and, if approved, would be implemented for future playoff games only. I've heard two versions of the proposal, and in this article I'll analyze both.
The version I heard goes like this: the team that loses the coin flip is always guaranteed at least one possession. If the coin-flip winner (which I'll refer to as the 'first team') scores and the second team matches the score, then the game reverts to the sudden death format. If the first team fails to score and the second team does, the second team wins. If the first team scores a field goal, and the second team scores a touchdown, the second team wins.
The other version, reported by ESPN, would guarantee the second team a possession only if the first team does not score a touchdown. In other words, if the first team scores a field goal, the second team gets one possession to match or exceed the score. If the first team scores a touchdown on its first possession, however, the game is over.
One very important consideration in the "response" format of football overtime is that the second team has the luxury of knowing what kind of score it needs to live on. In either version of the proposal, If the first team scores a field goal, the second team would never consider punting, and therefore would be far more likely to score than otherwise.
A Simple Model
Here is a simple model to illustrate just how big an effect this would be. A typical touchdown drive consists of 4 or 5 first downs, including the score itself. First downs are converted 67% of the time, moving the chains about 16 or so yards each conversion. This simple model makes for about a 70-yard drive, scoring a touchdown about 20% of the time (0.67 ^ 4 = 0.20). This makes sense because an offense typically scores a touchdown 20% of the time starting at its own 30.
If a team has all four downs available to it, how often could it score a touchdown? When a team uses a 4th down to convert, it essentially has two 3rd downs. Third downs of any distance in the NFL are converted 48% of the time. So in the 33% of series which go to a 4th down, an additional 16% of series will result in conversions (0.33 * 0.52 = 0.14), for a total conversion rate of 81%.
This wouldn't just mean a 14% increase in the chance of scoring a touchdown. It would potentially more than double the chance. Football drives are recursive, meaning the same process is repeated over and over. If you increase the rate of success for each sub-process, the overall success rate increases geometrically. Touchdown drives would increase from 20% of all to drives to well over 40% (0.81 ^ 4 = 0.43)!
Even if a team only needs a field goal and not a touchdown, it would still benefit from using its fourth downs, if necessary, prior to entering field goal range. And if they convert, they are still free to continue the drive seeking a touchdown.
I'll first look at the version of the proposal in which the second team always gets an opportunity to respond whether the first team scores a field goal or touchdown. The illustration below is known as an extensive form of a game, sometimes referred to as an event tree. All possible permutations are considered moving left to right. Each state of the game is represented by a box (the 'nodes'), and the probability of moving from one node to another is noted on each arrow (the 'edges').
If the two teams tie at the end of the first two possessions, the game basically reverts to the sudden death format we're already familiar with. We've already seen this movie, and we know how it ends. Ignoring the possibility of a tie (which would be slightly higher now), the first team wins a little more than 60% of the time. So no matter how it plays out, with turnovers or punts or a kickoff return, we can collapse the game into a sub-game of 60%/40% in favor of the first team.
In this case, I made some conservative assumptions based on typical drive outcome rates. A team that needs a touchdown to match will get it 40% of the time. A team that needs a field goal to match would match it 20% of the time while getting the touchdown 30% of the time for the win.
We can sum up all the total probabilities for the scenarios in which the first team wins. The actual probabilities are just rough estimates for typical drives, so this analysis submits a method for finding an answer rather than declaring an actual answer with much certainty. Feel free to replace my probabilities with your own. In any case, this estimate results in a 52%/48% outcome in favor of the first team, which would be a significantly smaller advantage than the current format.
(You might notice that when the first team does not score, the game effectively becomes a sudden death game, except now the second team has the 60/40 advantage.)
The other version does not reduce the advantage as much. If the second team cannot respond to a touchdown, it's not going to win as often. Here is the extensive form when the second team is guaranteed a response to a field goal by the first team.
In this version of the proposal, the advantage for the first team is 56%/44%, still smaller than the advantage in the current format.
Either way, the important point is that football is a recursive process, and its outcomes vary exponentially with respect to the sub-game outcomes. If a team's chance of converting a first down is increased by only a few percentage points, they would be able to score far more frequently. Second teams would be able to respond more easily than most people might think, including those on the NFL competition committee. I think the NFL will be surprised how often an overtime game under the new rules reverts to the sudden death format.