1. The initial drive of the first possession (A TD wins, a turnover or punt triggers Sudden Death (SD), and a FG triggers State 2.)
2. The team down by 3 now has one possession to match the FG (triggering SD) or score a TD to win.
3. Sudden Death
The possibilities are illustrated in the event tree below, along with some back-of-the-napkin transition probabilities I made back when the new rules were first proposed. (State 1 is "1st Poss". State 2 is the branch under "2nd Poss" that follows a FG in the 1st Poss. Sudden death is self-explanatory and occurs after a no-score in the 1st Poss or after a FG is matched in the 2nd Poss.)
The problem is that State 2 has never existed before. It will be a decade before there are enough examples to make any sort of reliable, empirical WP estimates. In this state, a team has all four downs to get into FG range, and once there it can score a TD to win. Plus, there is no urgency in terms of the clock. It’s a very weird situation.
State 2 also affects State 1, because the WPs of State 2 must be considered when valuing the FG option in State 1. In the new OT format, a FG doesn’t win. It only triggers State 2.
One important note before we continue: This analysis ignores time and the associated possibility of a tie. This is necessary to make the computations manageable. Although this assumption is not ideal, it is realistic. Although ties are going to be more frequent under the new format, they will remain rare. Additionally, it is far from certain how teams will typically value a tie. (See Jacksonville's 4th and 10 attempt from the Houston 47 with 2:36 remaining in week 11.) Three outcome possibilities complicate things far more than you might think. It reminds me of the Three Body Problem in my space dynamics class back in college--trying to compute the orbits of two objects, say the Earth and an object in orbit, are no problem. But throw in the Moon, and the math gets impossibly ugly.
So how do we crack the situation where a team needs a FG to survive, a TD to win, and there is no time pressure? We can look at the end-game of 4th quarters in which a team is down by 3. We can throw out all games that ended in expiration, and only look at combinations of field position and time remaining that realistically maximize the offenses chances of winning. For example, when we look at first down situations at a team’s own 20, at various times remaining, we chose the time in which a team roughly has the best chance of winning. That will be the point where time presses on them the least and the opponent does not have enough time to counter the drive. If we do this for all the various field positions, we can get a realistic estimate of how often teams in that kind of situation get a FG or score a TD.
Next we apply those FG and TD rates to estimate how often a team down by 3 points in OT will either win or trigger SD.
The most difficult analysis is to evaluate 4th down decisions in State 1, the initial possession of OT. This is because it must consider all the future possible game states. Using the standard 4th down conversion probabilities, punt distances, and FG success rates, we can generate the expected WP for going for it, punting, and FG attempts at each yd line and each to go distance. That analysis produces the chart below. The black lines are for FG attempts and punts. The colored lines are for conversion attempts at the various to-go distances. The blue line is 1 yd to go, and so on. Where the conversion lines are above the punt or FG lines, a team should go for it. Where either the punt or FG line is higher, a team should chose that option. (Click on the chart to enlarge.)
You may have noticed a few oddities about the chart. I apologize for the jittery lines. That's due to the win probability estimates for SD, which I already had on hand. Unfortunately, I rounded those numbers to 2 decimal places some time ago. Note that the more likely SD is the more jittery the line. For example, 4th and 1 is very smooth and 4th and 15 is the most jittery. The second oddity is the sudden drop-off in WP for 4th down attempts at the opponent's 20 yd-line and 10-yd line. Those discontinuities are due to the increasing difficulty of converting 4th downs inside the red zone. Obviously, the true estimates are going to be smooth and continuous. But it was easy to correct for that in the final analysis, which follows below.
Boiling down these numbers produces a handy cheat sheet. On or below the line a team should go for it.
There are two very interesting results. The first is that going for it gets better as a team is backed up in its own territory. This is because a punt triggers SD, which is a very bad situation to put yourself in while handing the opponent relatively good field position. Because it’s so bad, going for it at what would seem to be suicidal distances to go makes sense.
The second interesting result is that long FGs should not normally be attempted on the first possession. If a team misses a long attempt, which is obviously increasingly likely as distances get longer, SD is triggered with the opponent in relatively good field position. Oddly, the numbers suggest punting on 4th and 8+ is the right decision all the way up to the opponent’s 28-yd line! Try explaining that one in the post-game press conference.
What's left to be done? We still need to crunch the numbers for state 2 and 3. For example, down 3 points in OT (State 2), when should a team go for a conversion (or TD) instead of a FG? In SD, when should offenses pass up the long FG attempt in favor of a 4th down coversion? Actually, the numbers are already curnched. I just need to get around to writing everything up.