# When should a football team attempt a two point conversion instead of an extra point? A dynamic programming approach.

On Sunday November 10, 2019, the Carolina Panthers were down 14 against the Packers early in the 4th quarter. They scored a touchdown, putting them down by 8, and they went for a two point conversion.  The two point conversion did not succeed. This has been the subject of debate, with journalists both applauding and criticizing the decision.

I created a dynamic programming model to determine whether or not to go for a 2 point conversion. The dynamic programming model is based on Wayne Winston’s book Mathletics, which is a fantastic introduction to sports analytics. The state captures the team with possession, the score differential when they obtain possession, and the number of remaining possessions. When there is one remaining possession, it is the last possession. When there are three remaining possessions, the team with the possession has two scoring attempts. Each possession ends in a touchdown, a field goal, or no score. I assume half of all games end in a tie. The probabilities I used are based on average team statistics. I do not model other decisions, such as whether to go for it on fourth down, although these could further improve a team’s win probability.

The slides are below.

Bottom line: teams should go for two points when they score a touchdown and they are down 10, 8, 3, or 2 or up by 1, 2, 4, or 5 (including the points from scoring the touchdown) near the end of the game. These conclusions hold when there are at least two additional possessions in the game.

If you have the last possession: go for 2 when a touchdown on this last possession puts you down by 2.

If you just scored a touchdown but your opponent will have the last possession: go for 2 when a touchdown puts you down by 2 or up by 1, 4, or 5. You normally will want to go for 2 when a touchdown puts you up by 2 except in this situation, because missing the extra point means your opponent could win with a field goal.

Carolina went for two when down by 8 after scoring a touchdown. According to my math, Carolina made the right choice. However, the best strategy does not guarantee a win nor does it drastically improve the win probability.

We can examine the decision in more detail. When down by 8 with four possessions to go (which matches up with when Carolina went for a two point conversion), a team has one of two choices:

1. They could kick an extra point, which would give them a 11.3% win probability if successful (with probability 0.96) or a 7.9% win probability if not successful. Together, this yields a 11.2% win probability.
2. They could go for a two point conversion. If they succeed (with probability 0.48), they would have a 18.3% win probability. Otherwise, they would have a 7.9% probability of winning if not successful. Together, this yields a 12.9% win probability.

There are four things to keep in mind:

• Carolina improved their probability of winning by 1.7% by going for two.
• A good process does not guarantee a good outcome.
• Carolina was not likely to win using either approach.
• Carolina could have further improved their win probability by considering other decisions (who is playing, which plays are called, and whether to go for it on fourth down).

My conclusions are summarized in the chart below. For more reading: Benjamin Morris of 538 wrote an article about when to go for two here. My analysis is consistent with his, although we make different comparisons.

When to go for a two point conversion in NFL football