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As an active follower of the NFL from outside the US, I have always found it curious that teams overwhelmingly take a 1-point field goal rather than a 2-point conversion. Some reasons are given in the following post: Why do teams almost always pass on 2-point conversions?
I also wonder if the 2-point score is too little for the risk involved, so why not make the conversion be worth 3-points? As a thought experiment, I wonder what effect this would have on the game?
It really depends on what the 3 point play is gonna be in this hypothetical situation. The 2 point play is from the 2 yard line, but where is the 3 point play going to be from, the same distance, the 3 yard line, the 5? What happens to the field goals, are they also still worth the same amount as the post touchdown conversion? Since these criteria have profound impact on the game, I don't think we can discuss without further details.
Let's discuss the 2 point conversion (2PC).
In 2021 the kickers made 92.4% of the 1 point conversions (1PC). This is the lowest percentage in decades, mainly due to the kicks being pushed back 13 yards in 2015. This change was made as the kicks were basically automatic (99%+ conversion rate) at the 2 yard line, and the league wanted to add more randomness to the game, or as they explained it, more excitement. With the kicks not being so automatic anymore, the 2PC became more of a realistic choice. Specially since the same set of rules also changed the value of return of extra point conversion. A returned 1PC or 2PC would be worth only 2 points, as opposed to 6+1. It was treated as a normal touchdown before. So the 1PC got harder, and the penalty of horribly messing up a conversion drastically dropped. This made the percent of 2PCs more than double after the move of the extra point kick.
Statistically speaking, as mentioned before, 92.4% of extra point kicks were successful in 2021, while 49.6% of 2 point conversions were successful. So clearly the odds of scoring are much lower with the 2 point conversion. And since you're trying to score more than you opponents to win the match, it doesn't make it a favourable action during the game as it remains risky. Tho if it's done at the very end of the game with little time left, it may be worth the risk as converting it can either make you win or force your opponent to score again with a few seconds left.
But the 2PC rate doesn't take into account the weather conditions, the momentum, the fact that most conversions are made by teams desperate to catch their opponent or are offensive juggernauts, injury conditions or higher potential to get injured on 2PCs, higher odds of return in 2PCs (since passes can be intercepted, odds of fumbling), etc. And also we're seeing the worst 1PC kicking in recent history as well as 2PCs being high. The risk isn't worth it to go for 2PCs since if you miss it, your opponent can simply kick after their touchdown and gain a much easier point against you.
Now let's get back to the normal scenario of the beginning or middle of the game. As we saw, a single 2PC is not so rewarding. Let's consider 2 conversions. You're not guaranteed to score a touchdown twice, but if for example your team is down my 14+ points, you're gonna have to score 2+ touchdowns anyway, so you know you're gonna get at least 2 touchdowns to win. In this case you get up to 2 chances to try the 2PC. If the first one fails, you can try on after the second touchdown, and still come out with 14 points (same as kicking twice). If the first try succeeds, you can kick on the second try and come out with 15 points. This helps with the odds of winning and makes the 2PC more desirable when trailing.
You can show that statistically, but this is not the best medium to do so and I'm not gonna bore you with that. But the maths are far more favourable to go for 2 2PCs than just 1, as with 2 the risks are lower. The expected value for a conversion is the odds of the conversion times the points earned. Doing quick calculation you'll see a 2PC yeilds to higher value, but if you miss it, your opponent can kick and gain more points, so also need to think about the negatives. Hence why a 2PC is not often tried unless trailing.
I don't think we can make the same argument with 3PC. You could run the same logic and maths (which I didn't show) to argue if the 3PC is better than a 2PC or a 1PC. If a 3PC is going to be also on the 2 yard line, I expect to see a ton of 3PC, simply cause mathematically they're far more favourable since they yield in higher points. The reason I say we can't make this argument is I think a 3PC will uniquely negatively effect field goals. Why ever go for a kick on 4th and a couple of yards, if converting can give a chance to keep the drive alive to score a touchdown (6 points) and then try for a 3 point play as well? You'd see far less field goals, so the way the game is played will be changed. In my opinion the after-touchdown play needs to be of a lower point value than a field goals, to keep short and medium kicks around.
