As many of you already know, the NFL has changed the PAT (aka Extra-Point) from a snap from the two-yard line to a snap from the fifteen yard line, which effectively changes the try to roughly a 32-yard Field Goal equivalent rather than a seventeen yard attempt. There has been some talk in Sports Betting circles that this Rule change might change the value of certain Point Spreads and could also change the value, potentially, of key Over/Under bet numbers.
In my opinion, the bettors can relax.
While it is true that there should be more missed PATs, (Expected success rate of about 94%-rounded down-compared to the previous near 100%) that shouldn't be expected to change the strategy for when to go for two too much.
Initially, this seems counter-intuitive given the fact that the Two Point Conversion (based on a 50% success rate) should have a value of one point compared to a value of 0.94 points for the PAT. However, unlike Expected Value in gambling, the decision of whether or not to go for two still leaves an open-ended result whereas gambling decisions are closed when they are completed.
For a basis of comparison, let's take a look at the game of Craps. Imagine a player puts a $10 bet on the Don't Pass and any Point number is established, if the player were to then Place/Buy that Point for $10, then the player could no longer lose any money on the final result provided the player makes no further bets. In other words, the probability of losing becomes 0% and the probability of winning becomes contingent on whatever the Point Number is. The player could also Place/Buy for a lesser amount than the initial Don't Pass bet and lock up a small profit either way.
Of course, as any Craps player would immediately recognize, this is a very foolish way to play Craps. In addition to the fact that the Don't Pass bettor overcomes his initial disadvantage (7's, 11's on the Come-Out) when a Point is Established, given that a Point has been Established, the Don't Pass bet itself now has a hugely positive Expected Value which is always reduced by hedging with a Place/Buy bet on the Point that has its own House Edge.
In almost all cases, (other than hedging on life-changing sums of money) the Expected Value is going to be the key point in whether or not making a certain bet is a prudent thing to do. In all situations, gamblers should try to maximize their Expected Value.
However, maximizing the Expected Points, shall we say, is actually a poor decision in NFL Football.
The reason for this is because the decision of whether to attempt a PAT or go for a 2PT Conversion is not a closed-ended decision. There is still a game to be played afterwards. For an example, let's say that Team A is the first team to score in a game and that score happens to be a Touchdown, now we're going to look at the most common scenarios based only on the success rate of the PAT vs. the 2 PT Conversion.
The first thing that we notice is that Team A is going to make a decision based on the possibility that Team B's next score is going to be a Touchdown. Although, it is also worth noting that any failure to produce additional points by Team A will also put Team B in a position to tie the game with Two Field Goals.
We're going to round down and call the possibility of a successful PAT 94%, and whether or not Team A is successful, the decision that makes the most sense for Team B if they score next and it is a Touchdown, will be to attempt an extra point. For these purposes, we're going to assume that.
Team A Makes, Team B Makes: (.94 * .94) = .8836 or 88.36%
Team A Makes, Team B Misses: (.94 * .06) = .0564 or 5.64%
Team A Misses, Team B Makes: (.06 * .94) = .0564 or 5.64%
Team A Misses, Team B Misses: (.06 * .06) = .0036 or 0.36%
What we see here is that both Teams have an equal expected point value of 6.94 total points on the plays in question, and further, both teams have an equal probability of ending up tied 7-7, being up 7-6 or being down 6-7.
Team A also has the option of going for a Two-Point Conversion, which at a probability of success of 50%, has an Expected Value of one point. We immediately notice that the Expected Point Value is greater than with the PAT, but does that make it the right decision?
The answer is, "No," because this is not a closed proposition. A failed Two-Point Conversion attempt not only enables Team B to tie with two Field Goals, but it also makes it highly probable that they will take the lead if they score a Touchdown. Let's look at our scenarios:
Team A Fails, Team B Makes PAT: (.5 * .94) = .47 or 47%
Team A Fails, Team B Misses PAT: (.5 * .06) = .03 or 3%
Team A Succeeds, Team B Succeeds: (.5 * .5) = 25%
Team A Succeeds, Team B Fails: (.5 * .5) = 25%
What we take away from this is that, with a success rate of 50% and assuming Team B scores next and it is a TD, (or, really, if Team A scored a FG in the interim and then Team B scored a TD, Team B would still go for two) that Team A ends up with a probability of 25% to lead 8-6 after Team B's TD and a probability of 25% of being tied 8-8.
However, if the Two-Point Conversion for Team A is unsuccessful, now they have yielded a huge advantage to Team B. At this point, if Team B were to score a TD they have a 94% chance (47% probability overall) of taking a 7-6 lead and only a 6% chance (3% probability overall) of keeping it tied 6-6 with a missed PAT.
In other words, Team A would ultimately have a 25% chance of being ahead still even if Team B scores a TD, a 25% chance of it being tied 8-8, a 3% chance of it ending up tied 6-6 and a whopping 47% chance of Team B taking a 7-6 lead.
For this to ever be a correct decision, Team A would have to have a significantly greater probability (and granted, they might) of converting for two and Team B would have to have a significantly less than average probability (again, they might) of converting for two. However, if this is the case, then Team A probably also has a better offense, in general, so they normally shouldn't feel the need to press the issue by trying to put eight on the board.
As we clearly see, the probability of ending up with a Tie score of any kind assuming Team B's next score is a TD is about 88.72%. The probability of ending up ahead one point is 5.64% and the probability of Team B ending up ahead one point is also 5.64%. In other words, it's a wash and both teams are expected to score 6.94 points.
In the case of the Two-Point Conversion Attempt by Team A, we see that Team A has an average value of 7 total points scored, but at what cost? Team B has a conditional Points Scored value, they will also average 7 points scored if Team A successfully converts, but they will average 6.94 points scored if Team A fails, for an overall average of 6.97 points scored.
This would initially sound great for Team A, they have an Expected Score of 7 to an Expected Score of Team B's 6.97. However, we have to look at the probabilities and recognize that Team B gets to make its decision based off of Team A's known result.
Due to the fact that Team A has a 50% chance of failing on the Two-Point Conversion, Team B is going to go for the PAT and generate an average of 6.94 points on the play, but remember, that's compared to Team A's KNOWN value of six points scored. Given Team A's success rate of 50%, they will now have a KNOWN value of 8 points, and therefore, Team B will also go for two and have an Expected Value of Seven Points. We see that Team B has an advantage of .94 Expected Points compared to Team A's known points in one scenario and they have a disadvantage of one Expected Point compared to Team A's known points in the other scenario. In other words, by scoring first and going for two, Team A has an Expected Advantage of 0.06 points (if Team B scores a TD) compared to an Expected Advantage of 0 if they take the PAT.
However, in an open-ended game, Team A cannot ignore the probabilities. They have a 50% chance of failure which gives Team B a 94% chance of taking a one point lead if Team B scores a Touchdown. Overall, Team B has a probability of 47% of taking the lead in this scenario, a 25% probability of the score being tied 8-8, a 25% probability of being down 8-6, and a 3% probability of a 6-6 tie.
Ultimately, by going for two, Team A has a 25% probability of ending up ahead if Team B scores a TD, a 28% probability of being tied with them anyway, and a whopping 47% probability of ending up behind.
In gambling, most bets that we place are a closed decision. Furthermore, once the bet is resolved, that's it. If the Don't Pass hedger has a Point of Four Established, then either a four or a seven comes and he is on to the next bet.
Football is totally different in that it is completely open-ended. Neither team knows if or how many times the other team will manage to score, or what types of scores those will be, so the focus is more on the highest probability of staying in the lead at all times. The only thing that ends the game (other than going to OT) is the clock reading 00:00 in the Fourth Quarter.
Given that the focus is on getting in the lead, staying in the lead and not giving the opponent an easy opportunity to get in (or maintain) the lead, I would conclude that the vast majority of scenarios in which a Two-Point Conversion makes sense have not changed. The only question is going to be one of a Missed/Blocked PAT directly affecting the Line, and those scenarios are likely to be few and far between.