Posted by Mission146
Sep 16, 2015

Will the New PAT Rules Change NFL Betting?

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.

Option A PAT

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.

OPTION B TWO-POINT CONVERSION

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.

Known v. Expected Points

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.

Conclusion

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.

tooncestdc Sep 16, 2015

Very good points. Another thing to consider is that 2-point conversion plays are currently rare and very specific (exactly 2 yard endzone play, very little fear of a turnover), and therefore teams can afford to practice and perfect a small subset of plays that will rarely be seen by their opponents. If a team was to choose to go for 2 on all TD plays, they would have to start using a wider variety of plays that wouldn't be as well practiced to keep an element of surprise.

Keep in mind that last year's critical Super Bowl play was likely a 2-point conversion play that New England had prepared for and predicted.

tooncestdc Sep 16, 2015

The net point being that the conversion rate would likely drop well below 50.5% if they were commonplace.

Wizard Sep 17, 2015

This article makes a strong case that going for two points is the best play mathematically: Kick the PAT or go for 2? Math favors the 2-point conversion (link: www.milehighreport.com/2015/5/20/8628917/kick-the-pat-or-go-for-2-math-favors-the-2-point-conversion).

cyberbabble Sep 17, 2015

Starting at least 7-8 years ago, NFL statnerds have been publishing info that, in most game situations, coaches should go for fourth and short rather than punt. The coaches are aware the analysis, have commented publicly, and don't necessarily disagree. The coaches also punt too often.

The analysis is good but the coaches have to buy into it. It's unlikely the coaches will announce their plans ahead of time. Need to wait awhile to find out if there is any effect on points scored.

cyberbabble Sep 17, 2015

While I'm posting --
Denver 23 KC 24 with 36 seconds in the game.
Denver kicks a point and ties it.

KC gets the ball and fumbles, Den picks it up and scores.
Den 31 KC 24 with 26 seconds to go.

Lucky1 Sep 17, 2015

Although this high school, found it interesting as well http://profootballtalk.nbcsports.com/2015/08/14/nfl-teams-pick-the-brain-of-high-school-coach-who-never-punts/

Mission146 Sep 18, 2015

ToonCestDC,

I think that you bring up a good point, but I would imagine that it would only have a minimal effect on the actual conversion success rate, maybe a couple percent less. The reason that I say that is that, as it stands, teams already have a vast array of short-and-goal plays and defenses already prepare for short-and-goal plays in trying to prevent TD's. Honestly, if I were going to pull my most creative plays out of the bag, I'd do it when I was going for six, not two.

Wizard,

I actually agree that it is the best play mathematically, with respect to Expected Points. Although, the Article really took some liberties in rounding down the PAT success rate.

I actually presented a very similar argument above in terms of Expected Points and in terms of the Expected Point Advantage the go-for-two team would have over their opponents. The problem is that the Expected Points ignore everything else about football, perhaps notably on a first score that, a failure now enables the other team to tie with two FG's and quite probably take the lead if they score a TD.

The math also supports my theory for first score TD's which is, if the Team who scores first goes for two, then the most likely scenario if the next team also scores a TD is that (out of four possible scenarios) they will take a 7-6 lead. The first team leading 8-6 and the probability of an 8-8 tie are equally likely.

Cyberbabble,

That's true, but there's even more to this than Expected Points, in this, you also have to look at game strategy.

Lucky1,

A friend of mine who coaches Middle School football adopted the same philosophy, of course, a M.S. punter can kick for significantly less distance than your average H.S. punter.

DJTeddyBear Sep 18, 2015

What this article didn't discuss (unless I missed it) is that the rule change means you must declare your intentions.

As a result, the option of a two point sneak play is removed, and the possibility of salvaging a bobbled FG snap with a two point conversion is greatly reduced.

Wizard Sep 18, 2015

If you go for the two and make it you force the other team to either score twice to pull in the lead or match your accomplishment. I admit that the difference between 6 and 7 is big, but so is the different between 7 and 8. Furthermore, I think early in the game it should be about maximizing expected points.

I have to go on record as saying that I respectfully disagree with your point of view on this one.

Mission146 Sep 18, 2015

Wizard,

I'm usually not inclined to disagree with you, because that usually means I'm wrong about something.

Let me ask you this: Let's say we have a 24-24 tie in the 4th and a team scores with 2:42 left on the clock: Should they go for two?

The scenario is basically the same, a success forces the other team to match the accomplishment, but if you fail and the other team scores a TD, now they have a 94% chance of taking a one point lead with very little time on the clock, if any.

My point is that the beginning of the game is no different in terms of strategy because you don't know who will score again, how many times, or what those scores will be. It could be 6-0 until the middle of the 4th, then the other team scores, and now they have a 94% chance of taking the lead. All of that to have a 25% chance (assuming a TD) of getting an 8-6 lead if the other team fails to convert. In the meantime, even with eight, a TD and a FG put the other team in the lead, as does 3 FG's. I just have trouble seeing any upside other than Expected Points, but Super Bowls aren't won by scoring the most cumulative points in all your games.

Wizard Sep 21, 2015

End of game situations are different. My comment refers to early in the game. I suggest we agree to disagree.

Wizard Sep 25, 2015

I've spend three days on this and have presented my results in the last question of Ask the Wizard column # 293 (http://wizardofodds.com/ask-the-wizard/293/). Enjoy.