As we enter the last quarter of the regular season, the topic of playoffs (“Playoffs? Don’t talk about playoffs! Are you kidding me? Playoffs?”) comes to mind. This year, the race looks very interesting; we might finally see some of the more esoteric tiebreakers come into play. Since this is a website and a blog, we can look at different playoff scenarios, looking at the AFC.

Right now, the AFC has three teams at 10-2 (Cincinnati Bengals (CIN), Denver Broncos (DEN), New England Patriots (NE)). Those can’t all end at 10-2, because CIN and DEN play in Week 17. But some tiebreakers potentially become interesting.

Scenario 1: Denver wins out

If Denver wins all of its remaining games, it is the first seed. It holds the first tiebreaker (a head-to-head win) over New England, so New England has no chance of supplanting it. In this scenario, “Denver controls its own destiny” is the common nomenclature; as long as Denver wins, they can’t be replaced.

Scenario 1a: Denver wins out, but the Patriots lose a game, and Cincinnati wins all games except to Denver

In this scenario, Denver is still the first seed, but Cincinnati and the Patriots are now tied. They didn’t play each other, so the first tiebreaker is skipped. The second tiebreaker is “record in the AFC,” both Cincinnati and New England finish 10-2 (the Patriots remaining games are all in the AFC, so any loss there is the second AFC loss).
Now we move to the third tiebreaker: common games. The teams play Buffalo, Denver, Pittsburgh, and Houston as five common games (NE plays BUF twice; CIN plays PIT twice). Currently, the Patriots are 3-1, with one more game at Houston; CIN is 2-1, with games at Pittsburgh and Denver remaining. If the Patriots loss is not to Houston, then New England wins this tiebreaker, with a 4-1 record over Cincinnati’s 3-2 record. But what if the Patriots fall to the Texans?
Then we go to the fourth tiebreaker: Strength of Victory. This is calculated by taking the combined records of the teams that you defeated in a season, and comparing that to your opponent’s defeated teams’ records. In this scenario, Patriots victims have 81 guaranteed wins, and another 25 possible (some games in the future are between teams the Patriots have beaten, so no matter who wins, the Patriots get one win credit). The Bengals victims have 76 wins, with another 14 possible. 3 of both teams’ possibilities are Bills victories, so the Patriots hold a 5 game edge, with a 22 to 11 margin of possibility, so the Patriots likely take this tiebreaker (that’s only three Bills games, because in week 17, the Bills play the Jets, so that’s only upside for the Bengals, not the Patriots).

Scenario 2: The Patriots and the Bengals win out, Denver loses at least one

This scenario looks a lot like Scenario 1a, except that the Patriots don’t fall to Houston, and Cincinnati doesn’t fall to Denver. Both teams are at 10-1 in the AFC and 4-1 in common games, so we move to the fourth tiebreaker again. In this case, it is much tighter - Denver has a better record than Houston. Assuming Denver otherwise wins out, the Patriots project to a minimum strength of victory of 90 over the Bengals’ 89. The Patriots then have 22 more possible wins to the Bengals’ 14, again with 3 Bills games in common. That puts the Patriots at 19 possible games to 11, but only carrying a one game lead going in. Big edge to the Patriots.
What if they tie? The fifth tiebreaker is Strength of Schedule, so we only have to look at the strength of the opponents that defeated Cincinnati (Arizona and Houston) and New England (Denver and Philadelphia). Those are currently 15 and 16 win pairs, so it’d be tight if we get this far.
That’d take us to the sixth tiebreaker: The combined ranking in “Points for” and “points against” in the AFC. This was a lot closer before this last week; right now these teams are first (NE) and second (CIN) in points for, but first (CIN) and fifth (NE) in points allowed. Edge: Cincinnati (2+1 is lower than 1+5).
Really unlikely we’d get this far, but the seventh tiebreaker is just the sixth tiebreaker, but compared against the entire NFL. The Patriots are second and tenth (12), but the Bengals are fourth and first (5), so the Bengals still hold an edge.

Scenario 3: a three way tie

Scenario 3A: Denver beats Cincinnati, but loses any other game; NE loses one

Since Denver holds the head-to-head tiebreaker over both the Bengals and Patriots, they advance to the first seed. The Bengals and Patriots now break the tiebreakers as in Scenario 1A.

Scenario 3B: Cincinnati beats Denver; NE loses one, CIN loses one

Since there isn’t a clear head-to-head tiebreaker (CIN beats DEN who beats NE, but CIN and NE don’t play), we drop to the AFC tiebreaker. If Cincinnati’s other loss is to San Francisco, then the AFC records are sufficient (CIN 11-1 over NE 10-2, DEN 9-3), so Cincinnati takes the first seed. New England doesn’t take the second seed, as we start the tiebreakers over again, and Denver wins on the head-to-head tiebreaker. If Cincinnati’s other loss is to Pittsburgh or Baltimore, however, NE & CIN are tied at 10-2 over DEN 9-3, so we eliminate Denver, and continue to the next tie-breaker (actually, we start over, but fall right through to this point with NE & CIN).
Since common games is the next tiebreaker, it matters which games the teams lose to. Since New England plays Houston and Cincinnati plays Pittsburgh, if only one of those teams win, they win this tiebreaker. But if both teams win or both teams lose, we fall to the strength of victory tiebreaker.
Assume they both lose (otherwise, we assume the Ravens beat the Bengals, and I’m not ready to face that). The strength of victory really favors the Patriots: 81 with 25 possibles versus 77 with 19 possibles; again with three shared possibles. If they both win, we swap out the Ravens and Steelers, and either the Jets or Dolphins for Texans. That’s mostly neutral for the Patriots, but adds three games to the Bengals, putting this tiebreaker far more into the mix.
If the Patriots win those tiebreakers, then Cincinnati takes the second seed, but if Cincinnati wins the tiebreakers, then *Denver* takes the second seed, dropping the Patriots to the third seed.

We can get into even more complicated scenarios with deeper losses, but that’s a project for a later day. First, the Patriots and other teams need to Do Their Job.

As we look at NFL records, it’s sometimes tempting to create narratives around various teams, based on their win-loss records. Notionally, the AFC East is playing an easy schedule (against the NFC East and the AFC South) but how much should we question that effect on their rankings? Which divisions are more or less competitive, and which ones are stronger?

Starting with the Football Outsiders Defense-Adjusted Value Over Average (DVOA) team metrics, we can try to build a narrative about teams (all data behind this storytelling is from the 2015 Week 10 DVOA Ratings). Since these ratings include both a team value and a variance of that value, we can compare the normal distributions of two teams to answer the question, “How often will Team A beat Team B on a neutral field?”

A necessary question, of course, is “what does an average” division look like? An average divisions has four teams with DVOAs around 18%, 5.4%, -4.6%, and -18.8%, and variances in the 10-15% range. Given that average division, we can compare any NFL division to it, and calculate how many games we expect the NFL division to win against the average division (in a 16-game tourney). Since we’re going to use the average division from this year, the average of all NFL division expected wins will be 8. The strongest division so far is the AFC East, with 11.36 expected wins; the weakest is the AFC south, with 4.10 expected wins. Most divisions are within 1.5 wins of average.

We also would like to understand how competitive a division is. For this, we can simulate an intra-division round-robin of 6 games, and look at how distributed the wins are, on a scale of 0 (the top three teams have the same expected wins) to 1 (the top three teams have an impossibly far apart distribution of wins).

The Gauntlet Divisions

Four divisions are competitive, while also being stronger than average. A very competitive pair in the AFC West and NFC East, and a mildly competitive pair in the NFC West and AFC North. The Broncos wild start notwithstanding, Kansas City is potentially a better team, but right in the neighborhood, and the Raiders aren’t far behind. The NFC East is the coin-toss NFC East, with teams that are much closer to average than any other division. The NFC West has two very good teams (Cardinals/Seahawks at 2nd and 5th), as does the ADC North (Bengals/Steelers at 3rd and 6th).

The Cupcake Division

The AFC South, however, is interesting. No other division’s fourth best team is expected to win more than 0.55 games against their three division rivals (and that’s the Saints), but the Titans are expected to win 1.19. Given that with perfect parity, the fourth-best team can’t win more than 1.5 games, that’s a pretty surprising number. Any of these teams could win the division - but we’d expect all of them to be steamrolled by another division (the worst team in the AFC East has a higher DVOA than the best team in the AFC South).

The Cakewalk Division

Some divisions crown their winner early, not because the winner is necessarily that great, but because their competition is that weak. The NFC South obviously fits into this category; Carolina is really good, but Atlanta, the second-best team, is barely good enough to be the third-best team in an average division. Surprisingly, the NFC North also lands here, even though the Vikings at 7-2 are above the Packers at 6-3. In a weird scheduling quirk, while the Packers schedule has been pretty evenly distributed, the Vikings average opponent to date has been 9.7% worse than average (the second easiest schedule so far), while their future opponents average 9.1% better than average (the hardest schedule remaining). Unless the Vikings step up, that’s going to create a narrative of a late-season “collapse.”

The Tyrant Division

Or, as Rex Ryan might call it, the Bully Division. The AFC East expects to win a surprising 11.36 games in a 16 game tourney against an average division. Unfortunately for every team not the Patriots, the Patriots expect to win 2.84 out of 3 games when playing the Bills, Dolphins, and Jets. But there’s a tyrant atop this division, ruling with an iron fist, and they don’t intend to give up their throne easily. But the contenders in the shadows are good, and arguably better than many division leaders.

If you haven’t heard of DeflateGate by now, congratulations! I’m not sure how you avoided all mention of it; if you’re that person, feel free to go Google a summary.

If you ask a physicist to predict the winner of a horse race, the first words out of their mouth might be, “Well, if you assume a spherical horse…” Probably not very helpful, but most of the physics that the rest of us remember are in the spherical horse category; the rules work really well on paper, but are, at best, imprecise models when describing or predicting events in the real world.

Take the Ideal Gas Law (please!). PV = nRT; in a closed system filled with an ideal gas (far from a liquid), the product of the pressure exerted on the boundary and the volume enclosed by the boundary moves linearly with the product of the number of molecules of the gas and the temperature of the gas. R is just a constant, and it changes based on which units you want to measure your values in. A lot of ink has been spent on asserting what the Ideal Gas Law predicts should have happened to those footballs that night; but most of it assumes a spherical horse.

And when you assume a spherical horse, you get a little surprised when your horse has hooves and a tail. A simple running of the Ideal Gas Law with some assumptions about that night (Start temp=71F, end temp=48F, start air pressure is 12.5psi) projects a low end pressure of 11.32. And looking at the halftime measurements of the balls, even using the favorable Logo gauge, we see three balls are under that number - one as low as 10.90psi. That seems a little surprising, but only if you assumed a perfectly spherical ball. Some other factors might be at play. Here are a few.

Gauge calibration

From the Exponent Report in the Wells Report, page 28, we see that the Logo gauge doesn’t match true relative air pressure. A measurement of 12.5 on the Logo gauge corresponds to 12.17 on a true gauge. The measurement we’d expect to see at halftime (using the initial calculations) becomes 11.276 (corresponding to 11.01 of true pressure).

Measurement errors

There are two ways that the gauge might impact the measurement. Rounding error comes into play - the gauges only measure in twentieths of psi. Note that 11.276 above ought to round up to 11.3; but what if a ball didn’t start at 12.5, but instead was at 12.475? That drops our halftime prediction down to 11.25.

Of course, of less significant value is the air that leaks out every time the ball is measured - in this case, three times. From Exponent page 36, we see that every needle insertion removes 0.01 psi of air. Seemingly minor, but this 0.03 drops us to 11.22. That’ll round down to 11.2

Temperature, and a swamp cooler

Ever sweat, and then feel cooler as a breeze blows past? Or blow a fan across a bowl of water, and watch the air temperature drop? What’s happening is /evaporative cooling/ - as a liquid evaporates, it needs a substantial amount of energy to change states from liquid to gas. It draws that energy from its surroundings - cooling things down - to make its phase change. So even if the ambient temperature was 48 degrees Fahrenheit, a wet ball might exhibit as cooler.

And we see this in the chart on page 42 of the Exponent Report:



The wet balls do have a lower air pressure. Referencing back to the Ideal Gas Law, the means either the balls got larger (leather stretches, and may reduce force on the bladder), some gas escaped (unlikely, but if so, this whole scandal ought to be moot), or the temperature in the balls is lower - as evaporative cooling suggests. Unfortunately, Exponent doesn’t tell us how much cooler the balls effectively are, but we can eyeball the chart to guess. At 69 degrees (the two hour mark), the pressure appears to be about 14.48. After being on the field, we see the wet ball as having a pressure about 11.26, and the dry ball having a pressure around 11.36. That suggests that the air inside the wet ball was equivalent to about 45.46 degrees, while the dry ball was 47.39 degrees. Actual air temperature was 48 degrees; so the wetness of the ball causes around 2.5 degrees of temperature loss (or the equivalent growth of the ball).

Plugging in our estimated effective temperature in, we now expect a gauge pressure of around 11.09.

Actual air pressure

Each of these calculations has assumed that the ambient air pressure was 14.7 psi, which is the approximation for standard atmospheric pressure. Fortunately for our weather systems, air pressure changes; unfortunately for an argument in favor of the Patriots, it’s moving in an unhelpful direction that day:

At 3:45, the time of the initial measurement, the air pressure is 29.9 inches (of mercury!), which corresponds to 14.686 psi. By halftime (8:30), it’s 29.65 inches, or 14.56 psi. This should make our balls appear to get more “inflated” at halftime, and they do - we’re back up to 11.22.

Unknowns

And here’s where it gets hard to measure: the balls aren’t outside, but are inside Gillette Stadium. If pregame, the outside temperature was in the 70s, it’s possible that the building HVAC was barely running. As it drops to 48 degrees, if the HVAC is engaged at halftime, that’ll raise the ambient air pressure a bit, and lowering our anticipated halftime measurement.

And what was the temperature in the officials’ locker room pregame? Exponent measured it on February 7th, when the high was 28 degrees; with a crowd of people and a warmer day, was it a bit warmer? And since the balls were measured in the shower room, had anyone showered in there (warming the start temperature still further)? Had the shower room been at 74 degrees, and the halftime HVAC-assisted air pressure been at 14.725 psi, then we’d expect to see a lowest gauge reading at 10.90. But we’re now in the realm of “we didn’t measure it at the time, so we can’t now what happened.”

Were any of these balls fatigued and lose pressure? Again, not something that we’re likely to know. But there is enough uncertainty around the shape of these horses that we shouldn’t just expect them to be perfect spheres.

If you weren’t already convinced, then you’re probably not convinced further.