**SITE NEWS:**
We are moving all of our site and company news into a single blog for Sports-Reference.com. We'll tag all PFR content, so you can quickly and easily find the content you want.

Also, our existing PFR blog rss feed will be redirected to the new site's feed.

Pro-Football-Reference.com ยป Sports Reference

For more from Chase and Jason, check out their work at Football Perspective and The Big Lead.

## Re-handicapping the AFC

A couple of weeks ago, I tried to handicap the AFC playoff picture. I'll do it again today, with a little twist in the formula.

With all due respect to our Chiefs, Bills, Steelers and Titans fans, I'm going to leave them out of the equation today. Figuring out the odds with four teams (Jets, Jaguars, Broncos and Bengals) is difficult enough, without an NFL super computer. Random note: the second place teams in all four AFC divisions have the same record (8-6), and the third place teams in all four divisions *also* all have the same record (7-7).

Here are the remaining schedules for the AFC contenders.

Team 16 17

Cin @Den Pit

Den Cin SF

Jac NE @KC

NYJ @Mia Oak

Once again, it's time to replace those teams with their ratings from Jeff Sagarin.

Team 16 17

24.80 @21.05 22.64

21.05 24.80 9.47

29.95 29.93 @19.74

22.00 @20.69 10.54

Now we can calculate each team's chance of winning each game, using the formula:

Home team prob. of winning =~ 1 / (1 + e^(-.438 - .0826*diff)), where "diff" equals the home team's rating minus the road team's rating.

Team 16 17

Cin 0.47 0.65

Den 0.53 0.80

Jac 0.61 0.60

NYJ 0.42 0.80

From this, we could sum the weeks and get an expected number of season wins (8 + the number above):

Den 9.33

NYJ 9.22

Jac 9.21

Cin 9.12

As you can tell, that's pretty close. Things change pretty quickly around here, and Cincinnati went from being a playoff favorite to really being on the outside looking in...right?

Of course, the total number of expected wins is pretty irrelevant, because some teams have better tiebreaker scenarios than others. The four wildcard contenders play seven unique games over the last two weeks. Those games could end up in any of 128 different combinations. The most likely combination would be: Denver beats Cincinnati, Cincinnati beats Pittsburgh, Denver beats San Francisco, Jacksonville beats New England and Kansas City, the Jets lose to Miami and the Jets beat the Raiders. There's about a 9.5% chance the games go that way. If they do, the Broncos and Jaguars would be in with 10 wins, and the Jets and Bengals would miss out.

For those who took a hard look at the percentages above, you could probably guess the second most likely outgame: the same as before, except Cincinnati now topping Denver. In that case, the Bengals and Jaguars would make it.

The tiebreakers can get pretty complicated, and I can't promise you that I've done them 100% correctly. I'll give it my best. But suffice it to say, getting to 10 wins seems like a pretty safe bet (although the Jets would miss out if three teams get to 10).

Here are the odds that each team gets to X number of wins:

Wins 10 9 8

Cin 0.30 0.51 0.19

Den 0.43 0.48 0.09

Jac 0.36 0.48 0.16

NYJ 0.33 0.55 0.12

In terms of tiebreakers, the Jets look to be in the worst position because they'll lose out to the Jaguars via head-to-head, and will lose out to Bengals and Broncos because of a poor conference record. So the only way the Jets can make the playoffs is if they have a better record than two of the other teams. Of the 128 combinations, 36 of them would give the Jets a better record than two of the other three teams, and make the playoffs. The sum of the odds of any of those combinations occuring is just north of 27%, so the Jets are slightly better than average favorites to make it.

The Broncos are the opposite of the Jets; they seem very likely to make it if tied. If tied with the Jets or Jaguars, Denver would make it because of a better conference record. Even if Denver loses to Cincinnati, Denver will make it as long as they don't have a worse record than two other teams. There are 59 combinations where no one would have a better record than Denver, and another 39 where only one team would post a better record. The sum of the odds of any of those 98 combinations equals 74%.

Cincinnati would win a tiebreaker over the Jets and Jaguars, but might or might not against the Broncos. If Denver beats Cincinnati, Denver would get the tiebreaker. If Cincinnati beats Denver, Cincinnati would get the tiebreaker if neither the Jets nor the Jaguars have the same record as the Bengals and Broncos. If one of those teams do (i.e., a three-way tie), then Denver would be the first team in because of a better conference record, and would make it in over Cincinnati. (Then, depending on the record of the 4th team, Cincinnati may or may not get in.) I'll save you the grunt work, and just say there are 78 combinations that would put the Bengals in, and there's a 54% chance of that happening.

The Jaguars aren't in much better shape than the Jets; they'd beat out the Jets in a tiebreaker, but would not top Denver or Cincinnati. There are 56 combinations where they have two of the following three scenarios: the Jets don't have more wins than the Jags, the Jags have more wins than the Broncos, the Jags have more wins than the Bengals. The sum of those odds? 45%.

So for the two spots remaining, the Broncos lead the pack with a 74% chance of seeing the post-season. The Bengals and Broncos are neck and neck, with Cincinnati (54%) just a bit more likely than Jacksonville (46%) to make the playoffs. The Jets have only a 27% chance, but take solace in this, Jets fans: a win over the Fins increases the Jets' chances to 59.5%, and there's less than a 30% chance the Jets win out and don't make the playoffs. The real reason New York's penalized is that the Jets only have a 1/1000 chance of making the playoffs with 9 wins, by far the lowest of these final four.

Note: Just a reminder, these percentages all sum to 200% (two playoff spots). Of course, the assumption in all of this is that the Steelers, Bills, Titans and Chiefs all miss the playoffs.

This entry was posted on Friday, December 22nd, 2006 at 2:40 am and is filed under General. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.

Bills will control own destiny* in wk 17 with a wk 16 win vs Ten, a Miami win, a New England win, and a Cincinnati win.

==

* assuming Bills win "strength of victory" tie breaker vs Cin. Bills magic number is 8 out of 21 possible "weighting points" to be determined in several other games/

I should add, if the Jaguars beat the Patriots, a Patriots loss to Ten the last weekend would open up the possibility that the Jets could win the AFC East.

I really like this kind of analysis, Chase.

I put your individual game odds into my software (it's set up so any user can do this) and ran my Monte Carlo simulaitons. I ran two cases. The first should match your analysis very closely, as I gave all the 7-7 teams a forced loss so that they could finish no better than 8-8. I also gave NE a forced win over TN, to ensure NE finishes no worse than 11-5. Here are my WC odds for this scenario:

Denver: 68.4%

Cincinnati: 58.9%

Jacksonville: 45.4%

NY Jets: 27.4%

The differences in the Denver and Cincinnati odds are too big to be explained by the error in my Monte Carlo simulations (less than 1% for 5000 trials). I think the differences are that you did not consider the conditional probabilities correctly. If you break down the Denver scenarios using your numbers:

total odds of making a WC = 10/6 + 9/7 = 1*odds of 10/6 + x * odds of 9/7

where x is the conditional odds of making the playoffs if the team finsihes 9-7

Denver x =(.74-.43)/.48 = .645

So according to your numbers Denver's chances of missing at 9-7 are .355. The key is that if Denver finishes 9-7, there is a 78% chance they lost to Cincinnati (conditional probablity = 0.47*.8/(.47*.8+.53*.2 = .78). There are three independent conditions that, if satisfied, would eliminate Denver at 9-7. They are (with odds):

both Jets and Jags win out = 0.36 * 0.33 = 0.12

Jets win out but Jags don't and Denver loses to Cin = (0.33-0.12)*.78 = 0.24

Jags win out but the Jets don't and Denver loses to Cin = (0.36-0.12)* 0.78 = 0.19

total = 0.549 = x

This does not match very well with your x = 0.355. But it does match very well with my odds of Denver making a WC of .684:

x = (.684-.43)/.48 = 0.53

Initially I got the same answer you did (substituting 0.47 for 0.78 in the equations above) and would have never found the conditional probablity error without my unwavering confidence in my simulation software, so don't take this as a criticsm. It took me a couple hours to figure it out.

The other group of scenarios was based on the game odds that you used, but allowing the 7-7 teams to get in the picutre and leaving the AFC East race in play. I used my game odds for any that you didn't specify.

NY Jets 33.26

Denver 59.4

Cincinnati 57.94

Jacksonville 44.42

Buffalo 4.6

Pittsburgh 1.36

Kansas City 1.34

The Jets odds improve mostly because of the possiblity of winning the Division. Denver is hurt by tiebreakers with the 7 - 7 teams (probably KC). Everyone else stays about the same.

Thanks for the kind words, cdcox. You sound miles ahead of me as a computer programmer; I just used excel to come up with these, and may have had some errors.

As far as your first scenario...

I'm a bit confused as to whether you're giving Cincinnati 9 or 10 wins. Let's start with giving them 9.

1) Jets and Jags win out, Denver goes 9-7. I've got this down as 5.864% chance of happening.

2) Jets win out, Jags don't, Denver loses to Cincy. That gives three combinations: Cincinnati over Denver, Pittsburgh over Cincinnati, Denver over SF, Jets over Miami and Jets over Oakland must be in all of them. The other two games (Jac-NE, Jac-KC) can go three ways. Jacksonville can lose to NE, Cincy can lose to Pittsburgh, and Jacksonville could beat KC: That gives three teams 9 wins and the Jets 10. Probability? 1.03%. Same as above, but now Jac beats NE and loses to KC. Prob? 1.07%. Same, but now Jac loses to both NE and KC. Prob? 0.69%. Sum = 2.795% chance*.

3) Jags win out, Jets don't, Cincy over Denver. Three unique combos (same as above, except now we're looking at Jets losing to Mia/beating Oak, losing to Oak/beating Mia, losing both). Jets over Miami/lose to Oak = 0.40%. Jets over Oak/lose to Mia = 2.235%. Jets lose both = 0.56%. Total = 3.195%**.

*However, I don't believe this to be fully accurate. If Jac/Den/Cin are all 9-7, regardless of who wins the Den/Cin game, I think Denver is in. This reduces the chances to just 0.69%.

**However, I don't believe this to be fully accurate. Because if the Jets win 9 games, and Cincy beats Denver but loses to Pittsburgh, the Broncos would get in because in the three-way tie, Denver would break it. This reduces the chances to just 0.56%.

So by using your numbers (and NOT giving Denver the tiebreaker per my asterisk),

I've got the odds of Denver going 9-7 but missing the playoffs at 11.85%. In all of those scenarios above, I included Cincinnati losing to Pittsburgh (0.351%). Having Cincinnati beat Pittsburgh (0.649%), the sum of those three scenarios becomes 21.92%. Adding those together, that gives Denver a 33.77% chance of missing the playoffs at 9-7. Considering I've got the odds of Denver going 9-7 at 48.1%, that makes the odds of Denver going 9-7 and missing the playoffs = 16.2% total.

So I'm very confused as to where we at. Maybe you can help me out.

And yes, thanks for using your game odds. That's a nice touch, and I forgot how much Denver would be hurt by KC winning out. Of course Jets fans are unaffected by that scenario, because KC winning out means Jacksonville lost, so the Jets would be in with 10 wins anyway (and can't make it at 9 practically).

You still have the problem with conditional probability.

Take your first example under number 2) above.

The sequence of events is Cincinnati over Denver, Pittsburgh over Cincinnati, Denver over SF, Jets over Miami, Jets over Oakland, NE over Jacksonville and Jacksonville over KC.

You calculate the odds of this as:

0.47*0.35*0.8*0.42*0.8*0.39*0.6 = 0.01035

That would be correct if we were starting with a clean slate. But we weren't. We started with the precondition the Denver finished 9-7. That means that if Cincinnati beats Denver, Denver must beat SF, so the 0.8 for the chances of Denver beating SF must be changed to a 1.0.

We also have to recalculate Denver's odds over Cincinnati under the precondition that Denver finishes 9-7. There are two ways this can happen. Denver can beat Cin and lose to SF or vice versa. So the probability of Cincinatti winning, given that Denver finsihes 9-7 are

(L Cin)(W SF)/((L Cin)(W SF) + (W Cin) (L SF))

0.47*.8/(.47*.8+.53*.2) = .78

Your previous calculation:

0.47*0.35*0.8*0.42*0.8*0.39*0.6 = 0.01035

now becomes:

0.78*0.35*1*0.42*0.8*0.39*0.6 = 0.02238

If you repeat recalculation for every scenario where Denver goes 9-7 our answers should agree.

Now as far as the 3-way ties go, these are very tricky. In the scenario under consideration (Cincinnati beats Denver, Denver beats SF, Pitt beats Cin, Jax splits with KC and NE) at the conference tiebreaker, Denver and Cincinnati would both be 7-5 in the conference and Jacksoville would be 6-6. So Jacksonville would lose the tiebreaker and drop from consideration. Denver and Cincinnati are still tied. The tiebreakers state "Note: If two clubs remain tied after third or other clubs are eliminated, tie breaker reverts to step 1 of applicable two-club format." which means you would start the tiebreaker over with just Denver and Cincinnati. Now, Cincinnati's head-to-head tiebreaker would be decisive over Denver.

I make mistakes doing these calcs and tiebreaker considerations all the time. Normally, I run my software to get the right answer, then go back through and do the reasoning to explain the result.

Gotcha. Thanks man. I'll have to look at your software when I get a chance. Sounds good.

A quick note about the Chiefs -- since they have only 2 division losses and Denver has 3, if the two teams finish at the same 9-7 record the Broncos cannot get into the playoffs before the Chiefs. That's a *huge* variable in there but I can see why, for simplicity sake, you didn't include it.

How do these shake out if you use Doug's rankings?

booooooooooooooooooooooooooooorrrring

I dunno, monkeytime, that sure was an exciting game for me.

So given all this work, how in God's name are the Chefs and Jets in the playoffs?

The 16 game NFL season- it's FANTASTIC!