## Is Having A Bye Actually Harmful?

Or, "How you'll learn to stop worrying and love the bye"

One of the things that bothered me about Topher Doll's post on expanding the NFL playoffs was the assertion that the bye week is damaging to teams that receive it. This is not exactly an uncommon belief, but I've never seen anyone do any research to support the claim. And since basic math would suggest that the bye is highly beneficial, I find it baffling that so many people believe it.

So I'm going to do some research to find out one way or another. But before I do, let's do the algebra to find out what it would look like if the bye really is harmful.

WARNING: the algebra gets really, really messy. So I'm just going to tell you the answer, and put the actual math at the bottom if you're interested.

In order for a bye-week team to have a WORSE chance of REACHING the Super Bowl than a non-bye team, the bye must lower its odds to win each playoff game by 11%. And in order for a bye-week team to have a WORSE chance of WINNING the Super Bowl than a non-bye team, the bye must lower its odds to win each playoff game by 8.5%.

So how much does the bye ACTUALLY hurt those teams' chances to win? Let's look at what's happened in the playoffs and see if we can find out...

All records listed below are the records of teams WITH byes against teams WITHOUT byes ONLY.

Year Divisional Conference Super Bowl Overall
2012 3-1 0-1 0-1 3-3
2011 3-1 0-1 0-1 3-3
2010 2-2 1-1 0-1 3-4
2009 3-1 1-0 n/a 4-1
2008 1-3 1-0 1-0 3-3
2007 2-2 1-1 0-1 3-4
2006 2-2 n/a 0-1 2-3
2005 2-2 1-1 0-1 3-4
2004 4-0 n/a n/a 4-0
2003 2-2 1-1 1-0 4-3
2002 4-0 n/a n/a 4-0
'02-'12 Total 28-16 6-6 2-6 36-28

Sure, teams with byes have had a crappy run in the Super Bowl the past decade (2-6 against non-byes, 5-9 overall). But they dominate the divisional playoffs, only once having a losing record, and they break even in the conference championship. Overall, they have a winning playoff record. Far from suffering a disadvantage, they compound their advantage of not needing to deal with the wild card round by having a BETTER chance to win thereafter.

But let's expand the data set. Topher had good reason to stop at 2002, as expansion prior to that season changed the number of teams, which changes which teams will tend to qualify for the playoffs. But the current bye format has been in place since 1990 (the only change is that the #4 seed is now a division winner, not a wild card), so we can look further back to examine the effects of a bye.

Year Divisional Conference Super Bowl Overall
2001 3-1 1-0 n/a 4-1
2000 3-1 0-1 0-1 3-3
1999 3-1 0-1 1-0 4-2
1998 4-0 n/a n/a 4-0
1997 3-1 0-1 0-1 3-3
1996 3-1 1-0 n/a 4-1
1995 2-2 2-0 n/a 4-2
1994 4-0 n/a n/a 4-0
1993 3-1 1-0 n/a 4-1
1992 3-1 0-1 1-0 4-2
1991 4-0 n/a n/a 4-0
1990 4-0 n/a n/a 4-0
'90-'01 Total 39-9 5-4 2-2 46-15
'90-'12 Total 67-25 12-10 4-8 82-43

Some serious domination going on there in the divisional round, to the point where non-bye teams even reaching the later rounds was so infrequent that the results are almost meaningless. In case you forgot why it was so surprising (and awesome) when the '97 Broncos beat the Packers, you can see part of the reason in this chart (the whole NFC domination thing too, of course).

If you want to take all of this at face value and claim that byes make it harder to win in the Super Bowl specifically, I would say that you're focusing on way too small a sample to draw a meaningful conclusion. But suppose that's the case- you've only got a 33% chance to win the Super Bowl if you drew a bye and your opponent didn't. I'll take those odds, given that:

1. My odds of getting to the Super Bowl are WAY higher, so it's worth having worse odds of actually winning it once I'm there.
2. I'm pretty likely to face another team that drew a bye, in which case the effect is a wash.

And this is borne out in the data- 15 of the 23 Super Bowls in the data set were won by teams with byes, despite their poor record against non-bye teams.

However, while I think the declining level of domination as a bye team progresses through the playoffs makes sense, I don't think it's a bye week effect- it's an effect of playing better teams the farther you go. If we were going to see an effect from the bye itself, we'd expect to see the strongest impact on the divisional round, and that's the exact opposite of what we see.

----

One other claim I'd like to examine is the idea that, given identical records, a team is more likely to reach/win the Super Bowl without a bye than with one (this is something Topher Doll claimed in the comments in response to my objections). After all, the above data includes a lot of 14 win teams beating up on 10 win teams, or similar. Maybe, if you've got three teams at 12-4, it's better to be the one that avoids the bye.

Here's a summary of how teams with bye-worthy records (from the ‘perfect' ‘07 Patriots down to the 10-6 ‘06 Saints) have fared from 2002 on. You can see the entire list further down.

Lost in... Wild Card Divisional Conference Super Bowl Won Super Bowl
16-0 (Bye) 0 0 0 1 0
16-0 (No Bye) 0 0 0 0 0
15-1 (Bye) 0 1 1 1 0
15-1 (No Bye) 0 0 0 0 0
14-2 (Bye) 0 3 0 1 2
14-2 (No Bye) 0 0 0 0 0
13-3 (Bye) 0 8 4 4 1
13-3 (No Bye) 0 1 0 0 0
12-4 (Bye) 0 3 5 1 2
12-4 (No Bye) 5 4 2 0 1
11-5 (Bye) 0 1 3 1 0
11-5 (No Bye) 9 4 4 1 1
10-6 (Bye) 0 0 1 0 0
10-6 (No Bye) 19 9 0 0 2

I'm highlighting the 12-4 teams because that's the only place a meaningful comparison can really be drawn. And what we see, of course, is that the bye helps a great deal. Over one-third of the 12-4 teams without byes were eliminated in the Wild Card round, and over half of those remaining were eliminated in the divisional round (when they were guaranteed to be playing on the road). By contrast, 12-4 with a bye means you have an excellent shot at making the Conference championship, and almost half of those teams advanced to the Super Bowl.

The only place where you could even argue in favor of the idea that the teams without byes do better is at the 11-5 and 10-6 level. The 10-6 case is just silly, since there's only been one such team -- and moreover, it's clear that those two 10-6 SB champs are outliers, since no other 10-6 teams have made it out of the divisional round. As for the 11-5s, you still get twice as many bye teams making the Super Bowl percentage-wise (1/5 for 20% vs. 2 out of 19 for about 10%), though obviously the 11-5 bye team sample is still very small. Still, it makes sense, given that half of the non-bye 11-5 teams fall in the WC round.

Though I could have, I'm not going back past 2002 here, because gathering this data is frankly very tedious, and this was enough to prove the point. But if anyone's inclined to think that doing so would tilt the data back away from the bye, remember that the teams on byes have significantly stronger records pre-2002, so if anything it would probably just cement my case.

Full list of playoff teams with ‘bye-worthy' records, going back to 2002:

Team, Bye?, Lost in Round...

16-0 Teams
2007 Patriots, Yes, SB

15-1 Teams
2004 Steelers, Yes, Conference
2011 Packers, Yes, Divisional

14-2 Teams
2003 Patriots, Yes, Won SB
2004 Patriots, Yes, Won SB
2005 Colts, Yes, Divisional
2006 Chargers, Yes, Divisional
2009 Colts, Yes, SB
2010 Patriots, Yes, Divisional

13-3 Teams
2003 Chiefs, Yes, Divisional
2004 Eagles, Yes, SB
2005 Broncos, Yes, Conference
2005 Seahawks, Yes, SB
2006 Ravens, Yes, Divisional
2006 Bears, Yes, SB
2007 Cowboys, Yes, Divisional
2007 Colts, Yes, Divisional
2007 Packers, Yes, Conference
2008 Titans, Yes, Divisional
2009 Chargers, Yes, Divisional
2009 Saints, Yes, Won SB
2010 Falcons, Yes, Divisional
2011 49ers, Yes, Conference
2011 Patriots, Yes, SB
2012 Broncos, Yes, Divisional
2012 Falcons, Yes, Conference

2011 Saints, No, Divisional

12-4 Teams
2002 Eagles, Yes, Conference
2002 Buccaneers, Yes, Won SB
2003 Rams, Yes, Divisional
2003 Eagles, Yes, Conference
2008 Giants, Yes, Divisional
2008 Panthers, Yes, Divisional
2008 Steelers, Yes, Won SB
2009 Vikings, Yes, Conference
2010 Steelers, Yes, SB
2011 Ravens, Yes, Conference
2012 Patriots, Yes, Conference

2002 Packers, No, WC
2003 Titans, No, Divisional
2003 Colts, No, Conference
2004 Chargers, No, WC
2004 Colts, No, Divisional
2005 Jaguars, No, WC
2006 Patriots, No, Conference
2006 Colts, No, Won SB
2008 Colts, No, WC
2010 Ravens, No, Divisional
2011 Steelers, No, WC
2012 Texans, No, Divisional

11-5 Teams
2002 Titans, Yes, Conference
2002 Raiders, Yes, SB
2004 Falcons, Yes, Conference
2005 Bears, Yes, Divisional
2010 Bears, Yes, Conference

2003 Panthers, No, SB
2005 Bengals, No, WC
2005 Buccaneers, No, WC
2005 Giants, No, WC
2005 Panthers, No, Conference
2005 Steelers, No, Won SB
2007 Jaguars, No, Divisional
2007 Chargers, No, Conference
2008 Dolphins, No, WC
2008 Falcons, No, WC
2008 Ravens, No, Conference
2009 Eagles, No, WC
2009 Packers, No, WC
2009 Cowboys, No, Divisional
2010 Saints, No, WC
2010 Jets, No, Conference
2012 Colts, No, WC
2012 Seahawks, No, Divisional
2012 Packers, No, Divisional

10-6 Teams
2006 Saints, Yes, Conference

2002 Colts, No, WC
2002 Giants, No, WC
2002 49ers, No, Divisional
2003 Broncos, No, WC
2003 Ravens, No, WC
2003 Seahawks, No, WC
2003 Cowboys, No, WC
2003 Packers, No, Divisional
2004 Broncos, No, WC
2004 Packers, No, WC
2004 Jets, No, Divisional
2005 Patriots, No, Divisional
2005 Redskins, No, Divisional
2006 Jets, No, WC
2006 Eagles, No, Divisional
2007 Steelers, No, WC
2007 Titans, No, WC
2007 Seahawks, No, Divisional
2007 Giants, No, Won SB
2008 Vikings, No, WC
2009 Patriots, No, WC
2009 Bengals, No, WC
2009 Cardinals, No, Divisional
2010 Colts, No, WC
2010 Chiefs, No, WC
2010 Eagles, No, WC
2010 Packers, No, Won SB
2011 Falcons, No, WC
2011 Lions, No, WC
2011 Texans, No, Divisional

----

CRAZY ALGEBRA SECTION

When two teams with byes play each other, they'll have a .5 chance to win, since one has to win and one has to lose. Simple enough. Same thing when two teams without byes play each other.

When a bye team plays a not-bye team, however, let's say that the bye team suffers from the Bye Week Blues (whatever that may encompass). This results in the bye team having a chance to win of .5-x, where x is the effect of the Bye Week Blues. And likewise, the not-bye team has a chance to win of .5+x.

Obviously, I'm ignoring things like strength of the individual teams, or home field advantage. But we can just say that this is all wrapped up in x. After all, if the Bye Week Blues are outweighed by HFA or team quality, then the bye is clearly advantageous anyway.

If you think of each round of the playoffs as a coin flip, you can see what the big plus to the bye is: you don't have to make one of those flips. It's twice as hard to flip n coins in a row and get heads every time, than to flip n-1 coins in a row and get heads every time.

So the question is, how big does x need to be for the bonus of not playing in the wild card round to be washed out?

Let's start with chances to reach the Super Bowl. Call the chances of a non-bye team winning its conference PC(NB). We start in the wild card round, where our non-bye team plays another non-bye team, and thus has a simple .5 chance to win. Then we move on to the divisional round, where we know we're playing a bye team, so we have a chance of .5+x.

The conference championship week is a bit trickier, since we could be playing either a bye team or a non-bye team. Specifically, in .5+x cases, we'll be playing a non-bye team, and have a .5 chance to win. And in .5-x cases, we'll be playing a bye team, and have a .5+x chance to win.

So the total chances for a non-bye team to win its conference are:
PC(NB) = (.5) * (.5+x) * [((.5+x)*.5) + ((.5-x)*(.5+x))] = -.5x^3 + .375x + .125

What about PC(B), the chances of a bye team winning the conference? We go straight to the divisional round, where our chances are .5-x. Then we have the same setup in the conference round, only instead of .5 vs a non-bye team and .5+x vs. a bye team, we have .5-x vs. a non-bye team and .5 vs. a bye team. That gives us:

PC(B) = (.5-x) * [((.5+x)*(.5-x)) + ((.5-x)*.5)] = x^3 - .75x + .25.

We know that x has to be somewhere between 0 (no effect) and .5 (bye week teams always lose). So within that range, for what values of x is PC(NB) > PC(B)? We could do the algebra, or we could let Wolfram Alpha tell us that x needs to be at least .113037. In other words, the Bye Week Blues need to lower your chances to win by at least 11%, from .5 to .39, for a bye week team to have a worse chance of reaching the Super Bowl than a non-bye team.

What about winning the Super Bowl? Since there are 4 non-bye teams and 2 bye teams, we can multiply PC(NB) by 4 and PC(B) by 2 to get the chances of each type of team reaching the Super Bowl. So our would-be Super Bowl contestant faces a non-bye team with the following probability:

PSBO(NB) = 4 * (.5) * (.5+x) * [((.5+x)*.5) + ((.5-x)*(.5+x))] = 2 * (.5+x) * [((.5+x)*.5) + ((.5-x)*(.5+x))] = -2x^3 + 1.5x + .5

And they face a bye team with the following probability:
PSBO(B) = 2 * (.5-x) * [((.5+x)*(.5-x)) + ((.5-x)*.5)] = 2x^3 - 1.5x + .5

So for a non-bye team, their odds of winning the SB are PSB(NB) = PC(NB) * [(PSBO(NB)*.5) + (PSBO(B)*(.5+x))]
And for a bye team, their odds of winning the SB are PSB(B) = PC(B) * [(PSBO(NB)*(.5-x)) + (PSBO(B)*.5)]

So how big does x need to be for PSB(NB) > PSB(B)? The algebra at this point is completely nuts, but luckily Wolfram Alpha comes to the rescue again, and tells us that x needs to be at least 0.0853252. In other words, the Bye Week Blues need to lower your chances to win by at least 8.5%, from .5 to .415, for a bye week team to have a worse chance of winning the Super Bowl than a non-bye team.

This is a Fan-Created Comment on MileHighReport.com. The opinion here is not necessarily shared by the editorial staff of MHR

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