The playoffs are a funny bit of business, where people tend to assume the #1 seed has a really good chance of making it to the Super Bowl. That is, unfortunately, not even close to the truth. If you ignore home field advantage, then it becomes easy to see that in these circumstances, the #1 and #2 seeds have 1 chance in 8 of winning (0.125), whereas seeds 3-6 have a 1 in 16 chance of winning (0.0625). But since in the playoffs, there is a home field advantage (at least until you reach the Super Bowl), the actual odds from Seeds 1 to 6 vary quite dramatically.

For now, we’re going to assume a home field advantage of 0.60. From 2001 to 2010, 100 non-Super Bowl playoff games were played, and the home team won 60 of them. This year, the home team won every time, unless the visitor was named the New York Giants, leading to a record of 8-2. So, I guess, the running total now, from 2001 to 2011, has to be 68/110, or 61.8% or so.

That said, I’m still going to use 60% in my calculations below.

For the sake of making it easier to turn any calculations into code, we’ll assign the home field advantage to the variable U (for “upper”), and to 1 – U, we will assign the variable L (for “lower”). Given these assignments, we now have:

Temporary variables:

LL = L*L

T_{23} = U*L + L*U

T_{45} = LL*U + (1. – LL)*L

Calculations of playoff odds

Seed 1 = U*U*0.50

Seed 2 = U*T_{23}*0.50

Seed 3 = U*L*T_{23}*0.50

Seed 4 = U*L*T_{45}*0.50

Seed 5 = L*L*T_{45}*0.50

Seed 6 = L*L*L*0.50

T_{23} is necessary to calculate the second game of Seed 2 or the third game of Seed 3. In this game, these two teams could face Seed 1, Seed 4, Seed 5, or Seed 6. Critically, they will either face Seed 1, for which they would be the visiting team, or all others, for which they would be the home team. The odds therefore become (odds of Seed 1 winning)(vistor’s odds) + (1 – odds of Seed 1 winning)(home team odds).

T_{45} is necessary to calculate the third game of Seed 4 or 5. In this game, these two teams could face Seed 1, Seed 2, Seed 3, or Seed 6. As Seed 6 is the only team for which Seeds 4 and 5 would be the home team, it is easiest to calculate the odds of Seed 6 making it to the third game, and then subtract those odds for the probability of playing as the visitors. Since the odds of Seed 6 arriving at game 3 are L*L, you end up with the formula given above.

Choosing a value of 0.60 for the home field advantage, we end up with:

Seed 1 : 0.18

Seed 2 : 0.144

Seed 3 : 0.0576

Seed 4 : 0.05184

Seed 5 : 0.03456

Seed 6 : 0.032

The range, from 18% to about 3%, is considerably more broad than the naive 1/8 to 1/16 values. Home field has a marked effect on the ability of teams to reach and win the Super Bowl. But the sheer number of teams involved, 12, and the arrangement of the playoffs, means that a #1 seed has, with a HFA of 60%, about a 36% change of making it to the Bowl, and a 18% chance of winning.

Note: this link has a coded version of the calculations above.

February 10, 2012 at 4:20 pm

It’s not playoff odds, it’s winning SB odds right? U x U x 0.5 = .6 x .6 x .5 = .18

Can you provide table of actual winners by seed to see if logic/model works vs actual reality?

February 10, 2012 at 4:28 pm

I’d suggest that the Super Bowl is the ultimate playoff game, and yes, I’m calculating the odds to win the thing.

Tables of seeds over time: Those can be derived from Wikipedia articles, or perhaps Pro Football Reference.

And testing this as a baseline model? Perhaps when I also have more complicated models to test.

D-

February 10, 2012 at 4:42 pm

Current playoff rules only since 2002. http://wiki.answers.com/Q/What_seed_was_the_Super_Bowl_winners

add in 2010 Packers (#6 seed) and 2011 Giants (#4) totals are:

#1 1 win 11%

#2 3 wins 33%

#3 1 win 11%

#4 1 win 11%

#5 1 win 11%

#6 2 wins 22%

though clearly sample is too small for statistical significance, the location of the Super Bowl might drive “home field” adv a bit too (like 2006 when Steelers fans were ~90% of the stadium in Detroit)

February 10, 2012 at 5:19 pm

Incidentally, actually home win % in regular season since 2002 is 57.29%

February 10, 2012 at 9:18 pm

Yep, regular season percentages are less than playoff percentages. It’s interesting as a comparison, but my model is a playoff model, not a regular season model (for reasons I touch on here). Things that apply in the regular season don’t necessarily apply in the postseason.

D-

December 5, 2012 at 12:29 pm

[...] The methodology of these stats is discussed in previous posts in this series. If you’re wondering where I’m getting odds to go into the playoffs, see this post. If you’re wondering what chance your team has of winning in the playoffs, see this post on my logistic regression methods, based on studies of playoff games. How would your ranking in the playoffs affect your chances of getting into the Super Bowl? We studied that here. [...]