BCS Week- If We Had Playoffs

What would the playoffs look like if my proposal was in place today?

Because all 4 undefeated teams are in the BCS top 6, the 6 teams selected for the playoffs are just the BCS's top 6, in order.

1-Oregon

2-Auburn

3-TCU

4-Boise State

5-LSU

6-Stanford

Oregon and Auburn would get first round byes, then TCU would play Stanford and Boise State LSU in the first round. Here are the odds of each team making it to various rounds:

	Into Semis	Into Finals	Champion
Oregon	1	0.514	0.29
Auburn	1	0.295	0.085
TCU	0.503	0.355	0.174
Boise St	0.784	0.431	0.263
LSU	0.216	0.054	0.017
Stanford	0.497	0.349	0.17

Oregon would be the favorite thanks to the bye, with Boise St. right behind them. TCU and Stanford are fairly equal, and Auburn would have an outside shot. LSU would just enjoy getting to the playoffs.

This post wraps up BCS week, and it's time to move on and figure out how Auburn keeps winning. Plus, it's time for some basketball. Anyways, I hope everybody has enjoyed a full week of posts.

BCS Week- Playoff Proposal

So I've criticized the BCS, talked about the computers, and even done some research on playoff systems. If we're gonna get rid of the BCS though, we need a solid system to replace it. Here is what I think would be the best playoff system:

A Six Team Playoff

This is good in that it often selects the best team (yesterday's post). It also is small enough that other bowl games could exist around it, with it's five games being the same the BCS currently has. Most importantly though, it maintains the importance of every game in college football. Even the best team in the nation can't afford to lose a game and drop it's bye.

All Undefeateds Get In

If a team wins all of it's games, they've done all they can. All undefeated teams deserve a shot at the national championship. This doesn't disrupt the six team part normally, the most undefeated teams ever before was five. If there are more than seven undefeated teams though, the playoffs should be expanded.

A BCS Style Ranking Fills In The Rest

The BCS isn't perfect, but it's pretty good. Swapping out the computers will help make it better too. Usually there will be few undefeated teams, so the top 6 teams in the nation should get it, as determined by the BCS. The ranking system should also select the top two teams to get byes. There will be argument over this, but unlike the BCS angered teams will have a chance to put their money where their mouth is.

No Automatic Conferences

All teams are FBS teams already, all of them deserve to be treated equally at this point. Sorry SEC, I know you're whining. You're not one of the two best this year anyways.

That's it. It's not an overly complex system, it's basically just the NFL conference playoffs. Fingers crossed that someone from the BCS reads this and agrees. Unfortunately though, it looks like we might be stuck for the time being.

BCS Week- Which type of playoff is the best?

One of the goals of a playoff should be to reward the best team. This is why 1 seeds get to play 8 seeds, and why higher ranked reams get byes. Is there a way to maximize the chance the best team wins? With a couple reasonable approximations, I think there is.

Approximation 1: I'm going to assume the best teams wins at a constant percentage (X) of the time. This is not true, it depends on the strength of various teams, but it's a serviceable estimate.

Approximation 2: To do this I need to know where the "best team" is in the BCS standings. I'm going to assume that the likelihood drops off geometrically with each spot. So if the 1st team has a .5 chance of being the best, the second will have .25, the third will have .125, the fourth will have .0625. Of course, this has to be adjusted so that the sum equals 1, so the first team's chance is 1-R where R is the rate of drop off. Here's another example: 1st= .333, 2nd is .222, 3rd is .148, etc.

The chance that the best team wins in an N team is easy to calculate (using normal playoff systems and giving byes to the best teams). But instead of putting in tables with too much numbers like I normally do, I've broken out the markers and done a graph. If you can do this with software please let me know, it'd clearly be way better.

For the mathematically inclined there's a whole family of interesting curves that serve as many of the boundaries, all with the formula R=(1/X-1)^(1/odd).

So the more often the better team wins (farther right) and the less accurate BCS rankings are

(farther down) the more teams the optimal playoff has. Interestingly, at no point is the traditional playoff structure (powers of 2 like 2, 4, 8, etc.) the most efficient. Odd ones like 3, 5, and 6 swipe up large areas on the graph. What's important though are which values college football has for constants.

Using the Massey Ratings number 1 as the true "best" team, the drop off constant is .667. Using my ratings to find the true "best" team the best match is a constant of .429, and an average of the two gives a constant of .579 (yeah, that's not the arithmetic average, but it works).

Based off the calculations earlier in the week the better team wins about .85 of the time. That seems high though for a tournament with all the best teams in the nation. You could use my ratings to find a ratio between teams like Boise St and Auburn, and then you'd often get winning percentages around .7 or .75 (this year it's a little lower, the top teams are kinda bunched).

Using .85 and .579 as constants the most effective playoff system is a 6 team playoff.

Using .7 and .579 as constants the most effective playoff system is a 3 team playoff.

In general, 3, 5, and 6 team playoffs are the most effective for college football (hint for the BCS- NOT A 2 TEAM PLAYOFF). 3 and 6 are probably better than 5 because of the wide range of constants they cover. Which one of those two should be the playoff system in college football is a question for tomorrow.

BCS Week- BCS Computers

As part of it's rankings the BCS includes 6 computer rankings (all without margin of victory). Here's a breakdown of all 6, with some thoughts on they're accuracy. For those of you looking for dirt on the BCS, look at the last 3.

Sagarin Elo Ratings

Jeff Sagarin is THE big name in sports rankings systems, and without including him the BCS would be making a serious mistake. His Elo system is not his primary ranking, but it is the one that fits BCS requirements (the other is predictive). The Elo system is based off the rankings used by chess players, which has worked well over the years, but he doesn't give details.

Peter Wolfe's Ratings

Peter Wolfe is another big name, and he also has some very good rankings. His are probably the most mathematically sound, he uses the Bradley-Terry method to find the ratings that maximize the probability that the results of the real world happened. It's a great basis, but the one problem is that under that method all undefeated teams should have infinitely high ratings. Since this doesn't happen he presumably includes some limiting factor, but it's likely sort of arbitrary, reducing the mathematical rigorousness.

Ken Massey Ratings

Ken Massey doesn't give many details on his BCS ranking system, but it gives good results and his other ratings are documented as being mathematically solid. I'm guessing actually that his ratings are similar to mine, but I'm not sure. Even if they stunk though he'd deserve to be included for all his contributions to the rating community.

Anderson Hester Ratings

There is very little description for these ratings, but from what there is I see two problems.

1. They only include opponents record and opponents opponents record. This does not look deep enough into strength of schedule, and over rewards teams in weaker conferences (all conferences will have roughly .500 for both records, regardless of strength).

2. They include a conference rating as part of strength of schedule. Should New Mexico's losses reflect negatively on playing TCU? I don't think so.

Fortunately for the AH ratings the two flaws kind of cancel each other out. There are still better ways to rank teams though.

Colley Matrix Ratings

The Colley Matrix Ratings are great because they are mathematically sound and he even gives details on how the ratings work. They are not so good though, because they don't rank teams very well. He uses FBS games (and FCS vs FBS games in a weird way) only, but I've programmed his system and here's how the top 10 looks with all 730 teams:

1. Auburn

2. LSU

3. TCU

4. North Central

5. Boise St

6. Oregon

7. Stanford

8. St Thomas

9. Missouri

10. Wesley

Not only is LSU now ahead of TCU, but there are suddenly teams from other divisions like D3 St. Thomas up there. Does North Central deserve to be there? No, and this flaw severely limits the Colley Matrix method.

Richard Billingsly Rankings

Billingsly's rankings are by his own admission the least mathematically complex of the BCS rankings. They were also created through a process of his own tinkering. While it's neat that he was able to just mess with the games until it works, that involves way too much personal bias to be a part of deciding the national championship. The other, more glaring problem is that he uses the previous year's final rankings as part of his formula. While it's true that teams don't change too much between years, each season should be completely separate if you're evaluating the national champion of that year. He admits that the preseason rankings have an effect on undefeated teams. In week 8 of the BCS this year Missouri averaged 6th in the computer polls and was in the top 10 in all but Billingsley's, where they were unranked because they didn't play well last season. A system like that definitely should not be a part of the BCS.

So 3 systems (Anderson/Hester, Colley, Billingsly) should not be included in the BCS, who replaces them? Here are three of my favorites:

Pugh Ratings- These have great math behind them, and the work too.

Doktor Entropy Ratings-These are commonly cited as a good set of rankings, although they'd have to be adjusted to join the BCS.

Random Walker Rankings-Randomly ranking monkeys is a cool premise, with some neat theory behind it.

BCS Week- Philosophy

College football has always occupied an odd spot in the sporting landscape. Unlike pro football, players can't commit all their time to football (some do actually go to school), and unlike college basketball players can't play 40 times a season, or multiple times a week. Because of a limited schedule, it's traditionally been impossible to have a true champion. High school sports deal with this by having state champions, but as a nationwide sport college football had to settle on the "mythical national championship" selected by polls. Sometimes the MNC worked well, but occasionally it would result in multiple champions from various different polls. The BCS was established to help solve that problem, but who's satisfied with a 2 team playoff? Aren't there other qualified teams? How can a team win every single game they play, but still not win a championship? How to pick a championship team is a difficult task, but that's what I'm going to look at over the next week starting here.

What Do Championships Reward?

Many people would say that the best team in a league wins the championship. I would contend though that this isn't true. They actually reward the most successful team, and they should. A team that beats Washington by 30 points is probably better than a team that beats them by 5. Even when one team loses to another, they can often actually be better. 9-2 Virgina Tech lost to 6-5 James Madison, from the FCS. The Hokies are absolutely better, but they had an off day and an unlucky game. How much of a role does luck play? Here's a look at how often the better team wins in the 6 best conferences in football over the last couple years, using this method.

	2008	2009
ACC	0.76	0.5
Big 12	0.86	0.93
Big Ten	0.82	0.84
MWC	0.98	1
Pac-10	0.83	0.91
SEC	0.87	0.84
Average	0.85	0.84

So while luck plays a smaller role in college football than other sports, it still changes the result in 15% of top level games. That's where the difference between the best teams and the most successful teams enters. You can't reward Virginia Tech for being the best team, you have reward James Madison for winning.

What Ranking System To Use

The goal of playing a game though is just to win, not to win by any given amount. Take the example from earlier, where Team A beats Washington by 30, and Team B beats them by only 5. If you are predicting who wins games, you should pick Team A, but if you want to reward a champion you should rate both teams equally. Because of this, I think the BCS is right not to include margin of victory, and when selecting a champion I'll use my No MoV rankings. I'm going to criticize the BCS plenty this week but I'll credit them here, they do a pretty good job of focusing on the goal: winning football games.

College Football Top 25 11/21

It's the start of BCS Week here at Kobe, Tell Me How My Stats Taste, and in honor of a Big Game victory/mostly because I have free time over Thanksgiving break there will be a post each day on the BCS and NCAA football's championship system. We're going to start off the week the same way as always though, with the top 25.

Rank	Team	W	L	Rating	SOS
1	Boise St	10	0	663.2	32.73
2	Oregon	10	0	542.71	50
3	TCU	11	0	538.56	32.02
4	Stanford	10	1	531.33	75.83
5	Ohio State	10	1	397.1	39.95
6	Alabama	9	2	377.12	51.12
7	Oklahoma	9	2	287.72	69.93
8	Missouri	9	2	255.31	63.23
9	Auburn	11	0	224.18	56.08
10	Oklahoma St	10	1	222.03	45.13
11	Arkansas	9	2	221.07	54.33
12	Texas A&M	8	3	218.67	76.28
13	Nebraska	9	2	218.63	49.7
14	South Carolina	8	3	206.24	67.75
15	Iowa	7	4	199.57	48.81
16	Virginia Tech	9	2	194.33	45.4
17	Arizona	7	3	187.57	61.9
18	LSU	10	1	182.9	56.82
19	Florida	7	4	174.02	65.18
20	Oregon St	5	5	164.47	121.77
21	Florida St	8	3	160.52	48.31
22	Wisconsin	10	1	153.42	35.3
23	Arizona St	4	6	148.57	81.44
24	Southern Cal	7	4	146.51	103.48
25	Miami FL	7	4	132.74	56.64

That's right, with a dominating victory over a good Fresno State team the Boise State Broncos have moved into the top ranking slot. Other than that, the rankings have remained pretty steady, except for USC's drop. At this point in the season, there's been enough games played that only surprising games really alter the rankings very much. Even teams further back in the pack stayed pretty steady. Here's a look:

48

Texas

5

6

60.79

55.63

57

Washington

4

6

49.98

142.52

61

Iowa St

5

7

45.36

78.66

66

Northwestern

7

4

41.21

31.41

415

Georgetown DC

4

7

0.5

0.77

No MoV rankings can change a lot though still, because they are further from the true rankings. Here's a look at the top 10:

Rank	Team	W	L	Rating	SOS
1	Auburn	11	0	26.46	6.61
2	Oregon	10	0	24.27	6.06
3	TCU	11	0	24.11	6.02
4	Missouri	9	2	23.61	9.77
5	Stanford	10	1	23.29	7.49
6	LSU	10	1	23	7.4
7	Oklahoma	9	2	22.78	9.42
8	Boise St	10	0	22.59	5.64
9	Oklahoma St	10	1	21.91	7.04
10	Texas A&M	8	3	20.13	10.72

The big news here is that both TCU and Boise St. moved up in the rankings (which is true in the BCS too). At this point, they are shooting to do two things with the rankings: stay ahead of the one loss teams, and pass Auburn if they lose. Right now, I think they'll stay ahead of LSU, and I'm guessing one of them would pass Auburn but it will be very close. I also am convinced that Boise State will finish ahead of TCU in the BCS standings as long as they win out. Why that's true will be covered more in the upcoming BCS week.