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Conference champs being given an auto bid by KeoughCharles05

is the major problem, but not the only one.

I still think 12 is too many, in that it will frequently put a 3 loss team in the same single-elimination tournament as an undefeated team. But, getting rid of automatic designations for conference champions (or placing additional requirements on conference champs) would solve the worst aspects of the problem.

To demonstrate the issue with 12, here are teams in the CFP top 12 with 3 losses each year going back to 2010:

2019: 10-3 Wisconsin, 9-3 Auburn
2018: 10-3 Washington, 9-3 Florida, 9-3 LSU, 9-3 Penn State
2017: 10-3 Auburn
2016: 10-3 Wisconsin, 9-3 USC, 10-3 Colorado, 9-3 FSU, 9-3 Oklahoma State
2015: 9-3 Ole Miss
2014: 9-3 Ole Miss, 10-3 Arizona, 9-3 Kansas State, 10-3 Georgia Tech
2013: None (Actually would have been a really interesting year for taking the pure top 12 team approach).
2012: None (Same as 2013)
2011: 9-3 Baylor
2010: None

Do any of these teams have a legitimate claim to being the best team in college football that year? There might be one or two that whose losses were close, to highly ranked teams, had big wins in the regular season, and if they won three games against top 12 competition, could have a claim. Auburn in 2019 as an example. But for the most part, they're teams who lost the only games against top teams they played. Wisconsin 2019 lost twice to Ohio State, soundly, and to an awful Illinois team. Penn State in 2018 was throttled by Michigan, lost a nail-biter to Ohio State, and lost to a mediocre to bad Michigan State team. Ultimately, if the goal is to crown a champion, a 12 team field creates an unacceptable risk of an undeserving champ.

Some other notes...

It's interesting in how few 3 loss teams there were in the top 12 from 2010-2013, and how many there have been more recently. Ultimately though, when we take 12 teams, I think it ends up being unfair to the top teams to be placed in the same single-elimination tournament with teams who had demonstrably worse seasons.

One other thing I was intrigued by -- your critique of 2012 shows how badly awarding conference champs can distort the issue. Including Northern Illinois and Boise State in the field and giving FSU a bye disrupts what would otherwise have been a really interesting field:

BYES
ND (12-0)
Bama (12-1)
Florida (11-1)
Oregon (11-1)

Kansas State (11-1) vs FSU (11-2)
Stanford (11-2) vs Oklahoma (10-2)
Georgia (11-2) vs South Carolina (10-2)
LSU (10-2) vs Texas A&M (10-2)