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Transitive Rankings Post Week 7: Madness, or Possibly Sparta

As every college football message board debater knows, winning football games is transitive. In other words, if Team A beats Team B and Team B beats Team C, the fans of Team A are perfectly within their rights to talk trash to fans of Team C. Obviously, Team A could have completely wiped the floor with Team C, they just didn't happen to play each other. If Team D happens to lose to Team C, they are also deserving of as much scorn as possible. Thus one way of ranking teams just counts how many teams' fans are permissible trash talk targets (minus those in the reverse situation).

We've reached the unfortunate time of the year where we must deal with cycles (for example, the mess described below with Penn State and Minnesota).  My solution is to give the transitive win to the team with the shorter path (in this case Penn State).  If the two paths are equal length, both teams get a tie.  

First, straight transitive rankings.  For a given team, how many other teams do they have a (shorter) path to?  How many have a (shorter) path to them?

1 Michigan State 105-0 (#1 in the country)

The win over Illinois was nice, but the Spartan's previous win over Wisconsin is now worth a lot more.  The Spartans are the first Big Ten team to reach the national #1 rating.  

Next Week: At Northwestern.  MSU beat Notre Dame who beat Purdue who beat Northwestern.  

 

2 Wisconsin 104-1 (#2)

Unsurprisingly, beating Ohio State helps a lot here.  A bit more unexpected is the role of the Arizona State win.  The Sun Devil's lone FBS win over mercurial Washington looks a lot better now that the Huskies have beaten Oregon State (who beat Arizona) and USC. 

Next Week: At Iowa.  Wisconsin beat Arizona State who beat Washington who beat Oregon State who beat Arizona who beat Iowa.

 

3 Ohio State 103-2 (#9)

Curiously, OSU is rated higher both nationally and in the conference despite their loss.  The Miami win is finally paying some dividends, while Indiana, Ohio and Eastern Michigan all won this week as well.

Next Week: Purdue.  Ohio State beat Miami who beat Clemson who beat Navy who beat LaTech who beat USU who beat BYU who beat SDSU who beat AFA who beat Wyoming who beat Toledo who beat Purdue.  

 

4 Illinois 101-4 (#18)

Illinois does have 3 losses, but those teams (Missouri, MSU, OSU) have exactly 1 loss between them.  Meanwhile, their wins are against at least decent (if not great) competition.

Next Week: Indiana.  Illinois beat Penn State who beat Temple who beat Connecticut who beat Vanderbilt who beat Mississippi who beat Fresno State who beat USU who beat BYU who beat Washington who beat Oregon State who beat Arizona who beat Iowa who beat Michigan who beat Indiana.  

 

5 Nebraska 92-22-5 (#20)

Poor Nebraska.  Not only did they lose all time bragging rights to Texas, but they drop to fifth in the future Big Ten transitive rankings.  Texas' loss to UCLA stings the Huskers a bit, but their wins over KSU and Washington keep them in the top 25 nationally.  

Next Week: Oklahoma State.  Oklahoma State beat Texas Tech who beat Baylor who beat Colorado who beat Hawaii who beat Nevada who beat California who beat UCLA who beat Texas who beat Nebraska.  

 

6 Iowa 78-35-6 (#31)

Iowa's loss to Arizona looks a lot worse now that Oregon State, who beat Arizona, lost to Washington.  The resulting chains are long enough that they will be short circuited if Iowa keeps winning, but for now the Hawkeyes get knocked down a bit.  

Next Week: Wisconsin.  Wisconsin beat Arizona State who beat Washington who beat Oregon State who beat Arizona who beat Iowa.  

 

7 Northwestern 76-38-5 (#39)

The Purdue loss still stings a bit and the wins aren't helping that much.  The highest rated of Northwestern's victories is Vanderbilt, who is rated #71 in the country.  

Next Week:  Michigan State.  MSU beat Notre Dame who beat Purdue who beat Northwestern.

 

8 Michigan 73-39-7 (#45)

The Washington-Oregon State-Arizona sequence is hurting Michigan as well after their loss to Iowa.  The difference is that Michigan's best win is over Connecticut, which won't move them up too far.  

Next Week: Bye.  

 

9 Purdue 69-45-5 (#54)

The Toledo loss still hurts, but the win over Northwestern is enough to keep them in the top half of the country.  

Next Week: At Ohio State.  Ohio State beat Miami who beat Clemson who beat Navy who beat LaTech who beat USU who beat BYU who beat SDSU who beat AFA who beat Wyoming who beat Toledo who beat Purdue.

 

10 Penn State 53-62-4 (#69)

PSU had a rough bye week.  Both of their losses got worse this week.  Iowa's transitive troubles have been discussed already, but Alabama loss to South Carolina looks much worse now that South Carolina lost to Kentucky (who lost to Mississippi who lost to Vanderbilt).  As stated before, these long chains would be less of an issue if Penn State's had FBS wins other than Temple and Kent State ("Kent read, Kent write, Kent State").  

Next Week:  Minnesota.  Penn State beat Temple who beat Connecticut who beat Vanderbilt who beat Mississippi who beat Fresno State who beat USU who beat BYU who beat Washington who beat USC who beat Minnesota.  However, Minnesota beat MTSU who beat ULL who beat North Texas who beat FAU who beat UAB who beat UTEP who beat Rice who beat Houston who beat Tulane who beat Rutgers who beat Connecticut who beat Vanderbilt who beat Mississippi who beat Kentucky who beat South Carolina who beat Alabama who beat Penn State, so that's something the Gophers have going for them.  

 

11 Indiana 46-73 (#75)

Indiana also gets stung by the Washington-Oregon State-Arizona-Iowa chain from their loss to Michigan.  Like the others, a few decent wins would stop these long chains.  Right now, Indiana's wins over Arkansas State (#95), Western Kentucky (#108), and Akron (#117) don't help much.

Next Week: At Illinois.  Illinois beat Penn State who beat Temple who beat Connecticut who beat Vanderbilt who beat Mississippi who beat Fresno State who beat USU who beat BYU who beat Washington who beat Oregon State who beat Arizona who beat Iowa who beat Michigan who beat Indiana.  

 

12 Minnesota 39-79-1 (#81)

Like a lot of their Big Ten brethren, Washington has played a key role in Minnesota's position.  However, the Gophers get hit more directly from the USC loss.  (USC lost to Washington before it was cool).  

Next Week:  At Penn State.  See Penn State's discussion for the who beat who.  

 

 

Of course, this method isn't perfect. One flaw is that a transitive win over a good team (e.g. Wisconsin) should be worth more than one over a bad team (e.g. Western Michigan). A reasonable idea, so it deserves its own set of rankings.  In these rankings, MSU receives 100 win shares for their win over Wisconsin since they are the only team with a transitive win over the Badgers.  However, 97 teams have a transitive win over Western Michigan, so thats only worth 100/97~=1 win share for MSU (and the other 96 teams).  

National RankTeamWin-Loss Shares
1 Michigan State 380-0
6 Wisconsin 280-0
8 Ohio State 230-1
18 Illinois 185-3
20 Nebraska 156-22
32 Iowa 123-40
40 Northwestern 115-44
45 Michigan 107-44
54 Purdue 98-53
69 Penn State 63-78
75 Indiana 51-97
81 Minnesota 41-111

 

These rankings have converged to the point where they have the identical order to the first, with just a few spots difference in the national ranking.

 

 

Another perspective might be "Sure Team A and Team C haven't played and don't have a transitive chain, but Team A beat Team B by 14 points. Team C only beat Team B by 2. Therefore, Team A is better than Team C". While I might not agree with this idea, it is a reasonable line of thought. The final set of rankings considers the margin of victory along the shortest path of games from one team to another.  The "marginal win" goes to the team with the net better score along the path, with a home field advantage of 3 points and a maximum victory of 21 points.  This method also allows predictions which I've included below.

 

National RankTeamMarginal Win-Loss Record
10 Iowa 96-23
14 Michigan State 93-26
18 Ohio State 91-28
18 Nebraska 91-28
26 Wisconsin 88-31
34 Michigan 80-39
43 Illinois 74-45
58 Penn State 61-58
64 Northwestern 54-65
82 Purdue 45-74
88 Indiana 39-80
95 Minnesota 34-85

 

 

Predictions:

Wisconsin +3 at Iowa

Minnesota +4 at Penn State

Oklahoma State +2.5 at Nebraska

Indiana +6.5 at Illinois

Purdue +7.5 at Ohio State

Michigan State -2.5 at Northwestern

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