Transitive Rankings Post Week 9: Badgers on Top of the Mess

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).


The MSU loss to Iowa wiped all the remaining Big Ten teams out of the "transitively unblemished" group.  Now it's all about shorter paths.  


1 Wisconsin 108-11 (#10)

Wisconsin rises to the top.  Their wins over OSU and Iowa are better than the two best wins for anyone else and beating ASU looks decent as well (ASU beat Washington who beat USC, Oregon State, and Syracuse).  Their lone loss to MSU doesn't hurt much: Wisconsin only transitive losses are to the 9 above the fray teams, MSU, and Arizona (Arizona-Iowa-MSU-Wisconsin vs Wisconsin-ASU-Washington-Oregon State-Arizona).  

Next Week: At Purdue.  Wisconsin beat Ohio State who beat Purdue.  


2t Nebraska 102-17 (#13t)

Baylor beating Texas accounts for 7 of Nebraska's losses, mostly from the upper crust Big Ten (Iowa-Iowa State-Texas Tech-Baylor) and SEC (Arkansas-Texas A&M-Texas Tech-Baylor).  These chains are probably too long to survive if Nebraska keeps winning though.  Their last two games against Oklahoma State and Missouri are worth a combined 57 wins.  

Next Week: Iowa State.  Iowa State beat Texas who beat Nebraska, but Nebraska beat Kansas State who beat Iowa State.  


2t Iowa 102-17 (#13t)

The next 3 teams are tied.  Iowa's best win for transitive purposes is still Iowa State (out of conference affects transitive rights more because the closest connections to other out of conference teams likely passes through them).  ISU's wins over Texas, Texas Tech and Kansas (2-6, but beat GT so they help out for ACC teams) combine to give Iowa 40 transitive wins.  MSU is worth 27, Michigan is worth 21.  

Next Week: At Indiana.  Iowa beat Michigan who beat Indiana.  


2t Michigan State 102-17 (#13t)

Much like Iowa, MSU is 1-1 against the upper tier Big Ten teams.  They don't have the second loss, but they also don't have much for quality wins after Wisconsin (worth 37 wins).  Like everyone else at the top, they are trying to overcome the sins of Iowa's loss to Arizona.  12 of their losses come from the single game of Washington beating Oregon State (who beat Arizona).  

Next Week: Minnesota.  MSU beat Northwestern who beat Minnesota.  


5 Ohio State 90-29 (#30)

This spot is probably lower than almost every other ranking, but Ohio State just doesn't have many good wins, transitively speaking.  The Miami win is still decent, accounting for 38 wins (much of it low hanging fruit), but if asked to name OSU's second best win, most fans would be stumped.  According to transitivity it's actually Purdue, worth 15 wins.  

Next Week: Bye.


6 Illinois 79-40 (#42)

Illinois had been rather precariously perched in the unscathed group before this week.  They didn't have a loss to anyone who was outside of the lower group, but they didn't really have any quality wins.  Unfortunately for the Illini, two "upstream" teams lost this past week in Missouri and MSU.  Left to stand on its own, Illinois' wins over Penn State (though much more impressive after PSU beat Michigan), Purdue, and Northern Illinois were not enough to keep it in the upper tier.  

Next Week: At Michigan.  Illinois beat Penn State who beat Michigan.  


7 Michigan 73-46 (#46)

Losing to Penn State stings.  Losing to Iowa wouldn't hurt so much, but Michigan doesn't have a win against a team with a winning record.  Their best direct wins are over a 4-5 Notre Dame team (26 wins) and a 4-4-in-the-Big-East UConn (23 wins).

Next Week: Illinois.  Michigan beat Connecticut who beat Vanderbilt who beat Mississippi who beat Fresno State who beat Utah State who beat BYU who beat Washington who beat Oregon State who beat Arizona who beat Iowa who beat MSU who beat Illinois.  


8 Penn State 71-48 (#49)

Penn State effectively has the same transitive wins as Michigan.  They are ranked lower because UCF, Kentucky, and Toledo have shorter paths back to Penn State than Michigan has to them.  Other than Michigan, Penn State's wins are over a soft 6-2 Temple and an even softer 3-4 Kent State.  

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


9 Northwestern 56-63 (#61)

Northwestern has a mind-boggling 58 losses from the Purdue game (many of them embarrassing like Duke, Wake Forest and Tulsa).  A lot of these are long paths that could be short-circuited more effectively, but when the best wins are over 2-6 Vanderbilt and 2-6 Rice, that isn't going to happen.  Northwestern is perhaps as dependent as any team on a rather obscure result: Vandy over Mississippi (who later beat Fresno State and Kentucky).  This single game gives Northwestern 21 transitive wins.  

Next Week: at Penn State.  Northwestern beat Vanderbilt who beat Mississippi who beat Kentucky who beat South Carolina who beat Alabama who beat Penn State.  


10 Purdue 47-72 (#75)

Purdue is defined by their bad losses (ND worth 42 transitive losses and Toledo worth 24) and their decent-for-this-spot-in-the-rankings win over Northwestern (worth 36 transitive wins).  All their other games combined have a transitive record of 11-6.  

Next Week: Wisconsin.  Purdue beat Northwestern who beat Vanderbilt who beat Mississippi who beat Fresno State who beat Utah State who beat BYU who beat Washington who beat Arizona who beat Iowa who beat MSU who beat Wisconsin.  


11 Indiana 40-79 (#83)

Indiana's loss to Northwestern really hurts.  That game gave Indiana 45 transitive losses (43 of them from Purdue beating Northwestern).  Indiana's "quality" win remains Arkansas State by far, giving them 37 of their (though most from low hanging fruit). 

Next Week: Iowa.  Indiana beat Arkansas State who beat FAU who beat UAB who beat Southern Miss who beat Louisiana Tech who beat Utah State who beat BYU who beat Oregon State who beat Arizona who beat Iowa.  


12 Minnesota 33-86 (#89)

Minnesota's quest to be the lowest BCS conference team got substantially easier this last week.  Of the six teams lower than them last week, Virginia beat Miami, Duke beat Navy, Boston College beat Clemson, and Connecticut beat West Virginia.  Wake Forest previously had beaten Duke, so all 5 of those teams were lifted above the Gophers.  The only remaining team below Minnesota is Washington State, who continued their own quest to have no wins over FBS teams.

Next Week: At MSU.  Minnesota beat Middle Tennessee State who beat Louisiana-Monroe who beat FAU who beat UAB who beat Southern Miss who beat Louisiana Tech who beat Utah State who beat BYU who beat Washington who beat Oregon State who beat Arizona who beat Iowa who beat Michigan State.  



Wisconsin beats Purdue

MSU beats Minnesota

Penn State beats Northwestern

Iowa beats Indiana

Illinois beats Michigan


For what it's worth, the differences in order between these rankings and Sagrin's:

Nebraska and Wisconsin are flipped

Penn State and Michigan are flipped (though they are only 3 spots apart in both rankings)


Transitive rankings obviously aren't perfect.  One issue is that they unreasonably value long chains.  In some sense, the best information comes from the closest connection.  As an alternative, here are rankings according closest connection record.  These rankings consider only the shortest connection of the type "Team A played Team B played Team C".  If Team A beat Team B who beat Team C, then Team A gets a win over Team C.  If Team A beat Team B, but Team B lost to Team C, then the Team A vs Team B match-up is considered a no-result.  Confused?  Consider Purdue and South Carolina.  The shortest connection is Purdue-Northwestern-Vanderbilt-South Carolina.  Purdue beat Northwestern who beat Vanderbilt, but Vanderbilt lost to South Carolina.  So Purdue (and South Carolina) get a transitive tie.  

National Rank Team Closest Connection Record
4 Nebraska 64-6
12 Michigan State 51-3
14 Iowa 52-5
16 Ohio State 44-2
17 Wisconsin 42-1
36 Michigan 32-12
39 Illinois 27-10
41 Penn State 28-14
63 Northwestern 20-21
71 Indiana 8-15
76 Purdue 18-31
87 Minnesota 8-32

The rankings aren't much different, but they are more revealing in some ways.  For instance, Nebraska has the worst loss of anyone in the top 5, but they also have better wins.  


In the beginning, the point of the transitive rankings was to model which teams had bragging rights over others.  However, they have lost their way.  For instance, Arizona doesn't really have bragging rights over Wisconsin through Iowa because Wisconsin beat Iowa too.  For these rankings, to have a bragging rights win, a team has to have a bragging rights win over every team in the path and better bragging rights over each intermediate team than the team they beat.  Needless to say, this eliminates quite a number of would-be transitive wins.  

National Rank Team Bragging Rights Record
4 Wisconsin 70-1
7 Michigan State 62-1
11 Iowa 63-5
13 Ohio State 58-5
18 Michigan 44-8
32 Indiana 24-6
39 Penn State 26-15
40 Northwestern 31-21
53 Nebraska 11-11
74 Illinois 8-17
98 Minnesota 10-38
109 Purdue 7-50

These are obviously not "power rankings" in any sense of the word, but I still think they are interesting.  Wisconsin absolves itself of Iowa's loss to Arizona by beating Iowa itself, but Illinois has not.  Nebraska lost a whole lot of bragging rights in its loss to Texas.  

An obvious bonus is that these rankings have Northwestern higher rated than any other rankings I have seen (except the nonsensical coaches poll).  


The final set of rankings looks at the closest connection and takes the net score along that path to determine the winner.  Home field advantage is considered as 3 points and victories have a max margin of 21 points.  

Rank Team Margin Win-Loss Record
7 Iowa 102-17
12 Nebraska 98-21
15 Ohio State 94-25
23 Wisconsin 88-31
31 Michigan State 82-37
36 Illinois 78-41
38 Michigan 76-43
41 Penn State 74-45
79 Northwestern 48-71
86 Minnesota 38-81
86 Purdue 38-81
89 Indiana 37-82

For what it's worth, these rankings are close to Sagarin's margin aware: just flip Iowa-OSU, Minnesota-Indiana and resolve his (effective) ties.  


Margin based predictions:

Northwestern +8.5 at Penn State

Wisconsin -6.5 at Purdue

Iowa State +5.5 at Nebraska

Iowa -8.5 at Indiana

Minnesota +14 at MSU

Illinois -4 at Michigan

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