Smith Set Watch
In election methods, (according to Wikipedia as of this writing) the "Smith Set" is
the smallest non-empty set of candidates in a particular election such that each member beats
every other candidate outside the set in a pairwise election.
For the purpose of analyzing game results in the context of the schedule graph, we amend the definition to mean:
- Smith Set
-
the set of teams that
- are undefeated
- have only lost to undefeated teams
- have only lost to teams who've only lost to undefeated teams and/or teams who've only lost to undefeated teams
- ...
where ... means to extend the process defined in step 3 indefinitely
(i.e. include teams with no losses except to teams in categories 1,2,3, then teams with no losses except to teams in categories 1-4, and so on.)
If it appears that eventually this algorithm includes every team, it obviously can. Certainly all teams are
undefeated before any game is played, and after each team has played one game there's only { undefeated teams }
and { teams that lost to undefeated teams }. In fact, the set includes all teams up until there's at least one
A→B→C→...→A chain in the directed games graph. In 2009
that did not occur until the fifth weekend.
By the end of the year there may be a true Smith Set, with "beats" defined as "having a shorter or stronger
win path to every
team outside the set." For the time being, though, we're tracking "has not lost to any team outside the set," even if such teams have no wins outside the set, either.
Members as of 9 October
| Team | W | L | Conf | SSW | SS Wins over | Losses to |
| Colgate | 6 | 0 | Pat | 1 | Cornell; |
| Boise St | 5 | 0 | WAC | 3 | Oregon; Fresno St; UC Davis; |
| Cincinnati | 5 | 0 | BigE | 3 | Rutgers; Oregon St; Fresno St; |
| Iowa | 5 | 0 | B10 | 3 | Northern Iowa; Arizona; Penn State; |
| Alabama | 5 | 0 | SEC | 2 | Virginia Tech; Kentucky; |
| Auburn | 5 | 0 | SEC | 2 | Mississippi St; West Virginia; |
| LSU | 5 | 0 | SEC | 2 | UL Lafayette; Mississippi St; |
| Villanova | 5 | 0 | CAAF | 2 | Temple; William & Mary; |
| Wisconsin | 5 | 0 | B10 | 1 | Fresno St; |
| South Florida | 5 | 0 | BigE | 0 | |
| Florida A&M | 4 | 0 | MEAC | 2 | Delaware St; Howard; |
| Richmond | 4 | 0 | CAAF | 2 | Duke; Delaware; |
| Florida | 4 | 0 | SEC | 1 | Kentucky; |
| Kansas | 4 | 0 | B12 | 1 | Duke; |
| New Hampshire | 4 | 0 | CAAF | 0 | |
| TCU | 4 | 0 | MW | 0 | |
| Texas | 4 | 0 | B12 | 0 | |
| Montana | 3 | 0 | BSky | 2 | UC Davis; Northern Arizona; |
| Dayton | 3 | 0 | Pio | 0 | |
| Butler | 2 | 0 | Pio | 0 | |
| Team | W | L | Conf | SSW | SS Wins over | Losses to |
| Virginia Tech | 4 | 1 | ACC | 3 | Nebraska; Miami-Florida; Duke; | Alabama; |
| Nebraska | 4 | 1 | B12 | 2 | UL Lafayette; Missouri; | Virginia Tech; |
| William & Mary | 4 | 1 | CAAF | 2 | Central Conn; Delaware; | Villanova; |
| Georgia Tech | 4 | 1 | ACC | 1 | Mississippi St; | Miami-Florida; |
| Oregon | 4 | 1 | P10 | 1 | Utah; | Boise St; |
| Penn State | 4 | 1 | B10 | 1 | Temple; | Iowa; |
| Central Michigan | 4 | 1 | MAC | 0 | | Arizona; |
| Missouri | 4 | 1 | B12 | 0 | | Nebraska; |
| Northern Iowa | 4 | 1 | MVC | 0 | | Iowa; |
| Arizona | 3 | 1 | P10 | 3 | Central Michigan; Northern Arizona; Oregon St; | Iowa; |
| Central Conn | 3 | 1 | NE | 1 | Columbia; | William & Mary; |
| Miami-Florida | 3 | 1 | ACC | 1 | Georgia Tech; | Virginia Tech; |
| Rutgers | 3 | 1 | BigE | 1 | Howard; | Cincinnati; |
| Utah | 3 | 1 | MW | 0 | | Oregon; |
| West Virginia | 3 | 1 | BigE | 0 | | Auburn; |
| Columbia | 2 | 1 | Ivy | 0 | | Central Conn; |
| Cornell | 2 | 1 | Ivy | 0 | | Colgate; |
| Oregon St | 3 | 2 | P10 | 0 | | Cincinnati; Arizona; |
| Delaware | 2 | 2 | CAAF | 1 | Delaware St; | Richmond; William & Mary; |
| Howard | 2 | 2 | MEAC | 0 | | Rutgers; Florida A&M; |
| Team | W | L | Conf | SSW | SS Wins over | Losses to |
| Kentucky | 2 | 2 | SEC | 0 | | Florida; Alabama; |
| Northern Arizona | 2 | 2 | BSky | 0 | | Arizona; Montana; |
| Temple | 2 | 2 | MAC | 0 | | Villanova; Penn State; |
| UL Lafayette | 2 | 2 | SBC | 0 | | LSU; Nebraska; |
| Delaware St | 1 | 2 | MEAC | 0 | | Florida A&M; Delaware; |
| Duke | 2 | 3 | ACC | 0 | | Richmond; Kansas; Virginia Tech; |
| Mississippi St | 2 | 3 | SEC | 0 | | Auburn; LSU; Georgia Tech; |
| Fresno St | 1 | 3 | WAC | 1 | UC Davis; | Wisconsin; Boise St; Cincinnati; |
| UC Davis | 0 | 3 | GWest | 0 | | Fresno St; Montana; Boise St; |
|
The set is interesting more for the teams that are not included. There's no Ohio State for instance, because Southern Cal lost to a team outside the set.
This set can only get smaller - when a team is on the list because it only lost to team X on the list, when team X loses to a team not on the list, it and all of the teams on the list by virtue of "only" losing to X drop off. In fact, by the end of the year there may be no teams on the list. The usual case, though, is that there are a few who've only lost to each other, so I'll track this from here on out.