Pseudo-Smith Set After Week 11

© Copyright 2012, Paul Kislanko

In election method analysis, the "Smith Set" is the set of all alternatives that pairwise beat every alternative not in the set. It is axiomatic that the winner should be a member of the smallest such set.

I define a pseudo-Smith Set for Division 1 using the directed games graph as follows.

1. Find all the shortest A→B→...→Z paths between all team-pairs (A,Z). If the path from A to Z is shorter than that from Z to A, or the shortest length is the same but there are more paths A⇒Z than Z⇒A, say "A beat Z."
2. For each team A, count the number of teams Z for which A "beat" Z and call that WW. Similarly LL is the number of teams Z with a stronger Z⇒A chain than A⇒Z, and TT the number of teams Z that have the same number and length paths to A as A has to Z.
3. If there is no path in the directed games graph that connects A and Z in either direction, count it in UU (for unknown.)

Within this context, the pseudo-Smith Set consists of all teams with a nonzero UU count. There are 13 such teams, each of which has an A⇒... path to all 233 teams that have a zero in that column (none of which have such a path to any of the 13.)

The current list is below.

• You can explore the directed games graph at Division 1 Win Path Summary

• The SEC teams have only lost to each other, but by losing to each other in A→B→C→A fashion, they only have the four undefeateds plus Oklahmoa in their "unknown" column, while those teams have each other plus the SEC teams.