Tracking the Smith Set

© Copyright 2009, Paul Kislanko

What's a "good win" anyway?

An unambiguous definition of a "good win" (or for some teams, a "bad loss") is provided by the notion of a "Smith Set." That is a collection of teams that either have no losses or only have losses to each other. At the beginning of the season, every team is a member, since no team has lost to any other team. The set remains identical to the field (347 teams in 2009-10) until there is at least one "transitivity violation."
A transitivity violation is just a set of games connected by winnerloser in which the initial winner occurs more than once in the chain. I.e.
You can find all of the (shortest) win chains that exist in 2009 in the Second Order Winning Percentage index to the directed games graph.
Although every team begins the season as a member, the set isn't well-defined until every team has played at least one game. For the 2009-10 season, that didn't occur until the games of 30 November, and the initial Smith Set consisted of 138 teams.

The set can only become smaller as the season progresses, and may not even exist by the end of the year. Just one week after the field became connected, there were only 44 members.

We can use this concept to make precise (or at least unambiguous) the notions of "good win", "good loss", and "bad loss." "Bad win" only has meaning in the context of specific ratings that reduce a team's value because of the losing team's rating.

A "good win" is
one by any team over a team that's a member of the Smith Set.
A "good loss" is
one by one member of the Smith Set to another.
A "bad loss" is
one by a team in the Smith Set to a team outside of it.

"Bad losses" are especially interesting from this perspective, because when a team leaves the Smith Set so do all of the teams who were in it only because their losses were previously "good losses" (to teams in the set.) Florida's loss to Syracuse was a "good loss" because Florida remained in the set, but if Syracuse were to lose to St. Francis, Florida (along with Cal, UNC and Cornell) would drop out along with the Orange.

Tracking the Smith Set is a new report wherein I'll list the current members and upcoming games that will affect the membership status of its teams. Current members of the Smith Set are listed in italics, and the Dependent Teams column lists the teams that will drop out if the italicized team loses.

† The set can only become smaller as long as there's at least one undefeated team. When the last such team loses its first game, there can be a subset of 1-loss teams who've only lost to each other.