The Bad Loss Index

November 10, 2016

Texas A&M's loss to Mississippi State knocked Auburn, Arkansas and LSU out of the pseudo-Smith Set along with the Aggies. That currently leaves it at:

{ Clemson; Alabama; Michigan; Western Michigan; Washington; Louisville; Troy }
Strictly speaking, currently there is not a Smith Set when all of D1 is taken into account. You can add The Citadel (undefeated) and Chattanooga (only lost to The Citadel) to the above list, but undefeated Sam Houston does not have a "→ x → ... →" chain to all of the teams not in the expanded list. The Bearkats' nine wins only establish 16 win-chains -
PL#PW?Win Chains from Sam Houston(*) to...
11W∗→Nicholls State [0 0]
21W∗→Lamar→Southeastern Louisiana [0 0]
11W∗→Stephen F. Austin [0 0]
11W∗→Lamar [0 0]
11W∗→McNeese State [0 0]
11W∗→Houston Baptist [0 0]
41W∗→Texas Southern→Mississippi Valley State→Ark Pine Bluff→Alcorn State [0 0]
11W∗→Texas Southern [0 0]
11W∗→Abilene Christian [0 0]
21W∗→Texas Southern→Alabama A&M [0 0]
31W∗→Texas Southern→Alabama State→Jackson State [0 0]
21W∗→Texas Southern→Alabama State [0 0]
11W∗→Incarnate Word [0 0]
21W∗→Texas Southern→Mississippi Valley State [0 0]
25W∗→Lamar→Northwestern State [0 0]
∗→Incarnate Word→Northwestern State
∗→Abilene Christian→Northwestern State
∗→Nicholls State→Northwestern State
∗→McNeese State→Northwestern State
31W∗→Texas Southern→Mississippi Valley State→Ark Pine Bluff [0 0]
The only wins by these 16 teams are against each other, which prevents Sam Houston from having a path to all of the other teams with at least one loss.

I gather a lot of information from the directed games graph, and for teams not in the pseudo Smith Set I kind of like my "bad loss index" as a method for ranking (not rating) teams' resumes. For team A it is

number of teams B for which A has any A → ... → B chain
number of teams B for which A has a stronger A → ... → B chain than the correspondnig B → ... → A chains
This is zero for members of the pseudo Smith Set (also for teams that are winless or have only wins over winless teams, but those are easily filtered out - just set BLI to the number of teams B with a stronger B⇒A chain than A⇒B whenever there are more of those than A⇒B stronger than B⇒A.)

This report lists the teams by BLI For teams with the same BLI the order is essentially increasing number of other teams with a stronger win path to the team than the team to them.

The BLI quantifies notions such as "the best n-loss team" since it counts the number of teams that beat an opponent the team lost to except for those for which the team has a head to head (to head to ...) advantage. It really does more than that. It exposes the notion of just counting losses as an irrelevant metric since the best 3-loss team might have a better overall resume than many one or two -loss teams.

It also is the easiest way to sort out the loops in the directed games graph, like

Mississippi→Georgia Southern→South Alabama→Mississippi State→Texas A&M→Auburn→Mississippi
Mississippi State→Texas A&M→Auburn→Mississippi State
without having to examine all such paths.

© Copyright 2016, Paul Kislanko