WWP(A) = | |
∑ | F(A,t) |
^{t ∈ {teams}} |
Where:
with
F(A,B) = _{ρ} ∑ [ ( wp_{λ}(A,B) − wp_{λ}(B,A) ) × κ^{(1-λ)} ] ^{λ=1}
wp_{λ}(A,B) = the number of A→...→B chains λ →s long |
ρ = the maximum path length considered |
κ = ⌈ ^{ρ-1}√max { ∑wp() } ⌉ |
⌈ x ⌉ is the smallest integer ≥ x
Note that WWP(A) includes F(A,A) = 0. This is used to produce three alternative rankings based upon F (described below.)
Rank | Team rank by WWP |
WWP | WWP(team) |
Team | Team A's name, links to a detail page (see below) that supports analysis of its position in the DGG with respect to all other teams. |
Rec | Record against teams in the field. Some 1AA teams play non-D1 teams and a few do not play enough games against against the field to count as opponents for teams that are. |
Conf | Conference affiliation |
Self | The rank team A assigns itself in the ranking of F(A,team) values. Number of ranked teams minus this value is the number of teams ranked below team A according to team A. |
Best | The best rank by any team for team A. |
Maj | The best rank for which a strict majority (50% + 1) of teams rank team A this highly or better (the "Bucklin" rank with all teams' rankings considered as if they were "voters.") |
Worst | The worst rank for team A by any team. |
Borda | Ranking by Borda Count, the sum over all teams' ranks of the count of teams ranked worse than team A. |
The 2015 field includes 204 teams, each of which ranks all of them. To present the results all at once is impractical so we do so team-by-team. Each team A report has two sections:
#Votes | The number of teams that ranked team A at this rank |
Rank | The ordinal rank - ranks for which team A received no "votes" are not listed |
Teams rankings for team A | List of teams assigning team A the indicated ordinal rank. Team A's opponents are listed in bold and all other team names link to the team's ranking of all teams (part 2 of their report page) to make visible the teams ranked better or worse than team A. |
Rank | The ordinal rank based upon WWP Rating.
The method used to assign ranks is worth mentioning. Suppose there are three teams with the best F(Team A,team) value. Then all all three would be assigned rank 3, not "tied for first." The technical reason for this is that you can tell from the rank how many teams the tied teams are ranked strictly better than. With 128 teams, the tied "top 3" are each ranked better than 125 other teams. You could work out the same thing if each were ranked #1 only by knowing how many other teams were ranked #1. That's easy enough for three teams tied at the top, but not so much for 100 teams tied in the middle. |
Team | The team to which Rank is assigned by team A |
Rating | The contribution by the team listed in the Team column to team A's WWP. This is F(A,Team) and the sum of the values in this column is team A's WWP. |
Paths⇒ | The cumulative number of A⇒Team paths. |
Weighted⇒ | The ∑wp_{λ}(A,Team)×κ^{(1-λ)} half of the WWP formula. |
⇐Weighted | The ∑wp_{λ}(Team,A)×κ^{(1-λ)} half of the WWP formula.Rating = Weighted⇒ − ⇐Weighted |
⇐Paths | The cumulative number of Team⇒A paths. |
Cumulative Δpaths | ∑[wp_{λ}(A,Team)−wp_{λ}(Team,A)] for each path length λ through ρ. There is an exception for team A: for λ=1 to ρ the value in the λ column is wp_{λ}(A,A), indicated by enclosing the values with [] characters. [n] is nonzero when team A is involved in n λ-long "loops" in the directed games graph. For example A→B→C→A forms a loop with λ=3. Note that this example forms three such loops, since it could begin with any of A, B or C. |
© Copyright 2015, Paul Kislanko