Directed Games Graph Metrics: Weighted Win Paths (WWP)

October 10, 2015

Define the Weighted Win Paths metric as:
WWP(A) =  
F(A,t)
 t ∈ {teams}

Where:

F(A,B) =  ρ
[ ( wpλ(A,B) − wpλ(B,A) ) × κ(1-λ) ]
 λ=1
with
wpλ(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.)

Presentation

The WWP report ranks teams in descending order of WWP(team A), with additional analysis of the Directed Games Graph from the teams' perspectives. The column headings are:
RankTeam rank by WWP
WWPWWP(team)
TeamTeam 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.
RecRecord 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.
ConfConference affiliation
SelfThe 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.
BestThe best rank by any team for team A.
MajThe 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.")
WorstThe worst rank for team A by any team.
BordaRanking by Borda Count, the sum over all teams' ranks of the count of teams ranked worse than team A.

Team Analysis

The team pages linked-to by the WWP report demonstrate the teams' position within each "slice" of the directed games graph. To form a slice we start at a vertex (referebced as tean A below) and list all the other teams by how far away they are from the vertex as measured by the WWP function F(), beginning with the most negative (teams that have more weighted paths to the team than from it, through zero (where the "center" vertex falls, along with all teams for which there is no win path either to or from the team) on to the most positive values (representing teams for which there are more paths from team A than to paths.)

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:

  1. List of all the team A ranks by any team with a list of teams that placed team A at that rank.
    #VotesThe number of teams that ranked team A at this rank
    RankThe ordinal rank - ranks for which team A received no "votes" are not listed
    Teams rankings for team AList 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.
  2. Team A's rank assignment for all teams including itself. The presentation is designed to provide insight into how the rating is formed, and as a byproduct gives information about the "knots" in the directed games graph.
    RankThe 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.

    In fact, "how many teams is the team ranked better than" is exactly the Borda Count. When when the teams' sum over Borda Count for all teams' rankings is sorted in descending order and then ranks are assigned as described, we get the Borda rank reported in the WWP presentation described above. The Borda Rank is the same as the average rank in the sense that the sort by decreasing Borda Count will be identical to a sort by increasing average rank, with tied Borda counts corresponding to tied average ranks.

    TeamThe team to which Rank is assigned by team A
    RatingThe 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 ∑wpλ(A,Team)×κ(1-λ) half of the WWP formula.
    ⇐WeightedThe ∑wpλ(Team,A)×κ(1-λ) half of the WWP formula.
    Rating = Weighted⇒⇐Weighted
    ⇐PathsThe cumulative number of Team⇒A paths.
    Cumulative Δpaths∑[wpλ(A,Team)−wpλ(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 wpλ(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