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Head to Head Matchup Imbalance

A Head to Head Standings Rule may produce unexpected results if it is applied to more than two teams and the number of head-to-head matchups between the teams is not balanced. A short explanation of how this rule is implemented is in order.

The head-to-head rule is passed a collection of teams to rank and a collection of games on which the ranking is based.
Note that the collection of teams are teams that are tied using any earlier standings rules in the competition's standings scheme, so it will usually be smaller than all the teams being ranked.

The rule scans through the games and finds all games where one of the teams in the collection plays another team in the collection. For each pair of teams it looks at all the head-to-head matchups between the two teams. The team with the most wins is awarded 1 point and the other team is penalized 1 point. Ties have no effect. The result is a point count for each team that can be either positive or negative. The teams are then ranked according to each team's point count, and it generates ties where more that one team has the same point count.

If each team in the collection plays every other team in the collection the same number of times, the result should be as expected. However, if this is not true, and the number of head-to-head matchups is imbalanced, then the results can be unexpected. A particular example might make this clear.

Consider ranking three teams - A, B, and C. They played the following head to head games:

  • A vs. B
  • A vs. C
  • B does not play C - an imbalance
Now suppose A and B tied in their matchup game, and A wins the matchup with C. The fact that B did not play C introduces a problem - there really is no way to rank B in this matchup. B tied A in their matchup, so B's rank would relly depend on how it fared against C.

If B had beaten C it would be tied with A for first place, if B had lost to C, it would be tied with C for second place, and if B had tied with C, it would place second between A and C. Our logic implies the last, since no game appears like a tie.

For this to happen, there have to be more than two teams tied when the head to head rule is invoked, so it may not occur often. However, the only real solution to this is to make sure that all teams have at least one head-to-head matchup in the game schedule being ranked.