How can a seeded double elimination tournament with N teams be designed to ensure maximum fairness when constructing a loser's bracket?

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One approach to ensure maximum fairness in a seeded double elimination tournament is to use a modified bracket system called a "balanced double elimination" or "BDE" bracket. In a BDE bracket, the winner of the winner's bracket faces the winner of the loser's bracket in the championship game. This ensures that the best team from the winner's bracket and the strongest team from the loser's bracket compete in the finals.

To construct a BDE bracket, N teams are seeded as usual and placed in a traditional double elimination bracket. However, the bracket is modified to ensure that teams that lose in the winner's bracket face teams that have also lost in the winner's bracket, rather than teams that have already lost a game in the loser's bracket. This eliminates the possibility of teams having to play against weaker opponents in the loser's bracket.

To achieve this, the first round of the loser's bracket is divided into two parts. In the top half, the teams that lose in the winner's bracket play against each other, while in the bottom half, the teams that lose in the first round of the loser's bracket play against each other. The winners of each half then face each other to determine who advances in the loser's bracket.

The BDE bracket ensures that teams are only eliminated by teams of similar skill levels, resulting in a fairer and more balanced tournament.