How a double elimination bracket works
A team is out after its second loss, not its first. That one rule creates a second bracket, a grand final that can be played twice, and almost twice as many games.
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Two brackets
Everyone starts in the winners bracket, which is an ordinary single elimination bracket. Win and you stay in it. Lose and you drop into the losers bracket, where a second loss ends your tournament.
With 8 teams the winners bracket has 3 rounds and 7 games. The losers bracket has 4 rounds and 6 games. The team that wins the winners bracket has not lost; the team that wins the losers bracket has lost once.
Where losers drop
The losers bracket alternates between two kinds of round. In one, the teams already in it play each other. In the next, the survivors meet the teams that have just dropped from the winners bracket.
The order of those drops matters. If the teams dropping from each winners round are sent straight across, two teams that have just played each other can meet again one game later. A good bracket crosses them over so that a rematch can only happen near the end. In a simulation of a 16-team bracket published by tourneygeek.com, dropping in plain order produced about 2.6 avoidable rematches a tournament; the right order brought that to about 0.4.
You do not need to work this out. Each line in the losers bracket says which game its team comes from, for example "Loser of game 7".
The grand final and the reset
The grand final is the winners bracket champion against the losers bracket champion. If the unbeaten team wins, it is over.
If the team from the losers bracket wins, both teams now have one loss, so a second game decides it. That game is the reset (also called the "if necessary" game), and it is why the bracket shows a dashed box at the end. Some events skip it to save time and let the grand final stand on its own; say which before the first game.
How many games
How many games one team plays
At least two: lose twice and you are out.
At most, a team loses its first game, wins every round of the losers bracket, wins the grand final and plays the reset. With 8 teams that is 7 games; with 16 it is 9. Check that against the day: a team on that road plays far more often than the unbeaten one, with less rest.