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Double Elimination - How many rounds ?

In a single elimination tournament, you need
ceil(log2(players)) rounds.
(2 players, 1 round; 3-4 players, 2 rounds; 5-8 players 3 rounds, etc.)

In double elimination, players get a second chance.
After losing in the winners' bracket, they enter a
second bracket, and the winner of that emerges as a tournament finalist.
How many rounds are there in the losers' bracket ?
The losers' bracket (LB) is like the winners' bracket (WB) - sort of.
After each round, half the players are eliminated.
But ...
Some more players are then added from the winners' bracket (the first time losers).

For example for 8 players it looks like this.

The red arrows and numbers show movement from the WB to the LB. After each round in the WB, half the players move over to the LB, with half as many being added each time.

In our example, after round 1, 4 players go into each bracket.

After round 2, 2 players are eliminated from LB, and 2 more come over from WB.

After round 3, there are 2 players in LB, joined by 1 more from WB.

Round 4 determines the finalist from the LB, who then plays the other finalist from the WB. If the LB winner wins, then they replay, because the WB winner then only has one loss.

The formula for how many rounds are in the LB is: Let L2 = log2(players).
ceil(L2) + ceil(log2(L2))
ceil = ceiling function: next higher integer

The surplus is the extra rounds from bringing in more first-time losers.

4 players: 3 rounds (2 + surplus 1)

8 players: 5 rounds (3 + surplus 2)
16 players: 6 rounds (4 + surplus 2)

32 players: 8 rounds (5 + surplus 3)
64 players: 9 rounds (6 + surplus 3)
128 players: 10 rounds (7 + surplus 3)
256 players: 11 rounds (8 + surplus 3)

512 players: 13 rounds (9 + surplus 4)
1024 players: 14 rounds (10 + surplus 4)
2048 players: 15 rounds (11 + surplus 4)
4096 players: 16 rounds (12 + surplus 4)
8192 players: 17 rounds (13 + surplus 4)
16384 players: 18 rounds (14 + surplus 4)
32768 players: 19 rounds (15 + surplus 4)
65536 players: 20 rounds (16 + surplus 4)

131072 players: 22 rounds (17 + surplus 5)
262144 players: 23 rounds (18 + surplus 5)
524288 players: 24 rounds (19 + surplus 5)
1048576 players: 25 rounds (20 + surplus 5)
2097152 players: 26 rounds (21 + surplus 5)
...