Columns - Single and Double Elimination

Actual Number of Players

Log2(Np) - used to count number of rounds with 2-player games only

the expected value, defined as the most likely number of games played by the winner

the expected value, multiplied by 1.5x due to 3-player games

the expected value, multiplied by 2x due to 4-player games

the minimum number of games played by the winner. Winner gets maximum number of byes, and wins everything.

the maximum number of games played by the winner. Winner gets no byes, takes longest path

the recommended TGP grading for that number of players in that format, assuming 25 games played = 100%.

Multi-Round Games, expected number of wins to move on

Best 2 of 3

Best 3 of 5

Best 4 or 7

Double-Elimination, expected number of wins through the losers bracket.

Consider that player who crosses to losers bracket last and is guaranteed 3rd, has 50% chance of moving on.

That means every other player on loser's side collectively have 50% chance to move on.

Recurse that algorithm through the losers side, to get expected number of games played for the player who makes the final two

This gives EV=0.5*(1+0.5*(1+0.5*(1+ …)))) for number of rounds

In the limit, this provides an EV=2.

Thus Double Elimination is Number of rounds on winner side, +2, +0.5 for extra chance that winner of winner's bracket loses first game of final 2.

Knockout Tournaments

Formula for max games currently used by IFPA is for Random only. Swiss prevents that worst-case from ever happening.

The X-Strike Emulator can be used to calculate minimum and maximum number of games that might be needed for a certain number of players

Simply enter number of players, and worksheet emulates a match using two different profiles:

Minimum occurs when whoever has less strikes is assumed to win.

Winner finishes tourney with 0 strikes. Otherwise 50-50.

Maximum occurs whe whoever has most strikes is assumed to win.

Winner finishes tourney with one less strike than the number of knockout strikes. Otherwise 50-50.

The worksheet uses Floor/Ceiling functions to emulate those cases.

The emulator might not be exact, but should be at most +/- 1 error for any estimated value.

Use results on Brackelope Live to validate the Min/Max/Expected value of swiss.

The "minimum" games for Random (i.e. not swiss) is same as for Swiss, because shortest path in Random is if same matchups as Swiss.

Because the worst-case for Random is more, the EV which is calculated as average between best and worst-case, is higher for Random.

This is consistent with data on brackelope, where non-swiss generally runs a few rounds more than swiss with same number of players.

Example of some 3-strike data, swiss, from Brackelope Live. They show that expected value shown on "knockout" worksheet is relatively accurate:

In 3-player, swiss Maximum determined as Min+4. This is an approximation, but consistent with 2-player emulators.

Ladder - Single Game, aka Amazing Race.

2pl, 1ko: 2-player game, start with bottom 2 seeds, loser gets knocked out, winner continues, and join next higher seed.

3pl, 1ko: 3-player game, start with bottom 3 seeds, loser gets knocked out, other 2 continue, and join next higher seed.

4pl, 1ko: 4-player game, start with bottom 4 seeds, loser gets knocked out, other 3 continue, and join next higher seed.

4pl, 2ko: 4-player game, start with bottom 4 seeds, two losers gets knocked out, other 2 continue, and join next 2 higher seeds.

Ladder - Three-Game Matches

Same as Ladder, except each round is 3 games. Best 2-out-of-3 in case of just 2 players.