# Bracket Theory and Probability

So I apologize ahead of time if this isn't the correct venue for this question. I've been having a debate with a friend of mine for a few days now about what the best way to run a bracket is for a tournament I am having for a video game. No need to know anything about the video game at all, purely that there is not a single option of play. There are multiple characters and stages, which add variance as opposed to a game like chess, where there is only one thing at play, who goes first, and your mind.

Is there any way to attribute a number to measure accuracy in any way, shape, or form, to the following ways of running a tournament.

Log (base 2) of the # of entrants will give you the amount of rounds of Swiss you need to run.

Taking this # of rounds, I have a few different options.

1. Swiss->Double Elimination Bracket: I would use Swiss, and take the top 8 from those results and place them in a double elimination bracket.
2. Swiss->Single Elimination Bracket: I would use Swiss, and do the same, but place them in a Single Elim instead.
3. Only a Double Elimination Bracket.

In my mind, the difference in accuracy between #1 and #2 is not THAT much different, and I feel #3 is obviously the worst of the 3.

My friend is telling me that I am terribly sacrificing accuracy by doing #1, when I think the amount sacrificed is trivial.

How would I mathematically calculate these things?

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It's not clear exactly what you mean by "accuracy" here. Do you mean the probability that the player who wins is the player with the objectively highest skill? Do you care about relative rankings of second, third place, etc? –  Martin O'Leary Mar 29 '12 at 16:58
@MartinO'Leary - yes, sorry for not being clear. Your first statement is 100% correct. "probability that the player who wins is the player with the objectively highest skill". We do also care about relativity afterwards. The person I happen to be discussing this with is specifically talking about the rest of the people in the bracket. –  slandau Mar 29 '12 at 16:59
The simplest approach to this problem is probably simulation. I would sample player skills $S_i$ from a standard normal distribution, then model individual match-ups by testing $S_1 - S_2 > X$, where $X$ is a normal random variate with mean zero and variance $v$. The value of the parameter $v$ will determine the likelihood of weaker players beating stronger players, which you'll have to tweak until the results seem reasonable for your scenario. If you have a large body of match records, you can probably use this to guide your estimate, either in an ad hoc way, or by using a system like TrueSkill.
You can then set up a number of different tournament forms, and run a couple of thousand simulations on each. That should give you a rough idea of how the various different forms compare. Intuitively though, I think your initial ideas are probably right, but how much accuracy you 'sacrifice' in the Swiss/double-elimination system will probably depend strongly on the value of $v$.