Bracket Theory and Probability

Question

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?

Answer 1

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 $v$.