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[Tables]

(Warning: This topic will try to be vaguely mathematical but will also hand wave all proofs as being obvious by observation)

I've been thinking a little about this, as a concept. The idea is, normally when you run a knockout tournament you have two teams (which may be a single person e.g. Dominion), and after some process a winner is declared and proceeds to the next stage, where they play another team else until only one team remains. Extending that to three players is pretty trivial, change the word two to three and team to teams and you have a system. In fact you can do that for N teams, it's fine.

But what if you want some kind of fairness? For example, you have seeded teams and you want a fair seeding system (or as fair as possible, in the case that fairness is impossible). Or maybe for various reasons it's desirable to have two teams go through, not just one, or maybe even a double elimination system so one loss doesn't knock you out. This starts to get pretty complex in my mind. I did some browsing online and in my skimming, there's some interesting theory on seeding and running a tournament fairly with two players (or not, as it's impossible if you want certain desirable conditions), but nothing on three team knockouts.

Why am I thinking about this? Mostly, because I think it might be useful for Dominion tournaments, maybe. But also I think it's interesting for it's own sake.

So firstly I'll start with a simple ish case. You have N players, seeded, under the assumption that player x is more likely to beat y than z if y>z (for all x,y,z in [1,N]). You have a three player knockout system. Oh, so let's add the assumption that adding player c to a game between a and b doesn't affect the chance of a beating b. I think one desirable property is that a player x should have a greater or equal chance of winning with his current seed than he would with any greater numbered seed (in other words, sabotaging his own ability to decrease his seed could only decrease his chance of winning).

So what's the optimal seating? Well, I don't think there is one, but here's a system that looks okay:
With three: 1 2 3
With 9 (letters for ease of the 27 round):
1 6 9
2 5 8
3 4 7

With 27:
1 18 27
6 13 22
9 10 19

2 17 26
5 14 23
8 11 20

3 16 25
4 15 24
7 12 21

From there you work backwards (treat byes as having the lowest seeds during the first round). This system possibly achieves the criteria I listed above but also, I think, achieves another useful one, which is that it delays high seed meetings as long as possible, so while your chance of winning may not be maximised, your chance of going far is.

With four players, I can think of two possible options, and I don't know which is better. Here's what it looks like in the semi final:

1 8 9 16
2 7 10 15
3 6 11 14
4 5 12 13

OR

1 8 12 16
2 7 11 15
3 6 10 14
4 5 9 13

I think which of these would be better depends on the relative skills of the players. For example, if we assume that the top few teams get massively better than each other as seed improves (e.g. seed 1 has an 80% chance of beating seed 2, 90%+ seed 3 or below, while seed 2 has an 80% chance of beating seed 3 or below, and most other matchups will probably only be up to a 75% rate - not an unreasonable assumption in some sports, for example), I think the bottom option is better, it gives seeds 9 to 12 real incentive to raise their rating. On the other hand, if say probability of winning is pretty much inversely linear with seed, then the top option is fairer (mainly because of the teams in 8th-9th and 12th-13th on the bottom option, where the 8th best team would rather have the 9th bests seed and ditto for 12th).

So, uh, I think that's about as far as I got thinking about this. Mostly this was an as-I-went trail of thoughts, but I hope there's some interesting thoughts in here and if people have suggestions on running such a tournament, I'd like to hear it.

(I think the subjective 'best' option in terms of fairness and speed might be to seed the best K players where K is the number of games in the first round, place byes sensible (e.g. with the highest seeds) and then randomise all other players. I won't offer a defence of this view because I'm probably likely to change if presented with what seems like a better system)

[theory]

In researching what others did for such tournaments I looked all over the Internet for such discussion.  No luck, which is strange since you'd think there'd be an academic paper on this.

[Tables]

Probably because the correct solution is not to do it with 3+ players . Or otherwise use a swiss style tournament system?