Why are round robin tournaments limited to only 2 or 3 players per map?  I imagine there is a good reason for this, but the challenge is to be able to explain it in a way that I'll understand.

And it seems to me like there is kind of a gap in the tournament settings.  You can either play very few games or a ton of games in a tournament.  With Swiss or Elimination you max out at around 4-5 games per player.  And with round robin you end up with 15+ games per player, unless you want a very small tournament.  

Ideally I think a nice sized tournament would be around 50-100 people and about 6-10 games each.  Is there anyway to do this?  Maybe a double-elimination tournament?  Or how about some kind of combo tournament where there is Swiss system to move into the 'playoffs' where the top n^3 move on to an elimination tournament (for n-player games)?

Ozyman wrote:

Why are round robin tournaments limited to only 2 or 3 players per map?

The math was too hard for the math wizards to do.  Mwahahahaha!

...

Wait are those pitchforks?!?!?!

...

/e starts running.

So, the answer to your question is essentially what Yertle said.

Mongrel and I spent a lot of time figuring out the 3 player RR scheduling stuff and then got it implemented by tom.  We thought we had a way to extend to 4 player tournaments, but the scheduling complexities get pretty difficult.

Failing to be able to generalize the solution correctly, we (mostly Mongrel) spent a lot of time researching and we have decided that it is simply a hard problem although there are partial solutions.

For instance, I can schedule for squares, that is 16 players, 4 in a game, 25, 5 in a game, ... and I have thought to ask tom to implement this, but it seems too limited.  There are a few other special cases that can be done, but we have been trying to work on this problem and few others and have been making ground elsewhere.

There are more cases of three player tournaments that could be schedule as well, but they don't lead themselves to a nice start for the tournament.  What is mean by this is that for the 15, 21, 27, ... cases there is a round where everyone plays a game.  In the other possible cases there is no separation into rounds so not everyone would play a game immediately when the tournament started.

The other questions I will leave for others to answer.

Alpha did well, so I'll say a bit more. That 4 player tournaments can be constructed was shown by Hanani in 1961. For this site, the question is whether 4 player tournaments can be easily constructed by algorithm (Bose, 1931, for a few cases).

Easily constructed = lots of symmetry in the construction = prettiness.

This is what drew me in. A colleague and I found a nice algebraic way of constructing the "prettiest" tournaments, boiling the entire problem down to finding a single number. From this number, one could schedule an entire tournament. It was magic. But, the magic didn't occur for all cases, and we realized we had simply rediscovered Bose's cases, and a smidge more, in a very pretty way.

Not abandoned, but working on more important, fruit-bearing stuff at the moment.

You can imagine how the complexity explodes once you get to 5 player RRT's, 6 player RRT's...

Job security.

This seems like a good time for me to dust off the ole' World Cup format tournament suggestion I had awhile back...which would seem to fit the bill for medium-length/size tournaments.

The basic format would be: Group stage followed by a knock-out round (single elimination).

Group stage could consist of as many teams as you want, really, with the top 2/3/4/etc. advancing to the knock-out round.

How does the math look on that one?

~ATH

Ok - if you guys can't figure out it, and it sounds like you've done a literature search, then it either impossible or just nearly impossible. :)  The idea of allowing squares for larger #s I think is worth pursuing if it is not too complicated.  Sure it is pretty limited, but it's still a big step up from what we currently have.

ATH,

That sounds kind of like my suggestion of a few swiss system rounds & then the top players move to an elimination tournament.  I don't follow the world cup close enough to understand their play off system, but do you think that is a pretty close approximation?

The square scheduling is a pretty easy algorithm so I will put something together in the next couple of weeks and send it to tom.  I am not sure how much work it entails on his end however.

I'll take up 4 player games. Do you want the group stage as round robin? The number of players n that can support a 4 player RR schedule is

n = 4, 13, 16, 25,... or

n congruent to 1 or 4 mod 12.  4 players per group , since there's only one game to play in each group.

Given the size of the site, I could see something like this.

a 4-player-per-game tournament with 4*13= 52 contestants.

there are 4 groups with 13 players in each group. Do the RR, and take the top 4 players from each group. Randomize, then feed the 4*4 = 16 finalists into a 2 round single elimination bracket.

Mongrel wrote:

I'll take up 4 player games. Do you want the group stage as round robin? The number of players n that can support a 4 player RR schedule is

n = 4, 13, 16, 25,... or

n congruent to 1 or 4 mod 12.  4 players per group , since there's only one game to play in each group.

I am pretty sure that the 4 mod 12 cases could be split up into quasi-rounds like the current 3-player round robin tournaments (at least this is true for 16 and 64).  Certainly, the 1 mod 12 cases cannot be split up into rounds so when the tournament starts not everyone is guaranteed to be playing a game (depending on the settings for number of simultaneous tournament games).

Given the size of the site, I could see something like this.

a 4-player-per-game tournament with 4*13= 52 contestants.

there are 4 groups with 13 players in each group. Do the RR, and take the top 4 players from each group. Randomize, then feed the 4*4 = 16 finalists into a 2 round single elimination bracket.

I would prefer that the 16 player SE bracket be seeded from the opening round, but either way.

This could also finish with a 16 player RR so that everyone plays 4 games in the opener and then if they make it to the finals, they play 5 additional games.  (This could use the overall record to determine the winner or just use the rounds from the finals.  I prefer the former.)

I think it would be fun to have two round tournaments.  Round 1 is a RR style tournament and then Round 2 is one of the current styled tournaments (RR, SE, or SS) seeded by the top n players.  These I am sure would take tom a while to implement, but it might be worthwhile for more options with the number of players and length of tournament.

I only 80% understand what you guys are talking about, so just ignore me if I'm not making sense.

I don't understand why in a two-part contest you would use RR for the first part.  It seems to me that swiss-style would be better for that.  In RR you already play everyone, so I feel like the ranking from RR is pretty accurate, whereas for SS you only play a subset of opponents, so the ranking is more of an estimate and seems to make more sense to use it as a qualifying round and then finish up with SE.  RR tournaments are generally the longer tournaments, and SS the short ones, so why prolong the longer one even more by adding a second stage?

So I think the confusion is coming from an implicit assumption on my part (I think Mongrel as well).  I was thinking that with 52 players.  There will be 4 groups of 13 players (randomly seeded) that play a RR sub tournament (4 games, 4 players per game).  After those groups have finished their games, the top 4 from each group would advance to a 16 player RR tournament.

I think your suggestion also holds water.  Have x players (x divisible by 4) start in a SS sub tournament and then have the top 16 (or whatever) go into a RR (or SE or SS) tournament that determines the winner.

The other piece that may be confusing is the "disjoint rounds" - this is a term that I use that I do not think is standard.  By disjoint rounds, I meant that every player plays exactly one game in each round and by the end of the tournament everyone has played everyone else.  This can be done for 2 player RR tournament and some 3 player RR tournaments (the current system does so when possible).

The term "quasi-rounds" (definitely my term) that I used above means that there is a round where everyone plays in exactly one game (initial round) and then the next "rounds" are really groupings of games where not everyone plays in that round and some play more than once, but by the end of the tournament everyone has played everyone else in exactly one round.

The cases of 13, 25, 37, ... (1 mod 12) players for a 4 player per game RR tournament cannot be set up as disjoint-rounds or even quasi-rounds.  So there is not really even a nice way to split up the "rounds" for these cases.

The cases of 15, 21, 33, 39, ... (powers of 3 excluded from 3 mod 6) players for a 3 player per game RR tournament can be set up as quasi-rounds.  In these cases, there is an initial round and the next "rounds" come in groupings of 3 rounds where everyone plays 3 games in those 3 rounds.

Ozyman wrote:

I only 80% understand what you guys are talking about, so just ignore me if I'm not making sense.

You guys lost me at "The math was too hard for the math wizards to do.  Mwahahahaha!"  :)

This has probably been mentioned, but in a 2-player round robin, every possible pairing is made. So, the terrible brute force that always works is to schedule all C(n,k) k-player games for n players. This would work for smaller groups that want a larger tournament.  Every player would play C(n-1,k-1) games. This grows too quickly for most of us - for example, 6 players playing a 4-player RR of this kind play 10 games each , 7 players would play 20 games each, 8 players would play 35 games each.

Your constraint is to group each player with each opponent exactly once. For 4 players, can you construct tournaments where everyone is grouped with each opponent twice?

I don't think 1/4 mod 12 is too severe a restriction, but I can see why you'd want to lift it.

This is a different subject, but I would love the 2 players RR tournaments to have an option for 2 legs (so there would be 14 games for each of the 8 players). This is obviously because in many boards, the first player to start has a strong advantage.

Toto wrote:

This is a different subject, but I would love the 2 players RR tournaments to have an option for 2 legs (so there would be 14 games for each of the 8 players). This is obviously because in many boards, the first player to start has a strong advantage.

Agree. This is a very good idea.

Hugh wrote:

Toto wrote:

This is a different subject, but I would love the 2 players RR tournaments to have an option for 2 legs (so there would be 14 games for each of the 8 players). This is obviously because in many boards, the first player to start has a strong advantage.

Agree. This is a very good idea.

hmmm..  While you're at it, in three and four seat tourneys would it be possible to guarantee in your mathy solutions the full range of positions for players?  Just thought I'd throw in a extra dose of complexity to keep things interesting.

M57 wrote:

Hugh wrote:

Toto wrote:

This is a different subject, but I would love the 2 players RR tournaments to have an option for 2 legs (so there would be 14 games for each of the 8 players). This is obviously because in many boards, the first player to start has a strong advantage.

Agree. This is a very good idea.

hmmm..  While you're at it, in three and four seat tourneys would it be possible to guarantee in your mathy solutions the full range of positions for players?  Just thought I'd throw in a extra dose of complexity to keep things interesting.

Without giving it a lot of thought, I can say that there is one significant problem:
  9 players, 3 per game, 4 games each; it is impossible to have a start in each slot (1st, 2nd, 3rd) the same number of times for each player (this easy counting shows it would be impossible for all of the 3 player tournaments currently available).
  16 players, 4 per game, 5 games each; same problem.

It may be possible to ensure that each starting seat is held during the tournament, and possibly even with some global lower/upper bound on the number of times that starting seat is held by each player, but this would take some thought (this can easily be done for 3 players tournament based on how I created the rounds).