Looking for a maths genius

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Looking for a maths genius

Postby mjparrett » 15 Nov 2017, 15:21

Hi all :)

You may or may not have seen my thread looking to start a league... if not please check it out! (shameless plug)

http://www.playdiplomacy.com/forum/viewtopic.php?f=5&t=56547

Gathering a wee bit of interest. But I need a maths genius to help with the fixture list.

In my head the amount of people in the league will have to be a multiple of 7. But that may or may not be true. I am looking for a set of fixtures that:

1. Ensures everyone in the league plays all the other participants once, but;
2. No-one should play the same player 3 or more times (twice is acceptable)
3. Everyone should play the same amount of games (hence me thinking you want a multiple of 7)
4. The number of games per person should ideally be 3 or 4 each.

Is this doable. I don't mind if the odd person doesn't play everyone and only plays 18/20 of the other contestants. That (in my mind) is better than someone playing the same player in all 3 (or 3/4) of their games.

I feel like there should be a clever way of working this out, but it is escaping me. If you have any clever ideas please post of PM me!

Thanks
Mike
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Re: Looking for a maths genius

Postby Astrup » 15 Nov 2017, 15:57

I like the idea. I played in tournament for another board game at one point, where we did it a little differently, but it worked quite well. The idea is simple enough. Let's say you have 14 or 21 players in the league, and you earn a number of points based on your performance in a game (there's some fine examples in your main thread). The nice thing about this tournament was, that the top 7 placed players would play each other. 8-14 play each other. 15-21 play each other. After a round you have new scores. The next round the new top7 play each other and so on.

The good thing about this format is, that it keeps the tournament tight, as the top placed players always takes points from each other, and it takes a lot of skill and good results to stay on top. On the other hand it is easier to recover after a bad first round and get back in it. So it makes the tournament fun longer for more players, because everyone has a shot at the championship for a longer time.

Good initiative, and I am interested in being a part of the league
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Re: Looking for a maths genius

Postby jimbobicus » 15 Nov 2017, 16:37

If you have 21 people you can ensure nobody plays against the same person all 3 rounds:
Rd1: [1,2,3,4,5,6,7], [8,9,10,11,12,13,14], [15,16,17,18,19,20,21]
Rd2: [1,2,3,8,9,15,16], [4,5,10,11,12,17,18], [6,7,13,14,19,20,21]
Rd3: [1,10,19, +4 others], [2,11,20, plus 4 others], [3,12,21, plus 4 others].
But there will be luck of the draw as you play some people twice and others not at all. But that is unavoidable.
Also if you see logic I have done here you can extend this algorithm to other numbers or participants and rounds

Or alternatively, I think Astrup's idea is quite sensible.

But with either format, you also have the issue of needing a multiple of 7 as the number of people, all of whom are happy with the same deadlines (eg 48/24/24) or whatever it be.
Or a way round the multiple of 7... if you get lucky with number of people and rounds, you can fix it with byes. Eg 16 people, you have 8 rounds. In 1st round 1,2 don't play, in 2nd round 3,4 don't play etc. So you have 8 rounds and everyone plays 7 games.
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Re: Looking for a maths genius

Postby mjparrett » 15 Nov 2017, 16:47

Is it unavoidable though? I think with 4 rounds you could potentially play everyone at least once and not 3 times. Your suggestion (whilst sensible) means there are A LOT of people you won't play in the course of the league.

The problem with the other suggestion is that 7 rounds is too long (I think). I was looking for ideally 4. Can cope with 3 or 5 though.

Yes I agree the deadlines would be an issue. But only a few raised concern with 2/1/1. This is the tournament deadline structure and that still got enough people interested. I wouldn't be looking at rating or premium (although people with high NMR/surrender rate might not be welcome). ANYONE committed though can join so I think numbers wont be an issue.

I think there must be a way. I have got pretty close, with only 1 or 2 people missing out an opponent. Sadly though I also have 1 or 2 people playing someone in 3/4 games which I want to avoid. I think there should be a combination/permutation way of solving this mathematically which is what I am after.
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Re: Looking for a maths genius

Postby jimbobicus » 15 Nov 2017, 17:10

I gave you the best we can do with 21 players and 3 rounds. The algorithm I used is hopefully quite intuitive and I'm pretty sure you can't do better. As the number of rounds increases it becomes possible to design it so that you play everybody closer to the same number of times.
Eg if you had 21 players, 4 rounds, I expect you can do it so that for every person, you play each other play either once or twice (never 0 or 3 times)
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Re: Looking for a maths genius

Postby jay65536 » 15 Nov 2017, 17:29

Worth pointing out, don't see if it's been mentioned yet, but one reason the number of players must be a multiple of 7 is that if you want to have fewer than 7 games, you can't have a number of players that's not a multiple of 7 that assures all players will play the same number of games. (That's because 7 is prime.)

So 21 seems like the right number of players, and jimbo's idea seems pretty good. To the OP's complaint: it's actually not possible with 21 players and 3 rounds to play everyone once, as you can only see 18 total opponents over 3 games, and because you will have to see some opponents twice, that means more opponents you never see. 4 rounds would probably have the same problem, as you can now play 24 total opponents but you still have to face some people twice.

EDIT: I don't think you can fully solve all of your constraints unless you have 49 players and 7 games. Then I think a true round robin is possible.
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Re: Looking for a maths genius

Postby mjparrett » 15 Nov 2017, 17:53

Thanks Jay

So with 21 players, can you arrange it so everyone plays everyone once or twice (i.e. not none or three or more) in 4 rounds? I think it should be, but I can't work it out. I always end up with 1 or 2 people playing the same person 3 times. I think there might be a clever way of solving this but haven't got it yet. Take JimB's idea and extend it to 4 rounds and it soon becomes, well, a headfuck.
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Re: Looking for a maths genius

Postby jay65536 » 15 Nov 2017, 18:55

mjparrett wrote:Thanks Jay

So with 21 players, can you arrange it so everyone plays everyone once or twice (i.e. not none or three or more) in 4 rounds? I think it should be, but I can't work it out. I always end up with 1 or 2 people playing the same person 3 times. I think there might be a clever way of solving this but haven't got it yet. Take JimB's idea and extend it to 4 rounds and it soon becomes, well, a headfuck.


I think if you try to extend jimbo's idea to 4 rounds, it is still not possible to accommodate all your constraints. Here is why. Look at his round-by-round breakdown. After Round 2, player 1 has not played (10,11,12,19,20,21), in addition to others. But (10,11,12) AND (19,20,21) have all played each other twice. With 2 more rounds to go, if we want player 1 to play all those players, we can give him (10,19) in round 3 and (11,20) in round 4, but to add 12 or 21 to either of those boards means someone plays someone else 3 times.
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Re: Looking for a maths genius

Postby V » 15 Nov 2017, 20:32

An alternative idea for fixture setting.

There are many random number generators available online, in which you enter min & max values, then it will generate numbers for you.
If a willing non-participant spent a few minutes generating 3 or 4 batches of 7 numbers (maybe without even knowing which number was allocated to which participant) each round could be randomly generated for both opposition & countries.
I know some may get lucky & some less so, but it’s only for fun (& it would be seen to be fair).

The other suggestion about top 1-7, 8-14, 15-21 etc also looked pretty good (& easy)

Good luck with the maths,

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Re: Looking for a maths genius

Postby Astrup » 15 Nov 2017, 21:40

There's some good things here.

Just wanted to point out, that my suggestion doesn't require any specific number of rounds, which it seems like it was interpreted here somewhere. You just score a number of points per round and you play the people close to you in the league tabel. The important thing is just, that everyone is playing an extremely important game tournament-wise every game, because you are playing the people right next to you in the table. Some people you will play more times than others - doesn't really matter in this format.
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