Seating Arrangement Problem

[cabiness42]

This has a poker application but really is a math problem, so I post it here.

8 people want to play a series of 8 SNG tournaments.

Instead of randomly drawing for seats before each tournament, they want a system whereby each player is guaranteed to begin each tournament in each position.

The simple solution is that you seat all 8 players in the same order each time and start the button in a different place each time.

However, this results in having the same players on your left and right for each tournament, which is not fair given the differing skill levels of the players.

So, is there a mathematical way to ensure that every player starts from every position, yet doesn't have the same players on his/her right each time?

[pococurante]

Randomly draw the seats, then start the button in front of the person whose turn it is to be the button.

If you are trying to be perfectly fair, then keep track of every starting position each person gets, and don't allow them to get the same spot twice. For instance, if someone is UTG in game 1 and draws the same spot in game 4, make them switch with someone who hasn't been UTG yet. This is way more work than it's worth, no one really cares that much.

As far as a mathematical way to do this, I don't think there is one. If there is, it's more complicated than its worth.

[jay_shark]

It's not possible .

This is a simple pigeonhole principle argument . If player 1 has 2 and 8 beside him , then there are at most 5 players to choose from to be on his right or left and 7 sng tournaments which is impossible . In fact , this is not possible for 4 sng tournaments either since 5/2 = 2.5 and 2.5+1 < 4

[jay_shark]

I've also made the assumption that if player A is seated next to player B in sng i , then he cannot be seated next to him in any of the remaining sng's .

[tshort]

[ QUOTE ]
I've also made the assumption that if player A is seated next to player B in sng i , then he cannot be seated next to him in any of the remaining sng's .

[/ QUOTE ]

What's wrong with player B being to player A's left in one round and to his right in some subsequent round?

Edit: Regardless there will be a repeat in relative position between A and B. There are 7 places for to B to occupy once A has been placed in each various position relative to the starting button. In 8 tournaments, one of those "distances" must be duplicated.

[cabiness42]

Thanks for everyone's tips.

I ended up doing some simple trial and error, and got this far:

1) Everybody starts on the button exactly once
2) Everybody starts in the big blind exactly once
3) Everybody starts with each player on his/her immediate right at least once
4) Everybody starts with each player on his/her immediate left at least once


Some players do start in other positions more than once, and are the same distance from other players more than once, but I figure that those four criteria I listed are the most important for fairness.

[tshort]

Schostic generation of 8 x 8 latin squares would be a "fair" method of determining seating.

Latin squares are n x n tables with n symbols such that no column or row repeates a symbol. So, they would satisify your criteria of each person starting in each position relative to the button. Any given 8 x 8 latin square wouldn't guarantee fairness in relative position to other players. Randomly generating the latin squares would be a fair method of seating (although it could no longer be "fair" after the it is determined). Building in constraints to the random generation could further guarantee fairness.

[cabiness42]

Well, I'm pretty certain that I can't achieve fairness both in relation to the button and in relation to the other players. I figure the latter is more important, since position in relation to the button changes every hand but position in relation to the other players is fixed throughout the tournament.

[rufus]

[ QUOTE ]
Well, I'm pretty certain that I can't achieve fairness both in relation to the button and in relation to the other players. I figure the latter is more important, since position in relation to the button changes every hand but position in relation to the other players is fixed throughout the tournament.

[/ QUOTE ]

You can just high-card for the button on each SNG anyway....

If the number of players were prime, then you could assign a number to each player and step through increments:
1 2 3 4 5 6 7
1 3 5 7 2 4 6
1 4 7 3 6 2 5
and so on.

it's very easy to see that this will lead to a different 'local 3-some' each round. Since 8 is not prime, that won't quite work there. I like pococurante's suggestion of cutting/drawing lots for position each time around.

[pricklypete]

^^ makes sense