Creating a fair playing field for an odd number of players

[twoski]

Hello guys, i have a unique problem to solve.

Let's imagine you're playing a board game similar to Risk. The game can be played by anywhere from 2 to 22 players, and the game board will be resized accordingly based on the number of players. The board also wraps, so edge provinces connect to the province on the opposite end. This game also has one additional concept: For each player that is in the game, the game board has an additional resource province. These provinces need to be equidistant to all nearby players (for the purposes of fairness). If necessary, the board can have more resource provinces than the number of players, as long as the distribution is fair. The board should also be as close to square as possible. Rectangles are okay, as long as they are not too long and skinny.

Typically, the game board will have 16 provinces per player. For example, here's a drawing of a 4 player game board.

If we divided this into quarters, all 4 quarters would be the same. The green spots denote player starts, the pink spots denote the resource locations. Note that the resources are always placed equidistant from all players, and all players are placed equidistant to other players and resources. In short, this can be considered a fair layout since every player has the exact same starting conditions.

Now, what if we had 5 players? This makes it more difficult to create a similar layout. Fortunately with a little thinking, i came up with:

It's not ideal... There are 6 resource provinces for 5 players, and we had to bend the 16 provinces per player rule to accommodate 5 players. Still, it is pretty close to being fair. Every player has 2 resource provinces within 2 moves from their start position. The reality is that we need to bend the rules for cases such as this. As long as there are roughly 16 provinces per player, and at least 1 available resource province for each player, it is fine.

I have managed to solve this problem for the simple cases, but i am left with difficult cases now: 7, 10, 11, 13, 14, 17, 18, 19, 21, 22

It's worth mentioning again that we want to avoid layouts that are too rectangular. For 10 players, it is tempting to make make a layout consisting of 2 rows of 5 players. Unfortunately this layout is too rectangular, we want our map to be as square as possible.

If anyone could help me come up with layouts for the difficult cases i would be most grateful!

 

[jedishrfu]

A humorous answer would be to make one player the referree.

Perhaps you could treat the odd player as an alternate with some strategy as to who gets that player. One example would be to give it to the losing player when he has lost more than x points. It would make the game tougher to play but not too tough.

We had a similar real world example when my kids played soccer. I was a new coach and some of my players didn’t show up for a game so we were down three players. The other coach graciously loaned us three of his top players and we had a wild game. I think the other coach saw that it would be good for his team to play a challenging game and his three players to my team tipped the scales just right.

 

[twoski]

Well, unfortunately there's a lack of knowledge as to which player is being placed at which location. For all intents and purposes, it's considered random assignment. So we need to always account for all players and ensure they all have a fair start, which makes it difficult.

After much grumbling and head scratching i came up with a potential solution for 7 players.

Pink is the player starts, yellow is the resource provinces, cyan is just to help me denote the immediate provinces around players.

With this layout, you can see that 2 of the 7 players are slightly more boxed in... But there's not much we can do about that. Again we've ensured all players have 2 resource provinces nearby. In this layout, there are about 20 provinces per player. But that's a secondary concern, as long as the layout treats each player more or less the same i am happy.

 

[twoski]

And here is a potential 10 player layout. Not sure if the resource positions are totally fair though.

 

[jedishrfu]

If each nearest player can access the resource in the same number of steps doesn’t that make it fair?

 

[twoski]

I'm more concerned with players being able to steal other player's resources. For example, the 2 middle guys will have a more aggressive situation since their resources are pretty close together and thus, those provinces will be high value to them. Other guys might have an easier time since their resources are less contested.

 

[mfb]

Let’s say the first player out of n starts at (0,0), an arbitrary choice. You can get a perfectly even distribution if there is a vector (r,s) such that n*(r,s) is identical to (0,0) modulo the board size in x and y, respectively. Player k is at place k*(r,s). This only works if the board size is a divisor of the number of players. Bad luck with prime numbers. 7x7 gives 7 provinces per player, 7x14 gives 14, 14x14 gives 28. Resources can have the same pattern with a constant offset.

If you want different board sizes you'll need some compromise. Follow the pattern with non-integer r,s and round, for example.

Examples: