|December 24th, 2016|
The board has six regions, Brown, Purple, Red, Yellow, Blue, and Green, and in an N player game you randomly choose N connected regions to play with. In a three player game, for example, the options are:
- Brown, Purple, Red
- Purple, Red, Yellow
- Red, Yellow, Blue
- Red, Yellow, Green
- Yellow, Blue, Green
When selecting which regions to use, though, what does "randomly" mean? Here are three ways you might consider:
Choose three regions at random. If they're not connected, try again.
Choose a region at random. Then from the remaining contiguous regions, choose another region at random. Repeat for the third region.
Choose one of the five legal combinations at random.
While the first and third options give the same distribution—each of the five options 20% of the time—the second option does not:
|30%||Brown, Purple, Red|
|17%||Purple, Red, Yellow|
|16%||Red, Yellow, Blue|
|16%||Red, Yellow, Green|
|21%||Yellow, Blue, Green|
I think for this game "combinations" is the right thing to be selecting over, so would choose the third selection method above. But if you care more about regions than combinations, you might prefer the second method: with the even-20% method we have Yellow 80% of the time, while in the sequential method we only have it 70% of the time:
If an even distribution over regions is our goal, however, intentionally optimizing for it can do better:
What does this solution look like?
|50%||Brown, Purple, Red|
|0%||Purple, Red, Yellow|
|0%||Red, Yellow, Blue|
|0%||Red, Yellow, Green|
|50%||Yellow, Blue, Green|
Which shows, I think, that uniformly selecting over combinations of regions is what we want.