• Posts
  • RSS
  • ◂◂RSS
  • Contact

  • Region Selection in Italian Power Grid

    December 24th, 2016
    games  [html]
    On the France/Italy Power Grid expansion there are the usual six regions, but because of the way Italy is a peninsula there are fewer combinations than usual:

    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:

    Region Even-20% Sequential
    Brown 20% 30%
    Purple 40% 47%
    Red 80% 79%
    Yellow 80% 70%
    Blue 40% 37%
    Green 40% 37%

    If an even distribution over regions is our goal, however, intentionally optimizing for it can do better:

    Region Even-20% Sequential Region-optimized
    Brown 20% 30% 50%
    Purple 40% 47% 50%
    Red 80% 79% 50%
    Yellow 80% 70% 50%
    Blue 40% 37% 50%
    Green 40% 37% 50%

    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.

    Comment via: google plus, facebook

    Recent posts on blogs I like:

    The Gift of It's Your Problem Now

    Recently a security hole in a certain open source Java library resulted in a worldwide emergency kerfuffle as, say, 40% of the possibly hundreds of millions of worldwide deployments of this library needed to be updated in a hurry. (The other 60% also …

    via apenwarr January 1, 2022

    The container throttling problem

    This is an excerpt from an internal document David Mackey and I co-authored in April 2019. The document is excerpted since much of the original doc was about comparing possible approaches to increasing efficency at Twitter, which is mostly information tha…

    via Posts on December 18, 2021

    Experiences in raising children in shared housing

    Sometimes I see posts about people’s hope to raise children in a group housing situation, and it often seems overly optimistic to me. In particular they seem to expect that there will be more shared childcare than I think should be expected. Today I talke…

    via The whole sky October 18, 2021

    more     (via openring)


  • Posts
  • RSS
  • ◂◂RSS
  • Contact