::  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 Private Sector’s Role in Transit Innovation

The United States has long had private success and public failure – not just the sense of private affluence and public squalor, in which household income is high but the state of public services lags, but also in that the private sector is more productive…

via Pedestrian Observations June 17, 2019

Unintended pregnancy in folk songs

I’ve been listening to a lot of the Watersons and Waterson:Carthy this week. It’s reminded me how absolutely full British folk music is of songs about unintended pregnancy. Most commonly the result is unhappy motherhood: “But if I had kent that I now ken …

via The whole sky June 1, 2019

Programmer migration patterns

I made a little flow chart of mainstream programming languages and how programmers seem to move from one to another. There's a more common kind of chart, which shows how the languages themselves evolved. I didn't want to show the point of view of …

via apenwarr March 18, 2019

more     (via openring)

More Posts:


  ::  Posts  ::  RSS  ::  ◂◂RSS  ::  Contact