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72-game rollouts in XG

Posted By: Timothy Chow
Date: Monday, 23 May 2011, at 2:41 p.m.

In Response To: 72-game rollouts in XG (eXtreme Gammon)


Xavier wrote:


I did not see much value in rotating on the 2nd roll.

Also i did not see easy way to make it so that Rollout shorter than 1296 would not be biased. I am sure i can put in place some quasi-randomness (randomizing the 2nd roll while making sure each combination is hit at the end).


I thought about this a bit and unless I'm missing something, it shouldn't be hard to rotate on both the first and second rolls without causing biases in shorter rollouts. Create a random permutation of the 36 possible first rolls, and for each first roll, create a random permutation of the 36 possible second rolls, and interleave them.

For example, with coins instead of dice, first create a random permutation of HH HT TH TT, and then for each of those four possibilities, create a fresh random permutation of HH HT TH TT. Say your first random permutation is TH HH TT HT, and your four second-roll permutations are: 

 

first roll TH: TT TH HH HT
 

first roll HH: HH HT TH TT
 

first roll TT: TH TT HH HT
 

first roll HT: TT HH TH HT


Then you should run through the 16 possibilities for the two rolls as follows: 

 

TH TT
 

HH HH
 

TT TH
 

HT TT
 

TH TH
 

HH HT
 

TT TT
 

HT HH
 

TH HH
 

HH TH
 

TT HH
 

HT TH
 

TH HT
 

HH TT
 

TT HT
 

HT HT


If you're worried about the effect of rotating through the first-roll possibilities in the same order every time, then you can further randomize by running through the first-roll possibilities in a different random order each time (but keeping them associated with the same second roll).

Generating random permutations is easy using the Moses-Oakford algorithm (called "Shuffle" on page 2 of this article).

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