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Rollout strength  A mathematical model
Posted By: eXtreme Gammon
Date: Saturday, 18 February 2012, at 4:53 p.m.
In Response To: Rollout strength  A mathematical model (Maik Stiebler)
First of all, sorry for the math below, i know it may only interest a few of you.
However, if it is not clear, the goal of what i am doing here is to be able to tell what are best setting for Rollout. for instance the results I posted shows that there is basically no reason to use 1ply or 2ply RO: XGR++ will be stronger. Also full rollout under 2592 games are not as accurate as XGR++.
WARNING: MATH BELLOW
For XG, the level used to calculate the luck of each roll is the ply level used for checker play with the following exception:
1ply uses 2ply luck XGRoller uses 3ply luck XGroller+ and ++ uses 4ply luck. Stronger luck level allow the VR to converge faster.
Theoretically, a perfect level would make the rollout converge immediately (after one game roll)Here is the methodology used: I analyzed (using the first 1000 positions of the Depreli study) what a level consting of a full rollout using P ply and N game would play. I did that using 1ply and 2ply. I also modify the program to allow a rollout using 2ply to perform VR using 3ply. Also i've check using 3ply Red (that uses 3ply luck). I made test using 108,324 648,1296, 2592,5184 and 10368 games
Below is a graph of the results using 1/Sqrt(N) on the X axis and the PR in the Y axis
Making a linear regression shows the following
 the effect of the number of games is linear with 1/sqrt(N)
 increasing the ply in the VR makes the RO converge faster
 The effect of VR is only dependent of the level used in the VR as
 1ply and 2ply are roughly parallel
 "2ply using 3ply VR" and "3ply red" are roughly parallel
 infinite RO (X=0) do no depend of the level used in VR: "2ply" and "2ply using 3ply VR" converge to the same value.
These linear regressions give me 2 things:
the slope is the VR effect for 2ply and 3ply the intercept at X=0 gives the level of an infinite RO for 1ply, 2ply and 3ply red. now determining the formula for PR(Nply,infinite game) and VR(Nply) is not based on anything, i just pick a formula that made sense to me. For PR(Nply,infinite game) I've 4 points (knowing that PR=0 means PRro=0). a power function fitted well:
X axis is the Pr of the level, Y axis is the PR of an infinite RO with that levelfor the VR factor, I've only 3 point (PR=0, 2ply, 3ply), but i figure i'd also use a power function.

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