[ View Thread ] [ Post Response ] [ Return to Index ] [ Read Prev Msg ] [ Read Next Msg ]

BGonline.org Forums

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 1-ply or 2-ply 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:

  • 1-ply uses 2-ply luck
  • XGRoller uses 3-ply luck
  • XGroller+ and ++ uses 4-ply 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 1-ply and 2-ply. I also modify the program to allow a rollout using 2-ply to perform VR using 3-ply. Also i've check using 3-ply Red (that uses 3-ply 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
      • 1-ply and 2-ply are roughly parallel
      • "2-ply using 3-ply VR" and "3-ply red" are roughly parallel
    • infinite RO (X=0) do no depend of the level used in VR: "2-ply" and "2-ply using 3-ply VR" converge to the same value.

    These linear regressions give me 2 things:

  • the slope is the VR effect for 2-ply and 3-ply
  • the intercept at X=0 gives the level of an infinite RO for 1-ply, 2-ply and 3-ply red.

    now determining the formula for PR(N-ply,infinite game) and VR(N-ply) is not based on anything, i just pick a formula that made sense to me. For PR(N-ply,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 level

    for the VR factor, I've only 3 point (PR=0, 2-ply, 3-ply), but i figure i'd also use a power function.

    • Messages In This Thread

     

    Post Response
    Your Name:
    Your E-Mail Address:
    Subject:
    Message:

    If necessary, enter your password below:

    Password:
     

     

    [ View Thread ] [ Post Response ] [ Return to Index ] [ Read Prev Msg ] [ Read Next Msg ]

    BGonline.org Forums is maintained by Stick with WebBBS 5.12.