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Rollout strength - A mathematical model

Posted By: eXtreme Gammon
Date: Friday, 17 February 2012, at 3:33 p.m.

In Response To: Rollout strength - Your thoughts on a mathematical model (eXtreme Gammon)

I was hoping to get some idead from here but apparently this is not of much interest to anyone.

anyway, here is the formula I came with

(just in case the image file get deleted from my server here it is again)
PRro=0.03918×PRLevel^1.25740+(4.46108×PRLuckLevel^0.26730)/sqrt(N)

Where PRRO is the PR of a rollout using the settings:

  • Level: ply level used for checker and cube
  • LevelLuck: ply level used for calculating the luck in the VR
  • N: number of games rolled.
  • roll of double and non-double (see below)
  • no truncation when reaching Bear off (see below)

    For instance a XG2 3-ply RO 1296 games plays at a PR of 0.115 (level=levelluck=3-ply that plays at 0.451 PR), if you extend it to 5184, the PR is 0.064.

    notes:

  • The first term of the equation (before the + sign) is the level of an RO with an infinite number of games.
  • The second term of the equation is the effect of VR.
  • not using "roll of double and non-double" looses 0.009 PR on a 1296 RO using 1-ply. the effect should be much less on higher plies.
  • using truncation when reaching bear off database (11 point) looses 0.005 PR on a 1296 RO using 1-ply.
  • the formula seems to scale well (for money games) for a RO using a different level for checker and cube if using PRNchecker/Pcube=(PRN*5+PRP)/6

    The formula was derived from 1 and 2 ply results. however it match very well the observed PR for GnuBG 2-ply:

  • PR of the 1296 2-ply GnuBG RO measured using the Depreli 2012 study: 0.175
  • PR of 2-ply GnuBG using the Depreli 2010 study: 1.07
  • formula gives: PR=0.169 that's a 0.006 PR difference (3%)

    The formula also matches well "2-ply/3-ply Red Ro" and full "3-ply Red" RO

    Here is a table showing different results of the formula for different bots and level (if the bot is not indicated it is XG2)

    Level PR Luck Luck PR RO 108 RO 324 RO 648 RO 1296 RO 2592 RO 5184 RO 10368 RO 20736 RO 41472 RO 82944 RO inf.
    1-ply 2.063 2-ply 1.320 0.560 0.364 0.286 0.231 0.192 0.164 0.145 0.131 0.120 0.113 0.097
    2-ply 1.320 2-ply 1.320 0.518 0.322 0.244 0.189 0.150 0.122 0.103 0.089 0.078 0.071 0.056
    Snowie 3-ply 1.129 Snowie 3-ply 1.129 0.489 0.302 0.227 0.174 0.136 0.110 0.091 0.078 0.067 0.061 0.046
    Gnu 2-ply 1.068 Gnu 2-ply 1.068 0.479 0.295 0.221 0.169 0.132 0.106 0.087 0.074 0.064 0.057 0.043
    2-ply/3-ply red 0.988 2-ply 1.320 0.501 0.306 0.227 0.172 0.133 0.105 0.086 0.072 0.061 0.054 0.039
    3-ply red 0.640 3-ply 0.451 0.369 0.223 0.164 0.123 0.093 0.072 0.058 0.047 0.039 0.034 0.022
    3-ply 0.451 3-ply 0.451 0.361 0.215 0.156 0.115 0.085 0.064 0.050 0.039 0.031 0.026 0.014
    3-ply/XGR 0.421 3-ply 0.451 0.360 0.214 0.155 0.113 0.084 0.063 0.049 0.038 0.030 0.025 0.013
    4-ply 0.332 4-ply 0.332 0.330 0.194 0.140 0.102 0.075 0.056 0.042 0.033 0.025 0.021 0.010
    XGR+ 0.183 4-ply 0.332 0.324 0.189 0.135 0.097 0.070 0.051 0.037 0.028 0.020 0.015 0.005
    XGR++ 0.117 4-ply 0.332 0.322 0.187 0.133 0.095 0.068 0.049 0.035 0.026 0.018 0.014 0.003
    Before anyone ask: I am working on estimating the effect of truncation (it's not easy!)

    It is important to note that according to that model the Depreli RO level is between 0.030 and 0.050 PR. All the RO were compared to these with the assumption they are always right. What is the effect on the formula is unclear to me.

    Note that a full 3-ply 1296 game rollout appears as strong as a XGR++ analyze (0.115 and 0.117 PR respectively)

    Feedback is, as always, welcomed.

    Xavier Dufaure de Citres

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