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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)
PR_{ro}=0.03918×PR_{Level}^1.25740+(4.46108×PR_{LuckLevel}^0.26730)/sqrt(N)Where PR_{RO} 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 nondouble (see below) no truncation when reaching Bear off (see below) For instance a XG2 3ply RO 1296 games plays at a PR of 0.115 (level=levelluck=3ply 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 nondouble" looses 0.009 PR on a 1296 RO using 1ply. 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 1ply. the formula seems to scale well (for money games) for a RO using a different level for checker and cube if using PR_{Nchecker/Pcube}=(PR_{N}*5+PR_{P})/6 The formula was derived from 1 and 2 ply results. however it match very well the observed PR for GnuBG 2ply:
PR of the 1296 2ply GnuBG RO measured using the Depreli 2012 study: 0.175 PR of 2ply 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 "2ply/3ply Red Ro" and full "3ply 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)
Before anyone ask: I am working on estimating the effect of truncation (it's not easy!)
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. 1ply 2.063 2ply 1.320 0.560 0.364 0.286 0.231 0.192 0.164 0.145 0.131 0.120 0.113 0.097 2ply 1.320 2ply 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 3ply 1.129 Snowie 3ply 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 2ply 1.068 Gnu 2ply 1.068 0.479 0.295 0.221 0.169 0.132 0.106 0.087 0.074 0.064 0.057 0.043 2ply/3ply red 0.988 2ply 1.320 0.501 0.306 0.227 0.172 0.133 0.105 0.086 0.072 0.061 0.054 0.039 3ply red 0.640 3ply 0.451 0.369 0.223 0.164 0.123 0.093 0.072 0.058 0.047 0.039 0.034 0.022 3ply 0.451 3ply 0.451 0.361 0.215 0.156 0.115 0.085 0.064 0.050 0.039 0.031 0.026 0.014 3ply/XGR 0.421 3ply 0.451 0.360 0.214 0.155 0.113 0.084 0.063 0.049 0.038 0.030 0.025 0.013 4ply 0.332 4ply 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 4ply 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 4ply 0.332 0.322 0.187 0.133 0.095 0.068 0.049 0.035 0.026 0.018 0.014 0.003 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 3ply 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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