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a bias in rollouts

Posted By: bob koca
Date: Friday, 2 May 2008, at 3:02 p.m.

In Response To: a bias in rollouts (Matt Cohn-Geier)

Tom Keith's example is a good one. Matt Cohn Grier's comments have been good also. As you asked, I made a specific example to illustrate. It shows with actual results how a 36 game rollout is biased. For a longer rollout the idea is the same but the effect would be greatly diminished. In practice you would only worry about this effect if a) there were 2 or more results that ended up close in the rollout. b) the number of trials is not large enough for how close those plays ended up.

For example in the large rollout below, since one is quite confident that the best play won the rollout. The bias is extremely small.

Suppose you rollout a 43 opening played 24/20 24/21 at DMP using 36 games. There will be a large standard error, but it is not biased. The equity is just as likely to be low as it is high.

Now suppose though that what you did is take the top 4 candidates and roll them out 36 times. You look at which did the best and use that as your equity estimate. This gives a biased result. To illustrate I did that 10 times.

The summary of results (RO details given at end)

Trial 1: top play 24/20 24/21 with win % of 52.9575

Trial 2: top play of 24/20 13/10 with win % of 51.2005

Trial 3: 24/20 13/10 51.7436

Trial 4: 24/21 24/20 49.9576

Trial 5: 13/10 13/9 50.5379

Trial 6: 24/20 13/10 51.5579

Trial 7: 24/20 13/10 52.2451

Trial 8: 24/21 13/9 51.2286

Trial 9: 13/10 13/9 53.7690

Trial 10: 24/21 13/9 50.3245%

Based on a long rollout (last rollout below) I have that the equity is actually 50.1657 (dev of less than .1) with a best play of 24/21 13/9.

The technique though gives an average equity of 51.5521 Also note that 9 of the 10 trials resulted in an equity greater than the true equity.

____ ROLLOUT RESULTS. NO MORE THEORETICAL RAMBLINGS

TRIAL 1 Rollout 24/21 24/20 MWC: 52.95751%

0.529575 0.246290 0.016718 - 0.470425 0.142572 0.004245 CL 52.95751%

[0.019499 0.040208 0.011924 - 0.019499 0.015935 0.002320 CL 1.94993%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1001 and quasi-random dice

Play: 0-ply cubeful [expert]

Cube: 0-ply cubeful [expert]

2. Rollout 24/21 13/9 MWC: 51.39507% ( -1.56244%)

0.513951 0.174552 0.007460 - 0.486049 0.137096 0.041748 CL 51.39507%

[0.014655 0.026715 0.001628 - 0.014655 0.016167 0.021809 CL 1.46548%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1001 and quasi-random dice

Play: 0-ply cubeful [expert]

Cube: 0-ply cubeful [expert]

3. Rollout 13/10 13/9 MWC: 49.35821% ( -3.59930%)

0.493582 0.142789 0.104079 - 0.506418 0.250306 0.046523 CL 49.35821%

[0.016443 0.016237 0.044648 - 0.016443 0.044459 0.030444 CL 1.64426%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1001 and quasi-random dice

Play: 0-ply cubeful [expert]

Cube: 0-ply cubeful [expert]

4. Rollout 24/20 13/10 MWC: 47.79865% ( -5.15886%)

0.477987 0.192692 0.013348 - 0.522013 0.197415 0.004783 CL 47.79865%

[0.013975 0.026992 0.004838 - 0.013975 0.029453 0.004229 CL 1.39751%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1001 and quasi-random dice

Play: 0-ply cubeful [expert]

Cube: 0-ply cubeful [expert]

TRIAL 2:

1. Rollout 24/20 13/10 MWC: 51.20046%

0.512005 0.231598 0.017548 - 0.487995 0.180576 0.029720 CL 51.20046%

[0.016590 0.033418 0.006676 - 0.016590 0.040675 0.019235 CL 1.65898%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1002 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 24/21 24/20 MWC: 50.31250% ( -0.88795%)

0.503125 0.194302 0.022222 - 0.496875 0.224339 0.015549 CL 50.31250%

[0.013790 0.036184 0.014045 - 0.013790 0.043850 0.008972 CL 1.37901%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1002 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 13/10 13/9 MWC: 49.76306% ( -1.43740%)

0.497631 0.202245 0.035355 - 0.502369 0.187200 0.028584 CL 49.76306%

[0.015014 0.023659 0.013995 - 0.015014 0.032470 0.014258 CL 1.50137%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1002 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 24/21 13/9 MWC: 46.38186% ( -4.81860%)

0.463819 0.158487 0.007572 - 0.536181 0.224572 0.018994 CL 46.38186%

[0.019712 0.028231 0.002414 - 0.019712 0.030624 0.007794 CL 1.97121%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1002 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

TRIAL 3

1. Rollout 24/20 13/10 MWC: 51.74356%

0.517436 0.219470 0.010864 - 0.482564 0.215634 0.014652 CL 51.74356%

[0.009732 0.029363 0.003438 - 0.009732 0.034611 0.003351 CL 0.97316%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1003 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 24/21 24/20 MWC: 48.52721% ( -3.21636%)

0.485272 0.120482 0.005362 - 0.514728 0.188538 0.027620 CL 48.52721%

[0.014731 0.021159 0.016000 - 0.014731 0.029449 0.020416 CL 1.47307%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1003 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 13/10 13/9 MWC: 48.49795% ( -3.24562%)

0.484980 0.234772 0.019872 - 0.515020 0.184086 0.034069 CL 48.49795%

[0.016229 0.027737 0.011000 - 0.016229 0.025031 0.021755 CL 1.62291%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1003 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 24/21 13/9 MWC: 47.95841% ( -3.78515%)

0.479584 0.211532 0.017623 - 0.520416 0.173770 0.100386 CL 47.95841%

[0.020933 0.029004 0.009451 - 0.020933 0.033690 0.037567 CL 2.09334%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1003 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

TRIAL 4:

1. Rollout 24/21 24/20 MWC: 49.95757%

0.499576 0.190373 0.016197 - 0.500424 0.201848 0.042939 CL 49.95757%

[0.014960 0.039316 0.012841 - 0.014960 0.041583 0.021117 CL 1.49597%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1004 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 24/21 13/9 MWC: 48.73322% ( -1.22435%)

0.487332 0.177915 0.015520 - 0.512668 0.181516 0.011548 CL 48.73322%

[0.014974 0.030841 0.007650 - 0.014974 0.030047 0.002876 CL 1.49743%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1004 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 24/20 13/10 MWC: 48.56235% ( -1.39522%)

0.485624 0.153333 0.018710 - 0.514377 0.183034 0.035163 CL 48.56235%

[0.014709 0.015213 0.009293 - 0.014709 0.031900 0.018418 CL 1.47093%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1004 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 13/10 13/9 MWC: 47.54710% ( -2.41046%)

0.475471 0.211421 0.027845 - 0.524529 0.164655 0.014633 CL 47.54710%

[0.013492 0.032068 0.012056 - 0.013492 0.028719 0.004206 CL 1.34920%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1004 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

TRIAL 5

1. Rollout 13/10 13/9 MWC: 50.53792%

0.505379 0.195151 0.013812 - 0.494621 0.159383 0.016757 CL 50.53792%

[0.013477 0.033424 0.004588 - 0.013477 0.016740 0.004290 CL 1.34767%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1005 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 24/21 24/20 MWC: 50.33244% ( -0.20548%)

0.503324 0.146246 0.000220 - 0.496676 0.170008 0.025661 CL 50.33244%

[0.018416 0.019323 0.006951 - 0.018416 0.031462 0.011489 CL 1.84161%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1005 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 24/20 13/10 MWC: 49.20405% ( -1.33387%)

0.492040 0.179669 0.024168 - 0.507959 0.163036 0.009633 CL 49.20405%

[0.011450 0.017110 0.011390 - 0.011450 0.024236 0.004184 CL 1.14496%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1005 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 24/21 13/9 MWC: 48.93348% ( -1.60444%)

0.489335 0.183563 0.027652 - 0.510665 0.169076 0.081644 CL 48.93348%

[0.017593 0.024011 0.015085 - 0.017593 0.023050 0.039671 CL 1.75935%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1005 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

TRIAL 6

1. Rollout 24/20 13/10 MWC: 51.55786%

0.515579 0.175873 0.018704 - 0.484421 0.195321 0.016035 CL 51.55786%

[0.010906 0.034317 0.004450 - 0.010906 0.028165 0.005684 CL 1.09061%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1006 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 24/21 13/9 MWC: 48.87860% ( -2.67926%)

0.488786 0.172486 0.030511 - 0.511214 0.232973 0.007710 CL 48.87860%

[0.017375 0.020993 0.016326 - 0.017375 0.038912 0.002758 CL 1.73749%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1006 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 24/21 24/20 MWC: 48.84139% ( -2.71647%)

0.488414 0.161767 0.020807 - 0.511586 0.174350 0.033535 CL 48.84139%

[0.013985 0.016853 0.009813 - 0.013985 0.030162 0.020188 CL 1.39850%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1006 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 13/10 13/9 MWC: 48.33748% ( -3.22038%)

0.483375 0.199921 0.017772 - 0.516625 0.176265 0.024971 CL 48.33748%

[0.013695 0.035514 0.006196 - 0.013695 0.018455 0.015228 CL 1.36945%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1006 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

TRIAL 7

1. Rollout 24/20 13/10 MWC: 52.24508%

0.522451 0.169370 0.016774 - 0.477549 0.132479 0.027845 CL 52.24508%

[0.015869 0.027308 0.008093 - 0.015869 0.030544 0.026418 CL 1.58686%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1007 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 24/21 13/9 MWC: 49.94184% ( -2.30324%)

0.499418 0.185959 0.013560 - 0.500582 0.182281 0.021774 CL 49.94184%

[0.011617 0.024834 0.002760 - 0.011617 0.036260 0.017960 CL 1.16166%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1007 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 13/10 13/9 MWC: 49.21325% ( -3.03183%)

0.492133 0.191860 0.057763 - 0.507867 0.224097 0.034669 CL 49.21325%

[0.016455 0.017017 0.031033 - 0.016455 0.032084 0.019254 CL 1.64546%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1007 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 24/21 24/20 MWC: 48.48432% ( -3.76075%)

0.484843 0.144092 0.007047 - 0.515157 0.252245 0.020044 CL 48.48432%

[0.012696 0.020004 0.002932 - 0.012696 0.043277 0.011506 CL 1.26956%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1007 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

TRIAL 8:

1. Rollout 24/21 13/9 MWC: 51.22857%

0.512286 0.239896 0.054781 - 0.487714 0.198453 0.009521 CL 51.22857%

[0.019720 0.036073 0.025216 - 0.019720 0.024756 0.002505 CL 1.97201%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1008 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 24/21 24/20 MWC: 49.29839% ( -1.93018%)

0.492984 0.175928 0.014800 - 0.507016 0.206913 0.029461 CL 49.29839%

[0.016694 0.023512 0.004648 - 0.016694 0.040519 0.021190 CL 1.66943%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1008 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 13/10 13/9 MWC: 49.14498% ( -2.08359%)

0.491450 0.222063 0.051999 - 0.508550 0.186588 0.039035 CL 49.14498%

[0.017673 0.034640 0.025479 - 0.017673 0.046306 0.020006 CL 1.76733%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1008 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 24/20 13/10 MWC: 45.93947% ( -5.28910%)

0.459395 0.265624 0.035559 - 0.540605 0.220513 0.007618 CL 45.93947%

[0.018745 0.044928 0.014111 - 0.018745 0.045739 0.002149 CL 1.87451%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1008 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

TRIAL 9

1. Rollout 13/10 13/9 MWC: 53.76899%

0.537690 0.210272 0.016385 - 0.462310 0.194779 0.034684 CL 53.76899%

[0.018910 0.034226 0.006905 - 0.018910 0.026063 0.016077 CL 1.89096%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1009 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 24/21 24/20 MWC: 50.63165% ( -3.13734%)

0.506316 0.151956 0.011623 - 0.493684 0.170446 0.059804 CL 50.63165%

[0.010597 0.020586 0.003604 - 0.010597 0.025822 0.037315 CL 1.05972%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1009 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 24/21 13/9 MWC: 50.57812% ( -3.19087%)

0.505781 0.180818 0.057147 - 0.494219 0.189510 0.007240 CL 50.57812%

[0.011022 0.032897 0.035126 - 0.011022 0.039200 0.002953 CL 1.10224%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1009 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 24/20 13/10 MWC: 49.67561% ( -4.09338%)

0.496756 0.213567 0.034037 - 0.503244 0.161789 0.039951 CL 49.67561%

[0.015387 0.026449 0.018952 - 0.015387 0.017776 0.027432 CL 1.53868%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1009 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

TRIAL 10:

1. Rollout 24/21 13/9 MWC: 50.32450%

0.503245 0.230134 0.042749 - 0.496755 0.192765 0.016174 CL 50.32450%

[0.013702 0.037430 0.030328 - 0.013702 0.031467 0.009825 CL 1.37024%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1010 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 13/10 13/9 MWC: 49.39408% ( -0.93042%)

0.493941 0.215946 0.047190 - 0.506059 0.213947 0.016502 CL 49.39408%

[0.012905 0.023111 0.034783 - 0.012905 0.033140 0.005861 CL 1.29048%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1010 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 24/21 24/20 MWC: 47.89820% ( -2.42630%)

0.478982 0.172589 0.016491 - 0.521018 0.223219 0.000500 CL 47.89820%

[0.020168 0.058093 0.008905 - 0.020168 0.038084 0.014034 CL 2.01683%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1010 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 24/20 13/10 MWC: 47.68439% ( -2.64011%)

0.476844 0.224929 0.008678 - 0.523156 0.185893 0.032969 CL 47.68439%

[0.015538 0.043511 0.008574 - 0.015538 0.035733 0.023127 CL 1.55382%]

Full cubeless rollout with var.redn.

36 games, Mersenne Twister dice gen. with seed 1010 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

LARGE ROLLOUT:

1. Rollout 24/21 13/9 MWC: 50.16566%

0.501657 0.200846 0.034166 - 0.498343 0.191557 0.028966 CL 50.16566%

[0.000981 0.002012 0.001249 - 0.000981 0.002024 0.001144 CL 0.09810%]

Full cubeless rollout with var.redn.

10248 games, Mersenne Twister dice gen. with seed 1011 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

2. Rollout 13/10 13/9 MWC: 49.89588% ( -0.26978%)

0.498959 0.209382 0.035532 - 0.501041 0.204066 0.033546 CL 49.89588%

[0.001046 0.001942 0.001274 - 0.001046 0.002030 0.001186 CL 0.10457%]

Full cubeless rollout with var.redn.

10248 games, Mersenne Twister dice gen. with seed 1011 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

3. Rollout 24/21 24/20 MWC: 49.89251% ( -0.27315%)

0.498925 0.188348 0.030836 - 0.501075 0.180520 0.021722 CL 49.89251%

[0.001060 0.002076 0.001204 - 0.001060 0.001963 0.000967 CL 0.10600%]

Full cubeless rollout with var.redn.

10248 games, Mersenne Twister dice gen. with seed 1011 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

4. Rollout 24/20 13/10 MWC: 49.81109% ( -0.35457%)

0.498111 0.195648 0.031202 - 0.501889 0.192476 0.027695 CL 49.81109%

[0.001007 0.002021 0.001216 - 0.001007 0.002006 0.001130 CL 0.10072%]

Full cubeless rollout with var.redn.

10248 games, Mersenne Twister dice gen. with seed 1011 and quasi-random dice

Play: 0-ply cubeful

Cube: 0-ply cubeful

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