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Posted By: Nack Ballard
Date: Friday, 23 April 2010, at 9:03 a.m.

In Response To: 21\$-54 rollout data (equities, links) (Daniel Murphy)

I'd like to see some discussion of Tim's idea of averaging absolute equities (from meaningful rollouts by world class bots?) instead of averaging relative equities, and wonder what difference it makes.

For these rollouts I get

((10k*0.177) + (46k*0.1368) + (10k*0.175) + (10k*0.176) + (10k*0.167) + (10k*0.165) + (103k*0.158) + (19k*0.152))/218k = 0.156
and
((10k*0.165) + (46k*0.1340) + (10k*0.137) + (10k*0.136) + (10k*0.140) + (10k*0.149) + (103k*0.143) + (19k*0.145))/218k = 0.143
...
For a relative difference of 0.013

which ... what do you get by averaging the relative equities, 0.015 or 0.017? That is, no difference, considering that the SD of the relative equity in the longest of the rollouts is ~0.002.

If you average the absolute equities and subtract one total from the other, this should be the same result you get when you average the relative equities.

Below are the details for the 21\$-54 scenario so that you can verify. (I've translated the "k" numbers into multiples of 1296, rather than deal with five- and six-digit iteration numbers (like 103680, which is what 103k actually is).

(8*.0177) + (36*.01368) + (8*.0175) + (8*.0176) + (8*.0167) + (8*.0165) + (80*.0158) + (15*.0152) / 171 = 0.1562853
and
(8*.0165) + (36*.01340) + (8*.0137) + (8*.0136) + (8*.0140) + (8*.0149) + (80*.0145) + (15*.0145) / 171 = 0.1427777
...
So, the margin of absolute equity averages = 0.1562853 - 0.1427777 = 0.0135076 = 13.5076 thousandths.

Now we'll try it with relative equities, which I'll list in thousandths.

(8*12) + (36*2.8) + (8*38) + (8*40) + (8*27) + (8*16) + (80*13) + (15*7) / 171 = 13.5076. A lot simpler, yes?

There were two (or three) problems with your calculation, the main one being that the absolute equity of the R play in the second-to-last rollout is 0.145 (not 0.143).

Also, while admittedly not a big deal, a relative equity is slightly more accurate than a difference in absolute equities. An example is the third-to-last rollout. The absolute equities are listed as 0.165 and 0.149, but the margin is listed as 0.017. This means that the margin is closer to 0.017 than to 0.016 (even though I used the latter in my calculation to match yours). The fairest guess is actually 0.01675.

I think that Chase's no-Jacoby rollout should be omitted, as it was rerolled with the same seed on the same machine with the same settings (other than Jacoby). I'm inclined to think that Neil's reroll with seed 2 should be included (though I might exclude it later, depending on what we discover).

Removing the no-Jacoby doppelganger and using the slightly more accurate relative equities reduces the average to

(8*12) + (36*2.7) + (8*38) + (8*27) + (8*17) + (80*13) + (15*7) / 163 = 12.2.

Finally, I don't think all trials should be treated equally. Taking an extreme example, if you have [S R10] 400k from Bot A, and [S R16] 200k from Bot B, and you trust the bots equally, I think it is better to call the average [S R13] than [S R12]. Part of the consideration is that bots are going to converge to different margins if rolled out infinitely, and that effects exists to some degree even in short rollouts.

There's also the issue that it is not yet proven which of Snowie, Gnu and XG is the strongest bot. Moreover, each may have its areas of strength, even within the arena of early game positions.

Rather than assign a sophisticated series of weights (which I will do for a work less updatable like a book), for this forum I tend to do a straight average of each bot's average, then intuitively bump/round towards the bot having the largest number of trials and/or towards 4-ply. It saves a lot of time.

Here, Snowie is 12, Gnu is 2.7 and XG is (8*(38+27+17) + 80*13 + 15*7) / 119 = 15+. Averaging 12, 2.7 and 15+ and favoring the bot with most of the trials some but then shading it back towards the (narrow) 4-ply result yields a margin closest, I'd say, to 11 (thousandths).

It is possible that the Gnu rollout is corrupt (as I mentioned in the previous post), and if so that might raise 21\$-54 (S vs R) value by 2+ or so. If one of the other XG seed-2 rollouts turns out to be corrupt, that will lower value by 1+ or so.

Nack

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