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Subject: Re: PB-ON vs PB-OFF (final results)

Author: Ratko V Tomic

Date: 19:05:55 10/15/99

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>>If the guess rate is 50% (G=0.5) the Cn needs twice as much time
>> allocated. But with two Rebels,
>>G is probably 80% or greater,
>
> If you have any evidence to support this, please show it.  Otherwise,
> it is simply a guess.

When I wrote the above I judged/guessed that the pondering Rebel would predict
at least 80% of the moves of the non-pondering Rebel. But the results Ed gave
for the two matches is enough to compute this figure (within statistical error).

First we compute the rating difference for the two matches using the formula:

  D = 400*log(W/L)  where W=wins, L=losses

For the match 1 and 2 we get D1=78 and D2=49. If we denote the rating gain for
doubling the speed of program (or doubling the thinking time) as C, then the
rating difference corresponding to the thinking times ratio R will be:

  D = C * (R-1)   (eq-1)

If you set the time ratio R=2 (i.e. doubling the thinking time), the rating
difference is C (agreeing with the definition of C). If the 2 program instances
have equal time, i.e. R=1 the rating difference is 0, as we would expect.

Now we define Tp and Tn as times per move (on average) alloted (in the time
control settings) to the pondering Rebel (Cp) and to non-pondering Rebel (Cn)
and denote as G the guess rate, so that G=1 means Cp has guessed 100% of Cn's
moves. The total time/move Cp got for useful thinking is then:

   UTp=Tp+G*Tn   (eq-2)

since it will get all the time spent by Cn whenever it guesses the Cn's move.
The Cn has only the time alloted to it by the time control, hence UTn=Tn. The
time ratio R in (eq-1) is UTn/UTp. Applying this to the two matches and using
the fact that in the 1st match Tn=Tp and in the 2nd match Tn=2*Tp, we get the
following 4 equations:

  1st match:   R1 = 1+G1     78 = C * (R1-1)
  2nd match:   R2 = 0.5+G2   49 = C * (R2-1)

for the 5 unknowns, C, G1, G2, R1 and R2, where R1 and R2 are thinking time
ratios for the match 1 and match 2, and G1 and G2 guess rates for the two
matches. Note that due to different time control ratios in match 1 and 2 we
don't assume that guess ratios are same in the two matches. If we take that
doubling the Rebel speed gains it 100 rating points (this is a Newborn's figure)
against the slower Rebel, i.e. C=100, we get G1=78/C=0.78 which is quite close
to my guess of 0.8.

The 2nd match, as Ed noted, produced the unexpectedly large performance
difference. Using the above equations for the 2nd match gives G2=0.99, i.e. as
if pondering Rebel predicted 99% of the moves of the non-pondering (giving it
thus the large performance gain). If, on the other hand, we assume G2=G1=0.8
then the expected rating difference would be 30 ELO points (instead of the 49
ELO points in the match), i.e. the expected match result should have been 54 to
46 for the pondering Rebel. Given that the expected statistical uncertainty is
+/-7 points (i.e. sqrt(50)), the actual result (57:43) is well within the
statistical uncertainty for the given number of games, and therefore fairly
consistent with the 80 percent guess rate by the pondering Rebel.




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