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Subject: Re: The law of diminishing returns

Author: Andrew Dados

Date: 13:07:54 07/12/02

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On July 12, 2002 at 14:56:11, Ed Schröder wrote:

>Hi CCC,
>
>In Rebel I maintain a statistic file, on every iteration a counter is
>incremented with 1 (see column 2) representing the iteration depths Rebel has
>searched. When a new best move is found a second counter is incremented with 1
>(see column 3) representing how many times a new best move has been found on the
>given iteration depth, between brackets the percentage is calculated.
>
>As you can see the very first plies Rebel often changes to new best moves,
>however when the depth increases and increases the chance Rebel will change its
>mind drops and drops. From 16 plies on the chance a new better move is found is
>below 2%.
>
>I wonder what this all means, it is still said (and believed by many) that a
>doubling in computer speed gives 30-50-70 elo. That could be very well true for
>lower depths but the below statistic seem to imply something totally different,
>a sharp diminishing return on deeper depths.
>
>Interesting also is colum 4 (Big Score Changes), whenever a big score difference
>is measured (0.50 up or down) the percentage is calculated. This item seems to
>be less sensitive than the change in best move. However the maintained "Big
>Score Changes" statistic is not fully reliable as it also counts situations like
>being a rook or queen up (or down) in positions and naturally you get (too) many
>big score fluctuations. I have changed that and have limit the system to scores
>in the range of -2.50 / +2.50 but for the moment have too few games played to
>show the new statistic.
>
>Anyway the number of positions calculated seem to be more than sufficient (over
>100,000) to be reliable. The origin came from extensive testing the latest Rebel
>via self-play at various time controls.
>
>
>                 SEARCH OVERVIEW
>                 ===============
>
>Depth    Moves          Moves         Big Score
>       Searched        Changed         Changes
>
> 1     113768         0 =  0.0%        0 =  0.0%
> 2     113768     44241 = 38.9%    25329 = 22.3%
> 3     113768     34262 = 30.1%    10707 =  9.4%
> 4     113194     32619 = 28.8%     6939 =  6.1%
> 5     113191     30697 = 27.1%     6353 =  5.6%
> 6     108633     28516 = 26.2%     4954 =  4.6%
> 7     108180     25437 = 23.5%     3952 =  3.7%
> 8     102782     22417 = 21.8%     3265 =  3.2%
> 9      82629     15400 = 18.6%     2541 =  3.1%
>10      59032      9144 = 15.5%     1866 =  3.2%
>11      39340      5183 = 13.2%     1384 =  3.5%
>12      23496      2350 = 10.0%      982 =  4.2%
>13      12692       957 =  7.5%      522 =  4.1%
>14       6911       396 =  5.7%      369 =  5.3%
>15       4032       193 =  4.8%      179 =  4.4%
>16       2471        72 =  2.9%      127 =  5.1%
>17       1608        26 =  1.6%       58 =  3.6%
>18       1138        17 =  1.5%       42 =  3.7%
>19        921         6 =  0.7%       26 =  2.8%
>20        795         7 =  0.9%       22 =  2.8%
>21        711         1 =  0.1%       18 =  2.5%
>22        636         2 =  0.3%       10 =  1.6%
>23        574         5 =  0.9%       10 =  1.7%
>24        507         1 =  0.2%       12 =  2.4%
>25        451         3 =  0.7%        7 =  1.6%
>26        394         1 =  0.3%       10 =  2.5%
>27        343         2 =  0.6%        3 =  0.9%
>28        296         2 =  0.7%        7 =  2.4%
>29        269         0 =  0.0%        7 =  2.6%
>30        240         0 =  0.0%        6 =  2.5%
>31        217         0 =  0.0%        6 =  2.8%
>32        208         0 =  0.0%        7 =  3.4%
>33        187         0 =  0.0%        3 =  1.6%
>34        169         0 =  0.0%        1 =  0.6%
>35        159         0 =  0.0%        1 =  0.6%
>36        144         0 =  0.0%        0 =  0.0%
>37        138         0 =  0.0%        0 =  0.0%
>38        134         0 =  0.0%        0 =  0.0%
>39        125         0 =  0.0%        0 =  0.0%
>40        117         0 =  0.0%        0 =  0.0%
>41        105         0 =  0.0%        0 =  0.0%
>42         97         0 =  0.0%        0 =  0.0%
>43         83         0 =  0.0%        0 =  0.0%
>44         73         0 =  0.0%        0 =  0.0%
>45         67         0 =  0.0%        0 =  0.0%
>46         62         0 =  0.0%        0 =  0.0%
>47         55         0 =  0.0%        0 =  0.0%
>48         51         0 =  0.0%        0 =  0.0%
>49         49         0 =  0.0%        0 =  0.0%
>50         47         0 =  0.0%        0 =  0.0%
>51         46         0 =  0.0%        0 =  0.0%
>52         46         0 =  0.0%        0 =  0.0%
>53         44         0 =  0.0%        0 =  0.0%
>54         43         0 =  0.0%        0 =  0.0%
>55         43         0 =  0.0%        0 =  0.0%
>56         42         0 =  0.0%        0 =  0.0%
>57         41         0 =  0.0%        0 =  0.0%
>58         40         0 =  0.0%        0 =  0.0%
>59         40         0 =  0.0%        0 =  0.0%
>60         38         0 =  0.0%        0 =  0.0%

Thanks for that data.

There are few things that come to mind here...

First is endgame vs middlegame data - I believe since BF is drastically
different those 2 stages will produce different curves.

Then there is this effect of searching MUCH deeper when there is an easy move:
recapture or check evasion or simple forced move. Those moves will get 2+ plies
deeper in search and affect your statistic greatly: best move won't change.
(unless you auto-played with some easy-move algorithm and terminated search
sooner)

On the other hand in really complicated positions with big BF you are getting to
smaller depths. And those positions are really interesting.

I believe chess game consist of mostly 'easy' moves. That 2-3 'hard' moves in a
game make all the difference though.

-Andrew-



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