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Subject: Re: The law of diminishing returns

Author: Mark Young

Date: 21:24:49 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%.

A few points to think about.

1.) Positions were generated using self play. Meaning most games were likely
open positions were programs already play "like gods". Meaning it is much easier
to find the best move quickly at lower depths.

2.) The stronger the player the more depth is needed to see into the tactics of
the position. Meaning games played with 2200 rated players will see faster
diminishing returns then games played with a 2500 rated player.

3.) Closed positions will (or should) show slower diminishing returns.

4.) a 2% change is still enough to make a great change in a programs elo.
Meaning 1 change in 50 moves played. This could mean winning or drawing many
games that would other wise be lost to strong players. This could greatly
increase the programs elo in the long term.


>
>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%



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