Author: Christophe Theron

Date: 12:45:53 07/03/01

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On July 03, 2001 at 15:08:42, Christoph Fieberg wrote: >Description of the program I developed for simulating the result of a chess game >of two rated players. > >The result of the game depends on the performance factor and a draw factor. > >1.) Performance factor > >The performance factor depends on the rating difference and on the advantage >playing the white pieces. > >In order to determine the performance factor I analyzed nearly 100.000 games of >games with both players 2500+. The distribution for 1-0, Draw, 0-1 was (rounded) >31%-50%-19%. White had a success-rate of 56% (= 31 + 50/2) and Black of 44%. [I >would be happy if this distribution could be validated by others. It has to be >noted that this is an average distribution for players 2500+. On the top level >the success-rate is lower due to a higher number of draws.] > >The average opponent for White and Black had the same rating and therefore the >advantage playing white (if both players have the same strength) is 12%-points >(56-44) what can be expressed as 42 Elo-points (see formula below). Thus, for >calculation purposes White has to be added 21 Elo and Black substracted 21 Elo >in order to consider the advantage playing white. > >The following formula (I think used by FIDE) allows to calculate the performance >based on Elo-difference and vice versa. It is: > >Performance (%) = (1 / (1 + (10 ^ (-difference / 400)))) > >Example for calculating the likely performce of a 2600 player playing white >against a 2550 player: > >Rating 2600 – 2500; Advantage White => 2621 – 2479; Difference = 142; => 69.37%. >This means in 100 such games the 2600-player (who always has white) most >probably will win 69.5 - 30.5. [Note that in case that it is a match of 100 >games with changing colours there would be no white advantage and thus the >difference is only 100 => 64.01%. For specialists: Here the white advantage is >10.72%-points instead of 12%-points. This implies that according to the formula >it is as hard to get from 64.01% success rate to 69.37% as it is from 50% to >56%.] > > >2.) Draw factor > >The draw rate has to be seen in relation on the maximum number of draws >depending on the success rate and not in relation on the number of games played >(this is a important point which seems to be totally neglected in draw rate >considerations). > >Example: If there is a success rate of 90%, a player will get 90 points out of >100. It is clear that the maximum number of draws can only be 20 (80 wins, 20 >draws, 0 losses => 90 points). If there are really 20 draws the draw rate (here >depending on the success rate of 90%) therefore is 100% and not 20%. If a 90% >success rate results in 85 wins, 10 draws and 5 losses the draw rate would be >50%. > >The average draw rate based on the nearly 100.000 games of 2500+ players can be >calculated according to the following formula: > >Draw rate = Percentage of draws / ((100% - success-rate) * 2) > >As described above there were a success-rate of 56% and a 50% were draws. >=> draw rate = 50% / ((100% - 56%) * 2) = 56.82% > >This average draw rate of 56.82% now allows to calculate the distribution for >the result of a single game. > >Example: I will calculate the result of the game 2600 – 2500. The success-rate >taken into account the white advantage is 69.37% (see above). The average draw >rate is 56.82%. > >Therefore percentage of draws is (1 – (ABS(0.6937-0.5) * 2)) * 0.5682 = 0.3481 = >34.81% and the percentage of wins is (0.6937 – (0.3481 / 2)) = 0.5197 = 51.97% > >This means that we can expect in 1000 games from the 2600 player about > >520 wins / 348 draws / 132 losses (distribution 52% - 34.8% - 13.2%) > >[Check: (520 + 348 / 2) / 1000 = 69.4% success-rate; maximum number of draws = >612 (distribution 388 – 612 – 0) => draw rate = 348 / 612 = 0.5686 = 56.86%] > > >3.) Calculation the result with random numbers > >If the distribution for the result of a single games is known it is relatively >easy to simulate the outcome. In the example the distribution for a game 2600 – >2500 is > >52% for a win, 34.8% for a draw and 13.2% for a loss. > >(The calculation depended on the average success-rate of 56%, the average >advantage for the white player of 42 Elo-points and the average draw rate of >56.82% as shown above). > >The simulation-program creates a random number between 0 and 1 (e.g. 0.446). >In case the number is >= 0.48 (likelihood 52% as for a win) the 2600 player >wins. >In case the number is > 0.132 and < 0.48 (likelihood 34.8% as for a draw) the >game is a draw. >In case the number is <= 0.132 (likelihood 13.2% as for a loss) the 2600 player >looses. >As example the random number is 0.446 what means that the result of the game is >a draw. > > >4.) Simulation of tournaments and matches > >With the simulation of the result of a single game it is possible to calculte >whole tournaments and matches (the only difficulty is to program it). > >In respect of matches it has to be regarded that the main factors (average >success-rate, average white advantage, average draw rate) are average factors >and probably have to be adjusted to the level of the opponents. Also for >tournaments there schould be adjustments depending on the category of the >tournament. > >My vision is to find out the average factors for each level (from 2200 to 2850 >step Elo 50) and in another step to pre-calculate all distributions and to fill >a huge table for any possible game (where for example also the distribution of >2600 – 2500 = 52.0% / 34.8% / 13.2% would be stored). The speed of the >simulation would be enourmously accelerated. > > >I look forward to your comments. > >Best regards, >Christoph Very interesting, I hope you go on and publish your tables and the program used to generate them. Christophe

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