SPRT and narrowing of (elo1 - elo0) difference.

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Ajedrecista
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SPRT and narrowing of (elo1 - elo0) difference.

Post by Ajedrecista » Sat Oct 05, 2013 3:43 pm

Hello:

I toyed a little with elo0 (bayeselo_0) and elo1 (bayeselo_1) parameters of SPRT with my own SPRT simulator. I choosed the special case of elo0 = -elo1 and the worst case inside it: elo (bayeselo) = (elo0 + elo1)/2 = 0.

Code: Select all

alpha = beta = 0.05; 2000 simulations each time; bayeselo_fixed = 0.

--------------------------------------------------------------------

drawelo_fixed = 240:

bayeselo_0    bayeselo_1    <Games>/simulation              Shortest run                        Longest run

   -5            5                 10452               619   (+98  -163  =358&#41;         65633  (+13227  -13162  =39244&#41;
   -4.75         4.75              11250               781  (+197  -129  =455&#41;         74788  (+15043  -14975  =44770&#41;
   -4.5          4.5               12746               978  (+166  -238  =574&#41;         88990  (+17776  -17848  =53366&#41;
   -4.25         4.25              13916              1031  (+240  -165  =626&#41;         83352  (+16836  -16912  =49604&#41;
   -4            4                 15932              1364  (+316  -236  =812&#41;         88576  (+17830  -17911  =52835&#41;
   -3.75         3.75              18161              1510  (+342  -257  =911&#41;        110664  (+22325  -22239  =66100&#41;
   -3.5          3.5               21176              1566  (+269  -361  =936&#41;        162743  (+32579  -32671  =97493&#41;
   -3.25         3.25              23754              1961  (+447  -348 =1166&#41;        147703  (+29700  -29601  =88402&#41;

   -3            3                 28256              1870  (+404  -299 =1167&#41;        189855  (+38018  -38125 =113712&#41;

   -2.75         2.75              34418              2331  (+415  -532 =1384&#41;        240724  (+48381  -48498 =143845&#41;
   -2.5          2.5               40938              3020  (+676  -547 =1797&#41;        235075  (+47393  -47264 =140418&#41;
   -2.25         2.25              51968              4133  (+897  -755 =2481&#41;        315599  (+63554  -63697 =188348&#41;
   -2            2                 65912              5002 (+1108  -947 =2947&#41;        348841  (+70042  -69882 =208917&#41;
   -1.75         1.75              84304              6395 (+1184 -1367 =3844&#41;        571957 (+114802 -114985 =342170&#41;
   -1.5          1.5              116123             10039 (+2095 -1882 =6062&#41;        859506 (+171890 -172104 =515512&#41;

**********************************************************************************************************************

drawelo_fixed = 270&#58;

bayeselo_0    bayeselo_1    <Games>/simulation              Shortest run                        Longest run

   -5            5                 10981               851  (+179  -117  =555&#41;         71265  (+12609  -12546  =46110&#41;
   -4.75         4.75              12456               922  (+184  -119  =619&#41;         69056  (+12084  -12018  =44954&#41;
   -4.5          4.5               13897               831  (+113  -182  =536&#41;         71696  (+12627  -12558  =46511&#41;
   -4.25         4.25              14979              1093  (+170  -245  =678&#41;         79323  (+13787  -13714  =51822&#41;
   -4            4                 17130              1218  (+239  -162  =817&#41;        109297  (+18996  -19074  =71227&#41;
   -3.75         3.75              19609              1284  (+181  -264  =839&#41;        190861  (+33502  -33419 =123940&#41;
   -3.5          3.5               21761              1754  (+267  -356 =1131&#41;        134556  (+23532  -23621  =87403&#41;
   -3.25         3.25              25943              2043  (+408  -312 =1323&#41;        191905  (+33695  -33599 =124611&#41;

   -3            3                 30329              1611  (+235  -339 =1037&#41;        172176  (+30231  -30127 =111818&#41;

   -2.75         2.75              35943              2321  (+351  -464 =1506&#41;        225119  (+39352  -39239 =146528&#41;
   -2.5          2.5               43468              3835  (+592  -716 =2527&#41;        302031  (+52796  -52921 =196314&#41;
   -2.25         2.25              54740              2535  (+379  -517 =1639&#41;        304272  (+53063  -52925 =198284&#41;
   -2            2                 70596              4213  (+811  -656 =2746&#41;        465446  (+81160  -81005 =303281&#41;
   -1.75         1.75              91756              6607 (+1246 -1068 =4293&#41;        767735 (+134078 -134256 =499401&#41;
   -1.5          1.5              125512              9500 (+1562 -1769 =6169&#41;        778865 (+136041 -136248 =506576&#41;
I find slight similarities in the number (bayeselo_1 - bayeselo_0)*sqrt(<Games>/simulation) for each group of simulations with a fixed drawelo parameter. A rough approximation could be that reducing elo1 - elo0 by 2 means multiplying by 4 the average number of games before stop by LLR criteria. What do you think?

It is needless to say that my simulator is far from perfect (I wrote the shortest and longest runs of each simulation with the aim of let people decide the accuracy of my simulator) and other people should verify my simulated numbers.

Regards from Spain.

Ajedrecista.

Michel
Posts: 2038
Joined: Sun Sep 28, 2008 11:50 pm

Re: SPRT and narrowing of (elo1 - elo0) difference.

Post by Michel » Sat Oct 05, 2013 7:32 pm

Your conjecture is correct.

There are easy approximate formulas for the expected duration of an SPRT. See for example the function "sprt_characteristics" in this script

http://hardy.uhasselt.be/Toga/2sprt.py

If you do a Taylor series expansion of numerator and denominator around elo=(elo0+elo1)/2 you see that the expected duration in that case is proportional to 1/(elo1-elo0)^2.

Michel
Posts: 2038
Joined: Sun Sep 28, 2008 11:50 pm

Re: SPRT and narrowing of (elo1 - elo0) difference.

Post by Michel » Sun Oct 06, 2013 3:51 am

Actually here is a simple formula for the worst case (elo=(elo0+elo1)/2) expected duration of an SPRT.

The formula is of the form

Code: Select all

C/&#40;elo1-elo0&#41;^2
with

Code: Select all

C=AB/I
where

Code: Select all

A=ln&#40;&#40;1-alpha&#41;/beta&#41;
B=ln&#40;&#40;1-beta&#41;/alpha&#41;
I=2*bb*bb*&#40;1-Ld&#41;*Ld*Ld
where

Code: Select all

Ld=1/&#40;1+10^&#40;draw_elo/400&#41;)
bb=ln&#40;10&#41;/400
Example:

Code: Select all

If alpha=beta=0.05 and draw_elo=240 then

C=1020075     &#40;roughly 1,000,000!)

For a BayesElo difference of 6 this gives a worst case expected duration of

C/36=28335

Michel
Posts: 2038
Joined: Sun Sep 28, 2008 11:50 pm

Re: SPRT and narrowing of (elo1 - elo0) difference.

Post by Michel » Sun Oct 06, 2013 9:03 am

If you do a Taylor series expansion of numerator and denominator around elo=(elo0+elo1)/2 you see that the expected duration in that case is proportional to 1/(elo1-elo0)^2.
Actually the formula implies that the expected duration of an SPRT is always proportional to 1/(elo1-elo0)^2 as long as you keep the ratio (elo-elo0)/(elo1-elo0) constant. This applies for example to H0 (ratio=0), H1 (ratio=1) and worst case (ratio=1/2).

User avatar
Ajedrecista
Posts: 1396
Joined: Wed Jul 13, 2011 7:04 pm
Location: Madrid, Spain.
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SPRT: some distributions of the length of tests.

Post by Ajedrecista » Sat Dec 07, 2013 6:40 pm

Hello:

I modified my SPRT simulator: it can display now the distribution of the length of simulations in an output Notepad:

Code: Select all

alpha&#58;            0.0500
beta&#58;             0.0500

drawelo_fixed&#58;  240.0000

bayeselo_0&#58;      -1.5000
bayeselo_1&#58;       4.5000

bayeselo_fixed&#58;   1.5000
Elo&#58;              0.9627

Simulations&#58;  10000
 
========================

    1/10000     Passes&#58;     1     Fails&#58;     0     <Games>/simulation&#58;   75982
    2/10000     Passes&#58;     1     Fails&#58;     1     <Games>/simulation&#58;   60804
    3/10000     Passes&#58;     1     Fails&#58;     2     <Games>/simulation&#58;   41399
    4/10000     Passes&#58;     2     Fails&#58;     2     <Games>/simulation&#58;   33734
    5/10000     Passes&#58;     2     Fails&#58;     3     <Games>/simulation&#58;   34209
    6/10000     Passes&#58;     3     Fails&#58;     3     <Games>/simulation&#58;   34620

&#91;...&#93;

 9995/10000     Passes&#58;  4994     Fails&#58;  5001     <Games>/simulation&#58;   28597
 9996/10000     Passes&#58;  4994     Fails&#58;  5002     <Games>/simulation&#58;   28597
 9997/10000     Passes&#58;  4995     Fails&#58;  5002     <Games>/simulation&#58;   28602
 9998/10000     Passes&#58;  4995     Fails&#58;  5003     <Games>/simulation&#58;   28602
 9999/10000     Passes&#58;  4995     Fails&#58;  5004     <Games>/simulation&#58;   28600
10000/10000     Passes&#58;  4995     Fails&#58;  5005     <Games>/simulation&#58;   28599

Shortest simulation&#58;    1951 games &#40;simulation  3850&#41;.
Longest simulation&#58;   220405 games &#40;simulation  3251&#41;.

Average number of games per simulation&#58;   28599

Type I errors  &#40;false positives&#41;&#58;   0.00 %
Type II errors &#40;false negatives&#41;&#58;  50.05 %

There are  1211 simulations with score > 50% that failed SPRT.
There are    21 simulations with score = 50% that failed SPRT.

Estimated elapsed time&#58;   373.38 seconds.

Speed&#58;  765957 games/second.

Code: Select all

Shortest simulation&#58; 1951 games (+322 -423 =1206&#41; ---> fail.
Longest simulation&#58; 220405 games (+44338 -43623 =132444&#41; ---> pass.

Code: Select all

From    1000 to    1999 games&#58;     2 simulations (  0.02 %); accumulated&#58;   0.02 %.
From    2000 to    2999 games&#58;    40 simulations (  0.40 %); accumulated&#58;   0.42 %.

&#91;...&#93;

From   19000 to   19999 games&#58;   215 simulations (  2.15 %); accumulated&#58;  46.23 %.
From   20000 to   20999 games&#58;   245 simulations (  2.45 %); accumulated&#58;  48.68 %.
From   21000 to   21999 games&#58;   221 simulations (  2.21 %); accumulated&#58;  50.89 %.
From   22000 to   22999 games&#58;   204 simulations (  2.04 %); accumulated&#58;  52.93 %.
From   23000 to   23999 games&#58;   193 simulations (  1.93 %); accumulated&#58;  54.86 %.
From   24000 to   24999 games&#58;   205 simulations (  2.05 %); accumulated&#58;  56.91 %.
From   25000 to   25999 games&#58;   172 simulations (  1.72 %); accumulated&#58;  58.63 %.
From   26000 to   26999 games&#58;   171 simulations (  1.71 %); accumulated&#58;  60.34 %.
From   27000 to   27999 games&#58;   165 simulations (  1.65 %); accumulated&#58;  61.99 %.
From   28000 to   28999 games&#58;   139 simulations (  1.39 %); accumulated&#58;  63.38 %.
From   29000 to   29999 games&#58;   143 simulations (  1.43 %); accumulated&#58;  64.81 %.

&#91;...&#93;

From  127000 to  127999 games&#58;     3 simulations (  0.03 %); accumulated&#58;  99.41 %.
From  128000 to  128999 games&#58;     4 simulations (  0.04 %); accumulated&#58;  99.45 %.

&#91;...&#93;

From  219000 to  219999 games&#58;     0 simulations (  0.00 %); accumulated&#58;  99.99 %.
From  220000 to  220999 games&#58;     1 simulation  (  0.01 %); accumulated&#58; 100.00 %.

Number of finished simulations&#58; 10000.
------------------------

Code: Select all

alpha&#58;            0.0500
beta&#58;             0.0500

drawelo_fixed&#58;  270.0000

bayeselo_0&#58;       0.0000
bayeselo_1&#58;       6.0000

bayeselo_fixed&#58;   3.0000
Elo&#58;              1.7284

Simulations&#58;  10000

========================

    1/10000     Passes&#58;     1     Fails&#58;     0     <Games>/simulation&#58;   56247
    2/10000     Passes&#58;     1     Fails&#58;     1     <Games>/simulation&#58;   59612
    3/10000     Passes&#58;     1     Fails&#58;     2     <Games>/simulation&#58;   41418
    4/10000     Passes&#58;     2     Fails&#58;     2     <Games>/simulation&#58;   33629
    5/10000     Passes&#58;     3     Fails&#58;     2     <Games>/simulation&#58;   30384
    6/10000     Passes&#58;     3     Fails&#58;     3     <Games>/simulation&#58;   27274

&#91;...&#93;

 9995/10000     Passes&#58;  5006     Fails&#58;  4989     <Games>/simulation&#58;   30719
 9996/10000     Passes&#58;  5006     Fails&#58;  4990     <Games>/simulation&#58;   30720
 9997/10000     Passes&#58;  5006     Fails&#58;  4991     <Games>/simulation&#58;   30720
 9998/10000     Passes&#58;  5006     Fails&#58;  4992     <Games>/simulation&#58;   30718
 9999/10000     Passes&#58;  5007     Fails&#58;  4992     <Games>/simulation&#58;   30721
10000/10000     Passes&#58;  5008     Fails&#58;  4992     <Games>/simulation&#58;   30720

Shortest simulation&#58;    1412 games &#40;simulation  5747&#41;.
Longest simulation&#58;   300012 games &#40;simulation  1052&#41;.

Average number of games per simulation&#58;   30720

Type I errors  &#40;false positives&#41;&#58;   0.00 %
Type II errors &#40;false negatives&#41;&#58;  49.92 %

There are  2741 simulations with score > 50% that failed SPRT.
There are    23 simulations with score = 50% that failed SPRT.

Estimated elapsed time&#58;   419.15 seconds.

Speed&#58;  732912 games/second.

Code: Select all

Shortest simulation&#58; 1412 games (+196 -292 =924&#41; ---> fail.
Longest simulation&#58; 300012 games (+53212 -51820 =194980&#41; ---> fail.

Code: Select all

From    1000 to    1999 games&#58;     5 simulations (  0.05 %); accumulated&#58;   0.05 %.
From    2000 to    2999 games&#58;    19 simulations (  0.19 %); accumulated&#58;   0.24 %.

&#91;...&#93;

From   21000 to   21999 games&#58;   201 simulations (  2.01 %); accumulated&#58;  47.15 %.
From   22000 to   22999 games&#58;   215 simulations (  2.15 %); accumulated&#58;  49.30 %.
From   23000 to   23999 games&#58;   196 simulations (  1.96 %); accumulated&#58;  51.26 %.
From   24000 to   24999 games&#58;   205 simulations (  2.05 %); accumulated&#58;  53.31 %.
From   25000 to   25999 games&#58;   169 simulations (  1.69 %); accumulated&#58;  55.00 %.
From   26000 to   26999 games&#58;   177 simulations (  1.77 %); accumulated&#58;  56.77 %.
From   27000 to   27999 games&#58;   175 simulations (  1.75 %); accumulated&#58;  58.52 %.
From   28000 to   28999 games&#58;   172 simulations (  1.72 %); accumulated&#58;  60.24 %.
From   29000 to   29999 games&#58;   157 simulations (  1.57 %); accumulated&#58;  61.81 %.
From   30000 to   30999 games&#58;   150 simulations (  1.50 %); accumulated&#58;  63.31 %.
From   31000 to   31999 games&#58;   164 simulations (  1.64 %); accumulated&#58;  64.95 %.

&#91;...&#93;

From  127000 to  127999 games&#58;     2 simulations (  0.02 %); accumulated&#58;  99.12 %.
From  128000 to  128999 games&#58;     0 simulations (  0.00 %); accumulated&#58;  99.12 %.

&#91;...&#93;

From  299000 to  299999 games&#58;     0 simulations (  0.00 %); accumulated&#58;  99.99 %.
From  300000 to  300999 games&#58;     1 simulation  (  0.01 %); accumulated&#58; 100.00 %.

Number of finished simulations&#58; 10000.
------------------------

Code: Select all

alpha&#58;            0.0500
beta&#58;             0.0500

drawelo_fixed&#58;  270.0000

bayeselo_0&#58;       0.0000
bayeselo_1&#58;       3.0000

bayeselo_fixed&#58;   1.5000
Elo&#58;              0.8642

Simulations&#58;  10000

========================

    1/10000     Passes&#58;     0     Fails&#58;     1     <Games>/simulation&#58;  290883
    2/10000     Passes&#58;     1     Fails&#58;     1     <Games>/simulation&#58;  157972
    3/10000     Passes&#58;     1     Fails&#58;     2     <Games>/simulation&#58;  127151
    4/10000     Passes&#58;     1     Fails&#58;     3     <Games>/simulation&#58;  122171
    5/10000     Passes&#58;     1     Fails&#58;     4     <Games>/simulation&#58;  128136
    6/10000     Passes&#58;     1     Fails&#58;     5     <Games>/simulation&#58;  121786

&#91;...&#93;

 9995/10000     Passes&#58;  5035     Fails&#58;  4960     <Games>/simulation&#58;  122177
 9996/10000     Passes&#58;  5035     Fails&#58;  4961     <Games>/simulation&#58;  122170
 9997/10000     Passes&#58;  5035     Fails&#58;  4962     <Games>/simulation&#58;  122165
 9998/10000     Passes&#58;  5036     Fails&#58;  4962     <Games>/simulation&#58;  122159
 9999/10000     Passes&#58;  5037     Fails&#58;  4962     <Games>/simulation&#58;  122154
10000/10000     Passes&#58;  5037     Fails&#58;  4963     <Games>/simulation&#58;  122159

Shortest simulation&#58;    6475 games &#40;simulation  4217&#41;.
Longest simulation&#58;   933396 games &#40;simulation  2780&#41;.

Average number of games per simulation&#58;  122159

Type I errors  &#40;false positives&#41;&#58;   0.00 %
Type II errors &#40;false negatives&#41;&#58;  49.63 %

There are  2737 simulations with score > 50% that failed SPRT.
There are     8 simulations with score = 50% that failed SPRT.

Estimated elapsed time&#58;  1663.77 seconds.

Speed&#58;  734231 games/second.

Code: Select all

Shortest simulation&#58; 6475 games (+1252 -1029 =4194&#41; ---> pass.
Longest simulation&#58; 933396 games (+163750 -161637 =608009&#41; ---> fail.

Code: Select all

From    6000 to    6999 games&#58;     1 simulation  (  0.01 %); accumulated&#58;   0.01 %.
From    7000 to    7999 games&#58;     2 simulations (  0.02 %); accumulated&#58;   0.03 %.

&#91;...&#93;

From   91000 to   91999 games&#58;    42 simulations (  0.42 %); accumulated&#58;  49.32 %.
From   92000 to   92999 games&#58;    52 simulations (  0.52 %); accumulated&#58;  49.84 %.
From   93000 to   93999 games&#58;    64 simulations (  0.64 %); accumulated&#58;  50.48 %.
From   94000 to   94999 games&#58;    61 simulations (  0.61 %); accumulated&#58;  51.09 %.
From   95000 to   95999 games&#58;    56 simulations (  0.56 %); accumulated&#58;  51.65 %.
From   96000 to   96999 games&#58;    52 simulations (  0.52 %); accumulated&#58;  52.17 %.
From   97000 to   97999 games&#58;    51 simulations (  0.51 %); accumulated&#58;  52.68 %.
From   98000 to   98999 games&#58;    53 simulations (  0.53 %); accumulated&#58;  53.21 %.
From   99000 to   99999 games&#58;    38 simulations (  0.38 %); accumulated&#58;  53.59 %.
From  100000 to  100999 games&#58;    44 simulations (  0.44 %); accumulated&#58;  54.03 %.
From  101000 to  101999 games&#58;    42 simulations (  0.42 %); accumulated&#58;  54.45 %.
From  102000 to  102999 games&#58;    44 simulations (  0.44 %); accumulated&#58;  54.89 %.
From  103000 to  103999 games&#58;    42 simulations (  0.42 %); accumulated&#58;  55.31 %.
From  104000 to  104999 games&#58;    38 simulations (  0.38 %); accumulated&#58;  55.69 %.
From  105000 to  105999 games&#58;    35 simulations (  0.35 %); accumulated&#58;  56.04 %.
From  106000 to  106999 games&#58;    49 simulations (  0.49 %); accumulated&#58;  56.53 %.
From  107000 to  107999 games&#58;    43 simulations (  0.43 %); accumulated&#58;  56.96 %.
From  108000 to  108999 games&#58;    33 simulations (  0.33 %); accumulated&#58;  57.29 %.
From  109000 to  109999 games&#58;    59 simulations (  0.59 %); accumulated&#58;  57.88 %.
From  110000 to  110999 games&#58;    41 simulations (  0.41 %); accumulated&#58;  58.29 %.
From  111000 to  111999 games&#58;    53 simulations (  0.53 %); accumulated&#58;  58.82 %.
From  112000 to  112999 games&#58;    43 simulations (  0.43 %); accumulated&#58;  59.25 %.
From  113000 to  113999 games&#58;    40 simulations (  0.40 %); accumulated&#58;  59.65 %.
From  114000 to  114999 games&#58;    44 simulations (  0.44 %); accumulated&#58;  60.09 %.
From  115000 to  115999 games&#58;    44 simulations (  0.44 %); accumulated&#58;  60.53 %.
From  116000 to  116999 games&#58;    46 simulations (  0.46 %); accumulated&#58;  60.99 %.
From  117000 to  117999 games&#58;    45 simulations (  0.45 %); accumulated&#58;  61.44 %.
From  118000 to  118999 games&#58;    27 simulations (  0.27 %); accumulated&#58;  61.71 %.
From  119000 to  119999 games&#58;    37 simulations (  0.37 %); accumulated&#58;  62.08 %.
From  120000 to  120999 games&#58;    42 simulations (  0.42 %); accumulated&#58;  62.50 %.
From  121000 to  121999 games&#58;    34 simulations (  0.34 %); accumulated&#58;  62.84 %.
From  122000 to  122999 games&#58;    42 simulations (  0.42 %); accumulated&#58;  63.26 %.
From  123000 to  123999 games&#58;    38 simulations (  0.38 %); accumulated&#58;  63.64 %.

&#91;...&#93;

From  255000 to  255999 games&#58;    10 simulations (  0.10 %); accumulated&#58;  90.21 %.
From  256000 to  256999 games&#58;     7 simulations (  0.07 %); accumulated&#58;  90.28 %.

&#91;...&#93;

From  932000 to  932999 games&#58;     0 simulations (  0.00 %); accumulated&#58;  99.99 %.
From  933000 to  933999 games&#58;     1 simulation  (  0.01 %); accumulated&#58; 100.00 %.

Number of finished simulations&#58; 10000.
------------------------

Why I do this? Just to bring some (useful?) numbers to SF testing framework. In typical SPRT(-1.5, 4.5) and SPRT(0,6) tests I get less than 1% of simulations over 128000 games, which is the default maximum number of games set in SF testing framework. But I recently saw some SPRT(0, 3) tests where the default maximum number of games is 256000... I get around 10% of those simulations over 256000 games (99% of my simulations were of less than 466000 games). I set the worst case (the longest expected average of games) of elo = (elo0 + elo1)/2, so my numbers should be considered as a maximum.

I noted a curious thing in my results: in each of these three cases, the average (arithmetic average) number of games per simulation correspond to 62.8% or 62.9% (more less) of the accumulated distribution; moreover, median/average ratios of these three cases are between 0.755 and 0.764 at first approximation. I mean, those values (percentile of the average and the ratio median/average) seem to vary in tiny intervals from my inexperienced POV.

I hope that more people can verify my results. Sorry for my long post.

Regards from Spain.

Ajedrecista.

User avatar
Ajedrecista
Posts: 1396
Joined: Wed Jul 13, 2011 7:04 pm
Location: Madrid, Spain.
Contact:

Re: SPRT and narrowing of (elo1 - elo0) difference.

Post by Ajedrecista » Fri Oct 24, 2014 8:38 am

Hello Michel:
Michel wrote:Actually here is a simple formula for the worst case (elo=(elo0+elo1)/2) expected duration of an SPRT.

The formula is of the form

Code: Select all

C/&#40;elo1-elo0&#41;^2
with

Code: Select all

C=AB/I
where

Code: Select all

A=ln&#40;&#40;1-alpha&#41;/beta&#41;
B=ln&#40;&#40;1-beta&#41;/alpha&#41;
I=2*bb*bb*&#40;1-Ld&#41;*Ld*Ld
where

Code: Select all

Ld=1/&#40;1+10^&#40;draw_elo/400&#41;)
bb=ln&#40;10&#41;/400
Example:

Code: Select all

If alpha=beta=0.05 and draw_elo=240 then

C=1020075     &#40;roughly 1,000,000!)

For a BayesElo difference of 6 this gives a worst case expected duration of

C/36=28335
Sorry for bumping such an old thread but I have just noted a typo. Please take a look here:

https://groups.google.com/d/msg/fishcoo ... VpsxwFExEJ

Your C value and the average length C/36 of your example are correct, but you made a typo in the code box that contains the formula of I. The correct formula is:

Code: Select all

I = 2*bb*bb*&#40;1 - Ld&#41;*&#40;1 - Ld&#41;*Ld
You change one (1 - Ld) by Ld. I did not check the formulæ numerically and it was my error. Hopefully, better later than never! ;)

Thanks again for submit those expressions.

Regards from Spain.

Ajedrecista.

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