Ajedrecista wrote:Hello:

I have been looking for an adjust of this data and I think I have something decent. Here I go:

I looked into a Gompertz function (an example is

here) and I came with the following:

1.- I converted accumulated Elo gain (0, 144, 277,...) into score with the Elo model µ = 1/[1 + 10^(-Elo/400)].

2.- I used the TC values of 0, 1, 2 and so on in the horizontal axis, like it is seen in the cited paper just after equation 1.

3.- I used the numbers ln[ln(µ_1/µ_0)], ln[ln(µ_2/µ_1)], ..., ln[ln(µ_8/µ_7)] in the vertical axis. (Equation 4 of the paper).

4.- I did a linear regression with Excel to obtain beta and gamma parameters:

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```
Gompertz fit:
Fitted_µ = alpha*exp[-beta*exp(-gamma*TC)]
Linear fit of the 8 data points = m*TC + n ~ -0.64087232*TC - 0.51556368 (R² ~ 0.99744438)
(Equation 4): gamma = -m ~ 0.64087232
(Equation 4): beta = exp(n)/[exp(gamma) - 1] ~ 0.66489741
```

5.- By definition, alpha is the saturation level, so we can expect that max(µ) = 1 = alpha --> horizontal asymptote. If that:

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```
Fitted_µ ~ exp[-0.66489741*exp(-gamma*0.64087232)]
Converting fitted_µ into Elo gain (rounding up to the nearest Elo integer):
TC Elo Fitted Elo Elo - (fitted Elo)
1 144 151 -7
2 277 277 0
3 389 396 -9
4 490 512 -22
5 583 625 -42
6 656 738 -82
7 715 850 -135
8 766 961 -195
Average error = -61.5 Elo
```

6.- Equation 5 of the paper proposes the following:

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```
alpha_TC = exp[ln(µ_TC) + beta*exp(-gamma*TC)]
I obtain 8 values of alpha_TC. If I randomly choose alpha = average(alpha_TC) ~ 0.99310185
Fitted_µ ~ 0.99310185*exp[-0.66489741*exp(-gamma*0.64087232)]
TC Elo Fitted Elo Elo - (fitted Elo)
1 144 147 -3
2 277 270 -7
3 389 384 +5
4 490 489 +1
5 583 585 -2
6 656 668 -12
7 715 735 -20
8 766 785 -19
Average error ~ -7.1 Elo
```

I know that it sets the upper bound of 99.31% of score, that is, circa 863.3 Elo gain at most. But the average error has improved a lot.

Furthermore, I did not take into account error bars.

Bonus: if I continue giving increasing values of TC to fitted_µ ~ 0.99310185*exp[-0.66489741*exp(-gamma*0.64087232)], I get the next estimated Elo gains:

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```
Converting fitted_µ into Elo gain (rounding up to the nearest Elo integer):
Comparison TC Fitted Elo
5120 + 51.2 vs 2560 + 25.6 8 785
10240 + 102.4 vs 5120 + 51.2 9 818 (+33)
20480 + 204.8 vs 10240 + 102.4 10 838 (+20)
40960 + 409.6 vs 20480 + 204.8 11 849 (+11)
81920 + 819.2 vs 40960 + 409.6 12 856 ( +7)
```

I hope no typos. 818 (+33) should be understood as 818 - 785 = +33 Elo in (10240 + 102.4 vs 5120 + 51.2) and +818 Elo in (10240 + 102.4 vs 10 + 0.1).

It might be interesting to fit win ratio, draw ratio and lose ratio in similar ways.

**Last but not least:** thank you very much, Andreas.

Regards from Spain.

Ajedrecista.