Disclaimer: I open this thread just as a curiosity of how simple some things can be. If I am wrong in my next reasoning, please answer.

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

I reached by chance a Lichess blog about sharpness of positions:

Evaluating Sharpness using LC0's WDL

And I saw the WDL contempt of Lc0 according to that article:

Code: Select all

`contempt = {2/[ln(1/W - 1) + ln(1/L - 1)]}^2`

Then, I remembered the early days of Fishtest around 2013, where SPRT bounds were expressed in Bayeselo units instead of Elo. IIRC:

Code: Select all

```
bayeselo = 200*log10[(W/L)*(1 - W)/(1 - L)]
drawelo = 200*log10{[(1 - W)/W]*[(1 - L)/L]}
```

Code: Select all

```
drawelo/200 = log10{[(1 - W)/W]*[(1 - L)/L]}
drawelo/200 = log10[(1 - W)/W] + log10[(1 - L)/L]
drawelo/200 = log10(1/W - 1) + log10(1/L - 1)
drawelo/200 = [ln(1/W - 1) + ln(1/L - 1)]/ln(10)
drawelo*{[ln(10)]/200} = [ln(1/W - 1) + ln(1/L - 1)]/ln(10)
```

*Voilà!*The relation could not be simpler:

Code: Select all

`contempt = (2/{drawelo*[ln(10)]/200})^2 = {[400/ln(10)]/drawelo}^2 = (C/drawelo)^2`

Curiously, the constant C = 400/ln(10) is not that strange. While working with error bars in the past, I came across 1600/ln(10) several times (at least here, here, here, here and here).

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

Please note that contempt and drawelo values can be unbound, for example with extreme values of W and L such as W = L = 0 (drawelo → infinity; contempt → 0) in KB vs K and KN vs K endgames; or W = 1 and L = 0 (or W = 0 and L = 1) (drawelo → 0; contempt → infinity), for example in a position where the only possible move/s lead to checkmate, like in the following position, unlikely to be seen in a real game:

[d]6qk/8/5r1K/7P/7Q/8/8/8 w - - 0 1

**Qxf6+**(only),

**Qg7+**(only);

**Qxg7#**(only)

An even funnier position is this one: beware with which queen you capture... win or lose!

[d]3Q2qk/8/4qq1K/7P/7Q/8/8/8 w - - 0 1

**a)**White wins (perfect play):

Qhxf6+, Qxf6+; Qxf6+, Qg7+; Qxg7#

**b)**Black wins (white blunders):

Qdxf6+??, Qg7#

**c)**White wins (back-to-back blunders):

Qdxf6+??, Qxf6+??; Qxf6+, Qg7+; Qxg7#

Regards from Spain.

Ajedrecista.