I am a bit at loss with this argument. Doesn't the derivative set at the origin for the given by the model normalized, single-parameter logistic gives the full logistic (e.g. 75% at 200)? In fact, as I understood, the default Bayeselo (when calculating scale after mm) was compressing the default logistic to something that 75% meant 150 or 180 instead of 200, and when setting the scale to 1, 75% meant exactly that 200.
I think the scaling is required to bring ratings to the conventional arpad elo assumption of 200elo - 75% system. What you get from bayeselo originally is something larger, so you scale it down so that the slope's match at 50% winning percentage. So in the end you do not have 150 or 180 but 200 elos. I haven't followed the 'compression' topic closely but I assumed that was what was causing the problem. The problem with using scale=1 is for some models other than the default , the rating numbers may be bigger at first. I am not even sure we get somewhat comparable numbers with the default model and scale=1.
For example for current ccrl list I get -1031 to 731 elo using default, and -1534 to 1171 elo using modified model, both of them with scale=1. Infact at first I thought my modified model was wrong but it turns out it fits the data even better. But for comparison clearly it needs to be scaled by some 66% if we assume the first result is close to elo assumption. Btw eloDraw and eloAdvantage modify the model so that is why we don't get 200elo-75% out of the box.
Also, have not you shown that the draw model used by Bayeselo is not the optimal one (although this is not very relevant for the discussion, the model differences are small compared to that huge absolute scale variations)?
Kai
Well I would not call what I did anything formal and it still needs a lot of work to show that the model is indeed an improvement. Maybe if I can find some big databases with huge draw ratio like in reversi, the effect of the draw model would be measurable to a significant result.