1. The sigmoid function is the closest thing to linear that makes sense on probabilities⁺. A purely linear function would cross 0/100% which, while the sigmoid

1. The sigmoid function is the closest thing to linear that makes sense on probabilities⁺. A purely linear function would cross 0/100% which, while the sigmoid flattens exponentially as it approaches the extreme values. 2. The fit isn't as bad as the author claims. It looks like the biggest difference between the graphs is that the point differences are scaled differently (400 pts for 90% in elo vs 800 pts in the second graph). A quick and dirty overlay of the two graphs shows a reasonable fit: https://ibb.co/0YwYH9z https://ibb.co/0YwYH9z 3. I like observations about player psychology. Satisfying the players is more important than having the mathematically best ranking system. 4. Personally I like Whole History Ranking (https://www.remi-coulom.fr/WHR/ https://www.remi-coulom.fr/WHR/), but it's unlikely to be popular with players (the psychological criticisms the article makes apply to it as well, with some additional problems, like rank drifting without playing). KGS which uses ranking system similar to WHR (but more primitive) certainly draws a lot of criticism for its ranking system. If I had to design a mathematically optimal ranking system, I'd start with WHR and make parts of it trainable/fittable. ---- ⁺ Bayes' theorem turns into addition when applied to logarithmic probabilities and the sigmoid function converts from logarithmic probabilities to normal probabilities. This property is why it (or its multi category equivalent softmax) is used when predicting probabilities using logistic regression or neural networks.