I was considering sport disciplines for which there are multiple players at the event but rather than playing against each other, they do stuff, are assigned points and their final position is based on their relative scores.
Example would ski jumping (ignoring variable wind) or marathon running.
Also I wanted to do something similar to Elo rating so I assume that each players strength is represented by normal distribution with the same variance but mean is different, reflecting each player's strength. However instead - like in chess - comparing players results with expected result, I think I need to compare players position with expected position given the strength of the opposition.
So if I have players represented by means $u_1, u_2, ..., u_n$ (standard deviation $s^2$) I want to know what is expected position of player $i$ when the results are sorted from highest to lowest. And based on that adjust the ranking similarly to ELO.
Does it make sense and would it reasonably work? And if so - can someone give me a hint how to crack the formula (I'd like to work it out but I'm not sure at the moment how to calculate this).