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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).

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    $\begingroup$ This should be doable, but you must think about if you want to use the point scores (or timing, in marathon) directly, or only the ranking. Some kind of latent variable model. $\endgroup$ – kjetil b halvorsen Apr 12 '16 at 17:51
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  • $\begingroup$ After some thinking I can just make an average (I am not sure) of normal elo result (so score minus expected result) based on all 1on1 combinations. Should work. I think it would be fairer than to treat it as complete set of results 1 on 1 as the latter solution would differ in amount of points it offers depending on players number. Which for some sports makes sense but if you think about i.e. ski jumping - it's just one jump. It shouldn't boost your score in varying degree depending on how many other jumpers jumped badly on that day.@kjetilbhalvorsen what do you think? $\endgroup$ – nimdil Apr 27 '16 at 9:26
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If you're interested in use (more than in development), you should give a try to rankade, our ranking system.

Rankade is free and easy to use, and it's different from Elo ranking system (here's a comparison) because it can manage matches with 2+ players, and so it fits your needs, maybe.

Rankade's algorithm (ree algorithm, that is extremely complex and not in the public domain, for the time being) manages your expected position structure ("compare players position with expected position given the strength of the opposition"), and provides rankings, and stats, and graphs, and more.

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    $\begingroup$ Thanks. However depending on the external service with fairly nonspecific description of how it works (other than "extremely complex") is not s.t. I would use. $\endgroup$ – nimdil Apr 27 '16 at 9:22

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