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So ELO system has been implemented in many games like League of Legends, DOTA, CS and others, in which 50% of the players win and 50% lose, so, if we're not taking personal skill into account, every player has 50% chance to win.

How do I make ELO system work in a game that has, for example, 3 teams with 8, 3 and 1 people in them that have 60, 30 and 5% to win respectively (percentages are not adding up to 100% because a draw is possible)?

After the example is figured out, we should probably generalize the solution, that is make it work with N teams of n1, n2, ... nN teams of m1, m2, ... mN players with c1, c2, ... cN chances to win each.

Feel free to leave a note on non-ELO rating systems if they can be applied to this kind of game.

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    $\begingroup$ You give points for a win, draw and loss so that the sum of the out come is zero across all players in this game. Your ELO system predicts the expected points for each player based on their current rating (again adding up to zero across all players in this game) and then adjusts their rating in proportion to the difference between their actual outcome and their pre-game expected outcome. $\endgroup$ – Henry Jun 26 '15 at 19:08
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If you're interested in use (more than in development), you should give a try to rankade, our free-to-use ranking system for sports, game, and more.

It seems that rankade fits your needs, because it can manage any kind of match: one-on-one, faction vs. faction (two teams, which may be asymmetrical), multiplayer, multi-faction (so 3+ factions, as you're looking for), cooperative games, single player games, and so on.

Here's a comparison between most known ranking system, including Elo.

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