The ELO system was originally invented to rank Chess players against one another, by determining their relative skill levels. This allows it to be used for basically any 1v1 competition, and has been appropriated for many video games. The Pokemon Showdown ladders use a similar system.
The difference between two players' ELO ratings serves as a method of predicting a player's "chance of winning". A difference of 400 points means the higher player is, statistically, 10 times more likely to win.
This calculated "chance of winning" is used to determine how many ELO points a player should lose or gain from a single match.
For example (note these are only theoretical numbers and do not consider weighting factors):
Two people of even ELO battle. They have a 50:50 chance of winning. The winner will gain 50 points, and the loser will lose 50 points.
Two people with a 400 ELO difference battle. There is a 91:9 chance (approximately 10 times) of the higher player winning. If the higher player does win, they will gain 9 points, and the loser will lose 9. However if an upset victory occurs, the higher player will lose 91 points, and the lower player will gain 91.
The example above does not use a weighting factor. Most competitive systems, and this one, use a weighting factor that drops as the number of matches played increases. This means that players on their initial matches will lose or gain a larger number of ELO points than more established players. This allows them to roughly get to their correct spot on the ladder faster, and then settle into a more accurate spot as the weighting factor decreases.
In the URPG system, all players will start with an ELO of 1200. The weighting factor begins at 100, and reduces based on the number of battles until it eventually reaches a constant of 40.
More information can be found on Wikipedia, and several games that use an ELO ranking system.
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