Elo rating system

The Elo rating system is a method for calculating the relative skill levels of players in two-player games such as chess. It is named after its creator Arpad Elo, a Hungarian-born American physics professor. The Elo system was invented as an improved chess rating system, but today it is also used in many other games. It is also used as a rating system for multiplayer competition in a number of games and has been adapted to team sports including association football, American college football and basketball, and Major League Baseball.

In League of Legends the Elo rating of a player is used by the matchmaking in normal games and ranked games to find other players of a similar skill level to play with/against. Elo is not used for custom and Co-op vs. AI games. The Elo rating for ranked games is different for each queue types: 3v3 arranged, 5v5 solo and 5v5 arranged teams. The rating is only visible for ranked games after 10 games played in a certain queue type. A summoner's Normal game ELO remains hidden at all times and can only be guessed upon based off his or her win/loss ratio and the apparent skill of teammates and enemies.

The Math of Elo
The specific formulas used for Elo calculations in League of Legends are unknown. However, most Elo implementations share the same basics as that originally designed for chess. A brief summary is given below, for a more detailed discussion see Wikipedia

It is assumed that a persons performance varies from game to game in approximately a normal distribution and a persons Elo rating is the mean of that distribution. A person with a higher Elo will perform better on average than a player with a lower Elo. This score is determined entirely by win/loss statistics in relation to other players. For Player A and B with respective Elo ratings of Ra and Rb the expected outcome of the game for player A is given by the following formula.
 * $$Ea = \frac{1}{1 + 10^{(Rb-Ra)/400}}$$

For a difference of 200 points the team/player with the higher score is expected to win 75% of the time. The probability that player B will win is Eb = 1 - Ea. This 200 point standard is for Chess and may be different in LoL. After a game the actual outcome is compared to the expected outcome and each team/players rating is adjusted to bring them closer to where they should actually be. As a result, if a team was expected to win and does their score changes less than if they where expected to lose and instead won. Successive games should eventually bring each player/team to a point where they are expected to win 50% of the time against opponents of equal score.

A persons change in rating is linear to the difference between the expected outcome and the actual outcome. It is given by the following formula where Sa is the result of the game and is presumably 1 for a win and 0 for a loss.


 * $$Ra_{new} = Ra_{old} + K(Sa-Ea)$$

The magnitude of the score change is determined by a persons K value. In chess initially this K value is Big (25 for their first 30 games) resulting in large changes in Elo. This is so a player can rapidly find his correct place in the ranking system. As their number of wins and losses becomes more even this K value is reduced to prevent dramatic changes in Elo against evenly matched opponents (K = 15 to 10). This also prevents inflation in ratings at high Elo play. It appears that League of Legends uses a similar system of changing K values. Initially, K appears to be around 60 and levels out to 25 eventually.

All players start ranked play with an Elo of 1200 for their first 10 games at level 30. From there they are assigned a score and changes are made as normal.

Elo Hell
Elo hell is a supposed area of the ranked solo queue, usually said to be below 1200 rating. It is said to be populated with leavers, feeders and just generally poor players. This supposedly makes the life of any players queuing up a living hell. There is much debate about the existence and nature of Elo hell in the forums. Some players say it just an excuse used by un-skilled players as to why they are not ranked higher. Others claim that since they are constantly being placed with bad teammates, they cannot win enough games in a row to get to an appropriate Elo. However, this argument doesn't work because they are 'more likely to be placed opposite the bad teammate than teamed up with them because of the 5 enemy slots vs 4 teammate slots.