A High School Math Teacher Explains why the Elo System Works Almost Perfectly

  • As we begin with a new season, invariably we will continue to be flooded with threads complaining how the Elo system is an epic failure, along with a list of reasons why every single person is better than their Elo rating indicates.

    I'm going to try and clear a few of those things up, and hopefully everyone can get a grasp of why the modified Elo system used for League of Legends works very well.


    First, let's begin with an explanation of the Elo system. I think it would be good to cover what your Elo rating tells you, and what it doesn't tell you.

    An Elo rating IS NOT a measure of your skill level. Every single complaint or suggestion about how it is unfair because you might be more skilled but still lose points means you do not even understand what the system is trying to measure.

    The Elo system, and therefore your Elo rating, is a measure of "odds". The Elo system was designed to give a metric that can be used to determine a player's chances of winning. For example, two players with exactly the same Elo rating have a 50/50 chance of winning the match. This spread was originally designed logistically, such that a difference of 200 points, regardless of values, would give the higher player a 75% chance of winning (ignoring the possibility of a tie).

    It is important to note that there is no point on the scale where you are guaranteed to win. This is especially true of people who are in the middle section (a.k.a. roughly average). Your chances to win do not fluctuate very much, and therefore it is expected that you will lose about as many games as you win.

    WARNING! Scary Maths Incoming:

    To calculate my chances of winning, based on known information, let's look at an example. Since I like to play AD Carry, I will calculate my chances of winning against my favorite AD Carry player aphromoo.

    My Elo after the season 2 reset: 1270ish.
    aphromoo's current Elo: 2290ish

    The formula to determine expected score is given as 1/[1+10^((Rb - Ra)/400)]
    Where Ra is my rating and Rb is their rating.

    Based on the above Elo ratings, I would have a 0.28% chance of winning.

    Alternatively, aphromoo would have a 99.72% chance of winning.


    I can already tell that I'm setting myself up for massive whines by using an example of 1 player against 1 player to explain Elo, so I suppose now is the time to explain why the Elo rating works for League of Legends, even though it is a 5v5 game.

    It is impossible to measure your individual skill exactly. Your skill can only be inferred based on your ability to beat other players. The matchmaking system in League of Legends takes 10 players of similar Elo ratings and splits them into two groups with team averages as similar as possible. This is, indeed, a fair way of splitting people up into teams. Two teams with similar average Elo ratings will have approximately a 50% chance of winning.

    Here's another example to demonstrate:

    If I were to average the Elo of the 5 people I play with most often, we would have an average Elo of 1176.

    If I were to take the average Elo of the 5 members of team FeaR, they have an average Elo of 2159.

    Therefore, the expected chance of team FeaR winning would be 99.65%. As everyone expects, we would probably get completely crushed in every lane and it turns out that averaging the five Elo ratings of the players gives a good estimate of who would win a 5v5 matchup.

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    Now, it's time to address some complaints about the Elo system.


    Why does the thread title say "almost perfect" if everything works fine?

    Well, there is a slight problem with the Elo system as it is implemented in League of Legends. The problem, though, is not the one that you all think it is, nor is it a problem that has that big of an overall effect on your individual rating. Really, I shouldn't even call it a problem, but more like a potential slight improvement which is likely hard to code and probably exploitable by some of the more "clever" summoners.

    The League of Legends Elo system uses k-values (a.k.a. how the system determines how many points you gain or lose) that are mostly fixed values based on how many games you play. The only real way to improve on this would be to have a larger k-value for games which are completely one-sided stomps.

    To illustrate, a game where every lane gets beat, one team gets all the dragons and barons, and completely crushes the enemy team, even taking the nexus before the 20 minute surrender mark, the winning team is clearly significantly better than the losing team and should move up more than in a close game which drags on for 60-70 minutes.

    There are a number of things that the variable k-value could be based on, however I believe that a scale based on the difference in global gold and game length would be the most fair.


    What about leavers and trolls?

    Every player at every Elo has to put up with leavers and trolls. I genuinely believe that leavers should get an increased Elo loss while the rest of the team should get a slightly reduced Elo loss, but again it would have little overall effect on your personal score, but would rather move chronic leavers to a lower Elo at a faster rate.

    Also, over time, the enemy team will have as many leavers and trolls as your team, so it really balances itself out.


    I don't believe my Elo rating is accurate. (I think I'm better than it says I am).

    Play more games. Even with the above mentioned "flaw" with the k-value system, over time you WILL get to your true Elo. More than likely though, you will have to accept the fact that you are not as good as you think you are. Most people are, in fact, average.


    Finally, I will address what is perhaps the only legitimate complaint about the League of Legends Elo system.

    Many people around the starting Elo (1200) have not reached their true Elo yet because they have not played very many games. This problem would not be solved by measuring individual skill, nor can it be completely solved with k-values based on the rate that new players join the game.

    It is a problem that cannot ever be completely solved, but could be slightly alleviated by increasing the number of placement matches by at least double.

    It could be further alleviated by adjusting the placement matches to measure more statistics than a simple win/loss measure.

    Also, and this would absolutely help the situation, people doing their placement matches should not be allowed to duo queue, and should be completely segregated from the rest of the ranked population. I personally remember teaming with and against players who had ~100+ wins while I was doing my placement matches, and remember dominating at least three players who were currently doing their placement matches. I can only speculate, but I believe the match ups were directly related to duo-queueing.

    These changes would, again, have a minor effect on your true Elo rating after several hundred games, but would make it slightly more enjoyable to play ranked matches if your true Elo was actually around the 1200 mark.

    A better suggestion from later in the thread:
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    Originally Posted by Dominatius Maxim View Post
    More experience than the hundreds of games it takes to even get to level 30 to play ranked? I'd contend that if anything, players should be required to play a minimum number of Draft Normals before queuing up for ranked. A change like what you're suggesting does nothing for the integrity of the system and can actually work to worsen it's effectiveness.

    To summarize, the Elo rating system works very very well for League of Legends with a few minor flaws. The Elo rating does indeed give a fairly accurate measure of your chances to win against other rated opponents, and this accuracy extends very well to 5v5 matches.

    There are ways to make slight improvements to the actual matchmaking (i.e. the process of selecting and placing 5 players on each team) but these changes would have very small effects on individual Elo ratings. Certain system changes would make ranked slightly more enjoyable, but overall people are upset over the system because they believe they are better than they actually are.

    I hope I was able to help you all understand the Elo system a little bit better, and hopefully you all will direct those who need some clarification to this thread so that they may learn a little bit about math.

    P.S. I think it's time to settle a few non-legitimate complaints.

    1) If you don't like the Elo system, deal with it or play a different game.

    2) Because you don't like the Elo system, does not mean it doesn't FUNCTION in ranking players in a perfectly legitimate way. Worth noting about the system, the top 10% of players consistently rise to the top, the bottom 10% of players consistently fall to the bottom, and average people end up in the middle.

    3) Because you don't like the Elo system does not mean that any of the math surrounding the Elo system is wrong. Really, you should be complaining about the MATCHMAKING system, and that would go in a different thread than this one.

    Good luck with your complaint threads!
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    Originally Posted by Toprem View Post
    Answer this question then.... I was 1400 elo last season and now I'm in the 800's yet I have a higher KDA.. WHYYYY?
    There are far more factors that determine a game than your K/D/A, though certainly a good one doesn't hurt your chances. If you are playing against worse players, it's quite usual for you to actually perform mechanically better (better K/D/A)...so I'd be surprised if the situation was anything different. You might want to look to see if you can measure what intangibles are hurting your chances to win. Are you raging at teammates when they do poorly? Are you helping to lead your team when there is a leadership void? Are you encouraging your team to stick together? Are you contributing to your team's ward coverage, regardless of role? There are many, many ways you can improve your chances to win, and if your K/D/A is rising that doesn't mean you're a better player - in this case it likely just means your opponents are worse. You might have comparably great mechanics to your opponents - then what are you doing worse than them?

    By the law of large numbers, if you are overall better than the average player at your Elo, your Elo will rise over time (and vice versa). This is because you have only 4 allies and 5 enemies - meaning the enemy team is more statistically likely to have something that you are not. If you are never a leaver, then the enemy team over time will have more leavers. If you are never a rager, the same goes. This does not mean that you will never encounter them on your team, just that you will more likely face them. Conversely, if you are a leaver or rager MORE than the average, your team will have to carry a heavier share of them, and your teamplay/attitude will bring you down to the point where your superior mechanics balance out your (theoretical, please don't take this as an accusation that you ARE these things) negative attitude, leaving you in equilibrium.

    @sultryvoltron: Great analysis! I will caution, however, that there have been many modifications to our Elo system. I'll quote a post Zileas made back in March to address your point about it taking too long for players to reach the Elo they deserve.

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    Originally Posted by Zileas View Post
    Anyhow, I think one lesson this does give is that our K values are probably a bit low on 'matured' players.
    When we mentioned resetting system certainty on ranked players in between seasons, this is what we are referring to. K values are not a static number for us but can change based on a number of factors, though that instability flattens out as an account matures.
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    Originally Posted by Jimmy Rustling View Post
    When you say 'matures' are you referring to the time an account has existed, or its # of games played?
    Games played.

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    Originally Posted by Retsopmi View Post
    Are you short on reading comprehension? I never said winning lane = winning the game. I actually clearly stated that a good player might win their lane yet still lose the game due to the other 4 people feeding.
    Of course awesome players will sometimes lose games even when they play really well. However what Jozrael posted is accurate given a large number of games.