Is the Elo rating system accurate?

Is the Elo system rating accurate?

Given an absence of inspiration and interesting topics to talk about, I thought that for today's article it would be nice to revisit every whiner's favorite topic: Elo Hell!  No matter what skill rating you're at, you will find that you are still in what players call 'Elo Hell' due to dependence on your team's collective effort to win, rather than being able to singlehandedly influence the outcome of a game.  Today's post will look at Elo Hell as a statistical whole, rather than looking at an individual's contribution to winning.  We'll be investigating what it takes to fall into Elo Hell, the chances of winning a given game, and why you should stop whining about feeders and trolls.

How does the Elo rating system actually work?

We've heard a lot of talk about the Elo and what it suggests, but I'm sure many of you readers are uninformed as to how it actually works.  The Elo system originated as a simplistic method to compare the relative skill between different chess players, and is used today in dozens of games.  While there are more complex ways to compute the chances of winning a given game, the Elo system is prized because it is both simple and accurate.  Assuming that Riot's Elo system accurately reflects upon the Elo system used in chess, the chance that your team will win is:

 Percentage of winning a given game = 1 ÷ [1 + 10^(ΔElo/400)]

...And your chances of winning a game given your true skill rating are as follows:


But VVinrar, only people with a Ph.D. in Elo Hell Studies can read that table!  Help!

Luckily for you, "Elo Hell Studies" is a degree that only pretentious theorycrafters use as a joke.  It's not that hard to read at all!  The row variable is the difference between your true rating and the enemy team's average true rating, while the column variable is the difference between the rest of your team and the enemy team.  For example, if the average rating of the enemy team is 1200, your true Elo is 1300 (+100), and the average Elo of the rest of your team is 1225 (+25), then your team should statistically have a 57.1% chance of winning the game.

What does this suggest?  Even if you are more skilled than your enemy and you win your lane, you are still at the mercy of 9 other players, and in the grand scheme of things you have only slightly increased than chances of your team to win.  Conversely, if your entire team is dramatically less skilled than the enemy team, you have a much lower chance of winning.  If your team is an average of 400 true Elo lower than the enemy team, you have approximately a 9.9% chance of winning a given game.

If it wasn't already obvious, this implies that if your team's Elo is higher than the enemy team, you have a better-than-average chance of winning a game against them.

Elo Hell and You

But VVinrar... I used to be be 1400 and now I'm 1100, and it's all because of the trolls!

This may be true, but you have to remember, Riot's matchmaking system involves thousands of players who are at your exact skill level, and millions of players in its entire database.  Your Elo may not be precise for yourself as a player, but on a statistical whole, most players are exactly where they should be.  The system is very accurate, but it also has a pretty wide standard deviation (~200) which makes it hard to pinpoint player skill, especially for any given game where they are playing a role or champion that overstates/understates their predicted skill level.  There are trolls on each team, and I highly doubt your rating is low because of a few factors.  Remember, in all of the games you have ever lost, there has only been one common factor.

Using binomial probability, if you play 20 games where you are consistently 100 Elo higher than every other player in the game, you have approximately a 16% chance of dropping 12 games or more against your enemies.  Over the long term, this may average out and you will rise back to your true Elo; however, with a small sample size, you are basically at the whim of the matchmaking gods.  Especially if you have inconsistent play, it may take several dozen games to reach your true Elo; however, if you reached the triple digit mark of games, you've likely found your true skill level.  I hate to be the person who breaks the big news to you, but if you've been at a certain skill level for a few hundred games, you might actually belong there.

Common Complaints (Warning: Numbers)

But VVinrar, I fell 300 whole Elo.  It must be the trolls!

Believe it or not, a lot of players can luck into their top Elo.  For a fresh player jumping into ranked, you only need to go 6-4 to achieve a rating of 1300.  If you're a true 1100 Elo player, you have a 29% chance of getting 6 or more wins in your first 10 games against 1200 players, a 22% chance of getting 6+ wins against 1300 players, and a 15.6% chance of winning 6+ times against 1400 players.  Combine these numbers together, and what do you get?

The average 1100 player has a 6.4% chance of reaching 1400 and a 1% chance of hitting gold.

Well, these are the same 6.4% of players who are complaining about falling 300 Elo.  This means that in any given 1100 Elo game, there is a 10% chance that at least one out of the ten players has been at 1500 at one point and a 48% chance that at least one player has been at 1400.  And then guess who are the players complaining about the trolls? "Unlucky" bastards.

Note: Going 6-4 will only gain you 90 Elo instead of 100, but pretending it gets you 100 Elo is more convenient for these calculations. Also, these are approximate values only;  my math skills are too sucky for continuous calculations. 

Final Thoughts

Remember, no matter how good you are at League of Legends, you still have a chance of losing a game because of your teammates; however, if your own personal skill level should be enough to influence the odds of winning towards your favor if you are good enough, especially if you bring a 1-2 punch of a viable champion and solid gameplay.  If you don't win many games right away, don't be afraid!  The law of large numbers is on your side: eventually, you'll get back to your true skill level regardless of the feeders and trolls.  From a statistical perspective, you are only one player in a game controlled by dozens of internal and external factors, and you can't always singlehandedly determine the outcome of a game.  Given enough time, your rating should accurately reflect your true skill, but you need to give the system enough games to get you there.

TL;DR: The Elo system is accurate, but it has very high variance.  Don't worry about your Elo.  If you are actually below your true skill rating after a few games, you'll get there sooner or later.  If you've played more than a hundred games and you're still below your true Elo, you should introduce yourself to my friends Dunning and Kruger.

Quote from VVinrar

I hope you enjoyed reading today's post! I have a special request for the readers today: I would really appreciate it if you could take the simple time to visit and read my interview with Forbes @ If the article has a 'successful' amount of views, the author will publish the second part of my interview (which is much more interesting than this one) and it means a lot to me both personally and professionally if this happens.

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