The Elo Rating System for Chess and Beyond

The Elo Rating system is a method to rate players in chess and other competitive games. A new player starts with a rating of 1000. This rating will go up if they win games, and go down if they lose games. Over time a player’s rating becomes a true reflection of their ability – relative to the population.

My video was mostly based on A Comprehensive Guide to Chess Ratings by Prof Mark E Glickman

Below are some of the things I wanted to talk about, but cut so the video wasn’t too long!

Some explanations of the Elo rating system say it is based on the normal distribution, which is not quite true. Elo’s original idea did model each player’s ability as a normal distribution. The difference between the two players strengths would then also be a normal distribution. However, the formula for a normal distribution is a bit messy so today it is preferred to model each player using an extreme value distribution. The difference between the two players strengths is then a logistic distribution. This has the property that if a player has a rating 400 points more than another player they are 10 times more likely to win, this makes the formula nicer to use. Practically, the difference between a logistic distribution and the normal distribution is small.

Logistic distribution on Wikipedia
We replace e with base 10, s=400, mu=R_A – R_B and x=0 in the cdf.

For the update formula I say that your rating can increase or decrease by a maximum of 32 points, and I said there was no special reason for that. This value is called the K-factor, and the higher the K-factor the more weight you give to the players tournament performance (and so less weight to their pre-tournament performance). For high level chess tournaments they use a K-factor of 16 as it is believed their pre-tournament rating is about right, so their rating will not fluctuate as much. Some tournaments use different K-factors.

In the original Elo system, draws are not included, instead they are considered to be equivalent to half a win and half a loss. The paper by Mark Glickman above contains a formula that includes draws. Similarly the paper contains a formula that includes the advantage to white.

Another criticism of Elo is the reliability of the rating. The rating of an infrequent player is a less reliable measure of that player’s strength, so to address this problem Mark Glickman devised Glicko and Glicko2. See descriptions of these methods at

On the plus side, the Elo system was leagues ahead of what it replaced, known as the Harkness system. I originally intended to explain the Harkness system as well, so here are the paragraphs I cut:

“In the Harkness system an average was taken of everyone’s rating, then at the end of the tournament if the percentage of games you won was 50% then your new rating was the average rating.
If you did better or worse than 50% then 10 points was added or subtracted to the average rating for every percentage point above or below 50.
This system was not the best and could produce some strange results. For example, it was possible for a player to lose every game and still gain points.”

This video was suggested by Outray Chess. The maths is a bit harder, but I liked the idea so I made a in-front-of-a-wall video.

310 Comments

  1. So I have a chance at beating Magnus, sweet!

  2. I don't buy this. At a certain point Elo Maxes out. Like an I'm willing to bet a 2500 player will still fairly consistently beat a 3000 player.

    So that 400 points = 10x more likely is kinda bs.

  3. I saw the guy from numberphile and I clicked

  4. New ECF 4 figure rating system isn’t ELO then? It doesn’t follow the same formula for updating ratings.

  5. Neat summary, well explained. You gain 32 points 🙂
    (j/k… you gain 0 points because Singing Bananas is already at the top of the curve.)

  6. You're telling me that a 16 60 hung a piece on move 6??

  7. What happens to the elo if equal a.i. are infinitely pitted against eachother

  8. "I can do all things through Him who strengthens me." – Philippians 4:13

  9. Roger Cook actual inventor of chess rating system?

  10. My elo rating fide is like 1600 but i easily beat 2k guys.

  11. I really feel like there should be a nonlinear term in the update formula! A player who plays at an 800 level for their whole life but beats Carlsen once should definitely have a higher rating than someone who consistently plays at an 835 level!

  12. While this is all well, good and accurate, it is only a model. A player who is 800 points higher does not have a 1/100 chance of losing. The lower rated player would lose 100% of the time.

  13. Wonderful insights and understandable explanations.

  14. And now every gamer nows and cares about their elo :c

  15. I played chess for 4 hours today and my friend asked me to take elo chess, came out the score is 1290. Whats that mean? My friend didn’t give me any explanation 🙄

  16. The Cult of Johnny Del R. Souls who Died in April. says:

    Run that past me again huh, why isn't ⚽ like this?

  17. Given that you start with Elo in some online games when creating an account and all interactions are equal transactions, there is effectively economic inflation of Elo in that game

  18. Immensely informative. I genuinely thought ELO was an acronym but now I know

  19. Amazing that we still use this primitive method of evaluating skill.

    "Player win = skill–ooga booga" … even if that win was completely accidental.

    "Player lose = no skill–ooga booga" … even if that loss was a mouse slip or distraction from a dominant game.

    Absolute chaos. No accuracy whatsoever.

  20. What if you try and mix the concept of self-confidence into the equation? What is "elo-hell"? Theres Arpad Elo and Umberto Eco, and then theres someone like Brian Eno. See? Now I mentioned something you dont know. Is there a pattern there? Does the confusion create anxiety? Why am I so confident then, if theres no ground to stand on? Now try and beat me.

  21. I really love that Elo got adopted by competative Tetris.
    Never expected the system to show up in such a different kind of game.

  22. if 400 points means 10x difference, that means i could beat magnus carlsen if i played 100 000 games against him

  23. I've never had an 'elo' rating because I've never played chess online. This information has convinced me that I'm better off that way. If I start playing chess online, my elo will go down. So I'm happy to keep my nice 1000 elo.

  24. thank for explain, now i inderstand why is so hard to gain elo, in low elo you gain few points and need much more games than in high elo.

  25. Why they not use just the simple Laplace's Succession Method?
    P(win)=(n°previous winnings +1)/(total n° games played +2)

    Its starts for all at 50% (as if you know nothing about the players), and fastly converges to the relative position for each player… its a simpler method.

  26. 1:41 I like how he immediately starts to explain away bad results with external factors. What a gamer.

  27. In place of the 32 used as an example in this video, FIDE now assigns each player a K-factor of 40, 20, or 10 depending on level of experience and skill. Each player has his own K factor, so if players with different K factors play, the result is not a net zero, and rating points are added or subtracted from the overall pool.

  28. With some luck i can beat magnus. i got this right?

  29. I read "The Elo rating system for cheese" before I clicked on that video and now I am kinda disappointed 🙁

  30. Hello to you, Mr. Colossal Nerd 🤓🤓🤓

  31. What’s the formula for rating change before you have enough games for an established rating?

  32. Hungarian physicist, my friend, not American! 🙂 <3

  33. Great video! I loved the fact that the video explained everything from scratch.

  34. Hypothetically, if a 2822 player were to face a 2615 player, it would be an expected 23% chance for the lower rated to win.

  35. Sorry i didn't get where do 1000, 400, and 32 come from?

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