How Do Elo Ratings Work? The Maths Behind Chess Rankings
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When Magnus Carlsen's rating reached 2882 in 2014, the number meant something precise: that against a player rated 2682 he was expected to score about three points out of four, and against a club player rated 1882 to win essentially every game. The system that produces such numbers was designed by a physics professor and chess master, Arpad Elo, for the United States Chess Federation in 1960, and it has since been adopted, often without the name, by football, tennis, video games, dating apps and the rankings of universities and restaurants. It is simple enough to write on a napkin and subtle enough to have kept statisticians busy for sixty years.
The idea
Elo's insight was to treat a player's strength as a number that could be estimated from results and to treat each game as a measurement of it. A rating is a guess; a game is evidence; and after each game the rating is moved a little towards what the evidence suggests. Beat a much stronger player and your rating rises a lot, since the result was surprising; beat a much weaker one and it barely moves, since it was expected. Lose to a stronger player and you lose almost nothing. Over many games the guesses converge on values that predict results well, and the scale is anchored so that a difference of 400 points means the stronger player is expected to win about ten games in eleven.
The formula
Two steps, both short. First, the expected score of player A against player B is computed from the difference in their ratings: E equals one divided by one plus ten to the power of the rating difference over 400, where the difference is B's rating minus A's. That gives 0.5 for equal ratings, about 0.76 for a 200-point advantage and about 0.91 for 400. Second, after the game, A's new rating is the old rating plus K times the actual score minus the expected score, where the actual score is 1 for a win, 0.5 for a draw and 0 for a loss, and K is a constant that sets how fast ratings move. Some consequences:
- •A 1500 player who beats a 1700 player with K equal to 20 gains about 15 points; the 1700 player loses the same 15
- •The same 1500 player beating a 1300 gains about 5
- •Drawing against a player 200 points higher gains about 5, since a draw is better than the expected 0.24
- •Points are conserved between the two players, so the whole pool's average never changes through play, only through players joining and leaving
- •K is set high for new players, whose rating is a poor guess, and low for established ones; FIDE uses 40, 20 and 10
What the numbers mean
The scale has no fixed zero and no top; it is entirely relative, and a rating only means something against other ratings in the same pool. In chess the conventional bands are that 1200 is a beginner who knows the rules, 1600 a strong club player, 2000 an expert, 2200 a national master, 2500 a grandmaster and 2700 the world's top few dozen. The gap between adjacent bands, 200 to 300 points, corresponds to the stronger player winning roughly three games in four. Because the scale is relative, ratings in one era cannot be compared directly with another, and the argument over whether Carlsen is stronger than Kasparov or Fischer cannot be settled by comparing their peak numbers; inflation, deflation and the size of the pool all shift the scale over decades.
Where it falls short
Elo assumed that each player's performance varies around their true strength in a bell curve, which is close to right, and that the true strength is fixed, which is not; a rapidly improving junior is always underrated, and an ageing master overrated. The system also says nothing about how confident it is. A player who has played three games and one who has played three thousand can hold the same rating with very different reliability, and later systems address that: Glicko, used by many online chess servers, attaches a rating deviation to each player that shrinks with play and grows with inactivity, and TrueSkill, developed by Microsoft for Xbox matchmaking, extends the idea to team games with any number of players. All of them keep Elo's core, a number moved by the surprise of each result.
Beyond chess
The method has spread wherever pairs compete. FIFA's world football rankings switched to an Elo-based formula in 2018; the World Football Elo Ratings had been published unofficially since 1997 and predict results better than the old points table did. Backgammon, Go, Scrabble, table tennis and esports use variants, and statistical models of tennis and American football are built on the same expected-score idea. Elo's original paper noted that the method would work for any contest with a winner and a loser, and it has turned out to work for things that are not contests at all, from ranking the attractiveness of faces on a website in 2003, the origin of Facebook, to sorting which of two chatbot answers people prefer. The professor who designed it to settle arguments at chess clubs built a general instrument for turning comparisons into a scale.
Why a rating only makes sense inside its pool
Because points only move between players who meet, a rating is anchored by the games that link it to everyone else. Imagine two chess clubs on islands that never play each other. Each can have a 1600 player, but nothing guarantees the two are equally strong, since each island's numbers drift on their own. The moment players start crossing between the clubs, points flow from the weaker island to the stronger until the two scales line up. This is why a 1500 on one chess website is not the same as a 1500 in a national federation.
New players are the other weak point. A beginner has to be given a starting number before any evidence exists, and the choice matters: start everyone too high and their early losses hand free points to established players; start them too low and fast-improving newcomers take points from everyone as they climb. Many systems treat a new rating as provisional, calculated from the first batch of games as a whole, before switching to the ordinary update. Juniors improving faster than their rating can follow are a well-known source of drift, quietly draining points from the adults they beat.
Performance ratings and title norms
A rating moves slowly, but players often want to know how well they played in a single event. The answer is a performance rating: the rating that would have made your actual score the expected one. A quick approximation adds 400 times your wins minus your losses, divided by the number of games, to your opponents' average rating. Score six out of nine with no draws against opponents averaging 1800, and your performance is about 1800 plus 400 times three over nine, roughly 1933. FIDE uses a more precise table, but the idea is the same.
Performance ratings matter most for titles. To become a grandmaster, a player needs to reach an established rating of 2500 at some point and also earn norms, which are tournament results showing a performance of at least 2600 over at least nine games against suitably strong and varied opponents. Normally three norms are required. This combination means a title reflects both a sustained level and the ability to produce top results in demanding events. It also explains why ambitious players seek out strong international tournaments: a good score against weaker fields cannot produce the performance number the norm requires.
Adapting Elo for football
Using Elo outside chess means deciding what should count as a surprise, and football shows how. The World Football Elo Ratings add 100 points to the home team's rating before working out the expected result, because playing at home is a real advantage. Suppose a side rated 1900 hosts one rated 2000. With the home bonus the two are level, so each has an expected score of 0.5, and a draw moves neither rating at all. Without the bonus the home side's expected score would be about 0.36, and the same draw would count as a small upset in its favour and shift points.
The system also changes how far ratings move. Competitive matches use a larger K than friendlies, with World Cup finals weighted most heavily, since teams try harder when more is at stake. And unlike a chess game, a football result has a margin: a two-goal win counts for more than a one-goal win, a three-goal win for a little more again, with only small additions beyond that, so that thrashing a weak side cannot inflate a rating too far. Each adjustment is a modelling choice, judged as Elo judged his own system, by whether the ratings predict the next results better.
When the rating becomes the target
A number that carries prestige or prizes invites people to manage it rather than earn it. Many open chess tournaments award prizes by rating band, such as best player under 1600, and a player who deliberately loses rated games to stay just under a band's limit can then collect prizes against weaker fields. The practice is called sandbagging. US Chess responded with rating floors: an established player's rating cannot drop below a level tied to the highest rating they have reached, so a strong player cannot sink back into a lower prize section however many games they throw.
Protecting a rating is the opposite temptation. Because an Elo rating can only change through games, a player at a career peak can preserve the number by playing less, avoiding risky opponents or skipping events, and the rating then looks more certain than the evidence behind it. Glicko's growing uncertainty for inactive players is one answer; FIDE marks players as inactive after a period without rated games and leaves them out of its rankings of the top players. Online game companies face the same pressures, which is one reason many hide their matchmaking ratings from players altogether and show a separate, more forgiving rank instead.
The takeaway
An Elo rating is an estimate of a player's strength that moves after every game by an amount proportional to how surprising the result was: the expected score follows from the rating difference on a scale where 400 points means ten wins in eleven, and the rating shifts by a constant times the gap between the actual and expected score. Points are conserved between opponents, the scale is purely relative, later systems add a measure of uncertainty, and the same formula now ranks football teams, video game players and chatbots.