How Elo Ratings Work, and How to Rate a Whole Table

Elo is one of the few rating systems simple enough to work through by hand and good enough that it has survived seventy years of people trying to replace it. It runs chess, it runs most online game ladders in some modified form, and it works perfectly well for a board game group that wants to settle an argument.

Step one: the expected score

The scale is deliberately not linear. The first hundred points of advantage buy you a lot; the eighth hundred buys you almost nothing, because you were already expected to win nearly every game. This is the property that makes Elo self-correcting.

Step two: the update

This table is the part people find counterintuitive, and it is the heart of the system. Winning the game you were supposed to win earns 7.7 points. Drawing it loses you 8.3 — a draw against a much weaker player is a bad result, and Elo says so. Losing it costs 24.3, more than three times what winning paid.

The K-factor is the only real tuning knob

K sets how far a single result can move a rating. High K means the rating chases recent results and is noisy; low K means it is stable but slow to notice that someone has genuinely improved. Every rating system is a choice about that trade.

FIDE resolves it by making K depend on how much is already known about the player. Under the FIDE Rating Regulations, K is 40 until a player has completed 30 rated games, 40 for juniors in the year of their 18th birthday or younger while they remain under 2300, 20 for an established player, and 10 once a published rating has ever reached 2400. One more rule caps it: K multiplied by the number of games in a rating period may not exceed 700.

Note the wording on the last row: it is triggered by having ever reached 2400, not by currently being above it. A player who peaks at 2405 and drifts down to 2300 keeps K = 10.

For a casual group, K = 32 is a reasonable default and K = 16 is better once everyone has played a lot of games. The important thing is to pick one and leave it alone, because changing K mid-season makes earlier results and later ones incomparable.

More than two players

Elo was built for two-player games, which is awkward for a board game night where five people sit down to one game. The standard extension is to treat the table as a round robin: every pair of players is scored against each other as if they had played a head-to-head game, with the higher finisher taking the point.

Each player therefore plays n − 1 notional games. Divide the total change by that number of opponents and one evening counts as one game rather than as four, which stops a six-player game swinging ratings six times as hard as a two-player one.

The changes sum to exactly zero, which is the check that the extension is well formed. Ben and Cal started on the same rating and finished adjacent, so they swap 5.3 points — the entire difference between second and third place at equal rating. Ana gains despite being the favourite because she had to beat three players, not one.

What Elo does not know

• It does not model draws explicitly. Chess handles this by treating a draw as half a point, which works, but a game where draws are common and a game where they are impossible behave differently under the same formula.
• It has no concept of margin. Winning a board game by one point and by fifty are the same result.
• It assumes ratings are comparable. Two groups that never play each other will drift apart, and their numbers mean nothing against each other however similar they look.
• It has no uncertainty. A rating of 1500 after four games and 1500 after four hundred are treated identically, which is why newer systems such as Glicko carry a rating deviation alongside the rating.
• In multiplayer games it ignores everything about the table beyond finishing order — alliances, turn order advantage, who got attacked first.

None of these stop Elo being useful. They are the reasons to treat a rating as a summary of results rather than as a measurement of skill.

How FIDE actually computes the expected score

The formula at the top of this guide is the textbook logistic form of Elo, and it is what most online calculators and club systems use. FIDE does not use it directly. Its Rating Regulations convert a rating difference into a scoring probability using a published lookup table, table 8.1.2, which gives the probability to two decimal places for bands of rating difference.

For ordinary differences the two agree to within about a hundredth, so the logistic formula is a perfectly good way to understand the system. At the edges they part company, for a reason that has nothing to do with the table itself.

The 400-point rule

Rule 8.3.1 of the FIDE regulations, effective from 1 October 2025, says that a rating difference of more than 400 points is counted as though it were 400 for players rated below 2650. Above 2650 the actual difference is used. That flattens the expected score at 0.92 for any gap wider than 400, which is why the last two rows of the table above stop moving.

The effect is to make mismatched games slightly less punishing for the stronger player and slightly more rewarding. Without the cap, a player several hundred points above an opponent gains almost nothing for a win and risks nearly a full K on a single loss. Note that the rule is written for players below 2650; for players at 2650 and above the regulations use the actual difference in every case, so the uncapped column applies to them.

Rating a whole tournament: batch versus game by game

FIDE rates a tournament or rating period as a batch. Every game is scored against the player's rating at the start of the period, the differences between actual and expected scores are added up, and the total is multiplied by K once. Many club and online systems instead update the rating after every game. The two methods give slightly different answers from the same results.

The batch method gives about a point more here, because every game is scored against the lower, pre-event rating. Game by game, the rating rises after the first win, so each later game carries a slightly higher expectation and a slightly smaller reward. Neither is wrong; they are different conventions. FIDE's is simpler to audit, since the result does not depend on the order in which games were played.

How long it takes a rating to find the truth

The K-factor trade-off is easier to feel with a concrete case. Imagine a player whose real strength is 1700 but who starts at 1500 and plays only 1500-rated opponents. On average they score what a 1700 would against a 1500, 0.76 per game, while the system expects less and so pays them the difference. Here is how quickly the rating closes the gap, following the average path.

Halving K roughly doubles the number of games needed to catch up. This is the logic behind a higher K for new players and juniors: both are players whose rating is likely to be well away from their true strength, and at a K of 10 a gap like this one would take well over a hundred games to close. It is also why the same K would be wrong for an established master, whose rating is already close to the truth and only needs to track slow change.

Setting up Elo for a club or games group

• **Pick a starting rating and give it to everyone.** 1500 is a common convention. The number itself is arbitrary; what matters is that everyone starts in the same place.
• **Pick a K and write it down.** 32 for a new group is a reasonable choice, dropping to 16 once everyone has played twenty or thirty games. Changing K later makes old and new results incomparable, so decide in advance when it will change.
• **Consider a provisional period.** Treat a newcomer's rating as provisional until they have played a set number of games, and give them a higher K in that time, the same idea as FIDE's 40.
• **Record every result, not just wins.** Draws and losses carry as much information as wins, and a rating that only counts the games people remember is not a rating.
• **Decide how multiplayer games count** before the first one is played. The round-robin method described above is a reasonable default, divided by the number of opponents so one evening counts as one game.
• **Keep separate pools for separate games.** A chess rating and a board game rating measure different skills. Mixing results produces a number that means nothing.

Common misconceptions about Elo

• **An expected score of 0.76 means a 76% chance of winning.** It means 0.76 points per game on average, with draws counted as half. A player who draws a lot can have that expected score while winning far less than 76% of games.
• **Ratings measure absolute skill.** They measure performance relative to the pool. A 1500 in one club and a 1500 in another are only comparable if the two pools play each other.
• **You can protect a rating by not playing.** In a pure Elo system an inactive rating stays where it is, but it also becomes less accurate, and some systems apply inactivity rules. The rating reflects results, not merit.
• **Beating a higher-rated player always gains a lot.** It gains K times the gap between 1 and your expected score. Against a slightly stronger opponent that is only a little over half of K.
• **Everyone's rating goes up over time.** In a closed pool Elo is zero sum. If some ratings rise, others fall by the same total.

Performance rating: what a tournament says about you

A rating change tells you how far your rating moved. A performance rating tells you what rating your results in one event would justify on their own. FIDE's regulations include a second table, 8.1.1, that converts a fractional score p into a rating difference dp. Add dp to the average rating of your opponents and you have the performance figure. Many calculators, including ours, use a simpler linear version instead: the average opponent plus 400 points for each net win, divided by games played.

For middling scores the two agree within a dozen points or so. At the extremes they diverge sharply, and FIDE's own table shows why a perfect score is a special case: it lists a 100% score as 800 points only "notionally", because a perfect result is consistent with any rating far enough above the field. Treat any performance rating from a perfect or zero score as meaningless, whichever method produced it, and treat one from a handful of games with caution. Five games is a small sample for a number that is meant to summarise playing strength.

How often the underdog scores

Expected score is also a way of reading upsets. Using the lower-rated player's column of FIDE's table 8.1.2, here is what an underdog is expected to score per game at common rating gaps, and what that adds up to over a season of twenty such games.

A player 200 points below their opponents is expected to take nearly five points from twenty games, which means an upset is not an event but a routine part of a season. Expected score counts draws as half a point, so those points may arrive as a mixture of wins and draws. The practical lesson is that one surprising result says very little about either player. It is the running total across many games that the rating is built to read, and that is also what makes the system hard to game with a single good day.

Sources

• FIDE Handbook, B.02: FIDE Rating Regulations effective from 1 March 2024 — the K-factor rules including the junior rule and the 700 cap, the 400-point rule, and table 8.1.2, which FIDE uses in place of the logistic formula: https://handbook.fide.com/chapter/B022024
• The multiplayer extension is the standard pairwise round-robin generalisation, divided by the number of opponents so that one multiplayer game counts as one game. It is a convention rather than a published standard, and the calculator states as much.

Where to go next

Elo is one way of levelling an uneven contest; a handicap is the other. Bowling handicaps and darts checkouts works through why an 80% handicap deliberately leaves the better player ahead, which is the same design question Elo answers with its K-factor. If it is expected value you are interested in rather than ratings, pot odds and outs is the same reasoning applied to a single decision.

Common questions