What an Elo rating is
Elo ratings were developed by Arpad Elo to rank chess players, and the same idea, in many variations, now sits behind ranked ladders, matchmaking systems and some sports rankings. The core principle is simple: each player has a number, and the difference between two players’ numbers gives an expected result when they meet.
A higher number suggests greater strength, but the absolute value means little on its own. What matters is the gap between players, and how the number changes as results come in.
Turning a rating gap into an expected score
In the standard version, the expected score of a player is calculated from the rating difference using this formula:
expected score = 1 / (1 + 10^(-difference / 400))
The expected score is a number between 0 and 1. For a win-or-lose game it works like a probability of winning, and a draw counts as half a point in systems that allow draws. The 400 is a scaling choice made in the classic chess system, and other systems pick other scales.
Some illustrative values, rounded:
- Equal ratings (gap 0): expected score 0.50.
- A gap of 100 points: the higher-rated player’s expected score is about 0.64.
- A gap of 200 points: about 0.76.
- A gap of 400 points: about 0.91.
So even a very large gap does not make the result certain. A 400-point gap still gives the lower-rated player an expected score of roughly 0.09.
How a rating updates
After a game, the rating moves according to the difference between the actual result and the expected one:
new rating = old rating + K × (actual score − expected score)
K is a constant that controls how quickly ratings move. Take an illustrative K of 32 and a 100-point gap, where the favourite has an expected score of about 0.64.
- If the favourite wins, the actual score is 1, and the change is 32 × (1 − 0.64) = 32 × 0.36, about +11.5 points.
- If the favourite loses, the actual score is 0, and the change is 32 × (0 − 0.64), about −20.5 points.
The opponent’s rating moves in the opposite direction by the same amount, so the total stays the same. The key feature is that an unexpected result moves ratings more than an expected one. Beating a much stronger opponent gains a lot, and losing to a weaker one costs a lot.
Matchmaking numbers are not always plain Elo
Many games use rating systems that are inspired by Elo but add features. A common one is an uncertainty estimate: a new player’s rating is treated as less reliable and moves faster, then settles as more games are played. Others account for teams, the strength of each teammate, or the margin of victory. Some display one number to the player while using a different internal value for matchmaking. The exact formulas are often not published, so it is better not to assume details.
A common design goal is to pair players so that the expected result is close to even. If the system works as intended, a player’s win rate would hover around 50% regardless of how strong they are, because opponents keep getting matched to their level. That is why a ladder win rate says less about ability than people expect. Reading a win rate in this setting needs the context of the opponents faced.
Limits to keep in mind
- Ratings are estimates. They summarise past results, and a rating based on few games is uncertain. This is the same problem as small sample sizes, and the idea behind confidence intervals applies: a rating is better thought of as a range.
- Short run noise. Streaks of wins or losses happen in random sequences, so short run results can move a rating up or down without a real change in skill.
- Model assumptions. Elo assumes strength can be captured by one number and that the same formula applies at every level. Games with strong matchup effects, such as rock-paper-scissors dynamics between characters or strategies, strain that assumption, a topic related to mixed strategies.
- Different pools, different scales. A rating of 1500 in one system is not comparable to 1500 in another.
- Luck and skill both matter. In games with a large element of chance, ratings move more slowly towards the truth, since each game says less about skill. The distinction between luck versus skill affects how fast a rating system can learn.
A sensible way to read the numbers
Use a rating as a rough guide to relative strength, and use the gap rather than the absolute figure. Expect upsets: a 64% expectation for the favourite means the other player wins roughly one game in three. Don’t read too much into a single game’s rating change, and be cautious about comparing across seasons, regions or games. When your own rating falls after a run of losses, the maths is usually just doing what it was designed to do, and a few results are rarely enough to show that your skill changed. Taking breaks, setting limits on time spent gaming and avoiding frustrated play are more helpful responses than chasing the number.