Programming

Elo Rating System

ProgrammingAlgorithmsRanking

Learn how the Elo rating system turns wins and losses into skill ratings — expected scores, K-factors, zero-sum updates, and why beating a stronger player pays more.

Core Concepts

🎯

The Rating Number

A single number that estimates skill. New players start around 1000–1500 and the number moves after every game — up when you win, down when you lose.

Start ≈ 1500One Number

Use case: Chess titles — 2500+ is Grandmaster territory.

📈

Expected Score

Before the game, Elo predicts your result from the rating gap. Equal ratings mean 50/50. A 400-point gap means the stronger player should score about 91%.

0 to 1Logistic Curve

Use case: Win probability shown before a ranked match.

⚖️

K-Factor

The volume knob for rating changes. A big K makes ratings jump fast (good for new players). A small K keeps established ratings stable.

K = 40 NewK = 10 Elite

Use case: FIDE uses K = 40 for juniors, K = 10 for 2400+ players.

🔄

Zero-Sum Updates

Whatever the winner gains, the loser loses. Rating points are never created or destroyed in a game — they just flow between players.

+N / −NConserved

Use case: Keeps the rating pool stable across millions of games.

💥

The Upset Bonus

Beat someone 400 points above you and you gain a lot. Beat someone 400 points below you and you gain almost nothing. Surprise is what moves ratings.

Big SurpriseBig Reward

Use case: Why farming weak opponents barely raises your rank.

🌍

Beyond Chess

Arpad Elo built it for chess, but the same math powers online game matchmaking, football club rankings, and even “which photo is better” voting systems.

MatchmakingRankings

Use case: Pairing you against similar-skill players in ranked queues.

The Math (Only Two Formulas)

📈 Step 1 — Expected Score

E_A = 1 / (1 + 10^{(R_B − R_A) / 400})

E_A — how many points player A is expected to score (between 0 and 1).
R_A, R_B — the two players’ current ratings.
400 — the scale constant: a 400-point gap means the stronger player is expected to score about 10 times more.
E_A + E_B always equals exactly 1.

🔄 Step 2 — Rating Update

R′_A = R_A + K × (S_A − E_A)

R′_A — player A’s new rating after the game.
S_A — the actual result: win = 1, draw = 0.5, loss = 0.
K — the K-factor: the maximum points a single game can move a rating.
(S − E) — the surprise. Do exactly as expected and your rating barely moves.

📏 What a Rating Gap Means

Rating Gap	Stronger Player’s Expected Score
0	50%
100	64%
200	76%
400	91%
800	99%

One Game, Start to Finish

flowchart TD
    A["Two players matched
Ratings: R_A and R_B"] --> B["Compute expected scores
E_A = 1 / (1 + 10^((R_B - R_A)/400))"]
    B --> C{"Play the game"}
    C -->|"A wins
S_A = 1"| D["Surprise = 1 - E_A"]
    C -->|"Draw
S_A = 0.5"| E["Surprise = 0.5 - E_A"]
    C -->|"A loses
S_A = 0"| F["Surprise = 0 - E_A"]
    D --> G["Update both ratings
R' = R + K × (S - E)"]
    E --> G
    F --> G
    G --> H(["Ratings drift toward
true skill over many games"])

    style A fill:#1a1f2b,stroke:#00e5ff,color:#fff
    style B fill:#12161f,stroke:#ffab00,color:#fff
    style C fill:#1a1f2b,stroke:#b985f4,color:#fff
    style D fill:#0d1117,stroke:#3effa3,color:#fff
    style E fill:#0d1117,stroke:#ffab00,color:#fff
    style F fill:#0d1117,stroke:#ff5252,color:#fff
    style G fill:#12161f,stroke:#ffab00,color:#fff
    style H fill:#0d1117,stroke:#3effa3,color:#aaa

Watch a Rating Update Happen

Two players sit down. Alice (1500) faces Bob (1500). Their ratings are the only thing Elo needs to predict this game.

🙋‍♀️

Alice

1500

+16.0

🙎‍♂️

Bob

1500

-16.0

Expected Score

Alice 50%

Bob 50%

K-Factor 32E_A 0.50E_B 0.50Step 1 / 5

Try It Yourself

Elo Calculator

Player A Rating1500

Player B Rating1700

K-Factor

Result

Expected A24.0%

Expected B76.0%

A Change+24.3

B Change-24.3

A New Rating1524

B New Rating1676

Elo vs Modern Rating Systems

System	Simplicity	Uncertainty Tracking	Team Games	Adoption	Known For
Elo					Chess, the original — two formulas, one number
Glicko-2					Lichess — adds a “how sure are we” deviation value
TrueSkill					Xbox Live — Bayesian, built for team matchmaking

Resources

Title	Type	Link
Elo Rating System (Wikipedia)	web	Open
The Elo Rating System for Chess and Beyond	youtube	Open
The Chess Rating System Explained	youtube	Open
Elo Rating Algorithm — GeeksforGeeks	web	Open

Guide Contributors

ObydulSenior Software Engineer

Edit this page on GitHub↗

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Core Concepts

🎯

The Rating Number

A single number that estimates skill. New players start around 1000–1500 and the number moves after every game — up when you win, down when you lose.

Start ≈ 1500One Number

Use case: Chess titles — 2500+ is Grandmaster territory.

📈

Expected Score

Before the game, Elo predicts your result from the rating gap. Equal ratings mean 50/50. A 400-point gap means the stronger player should score about 91%.

0 to 1Logistic Curve

Use case: Win probability shown before a ranked match.

⚖️

K-Factor

The volume knob for rating changes. A big K makes ratings jump fast (good for new players). A small K keeps established ratings stable.

K = 40 NewK = 10 Elite

Use case: FIDE uses K = 40 for juniors, K = 10 for 2400+ players.

🔄

Zero-Sum Updates

Whatever the winner gains, the loser loses. Rating points are never created or destroyed in a game — they just flow between players.

+N / −NConserved

Use case: Keeps the rating pool stable across millions of games.

💥

The Upset Bonus

Beat someone 400 points above you and you gain a lot. Beat someone 400 points below you and you gain almost nothing. Surprise is what moves ratings.

Big SurpriseBig Reward

Use case: Why farming weak opponents barely raises your rank.

🌍

Beyond Chess

Arpad Elo built it for chess, but the same math powers online game matchmaking, football club rankings, and even “which photo is better” voting systems.

MatchmakingRankings

Use case: Pairing you against similar-skill players in ranked queues.

The Math (Only Two Formulas)

📈 Step 1 — Expected Score

E_A = 1 / (1 + 10^{(R_B − R_A) / 400})

E_A — how many points player A is expected to score (between 0 and 1).
R_A, R_B — the two players’ current ratings.
400 — the scale constant: a 400-point gap means the stronger player is expected to score about 10 times more.
E_A + E_B always equals exactly 1.

🔄 Step 2 — Rating Update

R′_A = R_A + K × (S_A − E_A)

R′_A — player A’s new rating after the game.
S_A — the actual result: win = 1, draw = 0.5, loss = 0.
K — the K-factor: the maximum points a single game can move a rating.
(S − E) — the surprise. Do exactly as expected and your rating barely moves.

📏 What a Rating Gap Means

Rating Gap	Stronger Player’s Expected Score
0	50%
100	64%
200	76%
400	91%
800	99%

One Game, Start to Finish

flowchart TD
    A["Two players matched
Ratings: R_A and R_B"] --> B["Compute expected scores
E_A = 1 / (1 + 10^((R_B - R_A)/400))"]
    B --> C{"Play the game"}
    C -->|"A wins
S_A = 1"| D["Surprise = 1 - E_A"]
    C -->|"Draw
S_A = 0.5"| E["Surprise = 0.5 - E_A"]
    C -->|"A loses
S_A = 0"| F["Surprise = 0 - E_A"]
    D --> G["Update both ratings
R' = R + K × (S - E)"]
    E --> G
    F --> G
    G --> H(["Ratings drift toward
true skill over many games"])

    style A fill:#1a1f2b,stroke:#00e5ff,color:#fff
    style B fill:#12161f,stroke:#ffab00,color:#fff
    style C fill:#1a1f2b,stroke:#b985f4,color:#fff
    style D fill:#0d1117,stroke:#3effa3,color:#fff
    style E fill:#0d1117,stroke:#ffab00,color:#fff
    style F fill:#0d1117,stroke:#ff5252,color:#fff
    style G fill:#12161f,stroke:#ffab00,color:#fff
    style H fill:#0d1117,stroke:#3effa3,color:#aaa

Elo vs Modern Rating Systems

System	Simplicity	Uncertainty Tracking	Team Games	Adoption	Known For
Elo					Chess, the original — two formulas, one number
Glicko-2					Lichess — adds a “how sure are we” deviation value
TrueSkill					Xbox Live — Bayesian, built for team matchmaking

Resources

Title	Type	Link
Elo Rating System (Wikipedia)	web	Open
The Elo Rating System for Chess and Beyond	youtube	Open
The Chess Rating System Explained	youtube	Open
Elo Rating Algorithm — GeeksforGeeks	web	Open

Guide Contributors

ObydulSenior Software Engineer