Glicko-2 Rating Calculator (one game)

Understanding the Glicko-2 rating calculator — concept and principle

The Glicko-2 system, designed by Mark Glickman, estimates a player's skill with two numbers instead of one: a rating r and a rating deviation RD that says how uncertain that rating is. This calculator updates both after a single game, showing how much a win, draw or loss should move you.

How it is calculated

1. Convert to the Glicko-2 scale: μ = (r − 1500)/173.7178 and φ = RD/173.7178, for you and the opponent (μj, φj).
2. Damp the opponent's influence by their uncertainty: g(φj) = 1/√(1 + 3φj²/π²).
3. Expected score: E = 1/(1 + exp(−g(φj)(μ − μj))).
4. Information from the game: v = 1/(g² E (1 − E)).
5. Pre-game uncertainty grows with volatility: φ* = √(φ² + σ²).
6. New deviation: φ′ = 1/√(1/φ*² + 1/v); new rating: μ′ = μ + φ′² g (s − E), where s is 1, 0.5 or 0.
7. Convert back: r′ = 1500 + 173.7178 μ′ and RD′ = 173.7178 φ′.

The full method also updates the volatility σ with an iterative step controlled by τ; here σ is held constant for simplicity.

The principle

Glicko treats your true strength as a bell curve centred on r with spread RD. Each game is evidence: the update is a Bayesian-style step where the size of the move depends on how surprising the result was (s − E) and how uncertain you were (φ′²). A player with a large RD moves a lot; a well-established player moves little. Results against an opponent with a large RD count for less, because g(φj) shrinks toward zero when we are unsure who you actually played.

Worked example

1. Inputs: r = 1500, RD = 200, σ = 0.06, opponent 1400 with RD 30, result Win (s = 1).
2. μ = 0, φ = 1.15129; μj = −0.57565, φj = 0.17269.
3. g = 0.99550; E = 1/(1 + e^(−0.99550 × 0.57565)) = 0.63947, so the expected score is 63.95%.
4. v = 1/(0.99550² × 0.63947 × 0.36053) = 4.37680.
5. φ* = √(1.15129² + 0.06²) = 1.15285; φ′ = 1/√(1/1.15285² + 1/4.37680) = 1.00970.
6. μ′ = 0 + 1.00970² × 0.99550 × (1 − 0.63947) = 0.36590.
7. New rating = 1500 + 173.7178 × 0.36590 = 1,563.56; new RD = 173.7178 × 1.00970 = 175.40.

How to read the result

A rough 95% range for your true strength is r ± 2RD, so 1563.6 ± 351 is still very wide after one game. A new player usually starts at RD 350; values around 50 or below indicate a well-established rating. The expected score counts draws as half a point, so it is not a pure win probability.

Common mistakes

• Comparing ratings from different systems or pools; a 1600 on one site is not a 1600 elsewhere.
• Running many games as separate single-game updates; Glicko-2 is meant to process all games of a rating period together.

Frequently asked questions

Why did my RD go down after a game?

Each result adds information (1/v), which shrinks uncertainty; RD grows only through σ.

How is Glicko different from Elo?

Elo uses a fixed K-factor for everyone. Glicko replaces it with RD, so the step size adapts to how reliable each rating is.

Sources: Mark E. Glickman, "Example of the Glicko-2 system", Boston University · Mark E. Glickman, "Parameter estimation in large dynamic paired comparison experiments", Applied Statistics 48 (1999)