Elo Expected Score
Probability player A beats player B given their ratings.
Formula
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Example
R_a=1800, R_b=1600: E = 1/(1+10^{-0.5}) ≈ 0.76
How it works
The Elo expected score turns two players’ ratings into the long-run chance that player A scores a win-equivalent result against player B. It works by comparing the rating gap: a higher-rated player is expected to score more often, and each 400-point difference corresponds to a tenfold change in the odds ratio used by the model. This is why the formula is smooth rather than all-or-nothing: even a weaker player still has a nonzero chance, just a smaller one.
When to use it
Use this when you need to estimate match expectations in Elo-based systems such as chess, esports ladders, ranking simulations, or any head-to-head competition using ratings. It is especially useful before a game to predict the likely score outcome or to compare how favorable a pairing is.
Worked example, step by step
Given: R_a = 1450, R_b = 1570 Step 1: Find the rating difference: R_b - R_a = 1570 - 1450 = 120 Step 2: Divide by 400: 120/400 = 0.3 Step 3: Substitute into the formula: E = 1 / (1 + 10^(0.3)) Step 4: Compute 10^(0.3) ≈ 1.995 Step 5: Evaluate the denominator: 1 + 1.995 = 2.995 Step 6: Final expected score: E = 1/2.995 ≈ 0.334 So player A’s expected score is about 0.33, or 33.4%.
Common mistakes to avoid
- Using the rating difference in the wrong direction, which flips the probability
- Treating E as a guaranteed win chance instead of an average score over many games
- Forgetting that the exponent uses base 10 and the 400-point scale matters
Did You Know?
A 400-point Elo gap means the favourite is expected to win about 91% of the time.
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