Elo Rating Update

Background

The Elo system updates two players' ratings after a match. The chess-rating heritage carried over directly to LLM-arena leaderboards: each side-by-side vote is a "match", winning model gains rating points, losing model drops. Anchor it to a starting rating (typically 1500) and you have ranked LMArena outputs.

Problem statement

Implement elo_update(rating_a, rating_b, score_a, K=32):

$$E_A = \frac{1}{1 + 10^{(r_B - r_A)/400}},\qquad r_A' = r_A + K(s_A - E_A),\qquad r_B' = r_B + K\bigl((1 - s_A) - (1 - E_A)\bigr)$$

score_a is $1$ if A won, $0$ if B won, $0.5$ for a draw. Return (new_a, new_b).

Input

• rating_a, rating_b — float, current ratings.
• score_a — float in $\{0, 0.5, 1\}$.
• K — float, the K-factor (default 32 for new ratings; 16 for stable).

Output

Returns a tuple (new_a: float, new_b: float).

Examples

Example 1 — equal ratings, A wins → A gains K/2

Input:  rating_a=1500, rating_b=1500, score_a=1.0, K=32
Output: (1516.0, 1484.0)

Example 2 — equal ratings, draw → no change

Input:  rating_a=1500, rating_b=1500, score_a=0.5
Output: (1500.0, 1500.0)

Example 3 — total rating is conserved

Input:  any single update
Output: (new_a + new_b) == (rating_a + rating_b)

Example 4 — upset → large swing

Input:  rating_a=1800, rating_b=1400, score_a=0.0, K=32
Output: a drops, b rises by ~30 points

Constraints

• score_a in $\{0, 0.5, 1\}$ (binary or draw).
• K > 0.
• Returns Python floats.

Notes

• K-factor. Higher K = faster adjustment. FIDE uses 40 for new players, 20 for established, 10 for masters. LMArena tunes K based on number of votes received.
• Bradley-Terry equivalence. Elo ratings are a logarithmic re-scaling of Bradley-Terry strengths: $\text{Elo}_i = 400 \log_{10}(r_i)$.