Rating
PL-Elo Exact — a custom three-player rating model (Plackett–Luce closed form + per-player σ tracking on the Elo scale).
Overview
Standard Elo assumes two players. Three-player tournaments need closed-form expected scores for all permutations of finishing order, including ties for second and third in cascading checkmate. PL-Elo Exact is the custom hybrid we use — Plackett–Luce closed form for expected scores, with per-player σ (uncertainty) tracking on the Elo scale.
The full system has 15 axioms (R1–R15). All five open questions (RQ1–RQ5) have been resolved. The model is locked; implementation across the server and client is scheduled but not yet shipped.
Key parameters
μ₀ = 1200(starting rating mean)σ₀ = 350(starting uncertainty)σ_min = 50(floor)β = 200(performance noise)K_max = 48,K_min = 16(K-factor derived from σ)R_display = max(0, μ − 2σ)— new players start at 500 displayed
What the system covers
- Closed-form expected score for each permutation of (1st, 2nd, 3rd) given three μ values
- K-factor derived from σ
- Zero-sum normalisation across the table
- Cascading checkmate handling: tie for 2nd/3rd scores 0.25 each
- Placement-only update (Snellman’s rule — never use mid-game metrics)
- Provisional gating (
σ > 200or fewer than 15 games) - Seasonal soft reset: decay 0.75, σ floor 175
- Inactivity drift:
σ × √(1 + 0.5 × days), back to provisional after 30 days
Status
Decided · 0/15 implemented as of 2026-05. Implementation lands alongside the ranked server release.
Coming soon — the full R1–R15 listing with worked numerical examples is being prepared.