"algo-rank-trueskill"

TrueSkill Rating System

Overview

TrueSkill (Microsoft Research) models each player's skill as a Gaussian distribution N(μ, σ²) where μ is estimated skill and σ is uncertainty. Supports teams and multiplayer (not just 1v1). Conservative rating = μ - 3σ. Uncertainty decreases with more games. Uses Bayesian inference via message passing.

When to Use

Trigger conditions:

• Rating players in team-based or multiplayer (3+ participant) games
• Building matchmaking systems that balance match quality
• When you need uncertainty estimates alongside skill ratings

When NOT to use:

• For simple 1v1 ranking with no uncertainty (Elo is simpler)
• For non-competitive ranking (product ratings — use Wilson Score)

Algorithm

IRON LAW: Skill Rating Has TWO Components — Mean AND Uncertainty
TrueSkill represents skill as N(μ, σ²). New players have high σ
(uncertain). After many games, σ shrinks (confident). The conservative
rating μ - 3σ ensures players are ranked by their LIKELY MINIMUM
skill, not their estimated average. Never use μ alone for ranking.

Phase 1: Input Validation

Initialize: μ₀ = 25, σ₀ = 25/3 (default). Collect match results with team compositions and finishing order. Gate: Valid match results, team compositions defined.

Phase 2: Core Algorithm

1. For each match, compute expected outcome from team skill distributions
2. Compare actual vs expected outcome
3. Update each player's (μ, σ) using Bayesian update:
   • μ shifts toward performance (up for winners, down for losers)
   • σ decreases (less uncertain after observing outcome)
   • Amount of update is proportional to σ (uncertain players change more)
4. Conservative rank = μ - 3σ

Phase 3: Verification

Check: σ decreases over time for active players. Top-ranked players by conservative rating win more than expected. Match quality metric (draw probability) is reasonable. Gate: Rating system produces intuitive rankings, σ converges.

Phase 4: Output

Return player ratings with uncertainty bounds.