algo-rank-trueskill
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ChineseTrueSkill Rating System
TrueSkill评分系统
Overview
概述
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.
TrueSkill（微软研究院提出）将每位玩家的技能建模为高斯分布N(μ, σ²)，其中μ为预估技能值，σ为不确定性。支持团队和多人模式（不仅限于1v1）。保守评分为μ - 3σ。随着对局次数增加，不确定性σ会降低。通过消息传递的方式采用贝叶斯推理。
When to Use
使用场景
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)
触发条件：
- 对团队类或多人（3名及以上参与者）游戏中的玩家进行评分
- 构建平衡对局质量的匹配系统
- 需要在技能评分之外获取不确定性估计时
不适用场景：
- 无需不确定性的简单1v1排名（Elo算法更简便）
- 非竞技类排名（如产品评分——应使用Wilson Score）
Algorithm
算法流程
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.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
阶段1：输入验证
Initialize: μ₀ = 25, σ₀ = 25/3 (default). Collect match results with team compositions and finishing order.
Gate: Valid match results, team compositions defined.
初始化：μ₀ = 25，σ₀ = 25/3（默认值）。收集包含团队组成和排名结果的对局数据。
校验点：对局结果有效，团队组成明确。
Phase 2: Core Algorithm
阶段2：核心算法
- For each match, compute expected outcome from team skill distributions
- Compare actual vs expected outcome
- 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)
- Conservative rank = μ - 3σ
- 针对每一场对局，根据团队技能分布计算预期结果
- 对比实际结果与预期结果
- 使用贝叶斯更新法更新每位玩家的(μ, σ)：
- μ会向实际表现方向偏移（获胜者上升，失败者下降）
- σ会降低（观察到结果后不确定性减少）
- 更新幅度与σ成正比（不确定性高的玩家评分变化更大）
- 保守排名 = μ - 3σ
Phase 3: Verification
阶段3：验证
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.
检查：活跃玩家的σ随时间推移逐渐降低。按保守评分排名靠前的玩家获胜次数高于预期。对局质量指标（平局概率）合理。
校验点：评分系统生成的排名符合直觉，σ趋于收敛。
Phase 4: Output
阶段4：输出
Return player ratings with uncertainty bounds.
返回包含不确定性范围的玩家评分。
Output Format
输出格式
json
{
"ratings": [{"player": "P1", "mu": 32.5, "sigma": 2.1, "conservative": 26.2, "games_played": 50}],
"metadata": {"initial_mu": 25, "initial_sigma": 8.33, "beta": 4.17, "tau": 0.083}
}json
{
"ratings": [{"player": "P1", "mu": 32.5, "sigma": 2.1, "conservative": 26.2, "games_played": 50}],
"metadata": {"initial_mu": 25, "initial_sigma": 8.33, "beta": 4.17, "tau": 0.083}
}Examples
示例
Sample I/O
输入输出样例
Input: Team [A(25,8.3), B(25,8.3)] beats Team [C(25,8.3), D(25,8.3)]
Expected: A,B μ increases ~2-3 pts, σ decreases ~0.5. C,D μ decreases, σ decreases. Conservative ratings adjust.
输入：团队[A(25,8.3), B(25,8.3)]击败团队[C(25,8.3), D(25,8.3)]
预期结果：A、B的μ值上升约2-3分，σ值下降约0.5。C、D的μ值下降，σ值也下降。保守评分相应调整。
Edge Cases
边缘情况
| Input | Expected | Why |
|---|---|---|
| New vs veteran player | New player μ changes more | Higher σ = more uncertainty = larger updates |
| 1v1 match | Degenerates to Elo-like behavior | TrueSkill reduces to simple case for 1v1 |
| Free-for-all (8 players) | All pairs compared | Multiplayer native support, unlike Elo |
| 输入 | 预期结果 | 原因 |
|---|---|---|
| 新手玩家 vs 资深玩家 | 新手玩家的μ值变化更大 | σ值越高（不确定性越强），评分更新幅度越大 |
| 1v1对局 | 退化为类Elo算法的表现 | TrueSkill在1v1场景下会简化为简单模式 |
| 自由对战（8名玩家） | 所有玩家两两对比 | 原生支持多人模式，与Elo算法不同 |
Gotchas
注意事项
- Computational cost: Message passing in factor graphs is more expensive than Elo. For millions of players, use approximations (EP truncation).
- Team skill aggregation: TrueSkill sums individual Gaussians for team skill. This assumes independence — correlated player skills (practiced teams) are undermodeled.
- Dynamic skill: σ only decreases. If a player's skill genuinely changes (improvement or decline), add a small drift term τ per time period to increase σ gradually.
- Partial play: If a player joins mid-game or leaves early, their contribution is ambiguous. Need partial-play weight extension.
- Patent status: TrueSkill was patented by Microsoft (expired 2024). TrueSkill 2 adds more features but check licensing.
- 计算成本：因子图中的消息传递比Elo算法更耗时。对于数百万玩家规模的场景，需使用近似方法（如EP截断）。
- 团队技能聚合：TrueSkill将个体高斯分布求和得到团队技能。这假设玩家技能相互独立——对于配合熟练的团队，其技能相关性未被充分建模。
- 动态技能：σ只会降低。如果玩家的技能确实发生变化（提升或下降），需添加一个小的漂移项τ，随时间逐步增大σ。
- 部分参与对局：若玩家中途加入或提前退出，其贡献难以界定。需使用部分参与权重扩展方案。
- 专利状态：TrueSkill由微软申请专利（2024年已过期）。TrueSkill 2增加了更多功能，但需注意许可问题。
References
参考资料
- For TrueSkill factor graph derivation, see
references/factor-graph.md - For matchmaking quality metrics, see
references/matchmaking.md
- 关于TrueSkill因子图推导，请参阅
references/factor-graph.md - 关于对局匹配质量指标，请参阅
references/matchmaking.md