Weng and Lin 2011 - Glicko Derivation and Comparison with Elo

Summary

The paper re-derives Glicko (Glickman 1999) in five steps (integrate out opponents under their priors, logistic-to-normal approximation, CLT, Taylor expansion) and reports a single-game version. Glicko’s rating-period update is justified when each player plays about 5-10 games per period; its one-game form is a weaker approximation than BT-full.

Overview

Prior skill $\theta \sim N(\mu ,{\sigma }^2)$; during a rating period the player plays $n_j$ games against opponent $j=\mathrm{1..}m$ with ${\theta }_j\sim N({\mu }_j,{\sigma }_j^2)$; outcomes $s_{jk}\in \{1,\frac{1}{2},0\}$. Glicko works on the Elo scale (base 10, divisor 400).

Main Content

Glicko approximation steps

(I) Approximate the opponents’ posterior by their priors: $P({\theta }_1..{\theta }_m\mid D)\approx {\prod }_j\varphi ({\theta }_j\mid {\mu }_j,{\sigma }_j^2)$ (Glickman: “integrating out the opponents’ strengths over their prior rather than their posterior”). (II) Define $P(D\mid \theta )={\prod }_{j,k}\int (\frac{10^{(\theta -{\theta }_j)/400}}{1+10^{(\theta -{\theta }_j)/400}}{)}^{s_{jk}}\varphi ({\theta }_j)d{\theta }_j$. (III) Replace each logistic cdf with a normal cdf of equal mean and variance so the integral closes: $\approx (\frac{10^{g({\sigma }_j^2)(\theta -{\mu }_j)/400}}{1+10^{g({\sigma }_j^2)(\theta -{\mu }_j)/400}}{)}^{s_{jk}}$ with $g({\sigma }^2)=1/\sqrt{1+3q^2{\sigma }^2/{\pi }^2}$, $q=\ln 10/400$; then approximate the likelihood by a normal $\varphi (\theta \mid \hat{\theta },{\delta }^2)$, ${\delta }^{-2}\approx q^2{\sum }_j{n}_{j}g({\sigma }_j^2{)}^2{p}_j(1-p_j)$. (IV) Obtain $\hat{\theta }-\mu$ from a first-order Taylor expansion of the score equation ${\sum }_{jk}g({\sigma }_j^2)s_{jk}=h(\hat{\theta })$. (V) Combine: ${\sigma }^{-2}\leftarrow {\sigma }^{-2}+{\delta }^{-2}$ and $\mu \leftarrow \mu +\frac{q}{{\sigma }^{-2}+{\delta }^{-2}}{\sum }_{j,k}g({\sigma }_j^2)(s_{jk}-p_j(\mu ))$.

Algorithm 5 (Glicko, single game)

$q=\ln 10/400$, $g({\sigma }^2)=(1+3q^2{\sigma }^2/{\pi }^2{)}^{-1/2}$, $p_j^{\ast }=1/(1+10^{-g({\sigma }_j^2)({\mu }_i-{\mu }_j)/400})$, $({\delta }_i^2{)}^{\ast }=[q^{2}g({\sigma }_j^2{)}^2{p}_j^{\ast }(1-p_j^{\ast }){]}^{-1}$; ${\mu }_i\leftarrow {\mu }_i+\frac{q}{{\sigma }_i^{-2}+({\delta }_i^2{)}^{\ast -1}}g({\sigma }_j^2)(s_{ij}-p_j^{\ast })$, ${\sigma }_i^2\leftarrow [{\sigma }_i^{-2}+({\delta }_i^2{)}^{\ast -1}{]}^{-1}$.

Caveat. With one game the likelihood is monotone, so its mode is $\pm \infty$ and the CLT step has no justification; the algebra survives but Glicko treats games in a rating period as simultaneous. The paper reports single-game Glicko error 33.88% on Head-to-Head, slightly worse than BT-full (32.53%).

Comparison with Algorithm 1. Both use BT. Glicko: precision-weighted (information-additive) update with $g(\cdot )$ attenuating opponents’ uncertainty; Weng-Lin: opponent uncertainty enters via $c_{iq}$ and update is applied per game and extends to multi-team ranks.

Examples

Glicko step for a new player

Setup. Player $\mu =1500$, $\sigma =350$; opponent ${\mu }_j=1400$, ${\sigma }_j=30$; one win ($s=1$). Solution. $q=0.005756$; $g(900)\approx 0.9955$ for ${\sigma }_j=30$ (variance 900); $p^{\ast }=1/(1+10^{-0.9955\cdot 100/400})\approx 0.640$; $({\delta }^2{)}^{\ast -1}=q^2{g}^2{p}^{\ast }(1-p^{\ast })\approx 7.6\times 10^{-6}$; ${\sigma }^{-2}=8.16\times 10^{-6}$; new $\mu \approx 1500+qg(s-p^{\ast })/(8.16+7.6)10^{-6}\approx 1500+131\approx 1631$. Interpretation. A high-$\sigma$ newcomer moves a long way after a single game, as with Weng-Lin; the step shrinks as $\sigma$ falls.

Connections

• Both reduce to Elo when variance terms are dropped. See Elo, Glicko and TrueSkill - Dynamic Skill Ratings (Concept Note).

See Also

Weng and Lin 2011 - Halo 2 Experiments and Results