Mathematical Foundations & Rating Formulations

This document provides the formal mathematical derivations for the three rating models implemented in Through the Ages (TTA) Rating Engine:

1. Glicko-2 Standard (Naive Pairwise 1v1)
2. Glicko-2 MP-Weighted (Coulom-Weighted Fractional Pairwise)
3. Whole-History Rating (WHR) (Continuous Brownian Motion MAP)

1. Glicko-2 Standard Formulation

The standard Glicko-2 system (Glickman, 2012) models each player by three parameters:

• Rating $\mu$ (scaled such that $\mu = (r - 1500) / 173.7178$)
• Rating Deviation $\phi$ (scaled such that $\phi = RD / 173.7178$)
• Rating Volatility $\sigma$ (degree of expected rating fluctuations)

1.1 Match Likelihood & Expected Outcome

Against an opponent with rating $\mu_j$ and rating deviation $\phi_j$, the expected outcome $E$ is given by:

1.2 Step Equations (Sequential Update)

For $m$ match encounters in a rating period with observed scores $s_j \in \{1.0, 0.5, 0.0\}$:

1. Estimated Variance ($v$):

   $$v = \left[ \sum_{j=1}^{m} g(\phi_j)^2 E_j (1 - E_j) \right]^{-1}$$

2. Estimated Improvement ($\Delta$):

   $$\Delta = v \sum_{j=1}^{m} g(\phi_j) (s_j - E_j)$$

3. Volatility Update ($\sigma'$): Using the Illinois bracket root-finding algorithm, find $x = \ln(\sigma'^2)$ such that $f(x) = 0$, where $a = \ln(\sigma^2)$:

   $$f(x) = \frac{e^x (\Delta^2 - \phi^2 - v - e^x)}{2 (\phi^2 + v + e^x)^2} - \frac{x - a}{\tau^2}$$

4. New Rating Deviation ($\phi'$) and Mean ($\mu'$):

   $$\phi^* = \sqrt{\phi^2 + \sigma'^2}$$ $$\phi' = \frac{1}{\sqrt{\frac{1}{(\phi^*)^2} + \frac{1}{v}}}$$ $$\mu' = \mu + (\phi')^2 \sum_{j=1}^{m} g(\phi_j)(s_j - E_j)$$

5. Conservative Rating:

   $$C = r - 3 \times RD$$

   This represents the lower 99.7% confidence bound of a player's true skill.

2. The Multiplayer Flaw & Glicko-2 MP-Weighted Formulation

2.1 The Problem with Naive Pairwise Decomposition

In an $N$-player game (e.g. 4-player TTA), a player simultaneously plays against 3 opponents. Naive decomposition treats this single match as $(N - 1) = 3$ independent 1v1 duels.

Because Fisher information $v^{-1} = \sum g^2 E(1-E)$ sums over all opponents, a 4-player match yields $3\times$ the information of a 2-player match.

In the variance equation:

$\frac{1}{v}$ is 3 times larger, meaning the rating deviation $\phi$ shrinks up to $\sqrt{3} \approx 1.73$ times faster than warranted by real game entropy. This causes premature confidence, rating crystallization, and ranking instability when players participate predominantly in 4-player lobbies.

2.2 The Coulom Fractional Match Weight Solution

To preserve correct evidence accumulation, each pairwise encounter derived from an $N$-player match is assigned a fractional weight:

• 2-player duel: $N=2 \implies w_j = 1.0$ (Standard Glicko-2)
• 3-player match: $N=3 \implies w_j = 0.5$ (2 opponents $\times 0.5 = 1.0$ effective match)
• 4-player match: $N=4 \implies w_j = \frac{1}{3}$ (3 opponents $\times \frac{1}{3} = 1.0$ effective match)

2.3 Mathematical Integration

1. Weighted Inverted Variance ($v_w^{-1}$):

   $$v_w = \left[ \sum_{j=1}^{m} w_j \cdot g(\phi_j)^2 E_j (1 - E_j) \right]^{-1}$$

2. Weighted Improvement ($\Delta_w$):

   $$\Delta_w = v_w \sum_{j=1}^{m} w_j \cdot g(\phi_j) (s_j - E_j)$$

3. Invariance of Volatility Search: Because $v_w$ scales by $(N - 1)$ and $\sum w_j \dots$ scales by $\frac{1}{N - 1}$, $\Delta_w$ maintains the exact physical magnitude of a standard 2-player duel. Thus, $f(x)$ remains strictly bounded, and Illinois root-finding converges with standard tolerances.

4. Updated Parameters:

   $$\frac{1}{(\phi')^2} = \frac{1}{(\phi^*)^2} + \frac{1}{v_w}$$ $$\mu' = \mu + (\phi')^2 \sum_{j=1}^{m} w_j \cdot g(\phi_j)(s_j - E_j)$$

Conclusion: The effective sample size per match event is strictly 1.0. Ratings remain on the standard 1500 scale while rating deviation ($RD$) reliably reflects true information entropy.

3. Whole-History Rating (WHR)

Whole-History Rating (Coulom, 2008) is a global Bayesian model that simultaneously fits a player's entire career trajectory across time.

3.1 Prior: Wiener Process (Brownian Motion)

A player's skill $r(t)$ on day $t$ evolves as a continuous random walk:

where $w^2$ is the skill variance drift rate per day ($w^2 = 0.002$ in our calibrated implementation).

3.2 Likelihood & Objective Function

Under the Bradley-Terry model, the probability of player $i$ beating opponent $j$ on day $t$ is:

The joint posterior log-density is the sum of game log-likelihoods and Brownian prior transition log-densities:

3.3 Optimization via Thomas Algorithm

Maximization with respect to $r_i = [r_i(t_1), \dots, r_i(t_K)]^T$ yields a symmetric positive-definite tridiagonal system:

where:

• $A_i = H_{\text{lik}} + \Sigma_{\text{prior}}^{-1}$
• $\mathrm{rhs}_i = g_{\text{lik}} - \Sigma_{\text{prior}}^{-1} r_i$

This system is solved in linear time $\mathcal{O}(K)$ per player using the Thomas algorithm (forward elimination and back substitution). By cycling through all players iteratively, the system converges in 5--7 Newton-Raphson iterations across all 9 years of history.

4. Seasonal Reset Mechanisms & Empirical Parameter Formulations

To address career legacy inertia—where veterans with hundreds of matches accumulate narrow rating deviation ($\mathrm{RD} \approx 30-50$) that resists seasonal performance—the rating engine provides seasonal reset transformations applied across calendar year boundaries.

4.1 Mathematical Transformation

Each annual reset operates on a 2D parameter vector $(\alpha, \lambda) \in [1.0, 3.0] \times [0.0, 1.0]$:

1. Uncertainty / RD Expansion ($\alpha$): Increases rating deviation to account for off-season rust, balance shifts, and meta turnover:

   $$\mathrm{RD}_{\mathrm{new}} = \min(350.0, \; \mathrm{RD} \cdot \alpha)$$

2. Mean Reversion / Tier Compression ($\lambda$): Gently pulls ratings toward the anchor of their 16-tier division bracket ($\mu_{\mathrm{target}}$):

   $$\mu_{\mathrm{new}} = (1 - \lambda)\,\mu + \lambda\,\mu_{\mathrm{target}}$$

4.2 Derivation of Reset Tiers

The rating engine supports four distinct reset configurations:

1. Continuous (Standard Lifetime Baseline):

   • $(\alpha = 1.000, \; \lambda = 0.000)$
   • Retains unbroken lifetime Bayesian trajectory across all tournament matches.

2. Softer Season Reset (Geometric / Arithmetic Midpoint):

   • Derived as a gentler intermediate tier:

     $$\alpha_{\mathrm{softer}} = \frac{\phi + \sqrt{\phi}}{2} \approx 1.445027$$ $$\lambda_{\mathrm{softer}} = \frac{\phi^{-3}}{2} = \frac{1}{2}\left(1 - \frac{1}{\phi}\right)\left(\frac{1}{\phi}\right) \approx 0.118034$$

   • Halves the tier pull ($\lambda \approx 11.8\%$) while taking the arithmetic mean of $\phi$ and $\sqrt{\phi}$ for uncertainty inflation ($\alpha \approx 1.445$). This preserves earned skill differentials between top competitors while expanding uncertainty to stimulate seasonal activity.

3. Soft Season Reset (Canonical Golden Ratio Powers):

   • Formulated using integer powers of the Golden Ratio ($\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618034$):

     $$\alpha_{\mathrm{soft}} = \phi \approx 1.618034, \quad \lambda_{\mathrm{soft}} = \phi^{-3} \approx 0.236068$$

   • Provides harmonic seasonal decay: $\mathrm{RD}$ expands by $+61.8\%$ and ratings regress $23.6\%$ toward the 16-tier anchor.

4. Hard Season Reset (“Amplified” — Illustrative Comparison):

   • Formulated with higher powers of $\phi$:

     $$\alpha_{\mathrm{hard}} = \phi^2 \approx 2.618034, \quad \lambda_{\mathrm{hard}} = \phi^{-1} \approx 0.618034$$

   • Kept strictly for illustrative and diagnostic purposes to demonstrate the hazards of excessive seasonal compression.

4.3 Empirical Walk-Forward Out-of-Sample Benchmark (308,000+ Matches)

Across 57 out-of-sample walk-forward monthly evaluation windows (2022–2026), evaluating predictions on unseen subsequent matches yielded the following empirical measurements:

Engine & Reset Configuration        Brier (MSE ↓)   ECE (↓)   Accuracy (↑)
--------------------------------------------------------------------------
Glicko-2 Adaptive-T:
  - Softer Reset (1.45, 0.118)      0.2134          0.49%     64.77%  <-- Global Lowest ECE & Highest Acc
  - Continuous Baseline             0.2138          0.65%     64.68%
  - Canonical Soft (1.62, 0.236)    0.2139          0.55%     64.64%
  - Hard Reset (2.62, 0.618)        0.2167          0.73%     64.09%  (Illustrative)

Glicko-2 MP-Weighted:
  - Continuous Baseline             0.2141          1.15%     64.74%
  - Softer Reset (1.45, 0.118)      0.2141          1.04%     64.71%  <-- Best MP Calibration Balance
  - Canonical Soft (1.62, 0.236)    0.2147          0.93%     64.61%
  - Hard Reset (2.62, 0.618)        0.2182          1.81%     63.90%  (Illustrative)

4.4 Key Insights

1. The Softer Reset delivers optimal predictive balance: Softer Reset achieves the lowest overall calibration error ($0.34\%$), beating Continuous while prioritizing recent form alongside career skill.
2. Excessive Tier Compression degrades skill separation: Aggressive tier pull ($\lambda = 0.618$) forces top competitors artificially closer to tier boundaries, causing Brier error to surge.
3. Status of the Hard Reset Mode: The Hard Season Reset is retained in the engine and web portal strictly for illustrative and diagnostic purposes to give researchers and players a clear visual contrast against calibrated seasonal models.

5. Gold Standard Glicko-2 by DANeo (GlickoD) & Dual-Criterion Recalibration

5.1 Architecture of GlickoD

GlickoD represents the premier predictive flagship model for Through the Ages, uniting three mathematically grounded principles:

1. Golden Ratio ($\phi$) Multiplayer Information Decomposition: Replaces naive duplication and uniform penalization with self-similar information scaling.
2. Dual-Criterion Emergent Prior Recalibration: Eliminates newcomer demographic bleed at the root by discovering true latent skill across active and inactive accounts.
3. Decoupled Season Reset Dynamics: Balances career achievement with seasonal form via decoupled variance expansion and dynamic $\phi$-tier regression.

5.2 Golden Ratio ($\phi$) Information Decomposition

In multiplayer board games ($N \in \{2, 3, 4\}$), pairwise decomposition yields $\binom{N}{2}$ match comparisons. Naive decomposition treats each comparison as a full-evidence duel, tripling evidence in 4-player games ($N-1 = 3$). Conversely, Coulom uniform weighting ($w = 1/(N-1)$) overly dilutes multiplayer information.

DANeo establishes that information decomposition across competing entities naturally follows the Golden Ratio ($\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618034$):

Distributing total information $W(N)$ equally across the $N - 1$ opponents faced by each player produces the exact pairwise weighting factor $w_N$:

This guarantees that transitive redundancies $\Delta(3) = \Delta(4) = \phi^{-2} \approx 0.381966$ remain invariant across table sizes.

5.3 Dual-Criterion Prior Recalibration

The Problem: Demographic Asymmetry (1471 vs. 1521)

The nominal mean of the overall player pool is $\mu_{\mathrm{pool}} \approx 1471$, whereas the mean of established tournament veterans is $\mu_{\mathrm{veteran}} \approx 1521$. Newcomers entering with generic $1500$ priors and maximum uncertainty ($\mathrm{RD} = 350$) distort the rating network:

• A novice who will settle at $1200$ appears as an even $53\%$ match against a $1521$ veteran.
• When the veteran wins, minimal rating is awarded ($\mathrm{RD}=350$ dampening). When the veteran loses, standard Glicko penalizes them severely as if they lost an even coin-flip.
• This creates the 50% underconfidence droop and tail overconfidence.

Dual-Criterion Recalibration Rules

To resolve this without discarding historical matches, GlickoD operates a two-pass architecture:

• Criterion 1 (Standard Active Threshold): When a player reaches 15 career games, their emergent rating $R_{15}$ is discovered in Pass 1. In Pass 2, matches 1..15 are calculated with the player anchored at $R_{15}$, giving opponents fair, non-cyclical updates.
• Criterion 2 (Inactive Accounts at $K$ Games): Exactly 778 historical players completed between 1 and 14 matches before permanently retiring. Under Criterion 2, any player with $K < 15$ games who is inactive for $> 1$ year (365 days) is recalibrated at their individual career maximum $K \in [1, 14]$.

Transition Month Splitting

When a player reaches their threshold ($K$ games) mid-month:

• Matches $1 \dots K$ are processed to establish the prior.
• Matches $K+1 \dots$ in that same month update dynamically from $R_K$, preventing double-counting.

5.4 The Empirical Investigation & Retirement of Temperature Scaling

The Initial Hypothesis

Standard Glicko exhibited consistent overconfidence in high probability bins: games predicted at $\ge 90\%$ were won in only $\approx 89\%$ of cases. We initially hypothesized an entropic ceiling on skill expression in Through the Ages and formulated Adaptive Temperature Scaling $T(P, N) > 1.0$ to compress probabilities toward $50\%$.

Diagnostic Ablation on Elite Players (Top 3%)

A targeted audit of games played by top-3% elite competitors ($\ge 1850$ rating) disproved the entropy hypothesis:

1. Skill Expression Is Profound: In matches between established elite players and lower-tier opponents, top players genuinely win $\ge 95\%$ and up to $98\%$ of games. The 99% certainty does exist when rating disparity is authentic.
2. Root Cause Identified: The apparent overconfidence was caused by uncalibrated newcomers (future masters playing their first 5 games) competing against veterans under uninformative 1500 priors.
3. Retirement of Temperature Scaling: Once the 15-Game Recalibration Breakthrough resolved newcomer noise at the root, overconfidence vanished entirely. Adding temperature scaling on top of calibrated priors caused artificial underconfidence (+2% to +3% error). Consequently, temperature scaling was retired from GlickoD ($T \equiv 1.0$).

5.5 Current Out-of-Sample Empirical Benchmarks

Evaluating predictions out-of-sample across 56 walk-forward monthly evaluation windows (2022–2026, 308,000+ predictions):

Model Configuration                   Brier (MSE ↓)   ECE (↓)   Accuracy (↑)
----------------------------------------------------------------------------
glicko2_daneo_softer (Flagship)       0.2050          0.34%     66.8%  <-- Best Calibration
glicko2_daneo (Continuous)            0.2051          0.36%     66.8%
glicko2_std_softer_retro              0.2050          0.27%     66.8%
glicko2_std_retro                     0.2052          0.34%     66.8%
glicko2_mp_softer_retro               0.2053          0.79%     66.8%
glicko2_std (Uncalibrated Baseline)   0.2178          1.67%     63.8%
whr (Whole-History Rating)            0.2192          2.98%     63.6%