Frequently Asked Questions & Model Guide

👑 The Full-Circle Journey: From Naive Glicko to WHR to GlickoD

The journey to the current GlickoD rating engine iteration involved several discoveries: Our starting observation was that standard Glicko-2, while effective overall, showed a steady overconfidence in high-probability predictions: when predicting a win chance of $X \ge 90\%$, the actual observed win rate in tournament play was closer to $\approx 89\%$.

We initially hypothesized an entropic ceiling on skill expression in Through the Ages—the idea that random card draws and multiplayer dynamics prevented true 99% certainty between experienced players. To compensate, we introduced Adaptive Temperature Scaling $T(P, N)$ to pull extreme probabilities inward toward 50%.

However, a targeted ablation study focused on top-3% tournament players revealed the real cause:

• The "Overconfidence" Came from Uncalibrated Newcomers: When skilled new players enter tournaments with default $1500$ priors and face Grandmasters ($1950+$), standard Glicko projects a $93\%+$ victory chance for the veteran. If the veteran wins only $75\%$ against this hidden expert, aggregate statistics falsely flag this as "favorite overconfidence."

• Elite Skill Expression Is Real: Once players are calibrated, top competitors facing lower-tier opponents genuinely achieve $\mathbf{\ge 95\%}$ and up to $\mathbf{98\%}$ win rates. Extreme certainty is well-founded when the rating gap is accurately measured.

• Retiring Temperature Scaling: When the 15-Game Recalibration addresses newcomer uncertainty directly, the apparent overconfidence vanishes. Applying temperature scaling on top of calibrated ratings actually hurt accuracy by artificially suppressing legitimate elite win rates. Temperature scaling was therefore retired ($T \equiv 1.0$), yielding tail calibration within $\pm 0.17\%$ of reality across 70–95% thresholds and an ECE of 0.24%.

Returning to sequential Glicko-2 with these empirical insights, GlickoD is built upon two core pillars:

1. 15-Game Emergent Prior Recalibration: When new players enter with generic ratings ($1500, \mathrm{RD}=350$), early matches distort veteran ratings. Active veterans average $\approx 1521$, while the pool mean is $\approx 1471$. Establishing emergent ratings at 15 games (or upon retirement after 1 year for inactive accounts with $K \lt 15$ games) and recalculating history against the calibrated prior removes newcomer distortion from the rating network. Newcomers see their ratings update dynamically after every match alongside a provisional ⏳ badge.

2. Golden Ratio ($\phi$) Multiplayer Information Decomposition: Winning a 4-player game is neither 3 independent duels (naive Glicko) nor 1 duel (Coulom fractional scaling). GlickoD deploys Golden Ratio information scaling $W(N) = \phi^{N-2}$ ($w_N = \frac{\phi^{N-2}}{N-1}$), giving $\approx 1.618$ effective match equivalents for 3-player and $\approx 2.618$ for 4-player games, accurately capturing table evidence without inflating certainty.

Additionally, applying a gentle annual soft season reset (expanding RD by $\alpha \approx 1.45$ and regressing ratings toward tier anchors by $\lambda \approx 0.12$) improves forward predictive accuracy, counteracting inactive rating inflation and ensuring recent form is balanced with career skill.

📐 Golden Ratio Multiplayer Information Decomposition

In board games with $N \in \{2, 3, 4\}$ players, a single table is conventionally decomposed into $\binom{N}{2}$ pairwise matchups:

However, treating all pairwise comparisons as independent full-weighted matches artificially duplicates information. In a 3-player game, if Player A defeats B, and B defeats C, the outcome of A defeating C is partially transitively redundant. Conversely, scaling by a naive uniform factor $w = \frac{1}{N-1}$ (the Coulom MP-Weighted approach) overly penalizes multiplayer games, stripping away valid statistical signal.

DANeo establishes that information decomposition across competing entities naturally follows the Golden Ratio ($\phi \approx 1.61803398875$), the fundamental mathematical constant governing self-similar recursive partitioning:

Dividing the total effective information $W(N)$ equally across the $N - 1$ pairwise comparisons faced by each player yields the exact pairwise weighting factor $w_N$:

Impact on 1-on-1 Match Equivalents: Comparing Decompositions

How does Golden Ratio weighting compare to alternative approaches in terms of effective 1-on-1 match evidence awarded to each competitor?

• Naive Pairwise Decomposition ($w = 1.0$): Yields $(N-1) \times 1.0 = N - 1$ match equivalents per player. In 3P, a player gains $2.0$ full 1v1 match equivalents; in 4P, $3.0$ match equivalents. This triggers an artificial information explosion: rating deviation ($\mathrm{RD}$) collapses twice or three times too quickly, exaggerating multiplayer noise.
• Coulom Fractional Weighting ($w = \frac{1}{N-1}$): Yields $(N-1) \times \frac{1}{N-1} \equiv 1.0$ match equivalent per player across all formats. While mathematically clean, this under-rewards multiplayer play: dominating three opponents simultaneously in a 4-player game provides strictly richer statistical evidence than winning a 1v1 duel, yet Coulom values them identically.
• Golden Ratio Decomposition ($w_N = \frac{\phi^{N-2}}{N-1}$): Yields $W(N) = \phi^{N-2}$ effective 1v1 match equivalents per player:
  $$W(2) = 1.000 \text{ eq.}, \quad W(3) = \phi \approx 1.618 \text{ eq.}, \quad W(4) = \phi^2 = \phi + 1 \approx 2.618 \text{ eq.}$$
  Each format awards certainty proportional to its combinatorial challenge, establishing the ideal bridge between 2-player duels and multiplayer tables.

Mathematical Intuition: Why the Golden Ratio ($\phi$)?

Why does $\phi$ serve as the optimal scaling constant? In any multi-entity ranking, each additional participant introduces diminishing new independent evidence because transitive constraints already link the table. The Golden Ratio is defined by the fundamental self-similar equation $\phi^2 = \phi + 1$ (or $\phi = 1 + 1/\phi$).

When subtracting effective evidence $W(N) = \phi^{N-2}$ from the nominal pairwise comparisons faced by each player $(N - 1)$, we discover an invariant information-balancing property:

Step-by-Step Algebraic Derivation:

1. The Golden Ratio Identity ($\phi^{-2} = 2 - \phi$):
   By definition, $\phi^2 = \phi + 1$. Dividing both sides by $\phi^2$ gives $1 = \phi^{-1} + \phi^{-2}$, which implies $\phi^{-2} = 1 - \phi^{-1}$.
   Dividing $\phi^2 = \phi + 1$ by $\phi$ gives $\phi = 1 + \phi^{-1} \implies \phi^{-1} = \phi - 1$.
   Substituting $\phi^{-1}$ into $\phi^{-2}$:
   $$\phi^{-2} = 1 - (\phi - 1) = 2 - \phi \approx 2 - 1.618034 = 0.381966$$
2. For 3-Player Games ($N = 3$):
   With $N = 3$, nominal faced comparisons are $3 - 1 = 2$, and effective evidence is $\phi^{3-2} = \phi^1 = \phi$.
   $$\Delta(3) = (3 - 1) - \phi^1 = 2 - \phi = \phi^{-2} \approx 0.381966$$
3. For 4-Player Games ($N = 4$):
   With $N = 4$, nominal faced comparisons are $4 - 1 = 3$, and effective evidence is $\phi^{4-2} = \phi^2 = \phi + 1$.
   $$\Delta(4) = (4 - 1) - \phi^2 = 3 - (\phi + 1) = 2 - \phi = \phi^{-2} \approx 0.381966$$

Does this hold for any $N \gt 2$?
No — it holds uniquely and exclusively for $N \in \{3, 4\}$!
In general, powers of $\phi$ expand using Fibonacci numbers: $\phi^k = F_k \cdot \phi + F_{k-1}$ (where $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, \dots$).
Therefore, $\Delta(N) = (N - 1) - (F_{N-2} \cdot \phi + F_{N-3})$ $= (N - 1 - F_{N-3}) - F_{N-2} \cdot \phi$.
For the coefficient of $\phi$ to remain strictly $1$ (yielding $2 - \phi$), we require $F_{N-2} = 1$. The only positive Fibonacci indices where $F_k = 1$ are $k = 1$ ($N = 3$) and $k = 2$ ($N = 4$). For $N = 5$, $\phi^3 = 2\phi + 1$, giving $\Delta(5) = 4 - (2\phi + 1) = 3 - 2\phi \approx -0.236 \neq 0.382$.

Because Through the Ages (and modern competitive board games) caps multiplayer formats strictly at 3-player and 4-player ($N \in \{3, 4\}$), the information discarded due to transitive table overlap is perfectly invariant across every multiplayer format in the game.

🔬 Lever 1: 15-Game Re-Calibration (Eliminating Newcomer Bleed)

When players enter the tournament pool, they begin with a nominal 1500 rating and high uncertainty ($\mathrm{RD} = 350$). Because active veterans average $\approx 1521$ while the overall pool mean is $\approx 1471$, games involving uncalibrated newcomers generate severe network noise, false upsets, and the notorious "50% underconfidence droop" in standard Glicko.

Under 15-Game Re-Calibration, a player's first 15 games establish their emergent prior $R_{15}$ (players with under 15 career games who have been inactive for over a year are also finalized at their career total). In the UI, newcomers see their ratings update dynamically after every match accompanied by a provisional ⏳ indicator, while in the rating engine, opponent updates are computed against the calibrated prior to eliminate newcomer distortion.

🎯 The Temperature Scaling Journey: Why GlickoD Retired Logit Compression

One of the most instructive chapters in the development of GlickoD was the rise and subsequent retirement of Adaptive Temperature Scaling $T(P, N)$.

In naive Glicko-2, extreme probability forecasts ($\gt 90\%$) appear to suffer from persistent overconfidence: when the model predicts $93\%$, actual tournament win rates hovered around $\approx 89\%$. We initially formulated the hypothesis that Through the Ages possessed an entropic ceiling on skill expression—that initial card rows, military tactics reveals, and multiplayer friction physically prevented a true 99% win probability from existing once players cross baseline competence. To bend predictions back toward the diagonal, we introduced thermal logit damping $T(P, N) = 1.0 + \alpha_N \cdot \delta^{\gamma_N}$.

The Diagnostic Breakthrough: Uncalibrated Newcomers vs. Real Elite Skill

When we performed a targeted ablation study isolating the top 3% of tournament competitors ($C \ge 1773$, Grandmasters and World Champions), the entropic ceiling hypothesis collapsed:

• The Subpopulation Asymmetry: In open tournaments (like the World Championship Series or Grand Slams), newcomers enter with default $1500$ priors. While below-average newcomers drag down the 50–65% bins (producing underconfidence), strong newcomers (future 1750+ masters) face Grandmasters ($1950+$) in early rounds. Standard Glicko calculates a naive $93\%+$ win chance for the GM against a "1500." When the GM only wins $75\%$ against this hidden shark, the aggregate data falsely records "favorite overconfidence."

• Elite Skill Expression Is Genuine: Once ratings are calibrated via 15-Game Recalibration (Option A), top-3% players facing lower-tier opponents genuinely win $\mathbf{95.7\%}$ of matches at $\gt 90\%$ confidence, $\mathbf{98.2\%}$ at $\gt 95\%$, and $\mathbf{100\%}$ at $\gt 99\%$! For instance, a 2000 GM facing an established 1475 average player has an expected $\Delta R \approx 525$, which yields a true, observed $\ge 95\%$ victory rate.

• Temperature Scaling Was Compressing Too Far: Because the retro prior already eliminated newcomer noise at the root, applying temperature scaling on top forced predictions too low, causing artificial underconfidence in elite matches (+2% to +3% prediction deficit).

Empirical Ablation: T-Enabled vs. T-Disabled (Across 388,493 Encounters)

Comparing GlickoD with temperature scaling enabled versus completely disabled ($T \equiv 1.0$) demonstrates that removing temperature scaling achieves superior calibration across every metric:

The Final Architectural Verdict

Temperature scaling was a well-intentioned bandage for a problem caused by uncalibrated cold-start priors. Once 15-Game Recalibration solved the newcomer problem at the root, temperature scaling became superfluous that actively degraded elite match predictions.

GlickoD operates with $T \equiv 1.0$ across all formats. Its calibration is anchored by its two core enhancements: Golden Ratio ($\phi$) multiplayer weighting and 15-game emergent prior recalibration.

🏆 Empirical Benchmark Scoreboard Across Formats

The table below documents the final match-time calibration metrics for GlickoD (Custom Glicko2 implementation for Multiplayer Boardgaming by DANeo) across all season reset modes on the full historical dataset (388,493 pairwise encounters):

🔬 Eliminating Newcomer Bleed: The 1471 vs. 1521 Population Mismatch

In competitive board game matchmaking, few problems are more damaging than uncalibrated newcomer noise. In standard Glicko-2, every newcomer enters the rating network with a generic placeholder rating ($\mu = 1500$) and maximum rating deviation ($\mathrm{RD} = 350$).

Across our empirical dataset of 388,493 pairwise matches, an important demographic asymmetry emerged:

• Nominal Pool Mean: $\mu_{\text{pool}} \approx 1471.0$ (when including casual players, dropouts, and one-off participants).
• Active Veteran Mean: $\mu_{\text{veteran}} \approx 1521.0$ (the true baseline skill of sustained competitors with $\gt 20$ tournament games).

Because active tournament competitors are systematically stronger than newcomers, a nominal contest between a $1521$ veteran and a $1500$ newcomer appears in standard Glicko as an almost perfectly even $53\%$ match. If that newcomer is actually an inexperienced novice with a latent skill of $1200$, the veteran's true probability of winning was $\gt 85\%$.

This mismatch triggers an acute systemic flaw: whenever the veteran wins, they gain virtually no rating (since the newcomer's RD is $350$, dampening their impact). However, if the veteran loses or finishes behind them in a volatile 4-player table, standard Glicko penalizes the veteran as if they lost an even coin-flip, artificially destroying their rating. This created both the infamous 50% underconfidence droop on calibration curves and widespread leaderboard deflation.

Why 15 Games? After 15 games, Bayesian Glicko-2 narrows a newcomer's uncertainty from $\mathrm{RD} = 350$ down to $\mathrm{RD} \approx 65–80$. At this point, the player's true latent skill $R_{15}$ is reliably unveiled, transforming them from a noisy statistical variable into an authentic competitive participant.

⚖️ The Transition Month Split: Preventing Newcomer Double-Counting

Online tournaments are compiled and rated in chronological monthly batches. This creates a challenging edge case: what occurs when a newcomer completes their 15th game halfway through a month?

• Flaw 1 (Freezing the entire month): If we freeze the newcomer's rating for the entire month they reach 15 games, matches 16 through 30 played during that same month are discarded or uncredited, denying the player their rightful rank gains.
• Flaw 2 (Updating the whole month normally): If the month is updated normally, matches 1 through 15 are counted twice: once to discover their 15-game starting anchor $R_{15}$, and a second time in that month's dynamic rating update, causing severe artificial rating inflation.

To achieve mathematical purity without discarding legitimate gameplay, GlickoD executes an exact transition split at match 15:

Empirical Case Study (Tianren):
In diagnostic benchmarking on top competitor Tianren, updating the transition month normally caused an artificial $+20.5$ point jump (leaping from $1972$ to $2025$) because his first 15 games were double-counted. Splitting the month cleanly at match 15 eliminated the artifact entirely, producing an authentic $+4.2$ point progression ($1976.78$, $\mathrm{RD} = 34.44$).

📐 Bayesian DAG Invariance: Why Opponent Updates Require a Calibrated Prior

A frequent question is: "Why do opponents calculate rating changes against a stabilized prior rather than a fluctuating provisional rating?"

In a Bayesian rating network, competitive history forms a Directed Acyclic Graph (DAG) across time. If a newcomer's rating dynamically fluctuates during matches 1 through 15 while opponents also adjust their ratings simultaneously, it triggers recursive feedback loops:

• Opponent A's rating update would depend on the newcomer's provisional rating after Game 3.
• The newcomer's provisional rating after Game 3 would depend on Opponent B from Game 2.
• Opponent B's rating would later shift after playing Opponent C, which retroactively invalidates Opponent A's update.

By discovering the newcomer's true emergent prior $R_{15}$ in Pass 1 and calculating opponents' outcomes against that stabilized baseline in Pass 2, the initial 15 games function as a pure, objective measurement probe. Opponents face an accurate skill estimate without cyclical feedback, preserving mathematical tractability and tournament integrity.

⏳ The Second Calibration Criterion: Historical Inactive Accounts (< 15 Games, > 1 Year Inactive)

What happens to players who enter the competitive scene, play 5 or 8 games, and then permanently retire?

In our historical archive of over 3,500 competitors, exactly 778 historical players completed between 1 and 14 matches before leaving competitive play. Under a strict, naive 15-game rule, these 778 accounts would never reach 15 games. Consequently, they would remain permanently anchored to the uninformative generic prior ($\mu = 1500, \mathrm{RD} = 350$) for eternity, continuing to inject demographic bleed into every opponent they ever faced.

To solve this, GlickoD implements an empirical Dual-Criterion Recalibration Architecture:

• Criterion 1 (Standard Active Threshold): Any player reaching 15 career games has their emergent prior $R_{15}$ established at match 15.

• Criterion 2 (Inactive Retirement at $K$ Games): Any player with $K \lt 15$ career games who remains inactive for > 1 year (365 days) after their final recorded match is also recalibrated. Their recalibration threshold is set dynamically to their individual career maximum $K \in [1, 14]$.

Their emergent skill at game $K$ becomes their fixed prior, purging decades of historical ghost noise from active players' records. When this second criterion was activated across the database, out-of-sample Brier error dropped from 0.2081 to an all-time record 0.2050, and Expected Calibration Error compressed to an astonishing 0.34%.

🔍 Frontend Display & The Provisional Calibration Indicator (⏳)

While the backend computational engine separates measurement from forward updates to maintain Bayesian DAG independence, the user-facing frontend operates with complete transparency:

• No Artificial Rating Freeze in UI: Every match played during a newcomer's provisional calibration phase displays its authentic rating change ($\Delta R$, e.g., +12.4, -8.1) directly in the match history table. Players always see continuous, responsive feedback after every tournament round.

• The Provisional Calibration Symbol (⏳): Players who are actively within their provisional phase (< 15 matches and active within the past year) are clearly distinguished by a golden hourglass badge:
  • On the Player Profile, an hourglass ⏳ appears next to the player's name and is accompanied by a status badge: ⏳ Prior Calibration (X/15).
  • On the Competitive Leaderboard, the ⏳ badge is rendered next to the player's name with an explanatory tooltip indicating their progress toward full calibration.

• Automatic Graduation: Once the 15th game is logged, the provisional indicator seamlessly disappears, reflecting the transition to full established status.

📊 Derivation of Calibration Bins: Normal Distribution Density & Probability Reliability Bins

Calibration in GlickoD operates on two mathematical levels: Probability Prediction Bins (verifying that a $70\%$ forecast realizes as a $70\%$ win rate) and Competitive Tier Bins (reshaping seasonal resets to respect the natural Gaussian skill distribution).

1. Probability Calibration Bins: The Reliability Curve (Soll vs. Ist)

To evaluate whether a rating model is well-calibrated, historical pairwise predictions $P \in [0.50, 1.00]$ are partitioned into $K = 10$ confidence intervals $B_1, \dots, B_{10}$ of width $5\%$:

Within each bin $B_k$, two values are evaluated against all $|B_k|$ matches falling into that confidence window:

• Expected Forecast (Soll): The average predicted probability $\bar{P}_k = \frac{1}{|B_k|} \sum_{i \in B_k} P_i$.
• Observed Win Rate (Ist): The actual empirical fraction of victories $\bar{y}_k = \frac{1}{|B_k|} \sum_{i \in B_k} y_i$.
• Expected Calibration Error (ECE): The match-weighted discrepancy across all bins:
  $$\mathrm{ECE} = \sum_{k=1}^{10} \frac{|B_k|}{N} \left|\bar{y}_k - \bar{P}_k\right|$$

The Flaw in Standard Glicko: In naive Glicko-2, extreme bins (e.g. $B_{10} = [95\%, 100\%]$) suffer from apparent tail drooping: the model expects $\bar{P}_{10} \approx 97.5\%$, but in uncalibrated tournament reality, the favorite only won $\approx 88.5\%$ of the time (a massive $-9.0\%$ gap). As discovered through our ablation study, this was an artifact of uncalibrated newcomers entering at 1500 and facing Grandmasters. GlickoD resolves this at the root through 15-game emergent prior recalibration and bilateral composite variance $g(c)$, aligning match forecasts directly with true player skill and reducing ECE from $3.8\%$ down to an unprecedented $0.24\%$ without requiring any artificial temperature damping.

2. Competitive Tier Bins: Normal Distribution Gaussian Tail Scaling

Why do seasonal calibration tier bins follow the 16-tier competitive hierarchy rather than uniform 100-point brackets?

In active tournament play, skill is not uniformly distributed across the rating line—it follows a bell curve (Gaussian normal distribution) centered near $\mu_0 = 1500.0$ with population standard deviation $\sigma_{\mathrm{pop}} \approx 175.0$:

• The Median Density Region ($1425 – 1575$): Over $70\%$ of all competitive matches occur between $1350$ and $1650$. If a seasonal reset arbitrarily deflates or compresses players near the middle, it destabilizes matchmaking for the vast majority of active competitors.
• The Extreme Tails ($\mu \ge 1900$ or $\mu \le 1100$): At the extremes ($|z| \ge 2.5$), player density drops below $1\%$. Here, unearned historical inertia allows inactive veterans to "camp" at top ranks for years unless an active variance expansion releases ratings back into competition.

To solve this, GlickoD establishes Gaussian Tail Scaling with Golden Ratio ($\phi$) Powers:

Mathematical Properties of the Gaussian $\phi$-Curve:

• Near the Median ($z \approx 0 \implies \mu \approx 1500$): $\tau(0) = 0 \implies \alpha(0) = 1.0$ and $\lambda(0) = 1.0$. Active players in Gold and Platinum experience minimal artificial dislocation, maintaining flawless day-to-day matchmaking stability.
• In the Elite Tails ($|z| \ge 2.5 \implies \mu \ge 1950$): $\tau(z) \to 1.0 \implies \alpha$ expands by $(\phi - 1)/2 \approx +30.9\%$ while regression $\lambda$ increases by $\phi^{-1}/2 \approx +30.9\%$. This gently taxes legacy camping, ensuring that active form and current tournament campaigns contest the highest podiums.