Archetypes first. Then synergy, uncertainty, and draft pressure.

The method presented in this website aims at constructing directional graph networks for MTG limited data. It tries to learn archetypes from decklists, then measures single-card strength, pair synergy, triplet motifs, and draft urgency inside each learned shell.

The formulas below describe the current implementation directly.

1. Archetypes from full decklists

Let \(X\) be the deck-by-card matrix built from the deck_* columns of the game export. Each row is one deck build and each column is one card. Cards that appear in more than 55% of decks are removed before clustering because they are too common to separate one archetype from another. A card that appears in almost every deck contributes very little information about which shell a deck belongs to.

The weighting used before clustering is a classic TF-IDF transform. TF-IDF means term frequency - inverse document frequency. Here the “term” is a card and the “document” is a deck. The first factor rewards cards that appear in a deck; the second factor downweights cards that are common across many decks.

• \(d\): one deck build.
• \(c\): one card.
• \(D\): total number of deck builds in the dataset.
• \(x(d,c)\): number of copies of card \(c\) in deck \(d\).
• \(df(c)\): number of deck builds that contain card \(c\).
• \(w(d,c)\): weighted deck feature used by the clustering model.

The weighted matrix is reduced with truncated SVD and then clustered with MiniBatch K-means.

2. Signposts

For archetype \(k\), the signpost statistic asks whether a card appears much more often inside that archetype than outside it.

• \(c\): one card.
• \(a_c\): number of decks inside archetype \(k\) that contain card \(c\).
• \(b_c\): number of decks outside archetype \(k\) that contain card \(c\).
• \(N_k\): number of decks inside archetype \(k\).
• \(N_{\neg k}\): number of decks outside archetype \(k\).
• \(\ell(c)\): smoothed log-odds contrast for card \(c\).

A positive value of \(\ell(c)\) means the card is concentrated inside the archetype. A larger value means that concentration is stronger.

The prior is centered on the card's global prevalence. In plain terms, the method starts from the card's average frequency in the whole format and then asks how much the archetype deviates from that average. This keeps the statistic stable when sample sizes are modest.

• \(p_{0,c}\): global deck prevalence of card \(c\).
• \(D\): total number of deck builds in the whole dataset.
• \(\alpha_c\) and \(\beta_c\): prior pseudo-counts that smooth the in-archetype and out-of-archetype log-odds.
• \(w_c\): rarity weight used inside the prior.

The rarity weight slightly decreases the influence of rare and mythic cards. The practical goal is to reduce the chance that a rare card rises to the top of the signpost list mainly because it is scarce and powerful.

• \(z(c)\): standardized signpost strength.
• \(signpost\_score(c)\): final signpost score used in the exported data.

3. Deck-level pairs and triplets

Once the archetypes are learned, the method asks which cards co-occur in completed decklists more often than independence would predict.

• \(a,b,c\): cards.
• \(\pi(a)\): fraction of decks in archetype \(k\) that contain card \(a\). The same definition applies to \(\pi(b)\) and \(\pi(c)\).
• \(O_{\mathrm{deck}}(a,b)\): observed number of decks in archetype \(k\) that contain both \(a\) and \(b\).
• \(O_{\mathrm{deck}}(a,b,c)\): observed number of decks in archetype \(k\) that contain all three cards.
• \(E_{\mathrm{deck}}\): independence expectation inside the archetype.
• \(L^{\mathrm{deck}}_{ab}\) and \(L^{\mathrm{deck}}_{abc}\): deck-level log-lift for the pair or triplet.

A positive deck-level lift means the cards appear together in finished decklists more often than independent deck inclusion would predict.

When the code finds a strong deck triplet, it pushes part of that information back into the pair graph. For each retained triplet, it computes a triplet quality score and then gives each of the three pairs inside that triplet a bonus equal to 12% of that quality. If several triplets contain the same pair, the code keeps the largest bonus for that pair.

4. Game-level pair synergy

The game-side statistics use the chosen game zone from the 17Lands game export. In the current implementation the chosen zone is the drawn zone, taken from the drawn_* columns. So the card-level rate here is a drawn-zone win rate, which is very close in spirit to a game-in-hand style statistic.

• \(p_k\): baseline game win rate of archetype \(k\).
• \(p_a\): win rate of games in archetype \(k\) where card \(a\) appears in the chosen game zone.
• \(standalone\_delta(a)\): card-level lift above the archetype baseline.

The expected pair win rate is computed in logit space. The reason is that independent probability factors multiply, so after moving to odds and then taking a logarithm they add cleanly.

• \(p_{ab}\): win rate of games in archetype \(k\) where both cards \(a\) and \(b\) appear in the drawn zone.
• \(\hat p_{ab}\): expected pair win rate under the additive log-odds model.
• \(synergy\_delta\_logit(a,b)\): amount by which the observed pair beats or trails its expectation in logit space.

5. Standard deviation and confidence

The uncertainty model starts from the usual binomial variance and propagates it to the synergy statistic.

• \(n\): number of samples supporting the corresponding rate.
• \(SE_{ab}\): standard error of the pair-synergy statistic.
• \(\sigma_{ab}\): signal-to-noise ratio for the pair effect.
• erf: function used to express confidence on a 0-100% scale.

The percentage shown on the site is the Gaussian central probability corresponding to the estimated signal-to-noise ratio. Higher confidence means the estimated effect is large compared with its estimated standard deviation.

6. Pair quality, triplet quality, and draft corroboration

The graph needs one ranking score to decide which links deserve visual emphasis. The score is a product of several terms. Each term measures a different ingredient: game-level lift, support, deck-level corroboration, triplet structure, draft corroboration, and confidence. A pair receives a large score when several of these signals agree.

• \(n_{ab}\): number of games where both cards appear in the drawn zone inside archetype \(k\).
• \(n_{abc}\): number of games where all three cards appear in the drawn zone inside archetype \(k\).
• \(triplet\_bonus(a,b)\): best bonus inherited by pair \((a,b)\) from retained deck triplets.
• \(draft\_pair\_events(a,b)\): number of projected drafts in archetype \(k\) that contain both cards.
• \(draft\_triplet\_events(a,b,c)\): number of projected drafts in archetype \(k\) that contain all three cards.

7. Draft pressure, ALSA, and wheeling

ALSA is measured globally across the whole draft table.

• \(N_c\): number of drafts where card \(c\) was observed.
• \(last\_seen_i(c)\): last pick at which card \(c\) was seen in draft \(i\).
• \(ALSA(c)\): average last seen at for card \(c\).
• \(ALSA_{sd}(c)\): empirical spread of the last-seen positions.
• \(\tau(c)\): decay scale used by the wheel model.
• \(wheel\_model(c)\): model-based probability that the card comes back when first seen on picks 1 through 4.

8. Exported scores

9. Network construction rules