fbpx

Joint Exploration: Adjusted Win Rates in Dominaria United Limited

If you’re a Limited Magic enthusiast, you have probably played enough Dominaria United to be well-acquainted with Wingmantle Chaplain and its team of offensively-inhibited support staff.

If you, like me, are interested in learning about the game from a variety of angles, including and perhaps especially from data, you are also likely familiar with Wingmantle Chaplain’s status as the “Mythic Uncommon” of the set, and Shield-Wall Sentinel’s reign at or near the top of the Game-in-hand Win Rate (GIHWR) rankings for commons, per 17Lands.

Others have done a more eloquent job than I can at explaining just how the defender archetype works to create these apparent statistical anomalies. Basically, Shield-Wall Sentinel’s tutoring ability lets it function as a de facto copy of any defender in your library, and when you have a strong build-around like the Wing Chap in your deck, and know it will be in your hand in essentially every game, you can really lean into building around it.

This is a particularly good example of card synergy, as I conceive of it: Card A and Card B are synergistic when including Card B in a deck with Card A makes Card A better. In this case, in a deck without other defenders, the Sentinel is a four-mana 1/3 defender, and the Chaplain is a four-mana 0/3 Defender – very weak! But each additional defender added to the deck increases the potential for card advantage, and the degree to which that card advantage accrues is the degree to which this set of cards has synergy.

There are lots of other examples, even within DMU, notably the cycle of signpost cost-reduction commons, which indicate generally what each color “wants” to do. Tolarian Terror, for example, works best when you run a large number of instants and sorceries, each of which marginally decreases the cost you’ll expect to pay for it.

This is all interesting, and part of what makes Magic such a compelling game, but it’s fairly remedial stuff. Deckbuilding 101 – find cards that work well with each other. Where it becomes interesting, however, is where it intersects with data-based inference about card quality. Wingmantle Chaplain and Shield-Wall Sentinel both have very high GIHWRs at their respective rarities – how should we interpret those win rates, especially as they compare to bombs like Sheoldred, the Apocalypse and efficient, flexible removal like Lightning Strike? Is Shield-Wall Sentinel “better” than Destroy Evil and Lightning Strike in some meaningful sense? How can we disentangle it from the other defenders for which it can tutor?

I argue that we just can’t do that with the 17Lands Card Performance data – the statistics there are extremely useful and full of information, but each card’s measured win rates are at least partially a product of the decks in which they are played. What to do?

 

 

Adjusted Win Rate

One option is to take the same data used to calculate those 17Lands win rates – the deck lists and records of tends of thousands of Limited decks – and apply our own model. In this case, this model is essentially a fancy linear regression model, which simultaneously controls and creates estimates for each card in each deck in the Public Data file, as well as the synergy of each card in each deck.

I have written at great length about the mechanics of the model elsewhere (calculating synergy, VOW, NEO, SNC), so I’m not going to get into that here, although I do recommend reading at least the first two of those. Here is the short version:

  1. Calculate a synergy score for each pair of cards, as a function of how much more often they co-occur in decks than we might expect at random.
  2. Fit a model that estimates each card’s contribution to a deck’s expected win rate – disaggregating each card’s power into “strength” and “synergy” coefficients, the latter of which reflect the degree to which cards get better when played with specific other cards. This model has the same goal as the Card Performance metrics on 17Lands, but improves on those by “statistically controlling” for the rest of the deck.

The strength coefficient attempts to measure how strong a card is, in terms of its contribution to expected wins, in a “synergy neutral” deck — a deck in which the card fits, with a mana base that can support casting it, but which isn’t necessarily all-in on supporting or feeding off of the card. The top of the Strength coefficient table, included below, is basically a list of the bombiest bombs in the format, generally all candidates for being picked first in any P1P1.

rnk card cost type rarity strength std_err
1 Archangel of Wrath C—A R 5.80 0.51
2 Sheoldred, the Apocalypse LC—PP M 5.73 0.87
3 Squee, Dubious Monarch LC—GN R 4.74 0.52
4 Wingmantle Chaplain C—HC U 4.40 0.27
5 Herd Migration S R 4.17 0.56
6 The Elder Dragon War E—S R 3.96 0.71
7 Sphinx of Clear Skies C—S M 3.95 0.90
8 Llanowar Greenwidow C—S R 3.61 0.65
9 Serra Paragon C—A M 3.43 0.78
10 Guardian of New Benalia C—HS R 3.35 0.58
11 Rith, Liberated Primeval LC—D M 3.04 0.80
12 Nishoba Brawler C—CW U 2.99 0.26
13 Quirion Beastcaller C—DW R 2.99 0.69
14 Yavimaya Coast L R 2.92 0.77
15 Fires of Victory I U 2.92 0.33
16 Leyline Binding E R 2.91 0.66
17 The Cruelty of Gix E—S R 2.89 0.55
18 Evolved Sleeper C—H R 2.84 0.66
19 Knight of Dawn’s Light C—HK U 2.84 0.39
20 Vodalian Mindsinger C—MW R 2.80 0.44
21 King Darien XLVIII LC—HS R 2.78 0.60
22 The Weatherseed Treaty E—S U 2.73 0.25
23 Anointed Peacekeeper C—HC R 2.67 0.72
24 Phyrexian Missionary C—PHC U 2.64 0.32
25 Ertai Resurrected LC—PHW R 2.63 0.63
26 Jaya, Fiery Negotiator LP—J M 2.61 0.95
27 Balmor, Battlemage Captain LC—BW U 2.59 0.52
28 Temporal Firestorm S R 2.58 0.75
29 Plaza of Heroes L R 2.57 1.09
30 Shivan Devastator C—DH M 2.56 0.89
31 Radha’s Firebrand C—HW R 2.46 0.74
32 Nemata, Primeval Warden LC—T R 2.42 0.61
33 Ivy, Gleeful Spellthief LC—FR R 2.41 0.92
34 Haughty Djinn C—D R 2.37 0.69
35 Shanna, Purifying Blade LC—HW M 2.37 0.93
36 Prayer of Binding E U 2.28 0.38
37 Valiant Veteran C—KS R 2.26 0.74
38 Drag to the Bottom S R 2.25 0.59
39 Yotia Declares War E—S U 2.25 1.13
40 Lagomos, Hand of Hatred LC—HS U 2.25 0.60
41 Protect the Negotiators I U 2.24 0.44
42 Defiler of Instinct C—PK R 2.20 0.63
43 Rona’s Vortex I U 2.19 0.46
44 Silver Scrutiny S R 2.17 0.62
45 Karplusan Forest L R 2.14 0.97
46 Tangled Islet L—FI C 2.13 0.18
47 Strength of the Coalition I U 2.11 0.39
48 Keldon Flamesage C—HS R 2.07 0.63
49 Danitha, Benalia’s Hope LC—HK R 2.06 0.55
50 Lightning Strike I C 2.03 0.17
51 Tatyova, Steward of Tides LC—MD U 2.01 0.43
52 Stenn, Paranoid Partisan LC—HW R 1.98 1.02
53 Micromancer C—HW U 1.96 0.28
54 Tear Asunder I U 1.94 0.33
55 The Raven Man LC—HW R 1.91 0.99
56 Caves of Koilos L R 1.90 0.89
57 Karn’s Sylex LA M 1.85 0.89
58 Threats Undetected S R 1.83 0.99
59 Radiant Grove L—FP C 1.76 0.16
60 Electrostatic Infantry C—DW U 1.75 0.31
61 Tolarian Geyser S C 1.74 0.12
62 Raff, Weatherlight Stalwart LC—HW U 1.73 0.30
63 Forest BL—F C 1.72 0.03
64 Yavimaya Iconoclast C—E U 1.72 0.35
65 Shivan Reef L R 1.72 0.77
66 Extinguish the Light I C 1.72 0.14
67 Sprouting Goblin C—GD U 1.71 0.37
68 Elas il-Kor, Sadistic Pilgrim LC—PKC U 1.71 0.42
69 Love Song of Night and Day E—S U 1.71 0.48
70 Take Up the Shield I C 1.65 0.23
71 Llanowar Loamspeaker C—ED R 1.65 0.64
72 Crystal Grotto L C 1.64 0.21
73 Mossbeard Ancient C—T U 1.64 0.34
74 Aether Channeler C—HW R 1.63 0.55
75 Llanowar Stalker C—EW C 1.60 0.59
76 Haunted Mire L—SF C 1.58 0.15
77 Keldon Strike Team C—HW C 1.58 0.23
78 Garna, Bloodfist of Keld LC—HB U 1.58 0.45
79 Wooded Ridgeline L—MF C 1.53 0.24
80 Baird, Argivian Recruiter LC—HS U 1.53 0.50
81 Defiler of Flesh C—PH R 1.52 0.65
82 Vohar, Vodalian Desecrator LC—PMW U 1.51 0.43
83 Citizen’s Arrest E C 1.51 0.22
84 Argivian Cavalier C—OK C 1.48 0.17
85 Astor, Bearer of Blades LC—HW R 1.47 1.34
86 Phoenix Chick C—P U 1.46 0.47
87 Knight of Dusk’s Shadow C—HK U 1.44 0.33
88 Sacred Peaks L—MP C 1.39 0.22
89 Shadow-Rite Priest C—HC R 1.36 1.04
90 Resolute Reinforcements C—HS U 1.35 0.32
91 Hurloon Battle Hymn I U 1.33 0.28
92 Mountain BL—M C 1.32 0.03
93 Balduvian Berserker C—KB U 1.32 0.41
94 Territorial Maro C—E U 1.31 0.34
95 Sulfurous Springs L R 1.30 0.91
96 Joint Exploration I U 1.30 0.46
97 Cult Conscript C—SW U 1.30 0.38
98 Swamp BL—S C 1.29 0.03
99 Idyllic Beachfront L—PI C 1.29 0.17
100 The Phasing of Zhalfir E—S R 1.29 1.14
101 Runic Shot S U 1.28 0.34
102 Contaminated Aquifer L—IS C 1.26 0.18
103 Tribute to Urborg I C 1.26 0.12
104 Defiler of Vigor C—PW R 1.26 0.63
105 Plains BL—P C 1.25 0.04
106 Island BL—I C 1.24 0.02
107 Sunlit Marsh L—PS C 1.24 0.25
108 Rivaz of the Claw LC—VW R 1.23 1.17
109 Monstrous War-Leech C—LH U 1.20 0.40
110 Molten Tributary L—IM C 1.20 0.18
111 Uurg, Spawn of Turg LC—FB U 1.19 0.46
112 Snarespinner C—S C 1.17 0.42
113 Tail Swipe I U 1.16 0.38
114 Thrill of Possibility I C 1.16 0.34
115 Essence Scatter I C 1.13 0.14
116 Soul of Windgrace LC—CA M 1.12 1.05
117 Battlewing Mystic C—BW U 1.12 0.42
118 Geothermal Bog L—SM C 1.12 0.19
119 Adarkar Wastes L R 1.11 0.91
120 Pilfer S U 1.11 0.73
121 Furious Bellow I C 1.10 0.31
122 Flowstone Infusion I C 1.10 0.30
123 Aggressive Sabotage S C 1.08 0.56
124 Rulik Mons, Warren Chief LC—G U 1.08 0.51
125 Slimefoot’s Survey S U 1.07 0.66
126 Battle-Rage Blessing I C 1.07 0.51
127 Braids’s Frightful Return E—S U 1.06 0.48
128 Destroy Evil I C 1.05 0.25
129 Phyrexian Espionage S C 1.04 0.13
130 Silverback Elder C—AS M 1.00 0.85
131 Jhoira, Ageless Innovator LC—HA R 1.00 1.35
132 Tidepool Turtle C—T C 0.98 0.90
133 Talas Lookout C—HP C 0.98 0.16
134 Inscribed Tablet A U 0.98 0.89
135 Coalition Warbrute C—MB C 0.97 0.31
136 Hero’s Heirloom A—E U 0.96 0.74
137 Choking Miasma S U 0.95 0.52
138 Liliana of the Veil LP—L M 0.93 0.97
139 Timely Interference I C 0.93 0.14
140 Benalish Faithbonder C—HC C 0.92 0.30
141 Magnigoth Sentry C—T C 0.90 0.20
142 Cleaving Skyrider C—HW U 0.88 0.47
143 Ertai’s Scorn I U 0.88 0.53
144 Bortuk Bonerattle LC—TS U 0.87 0.35
145 Impulse I C 0.87 0.15
146 Goblin Picker C—G C 0.87 0.42
147 Phyrexian Rager C—PH C 0.85 0.15
148 Frostfist Strider C—EG U 0.85 0.21
149 Coral Colony C—W U 0.84 0.43
150 Benalish Sleeper C—PHS C 0.83 0.25
151 Jodah, the Unifier LC—HW M 0.81 1.34
152 Meteorite A C 0.80 0.56
153 Splatter Goblin C—PG C 0.79 0.27
154 Weatherlight Compleated LA—V M 0.79 1.19
155 Bone Splinters S C 0.77 0.33
156 Join Forces I U 0.77 0.68
157 Jaya’s Firenado S C 0.76 0.40
158 Gaea’s Might I C 0.75 0.25
159 Ajani, Sleeper Agent LP—A M 0.72 0.88
160 Karn, Living Legacy LP—K M 0.70 0.94
161 Shadow Prophecy I C 0.68 0.26
162 Bite Down I C 0.67 0.20
163 Stall for Time I C 0.67 0.23
164 Thran Portal L—G R 0.66 0.84
165 Meria’s Outrider C—EA C 0.63 0.23
166 Rona, Sheoldred’s Faithful LC—HW U 0.62 0.50
167 Argivian Phalanx C—HKS C 0.62 0.31
168 Yavimaya Steelcrusher C—AW C 0.61 0.22
169 Flowstone Kavu C—K C 0.61 0.24
170 Defiler of Faith C—PH R 0.60 0.70
171 Battlefly Swarm C—PI C 0.60 0.45
172 Sengir Connoisseur C—V U 0.60 0.56
173 Relic of Legends A U 0.60 0.65
174 Floriferous Vinewall C—PW C 0.60 0.27
175 Vineshaper Prodigy C—ED C 0.60 0.21
176 Shore Up I C 0.58 0.32
177 Sunbathing Rootwalla C—L C 0.58 0.19
178 Aron, Benalia’s Ruin LC—PH U 0.57 0.45
179 Barkweave Crusher C—EW C 0.56 0.64
180 Urborg Repossession S C 0.55 0.16
181 Phyrexian Warhorse C—PH C 0.55 0.33
182 Tori D’Avenant, Fury Rider LC—HK U 0.55 0.62
183 Shield-Wall Sentinel AC—G C 0.54 0.20
184 Nael, Avizoa Aeronaut LC—ES U 0.53 0.34
185 Combat Research E—A U 0.53 0.78
186 Tura Kennerüd, Skyknight LC—HK U 0.52 0.43
187 Sheoldred’s Restoration S U 0.51 0.86
188 Radha, Coalition Warlord LC—EW U 0.51 0.54
189 Walking Bulwark AC—G U 0.50 0.34
190 Cut Down I U 0.50 0.52
191 Captain’s Call S C 0.50 0.24
192 Braids, Arisen Nightmare LC—N R 0.48 0.89
193 Phyrexian Vivisector C—PH C 0.48 0.29
194 Twinferno I U 0.46 0.61
195 Tattered Apparition C—S C 0.46 0.69
196 Negate I C 0.42 0.66
197 Salvaged Manaworker AC—C C 0.42 0.44
198 Samite Herbalist C—HC C 0.41 0.28
199 Mesa Cavalier C—HK C 0.40 0.28
200 Tolarian Terror C—S C 0.38 0.11
201 Hammerhand E—A C 0.36 0.75
202 In Thrall to the Pit S C 0.36 0.47
203 Blight Pile C—P U 0.33 0.29
204 Colossal Growth I C 0.33 0.30
205 Pixie Illusionist C—FW C 0.32 0.29
206 Eerie Soultender C—SC C 0.29 0.17
207 Bog Badger C—B C 0.28 0.27
208 Zur, Eternal Schemer LC—HW M 0.27 1.52
209 Yavimaya Sojourner C—T C 0.25 0.17
210 Queen Allenal of Ruadach LC—EN U 0.24 0.62
211 Toxic Abomination C—PZ C 0.21 0.37
212 Griffin Protector C—G C 0.18 0.24
213 Hexbane Tortoise C—T C 0.17 0.44
214 Chaotic Transformation S R 0.16 1.94
215 Artillery Blast I C 0.14 0.26
216 Clockwork Drawbridge AC—W C 0.13 0.41
217 Zar Ojanen, Scion of Efrava LC—CW U 0.12 0.44
218 Urza Assembles the Titans E—S R 0.10 1.70
219 Juniper Order Rootweaver C—HD C 0.00 0.27
220 Ghitu Amplifier C—HW C -0.01 0.23
221 Balduvian Atrocity C—PB U -0.01 0.45
222 Writhing Necromass C—ZG C -0.06 0.29
223 Deathbloom Gardener C—ED C -0.07 0.24
224 Warhost’s Frenzy I U -0.07 0.59
225 Soaring Drake C—D C -0.10 0.20
226 Impede Momentum S C -0.12 0.44
227 Urborg Lhurgoyf C—L R -0.12 0.74
228 Djinn of the Fountain C—D U -0.14 0.50
229 Dragon Whelp C—D U -0.15 0.51
230 Defiler of Dreams C—PS R -0.15 0.68
231 Hurler Cyclops C—C U -0.17 0.58
232 Charismatic Vanguard C—DS C -0.19 0.31
233 Heroic Charge I C -0.19 0.28
234 Timeless Lotus LA M -0.23 1.34
235 Linebreaker Baloth C—B U -0.24 0.52
236 Ratadrabik of Urborg LC—ZW R -0.25 0.98
237 Shalai’s Acolyte C—A U -0.26 0.67
238 Haunting Figment C—I C -0.28 0.23
239 Gibbering Barricade C—NW C -0.29 0.22
240 Rundvelt Hordemaster C—GW R -0.30 1.46
241 Automatic Librarian AC—C C -0.32 0.38
242 Vanquisher’s Axe A—E C -0.33 1.08
243 Volshe Tideturner C—MW C -0.34 0.53
244 The World Spell E—S M -0.35 1.45
245 Founding the Third Path E—S U -0.35 0.67
246 Meria, Scholar of Antiquity LC—EA R -0.36 1.29
247 Broken Wings I C -0.37 1.13
248 Viashino Branchrider C—VW C -0.38 0.39
249 Voda Sea Scavenger C—MR C -0.49 0.41
250 Temporary Lockdown E R -0.52 1.48
251 Coalition Skyknight C—HK U -0.55 0.63
252 Elvish Hydromancer C—EW U -0.64 0.44
253 Molten Monstrosity C—H C -0.70 0.34
254 Jodah’s Codex A U -0.72 0.41
255 Vodalian Hexcatcher C—MW R -0.78 1.48
256 Scout the Wilderness S C -0.80 0.29
257 Academy Loremaster C—HW R -1.00 1.23
258 Najal, the Storm Runner LC—EW U -1.06 0.38
259 Smash to Dust S C -1.06 1.11
260 Elfhame Wurm C—W C -1.07 0.32
261 Golden Argosy LA—V R -1.11 0.77
262 Academy Wall C—W C -1.49 0.23
263 Leaf-Crowned Visionary C—ED R -1.58 1.26
264 Vesuvan Duplimancy E M -2.04 1.09
265 Stronghold Arena E R -2.77 0.95

A couple of observations from this list:

  1. The top of this table is a list of bombs, all of which should be taken early, but it does look like the high end of Strength coefficients is a little lower than that of other recent formats.
  2. The cards that most benefit from association with Wingmantle ChaplainShield-Wall Sentinel, Clockwork Drawbridge, Walking Bulwark – suffer the most here in comparison with their 17Lands GIHWR. I call this a “coattail effect”, wherein appearing in decks with strong cards and good synergy can artificially inflate simple deck win rate statistics. The AWR model is designed to more correctly allocate credit to the cards that actually do the winning, and disentangle when cards are bad in isolation, but strong in combination.
  3. Defiler of Dreams – which has only a middling GIHWR – actually has a negative strength coefficient! My suspicion is that everything else blue in the format is trying to get you to play instants and sorceries, while the Defilers make your permanents cheaper.
  4. The other side, the cards with higher strength coefficients than GIHWR would predict, is a lot more heterogeneous. Plaza of Heroes, The Phasing of Zhalfir and Llanowar Stalker (?!?) are three such cards. Certain CFB-affiliated Limited grinders have long been singing the praises of Yotia Declares War, despite its relatively pedestrian GIHWR – their parsing of the format matches that of the AWR model, where it falls in the top 15 percent of cards, by AWR Strength!

Just as relevant to deckbuilding are returns to synergy, the estimation of which is perhaps the most novel aspect of this methodology. The easiest way to interpret these coefficients is something like: comparing a deck in which [Card X] has typical synergy to a deck in which [Card X] has Really Quite Good synergy — how much more does [Card X] contribute to winning? A look at the list will help clarify the intuition here.

rnk card cost type rarity synergy std_err
1 Wingmantle Chaplain C—HC U 1.31 0.26
2 Coral Colony C—W U 1.17 0.27
3 Tolarian Terror C—S C 0.88 0.12
4 Shield-Wall Sentinel AC—G C 0.80 0.19
5 Colossal Growth I C 0.79 0.25
6 Fires of Victory I U 0.75 0.21
7 Lagomos, Hand of Hatred LC—HS U 0.74 0.31
8 In Thrall to the Pit S C 0.73 0.29
9 Raff, Weatherlight Stalwart LC—HW U 0.71 0.23
10 Sheoldred, the Apocalypse LC—PP M 0.71 0.35
11 Baird, Argivian Recruiter LC—HS U 0.70 0.28
12 Battle-Rage Blessing I C 0.68 0.33
13 Balmor, Battlemage Captain LC—BW U 0.65 0.29
14 Walking Bulwark AC—G U 0.64 0.25
15 Drag to the Bottom S R 0.64 0.31
16 Phyrexian Espionage S C 0.62 0.16
17 Cult Conscript C—SW U 0.58 0.26
18 Blight Pile C—P U 0.58 0.29
19 Queen Allenal of Ruadach LC—EN U 0.58 0.32
20 King Darien XLVIII LC—HS R 0.58 0.33
21 Keldon Flamesage C—HS R 0.58 0.33
22 Tura Kennerüd, Skyknight LC—HK U 0.57 0.29
23 Choking Miasma S U 0.56 0.31
24 Karn, Living Legacy LP—K M 0.56 0.31
25 Hammerhand E—A C 0.56 0.37
26 Strength of the Coalition I U 0.56 0.26
27 Karplusan Forest L R 0.55 0.35
28 Floriferous Vinewall C—PW C 0.55 0.19
29 Love Song of Night and Day E—S U 0.55 0.27
30 Tail Swipe I U 0.55 0.31
31 Defiler of Flesh C—PH R 0.55 0.39
32 Electrostatic Infantry C—DW U 0.54 0.24
33 Nemata, Primeval Warden LC—T R 0.54 0.32
34 Aggressive Sabotage S C 0.54 0.31
35 Slimefoot’s Survey S U 0.52 0.33
36 Plaza of Heroes L R 0.52 0.37
37 Geothermal Bog L—SM C 0.52 0.18
38 Garna, Bloodfist of Keld LC—HB U 0.52 0.27
39 Phyrexian Warhorse C—PH C 0.52 0.22
40 Combat Research E—A U 0.52 0.30
41 Llanowar Greenwidow C—S R 0.52 0.31
42 Soul of Windgrace LC—CA M 0.51 0.30
43 Haughty Djinn C—D R 0.51 0.28
44 Captain’s Call S C 0.50 0.18
45 Ghitu Amplifier C—HW C 0.50 0.16
46 Thran Portal L—G R 0.50 0.34
47 Join Forces I U 0.50 0.29
48 Toxic Abomination C—PZ C 0.49 0.28
49 Academy Wall C—W C 0.48 0.19
50 Braids’s Frightful Return E—S U 0.48 0.32
51 Splatter Goblin C—PG C 0.48 0.20
52 Rona’s Vortex I U 0.48 0.27
53 Jaya, Fiery Negotiator LP—J M 0.48 0.37
54 Stronghold Arena E R 0.48 0.40
55 The World Spell E—S M 0.47 0.35
56 Herd Migration S R 0.46 0.35
57 Stall for Time I C 0.46 0.23
58 Scout the Wilderness S C 0.46 0.22
59 Shadow-Rite Priest C—HC R 0.46 0.35
60 Silver Scrutiny S R 0.45 0.35
61 Impulse I C 0.45 0.20
62 Defiler of Faith C—PH R 0.44 0.33
63 Extinguish the Light I C 0.44 0.18
64 Tori D’Avenant, Fury Rider LC—HK U 0.44 0.31
65 Urza Assembles the Titans E—S R 0.44 0.40
66 The Cruelty of Gix E—S R 0.43 0.30
67 Flowstone Infusion I C 0.43 0.16
68 Barkweave Crusher C—EW C 0.43 0.37
69 Nishoba Brawler C—CW U 0.43 0.21
70 Temporal Firestorm S R 0.42 0.31
71 Pilfer S U 0.42 0.32
72 Rulik Mons, Warren Chief LC—G U 0.42 0.30
73 Liliana of the Veil LP—L M 0.42 0.36
74 Squee, Dubious Monarch LC—GN R 0.42 0.28
75 Meria’s Outrider C—EA C 0.41 0.18
76 Hurler Cyclops C—C U 0.41 0.34
77 Phyrexian Missionary C—PHC U 0.41 0.23
78 Temporary Lockdown E R 0.41 0.46
79 Vanquisher’s Axe A—E C 0.41 0.35
80 Coalition Warbrute C—MB C 0.40 0.24
81 Citizen’s Arrest E C 0.40 0.19
82 Llanowar Loamspeaker C—ED R 0.40 0.32
83 Radha’s Firebrand C—HW R 0.40 0.34
84 Clockwork Drawbridge AC—W C 0.40 0.30
85 Shanna, Purifying Blade LC—HW M 0.39 0.32
86 Monstrous War-Leech C—LH U 0.39 0.31
87 Yavimaya Iconoclast C—E U 0.39 0.28
88 Hexbane Tortoise C—T C 0.38 0.26
89 Meteorite A C 0.38 0.36
90 Elas il-Kor, Sadistic Pilgrim LC—PKC U 0.38 0.28
91 Frostfist Strider C—EG U 0.37 0.26
92 Territorial Maro C—E U 0.37 0.25
93 Chaotic Transformation S R 0.37 0.33
94 Radha, Coalition Warlord LC—EW U 0.37 0.38
95 Mossbeard Ancient C—T U 0.37 0.22
96 Karn’s Sylex LA M 0.37 0.37
97 Evolved Sleeper C—H R 0.37 0.33
98 Rundvelt Hordemaster C—GW R 0.36 0.40
99 Radiant Grove L—FP C 0.36 0.17
100 Defiler of Dreams C—PS R 0.35 0.29
101 Leyline Binding E R 0.35 0.32
102 Founding the Third Path E—S U 0.35 0.34
103 Rivaz of the Claw LC—VW R 0.34 0.36
104 Warhost’s Frenzy I U 0.34 0.38
105 Benalish Sleeper C—PHS C 0.34 0.18
106 Vesuvan Duplimancy E M 0.34 0.34
107 Tolarian Geyser S C 0.34 0.15
108 Weatherlight Compleated LA—V M 0.34 0.38
109 Yotia Declares War E—S U 0.34 0.39
110 Archangel of Wrath C—A R 0.34 0.39
111 Sphinx of Clear Skies C—S M 0.33 0.31
112 Ivy, Gleeful Spellthief LC—FR R 0.33 0.33
113 Snarespinner C—S C 0.33 0.29
114 Shivan Devastator C—DH M 0.33 0.38
115 The Raven Man LC—HW R 0.33 0.31
116 Urborg Lhurgoyf C—L R 0.33 0.29
117 Phoenix Chick C—P U 0.33 0.25
118 Aron, Benalia’s Ruin LC—PH U 0.33 0.29
119 Yavimaya Steelcrusher C—AW C 0.32 0.19
120 Balduvian Berserker C—KB U 0.32 0.25
121 Golden Argosy LA—V R 0.32 0.31
122 Argivian Cavalier C—OK C 0.32 0.16
123 Shadow Prophecy I C 0.32 0.20
124 Tattered Apparition C—S C 0.32 0.40
125 Knight of Dusk’s Shadow C—HK U 0.32 0.25
126 Automatic Librarian AC—C C 0.32 0.21
127 Battlefly Swarm C—PI C 0.31 0.29
128 Jodah, the Unifier LC—HW M 0.31 0.34
129 Djinn of the Fountain C—D U 0.31 0.27
130 Llanowar Stalker C—EW C 0.31 0.29
131 Hero’s Heirloom A—E U 0.31 0.29
132 Jodah’s Codex A U 0.31 0.33
133 Mesa Cavalier C—HK C 0.30 0.22
134 Vodalian Hexcatcher C—MW R 0.30 0.35
135 Tribute to Urborg I C 0.30 0.17
136 Flowstone Kavu C—K C 0.30 0.21
137 Guardian of New Benalia C—HS R 0.30 0.35
138 Valiant Veteran C—KS R 0.30 0.35
139 Phyrexian Vivisector C—PH C 0.29 0.22
140 Leaf-Crowned Visionary C—ED R 0.29 0.40
141 Haunting Figment C—I C 0.29 0.18
142 Protect the Negotiators I U 0.29 0.30
143 Balduvian Atrocity C—PB U 0.29 0.29
144 Bone Splinters S C 0.29 0.20
145 Argivian Phalanx C—HKS C 0.29 0.21
146 Uurg, Spawn of Turg LC—FB U 0.29 0.32
147 Impede Momentum S C 0.28 0.27
148 Caves of Koilos L R 0.28 0.32
149 Ertai Resurrected LC—PHW R 0.28 0.35
150 Rith, Liberated Primeval LC—D M 0.28 0.41
151 Rona, Sheoldred’s Faithful LC—HW U 0.28 0.33
152 Nael, Avizoa Aeronaut LC—ES U 0.27 0.28
153 Vodalian Mindsinger C—MW R 0.27 0.32
154 Yavimaya Coast L R 0.27 0.35
155 Relic of Legends A U 0.27 0.33
156 Meria, Scholar of Antiquity LC—EA R 0.27 0.35
157 Ratadrabik of Urborg LC—ZW R 0.27 0.34
158 Bortuk Bonerattle LC—TS U 0.27 0.26
159 Keldon Strike Team C—HW C 0.27 0.20
160 Tatyova, Steward of Tides LC—MD U 0.26 0.26
161 Najal, the Storm Runner LC—EW U 0.26 0.30
162 Quirion Beastcaller C—DW R 0.26 0.32
163 Salvaged Manaworker AC—C C 0.26 0.22
164 Broken Wings I C 0.26 0.38
165 Magnigoth Sentry C—T C 0.26 0.18
166 Tidepool Turtle C—T C 0.26 0.36
167 Sulfurous Springs L R 0.26 0.38
168 Viashino Branchrider C—VW C 0.25 0.27
169 Thrill of Possibility I C 0.25 0.36
170 Danitha, Benalia’s Hope LC—HK R 0.25 0.27
171 Essence Scatter I C 0.25 0.14
172 Benalish Faithbonder C—HC C 0.25 0.29
173 Elfhame Wurm C—W C 0.25 0.25
174 Stenn, Paranoid Partisan LC—HW R 0.25 0.38
175 Ajani, Sleeper Agent LP—A M 0.25 0.34
176 Writhing Necromass C—ZG C 0.25 0.18
177 Threats Undetected S R 0.25 0.38
178 Astor, Bearer of Blades LC—HW R 0.24 0.36
179 Sengir Connoisseur C—V U 0.24 0.27
180 Micromancer C—HW U 0.24 0.23
181 Lightning Strike I C 0.24 0.17
182 Linebreaker Baloth C—B U 0.24 0.31
183 The Phasing of Zhalfir E—S R 0.24 0.30
184 Negate I C 0.24 0.32
185 Vohar, Vodalian Desecrator LC—PMW U 0.24 0.28
186 Runic Shot S U 0.23 0.31
187 Zar Ojanen, Scion of Efrava LC—CW U 0.23 0.33
188 Pixie Illusionist C—FW C 0.23 0.23
189 Deathbloom Gardener C—ED C 0.23 0.22
190 Serra Paragon C—A M 0.23 0.33
191 Heroic Charge I C 0.23 0.23
192 Inscribed Tablet A U 0.23 0.30
193 Gaea’s Might I C 0.22 0.21
194 Talas Lookout C—HP C 0.22 0.17
195 The Weatherseed Treaty E—S U 0.22 0.25
196 Twinferno I U 0.22 0.27
197 Resolute Reinforcements C—HS U 0.22 0.22
198 Tear Asunder I U 0.22 0.19
199 Jaya’s Firenado S C 0.22 0.26
200 Griffin Protector C—G C 0.21 0.19
201 Braids, Arisen Nightmare LC—N R 0.21 0.25
202 Tangled Islet L—FI C 0.21 0.16
203 Yavimaya Sojourner C—T C 0.21 0.20
204 Zur, Eternal Schemer LC—HW M 0.21 0.37
205 Jhoira, Ageless Innovator LC—HA R 0.21 0.33
206 Coalition Skyknight C—HK U 0.20 0.32
207 Shalai’s Acolyte C—A U 0.20 0.28
208 Aether Channeler C—HW R 0.20 0.32
209 Silverback Elder C—AS M 0.19 0.38
210 Crystal Grotto L C 0.19 0.22
211 Shivan Reef L R 0.18 0.33
212 Vineshaper Prodigy C—ED C 0.18 0.16
213 Dragon Whelp C—D U 0.18 0.25
214 Timeless Lotus LA M 0.18 0.37
215 Eerie Soultender C—SC C 0.18 0.15
216 Elvish Hydromancer C—EW U 0.17 0.27
217 Soaring Drake C—D C 0.17 0.19
218 Sheoldred’s Restoration S U 0.17 0.33
219 Wooded Ridgeline L—MF C 0.17 0.18
220 Timely Interference I C 0.17 0.17
221 Cut Down I U 0.16 0.21
222 Smash to Dust S C 0.16 0.34
223 Samite Herbalist C—HC C 0.15 0.20
224 Knight of Dawn’s Light C—HK U 0.15 0.24
225 Defiler of Instinct C—PK R 0.15 0.33
226 Joint Exploration I U 0.15 0.27
227 Phyrexian Rager C—PH C 0.15 0.15
228 Destroy Evil I C 0.14 0.19
229 Bog Badger C—B C 0.14 0.29
230 The Elder Dragon War E—S R 0.14 0.30
231 Goblin Picker C—G C 0.13 0.26
232 Voda Sea Scavenger C—MR C 0.12 0.29
233 Defiler of Vigor C—PW R 0.12 0.35
234 Shore Up I C 0.12 0.18
235 Prayer of Binding E U 0.12 0.22
236 Academy Loremaster C—HW R 0.12 0.39
237 Haunted Mire L—SF C 0.11 0.16
238 Furious Bellow I C 0.10 0.25
239 Idyllic Beachfront L—PI C 0.10 0.17
240 Ertai’s Scorn I U 0.10 0.32
241 Urborg Repossession S C 0.09 0.20
242 Hurloon Battle Hymn I U 0.08 0.23
243 Contaminated Aquifer L—IS C 0.06 0.12
244 Bite Down I C 0.05 0.18
245 Artillery Blast I C 0.04 0.23
246 Gibbering Barricade C—NW C 0.03 0.17
247 Adarkar Wastes L R 0.03 0.30
248 Sunlit Marsh L—PS C 0.03 0.14
249 Molten Monstrosity C—H C 0.02 0.28
250 Juniper Order Rootweaver C—HD C 0.01 0.24
251 Charismatic Vanguard C—DS C -0.01 0.31
252 Sprouting Goblin C—GD U -0.02 0.30
253 Plains BL—P C -0.02 0.03
254 Battlewing Mystic C—BW U -0.03 0.23
255 Molten Tributary L—IM C -0.03 0.15
256 Volshe Tideturner C—MW C -0.06 0.29
257 Cleaving Skyrider C—HW U -0.07 0.28
258 Sunbathing Rootwalla C—L C -0.09 0.14
259 Sacred Peaks L—MP C -0.11 0.17
260 Swamp BL—S C -0.12 0.05
261 Anointed Peacekeeper C—HC R -0.14 0.34
262 Take Up the Shield I C -0.15 0.16
263 Forest BL—F C -0.20 0.04
264 Island BL—I C -0.28 0.03
265 Mountain BL—M C -0.30 0.03

Another couple of observations:

  1. Wingmantle Chaplain is already great with “typical” synergy, as shown in the Strength coefficients above, but each marginal defender you surround it with also really pays off. Coral Colony is the same, but to a lesser extent. Shield-Wall Sentinel, GIHWR all-star at common, is shown here to really be only as strong as the defenders you run it with.
  2. As I’ve found previously, the uncommon signposts tend to be high-Synergy-coefficient cards, and this set is no different, lead by Lagomos, Hand of Hatred, Raff, Weatherlight Stalwart, Baird, Argivian Recruiter and Balmor, Battlemage Captain.
  3. There are several high-synergy commons of interest. First, the well-known Tolarian Terror, which can enter the game ridiculously early with enough instant and sorcery support. The high Synergy coefficients for Colossal Growth, In Thrall to the Pit and Phyrexian Espionage I’m attributing to the degree to which they are improved when you can kick them.

If you have read any of my other recent AWR analyses of recent formats (VOW, NEO, SNC), the preceding discussion will be familiar. However, I am pleased to introduce something new:

The Expected Win Rate Calculator

The Expected Win Rate Calculator is a way for anyone to apply the Adjusted Win Rate model to their own Limited decks!

A way to evaluate limited decks, in context!

The Calculator is pre-populated with a very strong WB Defenders/life gain deck, but let’s walk through an evaluation of a different deck, just as an example.

Here is a deck that the inimitable Luis Scott-Vargas claimed was, to that point, the best deck he’d had in the format:

Dominaria United Draft by Luis Scott-Vargas

 

Just looking at the list, its strength is obvious, but go ahead and copy that deck list and navigate to the Expected Win Rate Calculator.

Paste the decklist into the decklist field…

…and the Calculator should automatically populate with model-informed Expected Win Rates! Here is an example of what the output looks like:

Starting at the bottom and going section by section, here is how I would interpret all of this information:

  • The Expected win rate, by card section lists individual card contributions to the deck’s win rate. Each card in the deck is listed, along with its mana cost, rarity, a shorthand for its type (so Squee, Dubious Monarch is a LC—GN, or Legendary Creature — Goblin Noble), and a count of the number of copies in the deck. The xWR% strength column lists the card’s isolated strength coefficient directly from the AWR model, the xWR% synergy column lists the card’s AWR synergy coefficient × its synergy in this deck, and the xWR% column is the sum of those two numbers.
    • This LSV deck doesn’t really have any bombs – it’s just generally quite deep with moderately strong cards, almost all of which have positive synergy with the rest of the deck.
  • The Expected win rate section lists the total xWR% from synergy and strength for the deck, which is just the sum of the relevant columns in the by card table. This can give you a sense of whether your deck is a collection of good stuff, or a generally synergistic set of cards which mutually strengthen each other. xWR for average player is the contribution to expected win rate derived from the player’s overall win rate. You’ll note that just below the deck list text box there is a slider that allows you to adjust for a specific player’s actual win rate. By default, this is set to 50 percent, which for DMU translates to a mild bump of +3.6 xWR%. LSV would probably need to adjust the slider substantially to the right to get a better win rate forecast. Finally, Total xWR is the sum of the strength, synergy, and player xWR values. If the model is doing what it’s supposed to do, this Total xWR% is the proportion of games this deck would win if allowed to play a very large number of games against randomly drawn opponents.
  • Finally, next to the Expected win rate section is the illustration from the deck’s Most Valuable Card. This counts multiple copies of the same card, so in the LSV deck we’re looking at, 4 copies × 2.7 xWR% from Tolarian Geyser sum to 10.8 xWR% total from that card. There are going to be a lot of DMU decks for which the MVC image is Wingmantle Chaplain

What’s the point?

I like to think of this as a defense against Results-Oriented Thinking. Your skill as a drafter is distinct from your skill as a player, but generally we can only measure both at the same time, through wins and losses, and game outcomes are highly random and imprecise ways to measure deck quality and player ability.

The AWR model can help you better isolate your performance on those two dimensions, by evaluating your deck quality before you ever play a game. If you consistently draft high-xWR decks, you’re doing something right in the draft, but if you then consistently win games at a rate below expectation, it might be an indication you need to tighten up your play. Of course, the variance baked into deck randomization makes the second step of that inference pretty fraught, and I make no claim that the AWR model is perfect, but it’s at least a step toward understanding.

The AWR model can help guide your draft, to some extent, as it helps evaluate each card’s potential in the context of your deck. For example, one heuristic that the AWR model has taught me, over the course of several draft formats, is that quality removal, at common, is often underrated by 17Lands GIHWR. Cheap, flexible removal, like Lightning Strike in DMU and Murder in SNC, can be dragged down by association with relatively weak colors or archetypes. AWR, by dint of adjusting for every other card in the deck, plus returns to synergy, seems to more correctly (?) attribute high strength and low synergy coefficients to such cards, since they generally do their thing, regardless of the context.

In contrast, WotC designs the two-color uncommon draft signposts (#2CUDS) to point you in a clear direction, and another consistent finding across multiple formats is that they generally live up to their promise. It’s not necessarily the case that they’re all good cards, even in decks with lots of synergy, but it is typically the case that they get much better as synergy increases. So, by pack three, you should know very clearly whether any 2CUDS you encounter will fit in your deck, and AWR can give you a sense of how rewarding that fit can be.

What is this not?

This is not a pick order list – in fact, it’s quite the opposite, because the entire model is predicated on dynamic evaluation as your deck evolves. I also can’t totally endorse this as a guide to deckbuilding decisions.

I’m open to the possibility that you could choose 22 spells, 17 lands and then check several candidates for spell #23 to see which gives the highest xWR. But I don’t endorse using this to evaluate how many, or which kind, of lands to run, for example. In fact, mana bases and curves aren’t explicitly accounted for in the model at all, so it’s assuming you build those in a typical manner.

The model was trained on fully-formed 40-card decks and whether or not those decks won, so the closer our prediction task is to that process, the more realistic our predictions will be.

A couple of other cool things

If you are a 17Lands user, it’s trivial to copy your decklist to paste into the Calculator. Just navigate to any deck page (here is a recent Trophy deck), scroll all the way to the bottom, and click Click here for a text version. From there, just copy the main deck, and you’re good to go. (That  trophy deck list, for example, has an xWR of 59.3% for an average player.)

Also! You can use the Select format dropdown to pick any recent Limited environment for which 17Lands has released data. At the moment, that means you can evaluate decks from DMU, HBG, SNC, NEO, VOW, MID, AFR, STX, and KHM, though some format-specific features, like Strixhaven’s Lessons, are not fully modeled.

Thoughts? Questions? Critiques? Suggestions? Leave a comment below or talk to me on Twitter @MtG_DS. I’d love to hear what you think about the model, and where it seems to fall short. I’m very interested in improving it, so if you notice patterns in the decks it seems to model poorly, please let me know! (Although keep in mind that over a small number of games, variance can wash out any meaningful signal…) Also, if you use the Calculator to evaluate any of your own decks, please take a screenshot and @ me, especially if you’ve got an exceptionally strong, weak, or unusual deck. I really want to see how this thing works in the wild, outside of the cold confines of my computer!

 

1 thought on “Joint Exploration: Adjusted Win Rates in Dominaria United Limited”

Leave a Reply

Scroll to Top