Codex model - optimal picks

Nash equilibria / Tier list

Given a set of decks to choose from, if both players are picking a deck without seeing the other player’s deck first, how should we choose which deck to play?

For a set of decks, we can take the model’s simulated matchup samples, determine the Nash equilibrium for each sample, and take each deck’s mean weight over the equilibria to find the general Nash equilibrium. This takes proper account of the model’s uncertainty over the matchups. Using it to choose a deck is equivalent to Thompson sampling: for player skills, this would randomly choose a player to state as the best, weighted by the players’ probabilities of being the best. This approach is better calibrated.

Nash equilibria are given for three types of player: P1 when known to be going first, P2 when known to be going second, and Both for the more common case where the deck is chosen before a coin flip to determine who goes first.

Monocolour

Original cards

Player	Win probability
P1	0.548
P2	0.452
Both	0.500

The chance of winning before determining play order is 50%, as we’d expect if we’ve calculated things properly.

Deck	P1	P2	Both
MonoBlack	0.376	0.289	0.565
MonoRed	0.288	0.193	0.202
MonoPurple	0.029	0.295	0.095
MonoWhite	0.196	0.040	0.070
MonoBlue	0.082	0.108	0.048
MonoGreen	0.029	0.075	0.021

Black is dominant, as expected given that it’s known as a strong faction. Other decks have a larger presence when it’s known whether you’re going first or second. However, there is a lot of uncertainty: even Blue, considered weak due to an especially lopsided matchup against Black, hasn’t been cut.

Forum standard cards (v2.1)

Player	Win probability
P1	0.525
P2	0.475
Both	0.500

The chance of winning before determining play order is 50%, as we’d expect if we’ve calculated things properly.

Deck	P1	P2	Both
MonoRedv2	0.221	0.216	0.259
MonoBlackv2	0.148	0.194	0.165
MonoGreenv2	0.134	0.150	0.153
MonoPurplev2	0.102	0.205	0.145
MonoWhitev2	0.205	0.116	0.145
MonoBlue	0.190	0.118	0.134

Red is slightly dominant, but the weights are more even than in the original game. Given the low number of matches observed with the forum standard version, this is likely to change as we get more information.

Multicolour

Original cards

Win probability

Player	Win probability
P1	0.548
P2	0.452
Both	0.500

The first player seems to be less advantaged than in the monocolour game, though not by much.

Non-zero deck weights

Player	# with zero weight
P1	17
P2	12
Both	18

Few decks have been eliminated, so there is still a lot of uncertainty.

This deck Nash weight information allows us to give the model’s tier list for deck components with a simple measure: how much Nash weight does each component get?

Starter	P1	P2	Both
Black	0.251	0.265	0.317
Purple	0.072	0.250	0.165
White	0.216	0.089	0.145
Neutral	0.134	0.129	0.140
Red	0.147	0.090	0.088
Blue	0.112	0.089	0.087
Green	0.068	0.088	0.058

Spec	P1	P2	Both
Blood	0.221	0.196	0.231
Demonology	0.227	0.167	0.216
Necromancy	0.178	0.207	0.216
Finesse	0.214	0.158	0.204
Anarchy	0.231	0.157	0.183
Ninjitsu	0.177	0.148	0.169
Present	0.137	0.178	0.158
Bashing	0.137	0.168	0.157
Truth	0.124	0.163	0.146
Strength	0.134	0.158	0.139
Disease	0.144	0.123	0.138
Past	0.096	0.174	0.136
Future	0.102	0.174	0.129
Discipline	0.199	0.079	0.127
Peace	0.133	0.124	0.117
Growth	0.117	0.144	0.116
Law	0.124	0.116	0.115
Balance	0.097	0.133	0.108
Fire	0.121	0.123	0.107
Feral	0.087	0.110	0.088

Non-Disease Black components are dominant, as expected, and Blood is also very strong. Bashing and Ninjitsu rate unexpectedly high, despite being considered weak. This is likely to be due to them not being used often in tournament play: this results in high uncertainty for their strength in the model, so occasionally a simulated sample considers them to be strong components, rating highly in the sample’s Nash equilibrium.

Counters

Given the opponent’s choice of deck, but without knowing who will go first, how should we counter-pick?

Averaging over the matchup samples again, we find how likely each deck is to be the best counter-pick to each other deck.

Monocolour

Original cards

Taking the most-likely best counters, we get a five-colour cycle. Black counters the remaining deck, MonoGreen, that isn’t the most-likely-best counter to anything.

Black > Blue > Purple > White > Red > Black > Green

Forum standard cards (v2.1)

Taking the most-likely best counters, we see similar counters to the original game, except that Black is considered to be the best counter to White. However, there is less certainty about these being the best counters.

Multicolour

Original cards

The full counter table is too large to show here, so here are some highlights.

We get the following top counter-picks (i.e. highest probability of being best counter) for each opponent (there are a few ties):

Here is each deck’s counter with highest mean win probability:

Here are the top counter-picks for a few notable decks.

[Demonology/Necromancy]/Finesse, AKA Nightmare:

Deck	Probability best counter	Counter win probability
[Blood/Fire]/Growth	0.01300	0.655
[Blood/Fire]/Necromancy	0.01150	0.691
[Finesse]/Demonology/Feral	0.00950	0.631
[Truth]/Blood/Finesse	0.00875	0.594
[Peace/Truth]/Finesse	0.00825	0.586
[Fire]/Demonology/Discipline	0.00800	0.647

[Anarchy]/Growth/Strength, AKA Miracle Grow, which can do well against Nightmare:

Deck	Probability best counter	Counter win probability
[Ninjitsu/Strength]/Anarchy	0.00800	0.753
[Ninjitsu]/Anarchy/Present	0.00625	0.772
[Ninjitsu]/Anarchy/Law	0.00600	0.765
[Strength]/Balance/Truth	0.00600	0.742
[Ninjitsu]/Anarchy/Peace	0.00575	0.741
[Bashing]/Anarchy/Ninjitsu	0.00525	0.696

[Past]/Anarchy/Peace, AKA PPA:

Deck	Probability best counter	Counter win probability
[Bashing]/Blood/Present	0.00675	0.660
[Present]/Blood/Demonology	0.00575	0.674
[Bashing/Finesse]/Blood	0.00550	0.652
[Truth]/Blood/Demonology	0.00550	0.644
[Present]/Blood/Ninjitsu	0.00500	0.669
[Past]/Bashing/Demonology	0.00475	0.640

[Discipline/Strength]/Finesse, which has performed well in recent tournaments:

Deck	Probability best counter	Counter win probability
[Necromancy]/Future/Strength	0.00525	0.670
[Past]/Bashing/Ninjitsu	0.00500	0.694
[Future/Past]/Bashing	0.00450	0.686
[Future/Past]/Finesse	0.00400	0.696
[Future/Present]/Finesse	0.00400	0.747
[Future]/Balance/Blood	0.00350	0.675

[Demonology]/Anarchy/Balance, which has performed well against Miracle Grow:

Deck	Probability best counter	Counter win probability
[Necromancy]/Blood/Law	0.01100	0.743
[Demonology]/Discipline/Finesse	0.00675	0.704
[Finesse]/Anarchy/Present	0.00625	0.620
[Finesse]/Blood/Present	0.00625	0.689
[Ninjitsu]/Blood/Present	0.00600	0.634
Crashbarrow.dec	0.00575	0.691

[Demonology]/Discipline/Finesse, the deck with highest weight in the Nash mean:

Deck	Probability best counter	Counter win probability
[Blood/Fire]/Growth	0.00975	0.708
[Blood]/Finesse/Strength	0.00800	0.722
[Blood/Fire]/Necromancy	0.00800	0.726
[Future]/Blood/Fire	0.00725	0.697
[Past]/Finesse/Ninjitsu	0.00600	0.666
[Blood/Fire]/Future	0.00575	0.697

[Necromancy]/Bashing/Strength, the deck whose most-likely-best counter has the lowest mean win probability:

Deck	Probability best counter	Counter win probability
[Future]/Bashing/Fire	0.00525	0.522
[Present]/Blood/Peace	0.00425	0.527
[Finesse]/Demonology/Feral	0.00400	0.524
[Finesse]/Feral/Growth	0.00375	0.514
[Finesse]/Necromancy/Truth	0.00375	0.529
[Strength]/Finesse/Present	0.00375	0.507