Deck Chairs on the Titanic

WOTC printed [[Scapeshift]] in 2008, [[Valakut, the Molten Pinnacle:Valakut]] in 2009, and [[Primeval Titan]] in 2010. For eight years after that, they walked on eggshells to avoid further enabling land-based combo decks in Modern. As recently as 2018, fifty-six of the sixty cards in an Amulet Titan main deck were printed before the existence of Instagram1. Design philosophy finally changed about a year ago, and WOTC seems to be making up for lost time. After eight years of drought, we got [[Arboreal Grazer]], [[Castle Garenbrig]], [[Dryad of the Ilysian Grove]], [[Field of the Dead]], and [[Once Upon a Time]] all in the space of eight months.

One new card is exciting. Two is even better. But four new cards (plus a ban) can be overwhelming, especially for a synergy-driven deck like Amulet Titan. [[Field of the Dead:FOTD]] wants us to have more lands on the table, but half the point of bounce lands is that we can get to six mana with fewer lands. [[Castle Garenbrig]] works best when we play [[Breeding Pool]] over [[Gemstone Mine]], but that makes it harder to splash off-color sideboard cards. [[Dryad of the Ilysian Grove:Dryad]] took the place of colorless tools like [[Walking Ballista]], but without those tools it’s not clear if [[Ancient Stirrings]] is worthwhile. Finding a balance between so many competing incentives takes hours upon hours of playtesting – more time than we have to spare!

The Model

That’s where the computer comes in. My numerical model2 can read in a deck list, goldfish thousands of games, and tell us exactly how reliably that list can expect to cast [[Primeval Titan]] by turn three. Using the model, it’s possible to compare competing cards one-on-one (like [[Arboreal Grazer]] and [[Sakura-Tribe Scout]]) as well as measure the effect of card combinations that are more than the sum of their parts, like [[Castle Garenbrig]] and [[Dryad of the Ilysian Grove]]3.

The model doesn’t know anything about sequencing or strategy. Instead, it tries all possible combinations of legal plays. For example, let’s say we cast [[Ancient Stirrings]] and see [[Amulet of Vigor]], [[Forest]], [[Simic Growth Chamber]], and two other spells. An experienced player can generally eyeball the correct choice, but spelling out the decision explicitly for the computer is tedious and error-prone – a calculation based on what’s in our hand, what turn it is, and so on. Instead of all that, the model just makes three copies of the game. The first copy takes [[Amulet of Vigor]], the second takes [[Forest]], and the third takes [[Simic Growth Chamber]], then they each proceed independently from there. If any copy finds a line to get [[Primeval Titan]] on the table by turn three, it’s pretty safe to say a human player could have done the same. This approach is computationally intensive, but also straightforward and reliable.

Example Output Above is example output from the model. It knows how to track multiple colors of mana, transmute [[Tolaria West]], and use both mana abilities on [[Castle Garenbrig]].

The downside of brute force is that the computer essentially has superhuman “instincts” about the order of the deck. When a human player casts [[Ancient Stirrings]], they have to commit to a choice without knowing what their next card will be. The computer tries all options and keeps whichever works out best. Several corrections4 are included in the model to suppress non-human play patterns, but even so, numbers generated by the model include a percent-level bias in favor of five-card cantrips like [[Ancient Stirrings]], and a smaller bias in favor of the three-card [[Oath of Nissa]].

Open Questions

For the sake of brevity, we’re not getting into the numbers for [[Castle Garenbrig]] or [[Dryad of the Ilysian Grove]], both of which have seen near-universal5 adoption. Instead, we’re looking at three questions that (according to decklist aggregators) remain unresolved:

Simulations make use of the list below. The letter X represents [[Arboreal Grazer]] or [[Sakura-Tribe Scout]], Y is [[Azusa, Lost but Seeking:Azusa]] or [[Explore]], and Z is [[Ancient Stirrings]], [[Oath of Nissa]], or [[Once Upon a Time:OUAT]]. Put another way, we’re looking at all possible combinations of a one-drop ramp creature, an additional ramp spell, and a cantrip.

  Amulet Titan (March 2020)
4 X
4 Y
4 Z
4 [[Amulet of Vigor]]
4 [[Dryad of the Ilysian Grove]]
4 [[Primeval Titan]]
4 [[Summoner’s Pact]]
2 [[Pact of Negation:Other]] [[Engineered Explosives:Stuff]]
4 [[Castle Garenbrig]]
4 [[Misty Rainforest:Fetches]] and [[Breeding Pool:Shocks]]
4 [[Forest:Basic]] [[Snow-Covered Forest:Forests]]
3 [[Selesnya Sanctuary:Off-Color]] [[Golgari Rot Farm:Bounce]] [[Gruul Turf:Lands]]
5 [[Bojuka Bog:Off-Color]] [[Field of the Dead:Tapped]] [[Valakut, the Molten Pinnacle:Lands]]
4 [[Ghost Quarter:Off-Color]] [[Radiant Fountain:Untapped]] [[Hanweir Battlements:Lands]]
4 [[Simic Growth Chamber]]
2 [[Tolaria West]]

Note that the above list does not include [[Cavern of Souls]] or [[Vesuva]] due to computational complexity. We're also not accounting for small variations between deck lists. Some people play [[Boros Garrison]], [[Crumbling Vestige]], a third [[Tolaria West]], and so on. Wiggle room in the mana base introduces a potential percent-level bias across the board, but should not affect our ability to make relative comparisons.

Crunching the Numbers

A year ago, we might have looked at the odds of casting [[Primeval Titan]] by turn four. Recent additions have improved the deck’s speed and consistency to a point where that’s no longer interesting. The table below focuses on turn three. Turn two, rarely seen since ancient times, is again within reach as well.

X Y Z Turn 2 Turn 3 Turn 3 vs Removal
Grazer Azusa Oath 3% 26% 25%
Grazer Azusa Stirrings 3% 30% 29%
Grazer Azusa OUAT 4% 34% 32%
Grazer Explore Oath 3% 29% 27%
Grazer Explore Stirrings 4% 33% 30%
Grazer Explore OUAT 4% 38% 32%
Scout Azusa Oath 1% 20% 15%
Scout Azusa Stirrings 2% 24% 19%
Scout Azusa OUAT 2% 35% 18%
Scout Explore Oath 1% 18% 13%
Scout Explore Stirrings 1% 22% 15%
Scout Explore OUAT 2% 36% 15%

Statistical uncertainty ±1%. The difference between 25% and 26% is not significant, and the difference between 25% and 27% toes the line.

Above, columns X, Y, and Z show the different configurations of the deck, per the previous section. For each configuration, the value under Turn 2 is the percent of seven-card hands that can cast [[Primeval Titan]] on turn two. Similarly, the Turn 3 column shows the percent of seven-card hands that can cast [[Primeval Titan]] by turn three. Values for Turn 2 and Turn 3 assume a non-interactive opponent. The final column, Turn 3 vs Removal, assumes the opposite. For that case, the calculation is repeated assuming our opponent always has removal for our creatures, so we never get to untap with [[Azusa, Lost but Seeking:Azusa]], [[Dryad of the Ilysian Grove:Dryad]], or [[Sakura-Tribe Scout:Scout]]. This is particularly relevant when considering [[Dryad of the Ilysian Grove:Dryad]] as a potential replacement for [[Azusa, Lost but Seeeking:Azusa]]. [[Azusa, Lost but Seeking:Azusa]] ramps us from three mana to six on her own, even in the face of removal. [[Dryad of the Ilysian Grove:Dryad]] only gets us to five.

Conclusions and Caveats

[[Sakura-Tribe Scout]] may be the biggest loser from the [[Once Upon a Time:OUAT]] ban. The card is great on turn one but a pretty bad topdeck on turn two or three. [[Arboreal Grazer]] is similarly great on turn one – plus it nets mana immediately on later turns in combination with [[Amulet of Vigor:Amulet]] and a [[Gruul Turf:bounce land]]. [[Castle Garenbrig:Castle]] and [[Dryad of the Ilysian Grove:Dryad]] are bad for [[Sakura-Tribe Scout:Scout]]’s stock as well. The deck plays a lot more acceleration that it did a year ago, which means it’s less important to get multiple extra land drops out of a single card. Speed isn’t everything6 – but if speed is what we’re after, [[Arboreal Grazer]] is our best one-drop. It comes out ahead of [[Sakura-Tribe Scout]] even against a non-interactive opponent. Throw a few [[Lava Dart:Lava Darts]] into the mix and it’s not close.

Cantrips look straightforward as well. [[Ancient Stirrings]] outperforms [[Oath of Nissa]] across the board, but the difference isn’t huge. [[Ancient Stirrings:Stirrings]] is closer to [[Oath of Nissa:Oath]] than it is to [[Once Upon a Time:OUAT]], especially after we consider the model’s superhuman “instincts” about the order of the deck. If our sideboard is packed with [[Ashiok, Dream Render:planeswalkers]] and [[Collector Ouphe:creatures]] (rather than [[Grafdigger’s Cage:colorless]] [[Engineered Explosives:spells]]) then we could perhaps justify playing [[Oath of Nissa]] over [[Ancient Stirrings]].

Last up is [[Azusa, Lost but Seeking:Azusa]] versus [[Explore]], and it’s not looking great for our favorite Monk. In the first few turns of the game, she’s about on par with [[Explore]]. If the game goes long, we’d much rather draw a card with [[Explore]] as opposed to paying three mana for a [[Squire]]. Like [[Sakura-Tribe Scout:Scout]], [[Azusa, Lost but Seeking:Azusa]]’s selling point is that she provides multiple extra land drops on her own. That’s exactly what we wanted a year ago, but in combination with [[Castle Garenbrig:Castle]] and [[Dryad of the Ilysian Grove:Dryad]], it’s overkill.

Before the slew of new cards, about 25% of seven-card Amulet Titan hands could cast [[Primeval Titan]] by turn three. Lists often played [[Karn, the Great Creator:Karn]] or [[Trinket Mage]] to hold down the fort until turn four. With [[Once Upon a Time:OUAT]], that number was closer to 40% – after factoring in mulligans, players could expect to cast [[Primeval Titan]] by turn three more often than not. Now we’re somewhere in the middle. We’ll see if the deck continues to be all-in on turn three, or if it falls back on secondary threats. And, more importantly, we’ll see if WOTC continues to print new cards to boost the deck’s speed and consistency.

  1. Instagram was launched in October 2010. Cards most commonly played alongside [[Primeval Titan]] in Modern include [[Azusa, Lost but Seeking:Azusa]] (first printed in 2004), [[Sakura-Tribe Elder]] (2004), [[Farseek]] (2005), [[Simic Growth Chamber]] (2006), [[Search for Tomorrow]] (2006), [[Summoner’s Pact]] (2007), [[Tolaria West]] (2007), [[Vesuva]] (2007), [[Prismatic Omen]] (2008), [[Explore]] (2009), [[Oracle of Mul Daya]] (2009), [[Valakut, the Molten Pinnacle:Valakut]] (2009), [[Amulet of Vigor]] (February 2010), and [[Ancient Stirrings]] (April 2010). 

  2. The code (written in Python) is available on GitHub here. Pull requests welcome! 

  3. In the face of removal, [[Dryad of the Ilysian Grove]] only jumps us from three mana to five. [[Castle Garenbrig]] bridges the gap from five to six for [[Primeval Titan]]. [[Dryad of the Ilysian Grove:Dryad]] also makes all our lands Forests to ensure that [[Castle Garenbrig:Castle]] enters untapped. 

  4. To suppress non-human play patterns, the order of the deck is locked in as soon as the game begins. There are no mulligans. And any time the computer would search its deck for a card, instead it leaves the deck as-is and creates a new copy of that card out of thin air. This means no deck thinning, which introduces a small uncertainty (well under 1%). 

  5. For a rundown of how recent cards shook up Amulet Titan, check out this piece by Daryl Ayers. 

  6. [[Sakura-Tribe Scout]]’s ability can be activated at instant speed, which is a big deal in some matchups. It can deploy [[Bojuka Bog]] in response to [[Past in Flames]], or a bounce land to defend against land destruction.