Measure Twice, Valakut Once

A while back, I wrote a computer simulation of Titan Breach – a semi-mainstream [[Valakut, the Molten Pinnacle:Valakut]] deck in Modern. My writeup considered how different card choices affected the deck’s speed and consistency. In particular, I looked at [[Farseek]] versus [[Explore]]. Conventional wisdom prefers [[Farseek]] for its reliability. But if the goal is to land a T3 [[Primeval Titan]] as often as possible, it turns out [[Explore]] is a bit better. The risk of whiffing is more than made up for by the chance to draw a missing combo piece: [[Primeval Titan:Titan]], [[Through the Breach:Breach]], or [[Simian Spirit Guide:SSG]].

The question today is similar, but instead of [[Farseek]] versus [[Explore]] we’re looking at [[Cinder Glade]] versus [[Sheltered Thicket]]. The two lands fill a similar role: they’re both typed1 dual lands, and neither can be used to suspend [[Search for Tomorrow]] on T1. [[Cinder Glade]] is the more reliable mana source, typically entering the battlefield untapped from T3 onward. It’s widely played in both Titan Breach and Titan Shift. [[Sheltered Thicket]] always enters the battlefield tapped, which can make it an awkward topdeck in the early game. It’s rarely played.

Valakut Preference Poll A poll of the Valakut group on Facebook gives a sense for the conventional wisdom. [[Cinder Glade]] is strongly preferred over [[Sheltered Thicket]].

But, like [[Explore]], [[Sheltered Thicket]]’s text box includes three of the most important words in the game: “draw a card.” If we have [[Sheltered Thicket]] in our hand, we can pay a cost and discard it to draw something else. This doesn’t come up much in the early game, when resources are tight, but the flexibility is very powerful in the late game. In fact, I’d go so far as to say that [[Sheltered Thicket]] is clearly better than [[Cinder Glade]] after about T6. By that point we should have plenty of mana, so coming into play tapped or untapped is not really a concern. The question is: how much better is [[Cinder Glade]] in the early game?

The Model

The model2 we’re using is essentially the same one as last time. For each simulation, it shuffles up, draws an opening hand, then attempts all possible sequences of legal plays until it finds a way to get [[Primeval Titan]] onto the table. This strategy is computationally inefficient, but it’s guaranteed to find the best possible sequence of plays for every opening hand.

Sometimes the computer is a bit too good. For example, if it plays [[Explore]] and doesn’t like what it draws, it essentially gets to rewind and play [[Sakura-Tribe Elder]] instead. This effect is generally not a big deal – there honestly aren’t that many choices to make when goldfishing with a [[Valakut, the Molten Pinnacle:Valakut]] deck – but once in a while the computer’s superhuman “instincts” shine through. It’ll throw back a perfectly good seven-card hand in search of a faster six, or cycle instead of ramping to draw into [[Through the Breach]].

There are two corrections to minimize non-human play patterns. First, the model is allowed to mulligan to six, but no further. Second, there’s no shuffling. If you let it, the computer will sequence its shuffles to blind-draw into better cards. So once the game starts, the order of the deck is set. Any time we would fetch a [[Forest]] and shuffle, instead we leave the deck as-is and create a new [[Forest]] out of thin air. This means no deck thinning. Last time, we estimated this as a percent-level effect. Luckily, it does not grant an obvious bias in favor of [[Cinder Glade]] or [[Sheltered Thicket]].

Titan Breach

We compare [[Cinder Glade]] and [[Sheltered Thicket]] using the following list for Titan Breach, not so different from what we used the last time around. Notably, six slots are reserved for interaction: 3 [[Anger of the Gods]] and 3 [[Lightning Bolt]]. These could just as easily be [[Flame Slash]], [[Obstinate Baloth]], [[Reclamation Sage]], and so on. They don’t directly contribute to getting [[Primeval Titan:Titan]] on the table, so as far as the simulation is concerned they’re blanks.

Titan Breach
3 [[Anger of the Gods]]
4 [[Explore]]
3 [[Lightning Bolt]]
4 [[Primeval Titan]]
4 [[Sakura-Tribe Elder]]
4 [[Search for Tomorrow]]
4 [[Simian Spirit Guide]]
4 [[Summoner's Pact]]
4 [[Through the Breach]]
4 ???
2 [[Forest]]
8 [[Mountain]]
4 [[Stomping Ground]]
4 [[Valakut, the Molten Pinnacle]]
4 [[Wooded Foothills]]

We simulate 10k games using 4 [[Cinder Glade]] in the slots marked “???” and another 10k games with [[Sheltered Thicket]] in those slots. The other 56 cards remain unchanged. We also look at a pair of controlled variables: [[Taiga]] (which always enters the battlefield untapped) and “Slow Taiga” (which always enters tapped). This allows us to disentangle the advantage of cycling from the disadvantage of always entering the battlefield tapped. The numbers shake out as follows:

Lands 23-26 T3 ≥ T3.5 ≥ T4 ≥ T4.5
[[Cinder Glade]] 21% 29% 59% 82%
[[Sheltered Thicket]] 20% 27% 60% 84%
“Slow Taiga” 20% 27% 58% 81%
[[Taiga]] 23% 32% 61% 83%

T3 is the odds to [[Through the Breach:Breach]] on T3, which wins on the spot. T3.5 refers to hard-casting [[Primeval Titan:Titan]] on T3, which often stabilizes the board right away, but doesn't win until the next turn. Values are cumulative, so "≥T4" is the odds to at worst cast [[Through the Breach:Breach]] on T4. All values ±1%.

In terms of getting [[Primeval Titan:Titan]] on the table on T3, [[Sheltered Thicket]] is basically the same as “Slow Taiga.” This rings true. It’s technically possible for cycling on T2 to set us up for a big T3, but things have to line up perfectly – which happens in less than 1% of games.

We can also see that [[Cinder Glade]] is better than “Slow Taiga” across the board, but worse than [[Taiga]]. This is a nice sanity check. If [[Cinder Glade]] ever performed better than [[Taiga]], we’d know something was wrong with the simulation.

What’s striking is how close all the numbers are to one another. The difference between [[Taiga]] and “Slow Taiga” comes up once per 20 to 30 games. And [[Cinder Glade]] and [[Sheltered Thicket]] are even closer. In terms of getting [[Primeval Titan:Titan]] on the table on T3, [[Cinder Glade]] is better than [[Sheltered Thicket]] about once every 50 games. It just doesn’t matter that much whether lands 23 to 26 enter the battlefield tapped or not.

Titan Shift

We can run a similar experiment with Titan Shift, using the list below. As above, we run 10k simulations with [[Cinder Glade]] in the slots marked “???” then another 10k simulations with each of [[Sheltered Thicket]], “Slow Taiga,” and [[Taiga]].

Titan Shift
4 [[Anger of the Gods]]
3 [[Explore]]
3 [[Farseek]]
3 [[Lightning Bolt]]
2 [[Prismatic Omen]]
4 [[Primeval Titan]]
4 [[Sakura-Tribe Elder]]
4 [[Scapeshift]]
4 [[Search for Tomorrow]]
2 [[Summoner's Pact]]
4 ???
3 [[Forest]]
6 [[Mountain]]
4 [[Stomping Ground]]
4 [[Valakut, the Molten Pinnacle]]
2 [[Windswept Heath]]
4 [[Wooded Foothills]]

The numbers tell more or less the same story as we saw above for Titan Breach. [[Taiga]] and “Slow Taiga” behave similarly. The difference comes up about once every 50 games. [[Cinder Glade]] is about halfway between them – it’s only better than “Slow Taiga” about once every 100 games! And as the game goes on, the ability to cycle [[Sheltered Thicket]] becomes more and more of an advantage.

Lands 24-27 T4 ≥ T4.5 ≥ T5 ≥ T5.5
[[Cinder Glade]] 28% 64% 78% 91%
[[Sheltered Thicket]] 29% 66% 80% 93%
“Slow [[Taiga]]” 27% 63% 77% 91%
[[Taiga]] 29% 66% 80% 92%

T4 is the odds to [[Scapeshift]] on T4, which wins on the spot. T4.5 refers to hard-casting [[Primeval Titan:Titan]] on T4, which often stabilizes the board right away, but doesn't win until the next turn. Values are cumulative, so "≥T5" is the odds to at worst cast [[Scapeshift]] on T5. All values ±1%.

Caveats and Conclusions

The Titan Shift numbers suggest that by T5, [[Sheltered Thicket]] is better than [[Taiga]]. There’s reason to be skeptical. As noted above, the model plays each game every possible way and keeps the best outcome. If the model cycles [[Sheltered Thicket]] and doesn’t like what it finds, it essentially gets to rewind and do something else instead. In effect, the computer has superhuman “instincts” about when there’s something good on top of the deck.

Even so, it’s fair to say the conventional wisdom undervalues [[Sheltered Thicket]] – or perhaps overvalues [[Cinder Glade]]. In 98% of games we either won’t draw [[Cinder Glade]], or it’ll enter the battlefield tapped, or it’ll enter untapped at a time when it doesn’t matter. Maybe we’re playing it T2 off an [[Explore]], or it’s T3 and we only have a single ramp spell, or it’s T4 and we’ve already got all the mana we need. It’s only that other 2% of games where [[Cinder Glade]] enters the battlefield untapped at a crucial time. That’s once out of 50 games.

If this seems impossibly low to you, well, it seemed that way to me too. So I proxied up the deck and goldfished dozens of games with it. There were plenty of hands where [[Sheltered Thicket]] was awkward – a lone, tapped green source that prevented me from suspending [[Search for Tomorrow]] on T1. But that happens with [[Cinder Glade]] too. There was not a single game where [[Cinder Glade]] would have shaved a turn off the clock relative to [[Sheltered Thicket]].

On the other hand, I did find myself cycling [[Sheltered Thicket]] from time to time. The computer’s superhuman “instincts” make the effect hard to quantify, but it’s definitely worth something. Early on, it can help dig you out of a flood. In later turns, it finds you another threat or sideboard card that much faster. If you think it might save you bacon once out of 50 games, it’s probably worth a try.

  1. A typed [[Taiga:dual land]] has two basic land types, in this case Mountain and Forest, as opposed to a non-typed land like [[Kazandu Refuge]]. This is important because most of the spells in the deck require green mana, but [[Valakut, the Molten Pinnacle:Valakut]] only works with Mountains. 

  2. The original model, written in Python, is available on GitHub here. As an exercise, I went back and rewrote it in Go here. Forks and pull requests welcome for either repo! 

© Charles McEachern 2019 under CC-BY
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