Magic Math: Hogaak Can Be Played on Turn 2 in 60% of Games

Hogaak is dominating the Modern format right now. At Mythic Championship IV, it was the most-played deck with the highest win rate of all major archetypes. At Grand Prix Minneapolis, more than half of the Top 8 was on Hogaak, and the finals was a Hogaak mirror. This happened even though a majority of the Top 16 players ran 4 Leyline of the Void.

Hogaak, Arisen Necropolis

Hogaak can be consistently cast on turn 2 in Modern

For reference, here is a rough sketch of a typical Hogaak deck.

A Typical Hogaak list

4 Carrion Feeder
4 Gravecrawler
4 Stitcher’s Supplier
4 Satyr Wayfinder
4 Bloodghast
4 Vengevine
4 Hogaak, Arisen Necropolis
4 Insolent Neonate
4 Faithless Looting
4 Interactive spells
10 Fetch lands
10 Other lands

There are many reasons for the dominance of this archetype. It’s resilient, fast, and powerful. But above all, it’s consistent. Since you can either draw or mill Hogaak, you are extremely good at finding it every game. In fact, at Mythic Championship IV, several Hogaak players told me that they were casting turn-2 Hogaak at least 50% of the time, with some claiming it may be as high as 70% of the time.

So how consistent is Hogaak, really?

Out of curiosity, I wanted to determine out how often you would actually cast Hogaak on turn 2 with the list shown above. To do so, I set up an extensive computer simulation in Python. I assumed that you mulligan optimally in search of a turn-2 Hogaak and make reasonable gameplay decisions. Under certain modeling assumptions (which are described in detail in the second part of this article) I got the following answer:

You can cast a turn-2 Hogaak 59.4% of the time.

This is the average of 64.2% on the draw and 54.6% on the play. In my opinion, this combination of speed and consistency is way too powerful and format-warping for Modern, especially when answers like Rest in Peace are simply too slow. As much as I dislike bans, I believe Hogaak should be banned in the next B&R announcement on Monday, August 26.

But out of curiosity, let’s consider some alternatives to an outright Hogaak ban.

What if Hogaak cost one or two additional colorless mana?

Lodestone Golem

If Magic were a video game, then R&D could easily nerf cards by adding one or two colorless mana. For me, it’s easy to adapt my simulation code by adjusting Hogaak’s converted mana cost variable and then running the program again. The results are as follows:

  • If Hogaak would cost 8 mana, then you would still be able to cast turn-2 Hogaak 50.9% of the time.
  • If Hogaak would cost 9 mana, then you would still be able to cast turn-2 Hogaak 42.6% of the time.

So increasing the converted mana cost of Hogaak to 8 wouldn’t even change all that much, and even a 9-mana Hogaak would be castable on turn 2 surprisingly often. I will say that a 9-mana Hogaak would have made for a much more bearable design. But given the difficulty of changing the numbers on cardboard copies, we’re stuck with the way it was printed.

What if Hogaak players would have to start with one or more forced mulligans?

Hogaak, Arisen Necropolis

Suppose that, for whatever reason, R&D doesn’t want to ban the card, can’t adjust the mana cost, but instead forces every Hogaak player start the game with one or more forced mulligans (after which players may decide to keep or mull as usual). In that case, the turn-2 Hogaak probabilities would be as follows.

  • Start with mull to 6: 51.9%
  • Start with mull to 5: 41.7%
  • Start with mull to 4: 26.9%
  • Start with mull to 3: 10.9%
  • Start with mull to 2: 2.0%
  • Start with mull to 1: 0.1%

A forced mulligan to 6 wouldn’t even change all that much, and even a forced mulligan down to 5 would still result in a powerful strategy. This is all because of the London mulligan rule, which makes smaller opening hands so much better.

I found it hilarious to see that a turn-2 Hogaak is still theoretically possible after a mulligan to 1. Can you figure out how?

What if Faithless Looting is banned?

Faithless Looting

Suppose that instead of unrealistic cost adjustments or forced mulligans, R&D instead decides to ban Faithless Looting. That’s at least within the realm of possibility. I investigated the impact by tweaking the deck in my simulation: I cut 4 Faithless Looting and added 4 fetch lands.

As it turns out, the resulting deck would still be able to cast Hogaak 58.6% of the time. That’s almost no change at all. Weird as it may sound, if your sole goal is casting Hogaak on turn 2, then it doesn’t really matter if you run 4 extra fetch lands or 4 Faithless Looting in your deck. In any case, a Faithless Looting ban would be totally ineffective.

What if Stitcher’s Supplier is banned?

Stitcher's Supplier

Okay, now we found the real culprit. In Hogaak, Stitcher’s Supplier is basically an Ancestral Recall, Black Lotus, and Mox Jet all in one. And unlike Faithless Looting, Stitcher’s Supplier is specific to Hogaak, so its disappearance wouldn’t hurt other archetypes. Hence, I could imagine that R&D decides to ban the best support card to weaken the overall consistency of the deck.

If this were to happen and we would replace the banned card with 4 fetch lands, then my simulation found that the resulting deck would be far less consistent. Specifically, it would be able to cast turn-2 Hogaak 39.3% of the time.

That’s a huge reduction. The reduction is even larger than for a 9-mana Hogaak or a forced mulligan to 5. This just goes to show that Stitcher’s Supplier, especially when combined with Carrion Feeder, is a big driver of Hogaak’s turn-2 consistency. If R&D would decide that banning a new Modern Horizons card is not on the table, then Stitcher’s Supplier should be on the cutting block.

What decklist did I use?

 The remainder of this article will describe my methods and assumptions used to derive all the numbers above. The list I used for the simulation was already shown at the beginning of this article. It is a relatively stock version, but some elements require clarification.

4 Carrion Feeder
4 Gravecrawler
4 Stitcher’s Supplier
4 Satyr Wayfinder
4 Bloodghast
4 Vengevine
4 Hogaak, Arisen Necropolis
4 Insolent Neonate
4 Faithless Looting
4 Interactive spells
10 Fetch lands
10 Other lands

For the mana base, I used 10 Bloodstained Mire and 10 Gemstone Mine, and I assumed that Gemstone Mine can be fetched with Bloodstained Mire. This is obviously a simplification of reality. I chose this model because I didn’t want to keep track of colors and decide which land to fetch. Including these elements would complicate the gameplay logic substantially. Yet, since the deck’s colored mana consistency is high, it would improve accuracy only slightly. Given the time I had available to build a simulation, I believe the simplification is justified.

For the interactive spells, it doesn’t really matter what they are. I treat them as irrelevant for the purpose of casting turn-2 Hogaak. This is also a simplification because Lightning Axe could theoretically help fill the graveyard, but treating them as blanks seemed more reasonable in a goldfishing context.

How does the simulation approach gameplay decisions?

Sequencing spells with Hogaak is not easy. Due to all the branching paths and topdeck/mill possibilities, the deck is actually quite hard to play optimally. For the simulation, the best I could do was to implement a fixed set of if-then-else constructs that should yield reasonable gameplay decisions.

My general logic is as follows:

  • Lands: Play fetchlands over normal lands, and fetch a land right away.
  • Discards: Insolent Neonate is sacrificed immediately for simplicity. Both Faithless Looting and Insolent Neonate follow a fixed priority list to determine discards. If you hold a fetch land, then the discard priority order starts with Bloodghast, followed by Vengevine, then a non-fetch land, then an interactive spell, then Gravecrawler, then Insolent Neonate, then Carrion Feeder, then Faithless Looting, then Satyr Wayfinder, then Stitcher’s Supplier, then fetch lands, then Hogaak. If you don’t hold a fetch land, then lands are moved to the very end of the priority list to ensure you can hit your second drop if possible.
  • Turn 1 logic: After playing a land, cast Stitcher’s Supplier if possible. Otherwise, if you already hold Satyr Wayfinder and another land, then play a black creature (with Carrion Feeder preferred over Gravecrawler). Else, play a red card draw spell (with Faithless Looting preferred over Insolent Neonate). If none of that was possible, then play a black creature (with Carrion Feeder preferred over Gravecrawler) if you hold one and have the mana for it.
  • Turn 2 logic (playing Satyr Wayfinder): If we can play land, Satyr Wayfinder, and Hogaak for sure, then just do that. Alternatively, if we haven’t seen Hogaak yet and don’t hold Stitcher’s Supplier but casting Satyr Wayfinder would enable a castable Hogaak if it lurks in the top 4, then cast Satyr Wayfinder. Finally, in case Satyr Wayfinder is literally our only play and have no one-mana spells, then we have no choice but to cast Satyr Wayfinder.
  • Turn 2 logic (one-drop before land): If we’re not casting land, Satyr Wayfinder, then try to play one-mana spells in a fixed priority order. First, if we can cast Stitcher’s Supplier, then do so. Otherwise, if we have already seen Hogaak or control Stitcher’s Supplier, then cast Carrion Feeder. Otherwise, play a red card draw spell (with Insolent Neonate preferred over Faithless Looting if and only if we have already seen Vengevine). If none of that applied, then play a black creature (with Carrion Feeder preferred over Gravecrawler). Keep track of mana available along the way, of course.
  • Turn 2 logic (playing a land): Before playing a land, if we control Stitcher’s Supplier and Carrion Feeder and will have access to another black or green convoker but haven’t seen Hogaak yet, then sacrifice Supplier to Feeder. In any case, subsequently, play a land if possible and return Bloodghasts.
  • Turn 2 logic (one-drop after land): Play one-mana spells in the same fixed sequence as before the land drop, with two differences. First, we may now return Vengevines along the way. Second, if casting Gravecrawler enables us to cast Hogaak for sure, then prioritize Gravecrawler, either from graveyard or battlefield, over red card draw spells.
  • Cast Hogaak: If we have Hogaak in hand, control at least two black or green creatures for convoke, and have 7 or more creatures in play plus cards in graveyard, then we cast Hogaak from hand. Casting Hogaak from graveyard is done via a similar check, except we don’t count it as a card in the graveyard for delve purposes. If we couldn’t cast Hogaak yet but can still sacrifice Stitcher’s Supplier to Carrion Feeder, then try that in a last-ditch attempt to find Hogaak and possibly cast it.

An example game

Let’s see this in practice in an example game.

Suppose we are on the play and keep the following 5-card hand, after putting two cards on the bottom. (I’ll explain the mulligan process later.)

Gemstone MineBloodstained MireCarrion FeederFaithless LootingSatyr Wayfinder

On turn one, we would play Bloodstained Mire, fetch Gemstone Mine (because that’s how I modeled the mana base for simplicity) and cast Carrion Feeder. This is preferred over Faithless Looting because the program looks ahead and sees that we can cast Satyr Wayfinder on the next turn, which would make for a good sequence to potentially convoke Hogaak.

On turn two, suppose we draw Stitcher’s Supplier. Since we haven’t seen Hogaak yet, the program switches gears and realizes that playing Stitcher’s Supplier and possibly Faithless Looting is better than Satyr Wayfinder.

So we cast Stitcher’s Supplier. Suppose we mill Gravecrawler, Vengevine, and Faithless Looting. Since we still haven’t seen Hogaak yet and found another convoke-worthy creature in Gravecrawler, the logic dictates that we now sacrifice Stitcher’s Supplier to Carrion Feeder. Suppose we mill Bloodghast, Hogaak, and Bloodstained Mire. Perfect!

We then play Gemstone Mine, return Bloodghast, play an unnecessary Gravecrawler, and return Vengevine. We now control four creatures and have five cards in the graveyard, including Hogaak. Hence, we convoke Hogaak and do a victory dance.

How does the simulation approach London Mulligan decisions?

Due to the London Mulligan rule, I had to set up a completely new stochastic dynamic programming framework to optimize the entire mulligan process, including the decision which cards to put on the bottom after a mulligan. It wasn’t trivial, but I succeeded in coding a general approach that can take any deck. For those of you who like to dig through the code for the details, here is the Github link again.

In short, given a decklist, a setting to play first or draw first, and an optimization criterion (in this case, the probability to cast turn-2 Hogaak, which is evaluated via several simulations for every opening hand) the program works its way from small to large opening hand sizes. For each opening hand size, thousands of opening hands are randomly generated, and for each, their turn-2 Hogaak probability (under an optimal set of cards to put onto the bottom in case of a mulligan, determined by exhaustive enumeration) is compared to the corresponding expected turn-2 Hogaak probability after taking a mulligan. If the former number is larger than the latter, then it’s a keep; otherwise, it’s a mulligan.

This process is best explained via an example.

An example opening hand

Suppose that we are on the play. Also suppose that we have already determined that the success probability (i.e., the probability to cast turn-2 Hogaak) once you mulligan to 4 cards on the play is 19.7%. To interpret this correctly, note that 19.7% is not the success probability when you keep an arbitrary 4-card hand. It is the expected success probability once you decide to mulligan to 4 on the play (and will make optimal decisions on whether to keep or mulligan further).

In case you’re curious, the corresponding number would be 34.1% on the draw, making for an average of 26.9% if play-draw were randomized, but these numbers are irrelevant to this example. Since we’re on the play, all that matters is that we can expect a success probability of 19.7% once we mulligan to 4.

We now consider the situation where we are mulliganning to 5 cards. Let’s draw a random opening hand. Suppose we look at the following 7 cards:

Bloodstained MireBloodstained MireGemstone MineCarrion FeederFaithless LootingGravecrawlerSatyr Wayfinder

There are 21 different combinations of two cards that you may put on the bottom when counting Bloodstained Mires as distinct objects. It’s 16 combinations once you remove these duplicates. My program iterates over all of them one by one and estimates the turn-2 Hogaak probability for each. Sorted from best to worst, these probabilities are:

  1. Bottom Gemstone Mine, Gravecrawler: 34.4%
  2. Bottom Gemstone Mine, Faithless Looting: 34.2%
  3. Bottom Gemstone Mine, Carrion Feeder: 33.7%
  4. Bottom Faithless Looting, Gravecrawler: 32.5%
  5. Bottom Faithless Looting, Carrion Feeder: 31.5%
  6. Bottom Bloodstained Mire, Gravecrawler: 15.0%
  7. Bottom Bloodstained Mire, Faithless Looting: 15.0%
  8. Bottom Bloodstained Mire, Carrion Feeder: 14.4%
  9. Bottom Gemstone Mine, Satyr Wayfinder: 13.0%
  10. Bottom Gravecrawler, Satyr Wayfinder: 11.2%
  11. Bottom Bloodstained Mire, Satyr Wayfinder: 11.7%
  12. Bottom Carrion Feeder, Satyr Wayfinder: 11.1%
  13. Bottom Carrion Feeder, Gravecrawler: 9.6%
  14. Bottom Bloodstained Mire, Gemstone Mine: 8.1%
  15. Bottom Bloodstained Mire, Bloodstained Mire: 7.1%
  16. Bottom Faithless Looting, Satyr Wayfinder: 6.0%

As it turns out, bottoming Gravecrawler and Gemstone Mine is best, and the corresponding turn-2 Hogaak probability of 34.4% is higher than 19.7% (the success probability once you mulligan to four on the play). So if we keep, it’s far from a guaranteed turn-2 Hogaak, but it’s better than taking a mulligan. The resulting opening hand corresponds to the one from the earlier example game.

This is only one possible five-card hand. The program iterates over thousands of different five-card hands, finds the best possible bottom and corresponding turn-2 Hogaak probability for each, and then assigns to every hand a value corresponding to the maximum of that number and 19.7%. Finally, the average of these values over all five-card hands is determined, and that will become our success probability once you mulligan to five cards on the play. We use this number when we evaluate mulligans to six, and we conclude with running many seven-card opening hands (which have no bottoming decisions).

Afterwards, the same process is done again, starting from one-card opening hands and moving up, but now when drawing first. Finally, the success probabilities for seven-card opening hands under play first and draw first are averaged, which yielded the final answers shown at the beginning of this article.

Modeling restrictions

Any mathematical model, especially one involving simulation, comes with its restrictions. Some important ones to keep in mind:

  • I considered only one version of the deck. Obviously, if you replace the 4 interactive cards by Hedron Crab or Glowspore Shaman or Lotleth Troll, then consistency will get higher. I also don’t know if Insolent Neonate is better or worse than some of those cards. My goal was to get insightful numbers to support the B&R discussion, not to find the best version of a deck that will likely be banned in a week anyway.
  • The gameplay logic is decent, but I don’t guarantee it is optimal for every possible situation. I wish I had enough time to implement a full stochastic dynamic programming or deep learning approach to optimize gameplay, but Hogaak would probably be banned by the time I would have finished with that. In any case, a really good player could probably eke out a few extra percentage points by making better sequencing decisions.
  • The success probabilities are evaluated via simulation, and since my mulligan code isn’t optimized for speed, the current sample size is relatively small. It’s in the order of tens of thousands of hands per mulligan number and play-draw setting, with each simulated for several dozens of games for every possible bottom. As a result, there is a small band of uncertainty around the success probabilities; it is very well possible that the real success probabilities are one percent lower of higher than the numbers I provided.
  • I gave a turn-2 Hogaak probability of 39.3% for the case where Stitcher’s Supplier would be banned, but this number doesn’t tell the whole story. For example, instead of adding 4 fetch lands, many players might add 4 Hedron Crab to their deck and possibly improve the consistency numbers. By how much? Well, I don’t know because coding the gameplay and sequencing logic for a new card like Hedron Crab would be a massive undertaking. But no matter what the best replacement might turn out to be, we have seen a similar story after the Bridge from Below ban: the deck got rebuilt, and the card Hogaak itself remained a problem. This drove me to the conclusion that banning Hogaak itself would be safest.
  • In reality, there is more to Hogaak than casting an 8/8 on turn 2. If you have an opening hand that would guarantee you get to attack with two Vengevines on turn 2, then you would probably keep that in real life, whereas the program logic doesn’t care about that. As a result, the mulligan decisions could be considered a bit too aggressive. Some gameplay decisions are arguably too single-minded as well.
  • The mana base model is nice and simple, but the real-life failure cases where you draw Satyr Wayfinders and Blood Crypts, with no green source in sight, are not accounted for. I believe the impact of this is small, but it is another limitation.
  • Although I have tried to carefully check my code, it didn’t undergo rigorous testing, so it is possible that there are some lingering bugs or incorrect implementations. For example, imagine I forgot to include the line that checks the mana pool before casting Gravecrawler from the graveyard. In that case, the program would give no error message but simply treat it as a zero-mana creature. Small things like that are easy to miss, especially in an extensive simulation like this one, and they could skew the numbers a little bit.


The good news is that I have coded a general simulation framework that optimizes mulligan decisions under the London mulligan rule, and this will allow me to update some of my classic deck building articles in the coming months. In the meantime, I welcome any feedback or suggestions on my code and implementation. It is very much a work-in-progress.

The bad news is that I have confirmed that the speed and consistency of Hogaak is absurd. The ability to cast it on turn 2 in approximately 60% of the games warps the format, and given that counterplay appears ineffective, I believe R&D should take action. While a Stitcher’s Supplier ban might substantially reduce the turn-2 Hogaak probability, I would rather be on the safe side and ban Hogaak, Arisen Necropolis itself.

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