We’ve recently seen the rise of various combo decks in Modern, including Ad Nauseam, Collected Company, and Grishoalbrand. But how consistent are these decks, really? I ran the numbers and set up some simulations to provide some insights.

1. How dangerous is Spoils of the Vault in the Ad Nauseam deck?

For reference, here is the deck list that this question is based on.

Darien Elderfield’s Ad Nauseam – Top 8 at Grand Prix Charlotte 2015

The game plan of this deck is to play Ad Nauseam when you have either Phyrexian Unlife in play or Angel’s Grace active. This allows you to survive at a negative life total, so you can draw your entire deck. With your entire deck in your hand, you exile several Simian Spirit Guides, play Lightning Storm, and toss in enough lands to win the game.

The upside of this deck is that it doesn’t rely on creatures, so cards like Path to Exile, Terminate, and Pyroclasm are dead against you. This stands in stark contrast to Splinter Twin, Infect, and so on. The downside of this deck is that you cannot win if you fail to assemble the combo—you absolutely need to find a copy of Ad Nauseam every game, and the deck only contains 4 copies. You have Serum Visions and Sleight of Hand to dig for them, but that isn’t always enough. Players have been toying with Mystical Teachings in the past, but it’s too slow and expensive for Modern.

Instead, Darien Elderfield went with Spoils of the Vault. If you cast it while you have Phyrexian Unlife in play or Angel’s Grace active, then you will always be able to find Ad Nauseam and survive. However, there’s still a risk: If you exile both Lightning Storms before hitting Ad Nauseam, then there will be no win conditions remaining in your deck. (There may also be an issue if you exile too many lands or Simian Spirit Guides, but we’ll disregard this for now for simplicity. It is unlikely, and the impact of losing lands or Apes heavily depends on your mana situation and your opponent’s life total.)

• So what’s the probability of exiling all of your Lightning Storms if you cast Spoils of the Vault for Ad Nauseam?

Answer: 1/15, or 6.7%

There are (at least) two different ways to see this. Both rely on the observations that there are only 6 relevant cards in the deck (4 Ad Nauseam and 2 Lightning Storm) and that only their order matters.

Solution 1: Binomial coefficients teach us that the number of different orders for these 6 relevant cards is 6!/(2!4!)=15, and of these 15, only the one that has both Lightning Storms in the first two positions loses you the game. Since every order can occur with equal probability, the answer to the question is 1/15.

Solution 2: The probability that the first relevant card is a Lightning Storm is 2/6. After that, there are 4 Ad Nauseam remaining and 1 Lightning Storm, so the probability that the next one is another Lightning Storm is 1/5. Since both have to happen, we multiply the two numbers to find a probability of 2/30 or 6.7%.

Overall, this feels like an acceptable risk to me. If you see Spoils of the Vault as kind of a last resort that you only play if you have to, then it greatly adds to the consistency of the deck as you can count on them as Ad Nauseam numbers 5-7.

I also think that 2 Lightning Storm is the correct amount for the deck: With only 1 Lightning Storm, there would be a 20% probability of losing on the spot when casting Spoils of the Vault, which is too high. And with 3 Lightning Storm, the probability reduces to 2.9% but I don’t think that is a large enough reduction to compensate for the downside of drawing Lightning Storms in your opening hand too often.

• What’s the life-loss probability distribution if you look for, say, Angel’s Grace?

So far, I’ve been talking about looking for Ad Nauseam when you already have Angel’s Grace active. But what if it’s the other way round? Then suddenly the life loss from Spoils of the Vault might become relevant.

For concreteness, let’s say you were on the draw and it’s your turn 3, so you have 50 cards remaining in the deck. You already have Ad Nauseam in hand and, thanks to Pentad Prism, you’ll have 6 mana available on the next turn. However, all 4 copies of Angel’s Grace are still lurking in your deck. How likely is it to lose a certain amounts of life to Spoils of the Vault?

To start, the probability of losing 0 life is simple: That’s just the probability of Angel’s Grace being on top of your deck, i.e., 4/50 or 8%. You lose 1 life if you have both a non-Angel’s Grace on top (which happens with probability 46/50) and an Angel’s Grace as your second card (which, given that there was no Angel’s Grace on top, occurs with probability 4/49). The multiplication of these numbers comes down to 7.51%, so that’s the probability of losing 1 life. And so on. Here is the full result:

Spoils

By summing bars, we can infer from this graph that the probability of losing 4 life or less is 35.3%, that the probability of losing 7 life or less is 51.4%, and that the probability of losing 10 life or less is 64.3%. Conversely, the probability of losing at least 5 life is 64.7%, the probability of losing at least 8 life is 48.6%, and the probability of losing at least 11 life is 35.7%. So Spoils of the Vault is a little risky, but perfect for people who like to live dangerously.

And if you are not much of a risk-taker, then you can always cycle Spoils of the Vault if you have just scried with Temple of Deceit. By naming the card you left on top, you get to draw a card at no risk to your life total. Overall, I think Spoils of the Vault is a sweet card and a good addition to the Ad Nauseam deck.

2. How reliable is Collected Company in the Modern Company decks?

Collected Company is one the new standout cards from Dragons of Tarkir. Without it, there wouldn’t be Elf Company, Abzan Company, or Zoo Company. It can yield multiple creatures, allows you to dig deeper into your deck for combo pieces, and it does it all at instant speed!

I ran some general numbers on Collected Company before, but since we now have access to actual decks lists, we can analyze those to obtain more specific numbers. I coded a simulation that provides all kinds of interesting performance indicators, and I fed it the mana curves of the three best-performing Collected Company decks from Grand Prix Charlotte. Here are the results:

Michael Malone’s Elf Company deck (1st at Grand Prix Charlotte):

  • The deck has 16 one-drops, 7 two-drops, and 10 three-drops, making for a grand total of 33 hittable creatures
  • The expected amount of mana worth of creatures per Collected Company, assuming you always take the maximum, is 4.34. So, you usually get more mana out of it than you invested.
  • The expected number of creatures hit is 1.94. That’s quite reliable.
  • The probability of hitting two creatures is 94.6%, the probability of hitting one creature is 4.9%, and the probability of whiffing completely is 0.5%.

Ian Bosley’s Abzan Company deck (6th at Grand Prix Charlotte) and Paul Rietzl’s Zoo Company deck (10th at Grand Prix Charlotte):

  • We can group both decks together because they had the exact same mana curve: 9 one-drops, 10 two-drops, and 8 three-drops, which is 27 creatures in total.
  • In these decks, a Collected Company nets 4.11 mana worth of creatures on average.
  • The expected number of creatures hit is 1.83. It’s less than the Elf deck, of course, but still fairly reliable.
  • The probability of hitting two creatures is 85.7%, the probability of hitting one creature is 12.2%, and the probability of whiffing completely is 2.1%.

Based on these numbers, it is easy to understand the appeal of Collected Company: With the right creature base, it’s a reliable source of card and mana advantage.

3. How easy is it for a Grishoalbrand player to draw their entire deck?

 

To provide appropriate context, let’s start with a deck list again.

Jacob Wilson’s Goryo’s Vengeance – 11th At Grand Prix Copenhagen 2015

This deck plans to (i) get a Griselbrand into play via Goryo’s Vengeance or Through the Breach, (ii) chain Nourishing Shoals and draw cards until you find Through the Breach, Borborygmos Enraged, and several Simian Spirit Guides, (iii) ping your opponent to death with lands. The key innovation is Nourishing Shoal, which allows the deck to go off at instant speed and do all kinds of splice-onto-arcane tricks.

But how consistent is the kill?

I could tackle this question in various ways. If I had enough time, I would teach a computer how to goldfish the deck and ask it to derive the kill probabilities for turn 1, turn 2, and so on. However, teaching a computer how to Faithlessly Loot, when to exile Simian Spirit Guide, what card to play in any given situation and so on is very difficult and time-consuming.

Instead, I will fix a scenario and ask a sufficiently easy-to-analyze, yet insightful question. Specifically, suppose your opponent is on the play and went turn 1 Lightning Bolt you, turn 2 double-Bolt, putting you down to 11 life. Meanwhile, you had stacked the top of your deck with Faithless Looting, Griselbrand, Goryo’s Vengeance, Mountain, and Swamp—don’t actually stack your deck in real life please—which enables you to sneak Griselbrand in play on turn two. With Griselbrand on the battlefield, you attack and go up to 18 life. At this point, your deck contains 49 cards. We assume that Worldspine Wurm is the only card you can exile to Nourishing Shoal, we disregard Noxious Revival, and we ignore the additional card draw power of Faithless Looting and Manamorphose. Now here’s the question:

Given that your deck contains 4 Nourishing Shoals and 4 Worldspine Wurms, what is the probability that you can draw your entire deck and guarantee the Borborygmos Enraged kill?

Before reading further, take a guess. What do you think? Will these odds (under the assumptions I described above) be more or less than 50%?

Now, let’s dive in. The first thing to note is that 49 cards comes down to 7 Griselbrand activations. In our scenario, this would work as follows.

  • Activation 1: Our library gets down to 42 cards, and we fall to 11 life.
  • Activation 2: Down to 35 cards and 4 life. We need to have found Nourishing Shoal + Worldspine Wurm at this point, otherwise you “fizzle.” Assuming we do, we go back up to 15 life and continue.
  • Activation 3: Down to 28 cards and 8 life.
  • Activation 4: Down to 21 cards and 1 life. To be fair, at this point it is likely that you have already drawn enough pieces to win the game with Borborygmos Enraged. But if you want to draw your entire deck, as we stated in the original question, then you need to have found a second Nourishing Shoal + Worldspine Wurm pair to continue. Assuming you do, you go back up to 12 life and we activate Griselbrand again.
  • Activation 5: Down to 14 cards and 5 life. To continue drawing your entire deck, you need to have found a third Nourishing Shoal + Worldspine Wurm pair at this point. Assuming you do, you go back up to 16 life and we continue.
  • Activation 6: Down to 7 cards and 9 life.
  • Activation 7: Down to 0 cards and 2 life. Hurrah!

So, to draw your entire deck, there are three points at which you need to find a Nourishing Shoal + Worldspine Wurm pair. It would be possible to determine this likelihood analytically, but it would require careful conditioning, and my brain is at the moment too exhausted to perform the required analysis.

Typing up over 50 Modern deck lists, player interviews, Deck Techs, Pro Points system explanations, “What’s the Play” puzzles, and many other pieces of content for the text coverage page of Grand Prix Copenhagen last weekend—which I wholeheartedly recommend you check out if you haven’t done so yet—did take its toll on me.

Instead, I wrote a simulation. It lacks the beauty and elegance of exact analysis, but 10 million simulated games still get us close enough to the correct answer.

The result we got for the probability to draw our entire deck was: 40.3%! To reiterate, this number is obtained under the assumptions that Worldspine Wurm is the only card you can exile to Nourishing Shoal and that all card draw library manipulation spells don’t exist, so it will be slightly higher in reality. But even then, it was a bit of a surprise to me that you are not a favorite to be able to draw your entire deck.

On a positive note, being able to activate Griselbrand 4 times (which requires only one Wurm + Shoal pair to be in the top 20 unstacked cards of your deck) is much more likely, at 71.2% probability. And most of the time you will be able to kill from that point. But still, I would argue that my numbers support the insight that you are far from guaranteed a kill if you get Griselbrand in play—the chance of fizzling is real.