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Magic Math: Deploy the Gatewatch

Magic sets these days contain plenty of cards that are interesting to analyze from a mathematical standpoint. Eldritch Moon is no exception. This week, I’ll go over Deploy the Gatewatch and Lupine Prototype, and next week I’ll discuss Liliana, the Last Hope and several other cards.

Deploy the Gatewatch

This new Collected Company for planeswalkers has spawned quite a bit of discussion. Collected Company is a format-defining card, but Deploy the Gatewatch has a bunch of downsides—it costs 2 mana more, hits a card type that decks typically aren’t filled with, and can run into the rule that you can’t have 2 planeswalkers of the same type on the battlefield at the same time. In exchange, however, there is no mana cost restriction on the permanents you can hit and you get to look at 7 rather than 6 cards.

Let’s run some numbers.

What are the probabilities for a Standard deck?

When the card was revealed, I Tweeted out a picture of a simple spreadsheet.

Deploy the Gatewatch 60-card deck

The way to interpret this is as follows: You take a 60-card deck with a number of planeswalkers given in the A column, remove 1 Deploy the Gatewatch from the deck, and put it on the stack. The results are shown in the corresponding B, C, D, and E columns. The B column shows the probability of missing completely. The C column shows the probability of hitting exactly 1 planeswalker. The D column shows the probability of hitting 2 or more planeswalkers. Finally, the E column describes the expected number of planeswalkers that Deploy the Gatewatch will yield in the long term.

So for a deck with 16 planeswalkers, you’ll miss 9.4% of the time, get 1 planeswalker 28.6% of the time, and hit at least 2 planeswalkers 62.0% of the time. In expectation, that’s 1.53 planeswalkers per Deploy the Gatewatch. Assuming that the average planeswalker costs 3 or 4 mana, this doesn’t sound like a reliable enough payoff to me, especially when it doesn’t even take into account the notion that 2 Gideon and 5 lands should count as 1 hit, not 2. So in a 16-planeswalker deck, with these numbers, I’d rather just play Chandra or Sorin instead of trying my luck with Deploy the Gatewatch.

But once you go up to 20 planeswalkers, the numbers get more exciting, with the probability of a complete miss falling below 5% and the expected number of planeswalker hits rising above 1.7. These numbers are similar to a Collected Company in a deck with 23 creatures.

How about Commander?

When I Tweeted out that picture, I received requests for a similar table for Commander decks, so here you go. The table starts at the number of planeswalkers required for a 50%-or-higher probability of hitting 2 planeswalkers.

Deploy the Gatewatch 99-card deck

Based on this table, I would recommend at least 33 planeswalkers for regular Commander decks with Deploy the Gatewatch. But if your deck contains a lot of Brainstorm-style cards that aim to cheat Nicol Bolas and/or Ugin to the top of your deck, then you could do with less, although I would still be hesitant to go below 27.

What’s the impact of the planeswalker rule?

So far, I haven’t explored the adverse impact of the planeswalker rule that prevents you from having 2 planeswalkers of the same type in play simultaneously. To do that, you have to know how many 1-ofs, 2-ofs, 3-ofs, and 4-ofs you have of each planeswalker. In other words, you need to consider a particular deck composition. Since the issue is more prevalent with 60-card decks than with 99-card decks, I’ll consider the following Standard list.

4-Color Planeswalkers

This Abzan deck with a small red splash contains 22 planeswalkers, and I think that’s as far as you can realistically push it. This deck can surely be improved—for example, I could see a few Tamiyo or Nahiri instead of Liliana if you can fix the mana curve somehow. But the main purpose of this 22-planeswalker deck is to fix a deck composition that allows you to estimate the impact of the planeswalker rule.

For that, you can use the following formula, assuming that the board is empty when you cast Deploy the Gatewatch.

Deploy the Gatewatch probability

Click to deploy.

The results for my 22-planeswalker deck are as follows:

  • Hit 0 planeswalkers – 3.0%
  • Hit exactly 1 (unique) planeswalker – 18.6%
  • Hit 2 or more (unique) planeswalkers – 78.4%
  • Expected number of planeswalkers chosen – 1.75

Note: The numbers get worse if you take into account the possibility that Deploy the Gatewatch hits a planeswalker you already have on the battlefield. The likelihood of that depends on the matchup, on how much your opponent is pressuring your planeswalkers, and on how you sequence your planeswalkers, so I left it outside the scope of my calculation, but it is worth pointing out.

For comparison, if you ignore the planeswalker rule, as in the first spreadsheet picture, then you have the following probabilities:

  • Hit 0 planeswalkers – 3.0%
  • Hit exactly 1 planeswalker – 15.0%
  • Hit 2 or more planeswalker – 82.0%
  • Expected number of planeswalkers chosen – 1.79

So the planeswalker rule is a downgrade, but I wouldn’t call it a complete deal-breaker. Very roughly speaking, the numbers for a 22-planeswalker deck under the planeswalker rule are similar to the ones for a 20-planeswalker deck ignoring the rule. As I mentioned, those numbers were similar to the ones for a Collected Company in a 23-creature deck, which suggests that Deploy the Gatewatch is fine in my 22-planeswalker deck. Things get better once you turn to expected mana hit.

How Much Mana Can You Expect to Hit for this 4-Color Planeswalker Deck?

For any outcome of the top 7 cards, you can start by taking a Sorin if able, then an Ob Nixilis if able, then a Sarkhan if you haven’t taken 2 planeswalker cards already, and so on. This way, you maximize the converted mana cost of the planeswalkers you get. By multiplying the amount of mana worth of planeswalkers by the probability of seeing that outcome and subsequently summing over all outcomes, you can find the expected total converted mana cost of the planeswalkers that you take off of Deploy the Gatewatch.

The result of this calculation is that Deploy the Gatewatch will hit 7.78 mana worth of unique planeswalkers in expectation. That is pretty good. In the long term, Collected Company tends to break close to even on mana (it’s 4.3 mana in a deck with four 1-drops, ten 2-drops, and ten 3-drops) but Deploy the Gatewatch gets you ahead by almost 2 free mana. That’s value!

The Verdict

As I showed, in the right deck you can achieve a level of consistency where Deploy the Gatewatch is worth the 6-mana investment. The problem, however, is that a Standard deck with at least 22 planeswalkers just doesn’t seem very good right now. The 4-color deck I built as a reference, for example, doesn’t look that impressive to me. It’s hard to obtain an acceptable mana curve, consistent mana base, and synergy with an all-planeswalker deck. It’s much harder than with an all-creature deck, at least.

It might still be possible to run a 19-21 planeswalker deck with 2-3 Deploy the Gatewatch, leaving more room for other effects while still getting a sufficient payoff from Deploy the Gatewatch. Either way, I love the design on Deploy the Gatewatch, but I don’t expect it will become the new Collected Company.

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