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Magic Math – Should You Cut a Land With The New Mulligan Rule?

Starting with the Battle for Zendikar prereleases, the new mulligan rule (also known as the Vancouver mulligan) will go in effect. The rule says that after mulligans are completed, “any player whose opening hand has fewer cards than his or her starting hand size may scry 1.” In a previous article I showed that, for certain decks under certain criteria, the Vancouver mulligan would lead to more mulligans of sketchy 7-card hands, a higher chance of winning after a mulligan, and fewer non-interactive games.

But what I didn’t analyze yet is whether the new mulligan rule has an impact on how you should build your decks. In particular, should low-curve aggro or combo decks cut a land?

The intuition behind this idea is that decks such as Mono-Red Burn ideally draw 3-4 lands early on, followed by all spells for the remainder of the match. Cutting a land reduces the probability of mana flood while increasing the probability of mana screw. Yet, mana screws can be mitigated by the Vancouver mulligan rule by scrying away non-lands after keeping a 6-card hand with 1-2 lands.

To investigate this, I dug up the optimal decks for various goldfish formats from two years ago, fired up my computer, and determined the optimal deck configurations under the new mulligan rule.

Some Methodological Remarks

  • I will look for the “optimal” deck on a restricted domain to limit the computational time needed. I only evaluated decks (using the Vancouver mulligan) that could be constructed from the original optimal list (with the old mulligan rule) by swapping at most two cards. In other words, the maximum distance from the original optimum is four according to the Manhattan norm.
  • As explained in my article from two years ago, the criterion to evaluate decks is the average goldfish kill on the play in the formats where only each deck’s respective cards are legal. So, we are on the play against an opponent who doesn’t do anything, and the optimal deck is the one that minimizes the expected turn on which we deal the 20th damage.
  • The mulligan strategy and scry strategy is optimized for every deck via (stochastic) dynamic programming.
  • Evaluation of any deck is based on 10,000 simulations per opening hand to determine the optimal scry and mulligan strategy, and on 20 million simulations per deck once those had been determined.
  • My code is available here.

Now, on to the results.

The Savannah Lion, Putrid Leech, Lightning Bolt Format

The best deck under the old mulligan rule:

Average kill-turn under the old mulligan rule: 4.221.

Average kill-turn under the new mulligan rule: 4.181.

The best deck under the new mulligan rule:

Average kill-turn under the new mulligan rule: 4.181.

So, no change yet.

The Mons’s Goblin Raiders, Grizzly Bear, Gnarled Mass, Shock format

The best deck under the old mulligan rule:

Average kill-turn under the old mulligan rule: 5.675.

Average kill-turn under the new mulligan rule: 5.641.

The best deck under the new mulligan rule:

Average kill-turn under the new mulligan rule: 5.636.

There’s a small change here, but the number of lands remains unchanged.

The Lightning Bolt Format

The best deck under the old mulligan rule:

Average kill-turn under the old mulligan rule: 4.910.

Average kill-turn under the new mulligan rule: 4.872.

The best deck under the new mulligan rule:

Average kill-turn under the new mulligan rule: 4.872.

No changes here.

So far we’ve looked at decks with few lands and short games. Maybe we’ll see some changes in the number of lands when we consider decks with more lands or longer games.

The Pestermite and Splinter Twin format

The best deck under the old mulligan rule:

Average kill-turn under the old mulligan rule: 4.319.

Average kill-turn under the new mulligan rule: 4.255

The best deck under the new mulligan rule:

Average kill-turn under the new mulligan rule: 4.253.

The optimal deck now has one fewer land!

The same format as the first, but now the opponent starts at 40 life

The best deck under the old mulligan rule:

Average kill-turn: 5.830.

Average kill-turn under the new mulligan rule: 5.809.

The best deck under the new mulligan rule:

Average kill-turn under the new mulligan rule: 5.806.

The optimal deck now has one fewer land!

Conclusion

Some, but not all, of the optimal decks under the new mulligan rule had one fewer land than the optimal deck under the old mulligan rule. The differences in expected kill-turn are small—sometimes within the margin of error with the simulations—and you have to keep in mind that I am restricting my analysis to simple formats under a simple optimization criterion. So my analysis can’t provide a rigorous proof for an actual Magic format, but it provides nice insights and at least didn’t indicate that my intuition was wrong.

After running the numbers, my view on the matter is as follows: Don’t go out of your way to cut a land from your deck, but if you were having a real hard time deciding between, say, 23 and 24 lands before and rounded it up to 24 before, then you may want to round down to 23 under the new mulligan rule.

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