# Magic Math – The New Mulligan Rule

Last week, Wizards introduced a new mulligan rule with the stated aim of reducing the number of non-interactive games. 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.” For the time being, this rule will only be in effect at Pro Tour Magic Origins. However, if the evaluation of this test run is positive, then it seems likely that this Vancouver mulligan rule will go into effect everywhere else.

Some people aren’t affected by this new mulligan rule. Take Gerard Fabiano, for example. He’s a master, but he’s never taken a mulligan in his life, so the free scry will be irrelevant for him. For everyone else, it will be a nice consolation prize. Mulligans greatly reduce your chances of winning the game—depending on the matchup, your chances may fall to approximately 40% when you go down to 6 and 25% when you go down to 5—and they can often result in a frustrating mana screw or mana flood. Any attempt to reduce the number of non-interactive games seems fine to me.

But will it really work? I was curious to see how large the impact of this new mulligan rule would be, so, I fired up my computer, ran the numbers, and found a bunch of interesting things.

## 1. Delver of Secrets becomes better, but not by much

Let’s get this out of the way first. Consider this deck:

## A Legacy Delver Deck

The deck will be a nightmare to play against but, more importantly, it captures all the key ingredients to determine the probability that Delver of Secrets transforms right away. I coded a simulation for this deck based on the following assumptions:

• For both the old and the new mulligan rule, I simulated 10 million games, always on the play.
• If an opening hand has no lands, always mulligan. If an opening hand has at least one land, then mulligan in 15% of the cases. (This 15% is my best guess, based on experience, for the number of land-containing opening hands that you should still mulligan. As I show later, more aggressive mulliganing can pay off under the new mulligan rule, but exactly how much is unclear for this particular Delver deck, so I imposed 15% throughout for ease and clarity of comparison.)
• If an opening hand doesn’t contain a Volcanic Island, then you have to use Scalding Tarn, which ruins the free scry.
• With the free scry, you always keep any sorcery or instant on top and bottom everything else.

Given these assumptions, the probability to immediately flip a turn-1 Delver (averaging over all games, with or without mulligans, as long as we’re on the play with a turn-1 Delver) is as follows:

• Under the old mulligan rule: 49.6%.
• Under the new mulligan rule: 52.8%.

So there is a difference, but it’s only a couple percent. This makes sense because you only get to scry if you mulligan in the first place, which is not all that common. And once we add the games where we’re on the draw and/or don’t draw Delver, then we can conclude that the new mulligan rule will only impact Delver transformations once every hundred games or so. Multiplying that with the small probability that a transformed Delver is actually the difference between winning and losing, we see that it’s a minor bonus only.

## 2. Overall, decks can kill slightly faster

For the remainder of this article, I dug up several decks from my earlier series (part 1, and part 2) on optimal decks for various goldfish formats. These are the decks in question:

The best deck for the Savannah Lion format:

The best deck for the Grizzly Bear format:

Noncreature Spells14 x Shock

The best deck for the Lightning Bolt format:

The best deck for the Splinter Twin format:

As explained in the original articles, these decks provide the fastest average goldfish kill on the play in the formats where only each deck’s respective cards are legal. I derived them two years ago using computer simulation and enumerative optimization. Today, I will adhere to the same optimization criterion as back then. So, we are on the play against an opponent who doesn’t do anything, and our aim is to minimize the expected turn in which we deal the 20th damage. With this in mind, I determined an optimal mulligan strategy and, if applicable, optimal scry strategy for each deck via dynamic programming. Although my code has become messy, my implementation is here for reference.

With all that out of the way, the first thing I did was to analyze how the new mulligan rule influences the average kill-turn. Here are the results:

(Click to enlarge.)

Yup, the games are over sooner. These numbers are of course based on specific decks and criteria, so my findings need not generalize to all formats. Nevertheless, my focus on simple decks and well-defined optimization criteria enables me to obtain sharp numbers.

Overall, the differences are relatively small (so I’d be surprised if the new mulligan rule influences the metagame and/or card choices too much in Vancouver) but you can still notice the boost to the average kill-turn after a mulligan. When it comes to beating a goldfish as quickly as possible, this free scry can be worth almost half a card. The largest relative benefits are seen for the Splinter Twin format.

## 3. You should keep fewer sketchy 7-card hands

In general, you should mulligan a 7-card hand if its win percentage is lower than the one you expect if you mulligan down to 6. Since mulligans improve under the new scry rule, the sketchiest 7-card hands should instead be mulliganed.

My simulation optimization confirmed this. I’ll discuss the optimal mulligan frequency later, but for now I want to make things concrete by providing several example hands that should be kept under the old mulligan rule but that should be tossed under the new one.

The optimal mulligan strategy for the deck with Lightning Bolts and Mountains was not affected.

## 4. You should keep risky 6-card hands slightly more often

Interestingly, for my decks, an exhaustive search revealed no 6-card opening hand that would be kept under the old rule but mulliganed under the new rule. The reverse, however, did occur.

For concreteness, here are some example hands that (according to my computer) should be mulliganed under the old mulligan rule but that should be kept under the new one:

It’s not hard to see what’s going on here: These hands are looking for a specific card (a land) so scrying is almost as good as drawing a card. The numbers indicate that these hands go from mulligans to keeps under the Vancouver rule.

## 5. Overall, there will be more mulligans

So far, we’ve seen an effect that increases mulligan frequency and an effect that decreases it. But how large are these effects? Well, as it turns out, point 3 overshadows point 4. Take a look at the numbers:

With the exception of the Lightning Bolt format, the optimal mulligan frequency becomes higher across the board. It is likely that rounds will go a little longer as a result. Let’s take the Savannah Lions format, where the expected number of mulligans per game grows by 0.13. If we assume that a mulligan takes a minute, that scrying takes ten seconds, and that each match takes 2.5 games, then on average the Vancouver mulligan rule will add 29 seconds to the clock per match. That’s acceptable to me, but it’s still a downside.

## 6. The number of non-interactive games decreases

This is a tricky one. How do I formalize “non-interactive game” in a goldfish format that is by definition non-interactive? The best I could come up with is to define “non-interactive game” as a game where you are unable to deal 20 damage by turn 6. That is, games with a kill-turn of 7 or higher. Since the decks are designed to win as quickly as possible with no regard for late-game presence, this turn threshold is a reasonable indicator of a game where you had no chance against a real opponent.

Under this definition, the number of non-interactive games indeed decreases by a small percentage under the new mulligan rule. This is not an obvious outcome. I actually feared that the increase in mulligan frequency would overshadow the kill-turn improvement for mulligans. That is, if the new mulligan rule would encourage more frequent mulligans into hands whose kill-turn has a better expected value but a worse variance, then the new mulligan rule might backfire. However, at least for the four decks I considered, this was not the case. The number of non-interactive games reduced, and I hope to see a similar effect at the Pro Tour in Vancouver.

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