# The London Mulligan Rule Mathematically Benefits Strategies That Rely on Specific Cards

At Mythic Championship II in London, Wizards of the Coast will try out a new mulligan rule that should make mulligans less punishing than the current Vancouver mulligan rule.

This new rule is as such: “When you mulligan for the Nth time, you draw seven cards, then put N cards on the bottom of your library in any order. So, for example, let’s say you’re taking your second mulligan of a game, what we often call a mulligan to five. You will draw seven cards, select two, and place those two on the bottom of your library in any order. Then you will decide whether to keep or mulligan again.”

R&D discussed this rule before. Several years ago, they dismissed it because “combo decks got a huge advantage” and it “encouraged big changes in deck building.” But they tested the rule internally for the past 4-5 months, and it’s been playing great in Standard and Limited, reducing the number of non-games. (They haven’t really tested it in Modern, though.)

They are aware that there are risks associated with this mulligan, especially in formats with larger card pools. For example, as I will quantify in this article, combo decks can find their pieces more consistently. These negative aspects are more likely to come up in non-rotating formats with powerful combos. But this is exactly why a competitive Modern event is the perfect opportunity to test the new rule. With hundreds of competitors trying to break it, it will act as a true stress-test.

For the time being, this rule will only be tested at Mythic Championship II in London. So in this article, I will refer to it as the “London mulligan.” If things play out well at the event and the evaluation is positive, then this mulligan rule will go into effect globally in all formats. So let’s go over the implications and run some numbers.

# Which Modern Decks Benefit Most From the London Mulligan Rule?

Decks that can assemble a winning opening hand with only four cards will benefit greatly from this new mulligan rule. Two examples are Dredge and Tron.

This four-card opening hand can easily win games. All the Dredge deck needs is a land, a discard outlet, and a dredge card. Additional cards in hand are largely irrelevant. In fact, it can even be an advantage to put Narcomoeba or Creeping Chill from your hand to the bottom of your library because (after shuffling with a fetchland) it allows you to dredge them into your graveyard later on.

With aggressive mulligans, as I will show later in more detail, the probability of getting an opening hand with two lands, a card draw spell, and a dredge card will shoot up from 45% under the Vancouver mulligan rule to 70% under the London mulligan rule.

Tron’s dream start of turn-3 Karn will become much more likely to assemble if you get to look at seven cards each time. Generally speaking, synergy-based or combo decks that can win with a specific combination of cards will find their key cards more consistently. As I will show later in more detail, the probability of finding an opening hand that guarantees a turn-3 Karn Liberated or Wurmcoil Engine goes from 16% under the Vancouver mulligan rule to 33% under the London mulligan rule.

# Which Modern Decks Won’t Benefit as Much?

Decks that care about quantity of resources, either because they are not looking for specific cards or because their game plan is centered around efficient one-for-one trades, will benefit less from this rule than other decks.

In Burn, nearly every card is either a Lightning Bolt or a Mountain—most cards are interchangeable. Having a creature in your opening hand is nice, but since you can win the game by casting seven Bolts, it’s more about quantity than quality. This means that mulligans are very punishing. Even the perfect four-card hand will have trouble finding enough Bolts to deal 20 damage.

For Humans, this is a good mulligan to four. But it’s nowhere near as degenerate as what combo decks can assemble. Humans is already consistent, with many redundant pieces. So the deck won’t benefit as much from better card selection. Moreover, Champion of the Parish and Thalia’s Lieutenant require a critical mass of Humans to trigger them, so they get much worse when you start with fewer cards.

Another aspect is that Humans has interactive cards like Reflector Mage or Thalia, Guardian of Thraben that can range from amazing to useless in various matchups. In game 1, when you don’t know what you are playing against, you can only guess which card to put on the bottom after a mulligan. Purely proactive decks don’t have this problem.

# How Often Will You Begin the Game with a Leyline on the Battlefield?

While Dredge gets more consistent, opponents will also be able to find sideboard cards (such as Leyline of the Void) more often. Let’s find out how often.

Consider a 60-card deck with four Leylines. A basic application of the hypergeometric distribution reveals the following:

 Vancouver probability of Leyline in opening hand London probability of Leyline in opening hand Always keep 7 39.9% 39.9% Willing to mull to 6 61.1% 63.9% Willing to mull to 5 72.8% 78.3% Willing to mull to 4 79.5% 87.0% Willing to mull to 3 83.4% 92.2% Willing to mull to 2 85.5% 95.3% Willing to mull to 1 86.5% 97.2%

Under either mulligan rule, you have a 39.9% to find at least one of your four Leylines in your starting seven. But if you are willing to mulligan down to a certain number of cards, then the odds of finding a Leyline go up dramatically.

For instance, if you always mulligan any seven-card or six-card hand without a Leyline, but always keep any five-card hand, then this corresponds to the “willing to mull to 5” row. Under this strategy, the probability that you begin the game with a Leyline on the battlefield is 72.8% under the Vancouver mulligan rule and 78.3% under the London mulligan rule.

If you are willing to mulligan into oblivion, then you were only 86.5% to find a Leyline under the Vancouver mulligan rule, with an average starting hand of 5.22 cards. Under the London mulligan rule, you’re 97.2% to find a Leyline, with an average starting hand of 5.55 cards. Another way to put it: the odds of missing seven hands in a row with the Vancouver mulligan is 13.5%, but the odds of missing seven hands in a row with the London mulligan is only 2.8%. That’s a huge difference.

Although these numbers are motivated by Leylines, they also apply to any other specific 4-of that you would like to see in your opening hand. For example, Amulet of Vigor, Hardened Scales, Eldrazi Temple, Lotus Bloom, Goryo’s Vengeance, Stony Silence, Chancellor of the Annex, or Gemstone Caverns. Under the London mulligan rule, it becomes more reasonable to aggressively mulligan toward them.

# What if We Add Serum Powder to the Mix?

Serum Powder becomes much better under the London mulligan rule because you have a higher likelihood of drawing it after a mulligan, you have more choice in which cards to exile, and you can tuck it on the bottom when you keep your hand.

The announcement clarifies how the mulligan rule interacts with Serum Powder: “For example, if you mulligan twice, then the third set of seven cards that you see contains a Serum Powder, you’ll first have to put two cards from your hand on the bottom of your library. Then (assuming you still have the Serum Powder in your hand), you can choose to exile your hand and draw five new cards, mulligan again, or keep your hand.”

Consider a 60-card deck with four Serum Powder and four key cards. The key cards could be Bazaar of Baghdad, Leyline of the Void, Eldrazi Temple, Goryo’s Vengenace, or any other card you want to have in your opening hand. For expositional ease, I will use Bazaar throughout this section. The key question is how often you will find Bazaar if you’re willing to mulligan into oblivion.

The required calculation is not straightforward. Once you use Powder, it changes the contents of your deck and potentially the bottom of your deck, which affects the probability of finding subsequent Powders or Bazaars. There are also choices to make. For example, if your seven card opening hand is four Serum Powder and three other cards, do you use Serum Powder or take a regular mulligan? The mathematical technique to model and solve this problem is called stochastic dynamic programming.

An implementation was coded by TheElk801. After I verified the general correctness of his approach and resolved some small issues, the result I got was that the probability of ending up with Bazaar in your opening hand is 99.25%. Bobby Fortanely, who correctly used stochastic dynamic programming calculations for Vintage Dredge under previous mulligan rules, claimed that it was 99.64% on Twitter. To approach the problem from a different angle, I coded a quick simulation under the simplifying assumption that you always use Powder if you don’t hold Bazaar. Simulating 10 million games yielded 99.26%, with an expected starting hand size of 6.12 cards.

These tiny differences could result from coding errors and/or in the handling of edge cases. This new mulligan rule combined with Serum Powder makes things very complicated, and it’s easy to make a mistake somewhere. But all above-quoted calculations agree that you will find Bazaar in your opening hand at least 99.25% of the time, which is an incredible amount of consistency when you compare it to the current 94.17%. If the London mulligan rule is implemented, Bazaar of Baghdad should be restricted in Vintage.

Coming back to Modern, there are already fringe decks that use Serum Powder to dig for Goryo’s Vengeance, Eldrazi Temple, or Leyline of the Void. Unlike Vintage Dredge, they may not be willing to mulligan into oblivion. For such decks, my implementation yields the following numbers under the London rule:

 Leyline in opening hand, deck with 0 Serum Powder Leyline in opening hand, deck with 4 Serum Powder Always keep 7 39.9% 54.1% Willing to mull to 6 63.9% 78.4% Willing to mull to 5 78.3% 89.6% Willing to mull to 4 87.0% 94.8% Willing to mull to 3 92.2% 97.4% Willing to mull to 2 95.3% 98.6% Willing to mull to 1 97.2% 99.3%

The first column is included for comparison and is taken from the previous Leyline table. The second column is the interesting one, as it shows the impact of adding four Serum Powder. For Modern, it means that a four Serum Powder deck that is willing to mull down to five cards in search of a 4-of Eldrazi Temple or a 4-of Goryo’s Vengeance will be successful nearly 90% of the time under the London rule. That is a scary amount of consistency.

# What if You Have Nature’s Claim to Answer Leyline?

Let’s disregard Serum Powder and consider the following thought experiment. We have a matchup between Dredge and Mardu. After sideboard, Mardu brings in four Leyline of the Void, while Dredge boards four Nature’s Claim. Both decks have zero Serum Powder. Players may mulligan, but they will keep the first hand containing at least one copy of their key sideboard card.

Subsequently, we look at the opening hands kept by both players after mulligans. If the number of Leylines held by the Mardu player is larger than the number of Nature’s Claims held by the Dredge player, then the Mardu player will win. Otherwise, if the number of Leylines is smaller than or equal to the number of Claims, then the Dredge player will win.

Clearly, this is an overly simplified model of reality. It doesn’t consider future draw steps or the fact that the Dredge player might still lose if they keep a poor hand with Nature’s Claim. But it does capture a key interplay that may often occur in London.

By convoluting the Leyline and Claim probability distributions, I determined that under the London mulligan rule, the probability that the Mardu player will win (i.e., the probability of drawing more Leylines than Claims) would be:

• 26.14% if both players always keep their opening seven
• 28.56% if both players are willing to mull to six in search of their key card
• 25.23% if both players are willing to mull to five in search of their key card
• 21.50% if both players are willing to mull to four in search of their key card

So while bringing in Leyline will help, it will rarely be enough to beat an opponent who aggressively mulligans toward an answer. On the whole, combo decks come out ahead.

# How Often Can a Tron Deck Guarantee Turn-3 Karn or Wurmcoil?

Consider the following model of a Tron deck:

Suppose we use the following mulligan strategy, based on all hand combinations that guarantee a turn-3 payoff card. We keep if and only if our hand contains at least:

• One of each Tron piece and a payoff card,
• Two Tron pieces, an Expedition Map, and a payoff card,
• Two Tron pieces, a Chromatic artifact, a Sylvan Scrying, and a payoff card.

The probability of finding any such good opening hand in your first seven cards is 9.64%. To determine this, my code enumerates all feasible starting hands and determines their likelihood via the multivariate hypergeometric distribution, which I explained in this recent article. But that’s only for a single seven-card hand.

If you are willing to mulligan all hands without a guaranteed turn-3 payoff card but always stop at four cards, then under the London mulligan rule you will end up with a hand that guarantees a turn-3 payoff card 33.32% of the time. You would keep 4.54 cards on average. By contrast, this used to 16.10% be under the Vancouver mulligan rule with an expected opening hand size of 4.39 cards. Full numbers are below.

 Vancouver probability of good opening hand London probability of good opening hand Always keep 7 9.64% 9.64% Willing to mull to 6 14.11% 18.34% Willing to mull to 5 15.75% 26.21% Willing to mull to 4 16.10% 33.32%

Note that my definition of “good opening hand” is quite narrow, and the mulligan strategy that these numbers are based on is extremely aggressive. It doesn’t consider Ancient Stirrings, and it would mulligan natural Tron if there is no payoff card to go with it. It only keeps hands where a turn-3 payoff is guaranteed.

I realize this strategy need not optimize the probability of actually casting a payoff card on turn 3, but the stochastic dynamic programming approach required to take these additional aspects, and potential draw steps, and scrying and sequencing and gameplay decisions into account requires a ton of extra effort regardless of mulligan rule. (As a remark, I ignored the Vancouver scry because I only considered opening hands. This means that my comparison between the two mulligan rules is not entirely fair. But keep in mind that the scry doesn’t change the information you have when you make the decision of whether or not to keep your opening hand.)

Anyway, under my definitions, the increase in consistency caused by the London mulligan rule is huge. If you’re willing to mulligan down to four, then the probability of finding an opening hand that guarantees a turn 3 Karn Liberated or Wurmcoil Engine more than doubles.

# How Often Will You Guarantee a Two Card Combo?

Consider the following model of a combo deck:

• 20 lands
• 8 copies of combo piece A
• 8 copies of combo piece B
• 24 other spells

Lots of Modern decks can be modeled in this way. For example:

The specifics of the deck don’t really matter, but the mulligan strategy does. I assume that we keep if and only if our hand contains at least one of each combo piece and at least two lands. This is what I define as a good opening hand.

By once again enumerating all feasible starting hands and determining their likelihood via the multivariate hypergeometric distribution, my script gave the following results:

 Vancouver probability of good opening hand London probability of good opening hand Always keep 7 26.28% 26.28% Willing to mull to 6 38.49% 45.65% Willing to mull to 5 43.55% 59.93% Willing to mull to 4 44.96% 70.46%

If you use this mulligan strategy but keep any four card hand, then you would keep 5.08 cards on average under the Vancouver rule, compared to 5.32 cards on average under the London rule.

So under the London rule, you will be able to sculpt a perfect opening hand (i.e., a hand with one of each combo piece and at least two lands) substantially more often, and you end up with larger hands on average. If you mulligan aggressively, you’re 70.46% to end up with such a perfect opening hand. Given that this number does not even consider future draw steps or card selection spells, combo decks can be expected to carry out their plans quite reliably in London.

# Can You Cut a Land From Your Deck?

As I’ll show, under the new mulligan rule, you may want to run more lands to optimize your probability of getting a keepable opening hand.

Of course, this is all based on the specific definition of “keepable hand”. In an insightful Reddit comment by aldeayeah, a “keepable hand” was defined as one with at least two lands and two spells. For my analysis, I slightly tweaked this definition to exclude hands with four or more lands. For the context of an aggro deck whose mana curve tops out at 3-drops, which is typical for Modern, such hands may flood out and therefore may deserve a mulligan. More importantly, by excluding 4+ land hands, the optimal deck configuration won’t be the obvious symmetric 30 lands and 30 spells, which allows us to get some actual insight on deck construction.

So suppose that we play an aggro deck with 21 land and 39 spells, and that we mulligan until we get a keepable hand containing two or three lands and at least two spells. My script yields the following:

 Vancouver probability of keepable opening hand London probability of keepable opening hand Always keep 7 59.6% 59.6% Willing to mull to 6 83.4% 89.4% Willing to mull to 5 92.2% 97.6% Willing to mull to 4 94.7% 99.5%

Suppose we define a non-game as a game where a player has to mull down to four in search of a keepable hand. Then given this deck and mulligan strategy, the number of non-games was roughly 1 in 19 under the Vancouver mulligan but will be roughly 1 in 190 under the London mulligan rule. That’s a major improvement.

Next, suppose that players are building their decks with the sole goal of minimizing the amount of non-games. That is, they want to optimize the probability of a keepable hand when willing to mull to 5. By enumerating over all possible land counts, my script determined that under the London mulligan rule, the probability of a keepable hand is indeed optimized with 27 lands. Under the Vancouver mulligan rule, however, this probability was optimized with 25 lands.

Now, real aggro decks in Modern don’t run 25 lands. This is because deck construction is not only about opening hand optimization but also about flood protection in the late game. Also, after a mulligan under the Vancouver rule, you could keep one-landers in the hope of scrying towards another land. These factors were not considered in this preliminary analysis.

Nevertheless, there is a difference, and it indicates that under the London mulligan rule, decks should run more lands than before. This is counter-intuitive, and requires further research.

[Editor’s note: The original version of this article had incorrect numbers for hands with two or three lands and at least two spells, and an incorrect conclusion regarding land counts.]

# Conclusion

Several years ago, the London mulligan rule was dismissed because “combo decks got a huge advantage” and it “encouraged big changes in deck building.” My article unsurprisingly confirms this, but its main contribution is to show concrete numbers and examples. I used simplified models that I can improve over the next few weeks, but I believe these first steps are already insightful. My takeaway is that for decks that rely on specific cards, the differences between the Vancouver rule and the London mulligan rule seem worryingly large.

At the same time, the amount of non-games (where one player never finds a decent number of lands) will be reduced, and mulligans will be less punishing. That will be good for the game. At this point, I am not sure how to weigh these advantages and disadvantages. I can run all the numbers on opening hands I want, and I may provide more detailed analysis in the future, but ultimately it will come down to how the games play out in practice.

At worst, we have one miserable tournament. At best, we get an improvement to the game. Although I am a little worried, it’s worth trying as a test, and I am looking forward to the Mythic Championship in London.

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