(Grenzo, Dungeon Warden | Art by Lucas Graciano)

The Math Behind the Mulligan

Feel like your deck just is not coming together? Welcome to Solve the Equation, where we take a look at the numbers and see what's making your deck and in-game decision-making fall flat.

Today, we're going to take a deep-dive on how we can use math to take advantage of the mulligan. The first decision we make in a game of Commander can make the difference between winning and losing. Many articles have been written about how to mulligan and what types of hands to keep, but my goal with this article is to argue why you should mulligan more often than you currently are.

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The London Mulligan

A few years back, Magic adopted a new mulligan rule, and the Commander Rules Committee confirmed that the rule would also apply in Commander:

Commander will be adopting the London Mulligan (http://bit.ly/314vgT7) at the same time as other formats, with the standard first-mulligan-is-free for games with more than two players (CR103.4c).

So how does the London Mulligan work?

03.4. Each player draws a number of cards equal to their starting hand size, which is normally seven. (Some effects can modify a player's starting hand size.) A player who is dissatisfied with their initial hand may take a mulligan. First, the starting player declares whether they will take a mulligan. Then each other player in turn order does the same. Once each player has made a declaration, all players who decided to take mulligans do so at the same time. To take a mulligan, a player shuffles the cards in their hand back into their library, draws a new hand of cards equal to their starting hand size, then puts a number of those cards equal to the number of times that player has taken a mulligan on the bottom of their library in any order. Once a player chooses not to take a mulligan, the remaining cards become that player's opening hand, and that player may not take any further mulligans. This process is then repeated until no player takes a mulligan. A player can take mulligans until their opening hand would be zero cards.

-Magic: the Gathering Comprehensive Rules

How did this rules update change the game? Strategies that rely on a single or a few key cards gained a huge boost with the update.

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How Does this Work in Commander?

Most formats reward deckbuilding that prevents the need to mulligan. Going down to six cards can be the difference between a win or loss. Commander throws a wrench in the math here for three main reasons:

• You have 1-2 cards in the command zone that you gain access to inherently via the format.
• You have 3 opponents, so the game will likely last longer, giving you a better opportunity to catch up and meaning that statistically there'll be less hate coming your way.
• Finally, and most importantly, you have a FREE mulligan.

Now, I am not encouraging you to throw 15 six-drops in your deck because you have a free mulligan, but the mulligan rule should factor in to how you build your deck.

Your commander makes a big difference, as it will always fill out a spot on your curve if you have nothing else to do. In this example, let’s say your commander is Kess, Dissident Mage. She costs four mana, so she can be cast as soon as you hit that benchmark. How soon can you hit it? Well, you need four mana in three colors. A card like Commander's Sphere is not the type of ramp we want. This can make our curve very awkward if we want Kess, Dissident Mage on the battlefield as soon as possible. We want to ramp on turn 2 and play our commander on turn 3.

How do we achieve this? We need 2-mana ramp in our deck. Without green, our options are limited or underwhelming, but there are some good options, such as Arcane Signet, Wayfarer's Bauble, Izzet Signet, Dimir Signet, and Rakdos Signet. However, we don’t really want to stuff our deck full of these cards because a Charcoal Diamond towards the end of the game feels bad.

The good news? We don't need to fill our deck with these ramp spells if we're willing to take advantage of the mulligan. A few months ago, we took a look at probability on two separate occasions starting with my article here. Once again, probability is the key to our mulligan decisions. Unlike in my article here, the math to calculate your odds on a mulligan are more straightforward, and you can see how your deck stacks up with this tool.

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Let’s Take a Look at an Example

If we have eight ramp spells in our deck?

• Odds of a ramp spell being in our opening hand
  • ~46%
• Odds of a ramp spell being in our opening hand (given two chances)
  • P(A U B) = P(A) + P(B) – P(AB), or Probability of (A or B)
  • Don’t want to take your calculator out? Here's an online tool for calculating probability!
  • ~71%

Now, I can’t recommend relying on the mulligan in just any deck. This strategy still requires responsible deckbuilding in a few key areas:

• Although the mulligan makes expensive spells in your opening hand less of an issue, you still can’t have too many of them. These cards will feel like you went down a card when they sit rotting in your hand. I promise, you don't need as many of these finishers as you think. Instead…
• Play tons of card draw! Card advantage is good at basically every point in the game. I've thrown cards in decks that I draw and regret putting in, but card draw spells are the lowest risk spells to put in your deck. 
• Finally, and most importantly, play enough lands. I am an advocate for aggressive mulliganing in Commander, but I am also an advocate of playing 38+ lands. I rarely ever dip below 36 lands. The less lands you plav, the less aggressive you can be on mulligans because you need your opening hand to function.

These may feel like hoops to jump through during deckbuilding, but these have become my basic deckbuilding heuristics. When I follow these, my deck runs smoother more often. The ability to mulligan more freely is a bonus.

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Taking Advantage of the Mulligan

Some decks really loved the mulligan rule update. There are entire decks in the Eternal formats (i.e.: Legacy, Vintage) that are built around taking advantage of this ruling: combo decks. Combo decks have a huge boost from this ruling because they're looking for particular cards to make their strategy work. Dredge decks can throw away their whole hand just to find a Bazaar of Baghdad. A 100-card singleton format makes our odds of finding a particular card less likely, but as you go up in power level and trend towards cEDH, the mulligan becomes a useful tool to make sure you have cheap interaction, such as Force of Will, to keep the decks around you in check or to try to win as early as possible with a Dockside Extortionist.

One deck in particular loves the London Mulligan as both a competitive deck and a casual deck, and that's Grenzo, Dungeon Warden. Grenzo, Dungeon Warden utilizes the bottom of your library like a second hand, so you don't mind if your mulligan puts a few cards on the bottom. He's also cheap and comes down early, so this deck is much less reliant on ramp spells, which means that this deck lets you break a bunch of the rules we established earlier. This commander can reward unique and risky deckbuilding. We can worry less about keeping creatures in our opening hand because if we can get our strategy online, we'll have access to them off the bottom of our library.

Is this reliable? No, not always. It is vulnerable to removal, but is it explosive? Absolutely! We can trim lands and focus on keeping a hand once we draw one with enough lands and stick the creatures on the bottom of our library to access later. Say we run 28 lands (I am not condoning this madness, build at your own risk!) The likelihood of 3+ lands is only ~31%. But what if we had a chance to see three 7-card hands? 

• Odds seeing two 7-card hands = 52%
• Odds seeing three 7-card hands = 67%

Those odds are better than the 54% chance we have 3+ lands with our 38-land deck on the first draw.

Even with the 67% chance, this would only require us to throw one card on the bottom, which you can pick so Grenzo, Dungeon Warden can get it back in a few turns. Do I recommend you purposely mulligan down to three cards so you can stick a bunch of creatures on the bottom? No, it's too risky and may or may not make your friends hate you when it takes ten minutes to start a game. That being said, I do think the mulligan should be something you consider in deckbuilding when a commander allows you to leverage it as blatantly as Grenzo, Dungeon Warden does.

Now that we can go down on lands we can pack the deck with loads of the fun stuff: creatures! Grenzo, Dungeon Warden wants low-power creatures regardless of their mana value. What creatures generally have low power? Goblins! Not only do they have lower power and interesting abilities, but they are on-theme with our commander being a Goblin.

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Grenzo, Our Goblin Guide

We want to make sure Grenzo, Dungeon Warden can play most of, if not all, of our creatures off the bottom of our deck. The sweet spot for Goblins is three mana, which means that we want to be making Grenzo, Dungeon Warden a 3/3 or more each time we cast him. This allows us to play the best Goblins that can make more expendable bodies. Krenko, Mob Boss, Siege-Gang Commander, and Sling-Gang Lieutenant can make a ton of Goblins in a hurry. Combine these with Impact Tremors effects and Blood Artist effects to kill opponents quickly.

We can even combo relatively easily. There are multiple Goblin tutors, such as Goblin Matron, not to mention that Grenzo is in colors with the best access to efficient tutors, such as Gamble and Demonic Tutor. Use these cards to search up Kiki-Jiki, Mirror Breaker or Zealous Conscripts, which go infinite and can also both be played off the bottom of our library.

Check out the deck here!

[Commander]

*1 Grenzo, Dungeon Warden

[/Commander]

[Creatures]

*1 Beetleback Chief

*1 Blood Artist

*1 Brash Taunter

*1 Cabal Slaver

*1 Caterwauling Boggart

*1 Conspicuous Snoop

*1 Dark-Dweller Oracle

*1 Dockside Extortionist

*1 Epitaph Golem

*1 Frogtosser Banneret

*1 Gempalm Incinerator

*1 Goblin Assassin

*1 Goblin Chieftain

*1 Goblin Chirurgeon

*1 Goblin Cratermaker

*1 Goblin King

*1 Goblin Matron

*1 Goblin Piledriver

*1 Goblin Rabblemaster

*1 Goblin Sledder

*1 Goblin Trashmaster

*1 Goblin Warchief

*1 Ignition Team

*1 Judith, the Scourge Diva

*1 Junktroller

*1 Juri, Master of the Revue

*1 Kiki-Jiki, Mirror Breaker

*1 Krenko, Mob Boss

*1 Krenko, Tin Street Kingpin

*1 Legion Loyalist

*1 Legion Warboss

*1 Mad Auntie

*1 Mogg War Marshal

*1 Munitions Expert

*1 Murderous Rider

*1 Pashalik Mons

*1 Purphoros, God of the Forge

*1 Reito Sentinel

*1 Siege-Gang Commander

*1 Skirk Fire Marshal

*1 Skirk Prospector

*1 Sling-Gang Lieutenant

*1 Torbran, Thane of Red Fell

*1 Viscera Seer

*1 Warteye Witch

*1 Wort, Boggart Auntie

*1 Zealous Conscripts

*1 Zulaport Cutthroat

[/Creatures]

[Lands]

*1 Auntie's Hovel

*1 Blightstep Pathway

*1 Blood Crypt

*1 Bloodstained Mire

*1 Cabal Coffers

*1 Castle Embereth

*1 Command Tower

*1 Dragonskull Summit

*1 Evolving Wilds

*1 Fabled Passage

*1 Luxury Suite

*7 Mountain

*1 Prismatic Vista

*1 Shadowblood Ridge

*1 Smoldering Marsh

*1 Sulfurous Springs

*5 Swamp

*1 Temple of Malice

*1 Urborg, Tomb of Yawgmoth

*1 Zhalfirin Void

[/Lands]

[Enchantments]

*1 Braid of Fire

*1 Fires of Invention

*1 Impact Tremors

*1 Shared Animosity

[/Enchantments]

[Artifacts]

*1 Ashnod's Altar

*1 Coat of Arms

*1 Heartstone

*1 Mind Stone

*1 Rakdos Signet

*1 Skullclamp

*1 Sol Ring

*1 Teferi's Puzzle Box

*1 Tel-Jilad Stylus

*1 The Cauldron of Eternity

[/Artifacts]

[Instants]

*1 Brightstone Ritual

*1 Chaos Warp

*1 Terminate

*1 Valakut Awakening

*1 Vampiric Tutor

[/Instants]

[Sorceries]

*1 Demonic Tutor

*1 Gamble

[/Sorceries]

[/Deck]

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What do you think of this deck? How often do you mulligan? Do you think I'm right, or is relying on the mulligan a bad idea? Let me know in the comments below.