andrew w. moore | blog

This post is a bit of theorycrafting about building limited decks in Magic: the Gathering. Consider this hypothetical:

You’ve just finished a booster draft with 8 of your friends. During the draft, you were able to pick a 5UU bomb that greatly shifts the game in your favor when you cast it. The rest of your picks support an aggressive white-red deck with a low curve, combat tricks, and some removal. However, you decide to include the off-color card due to its power. Your 40-card deck has 5 mana sources that produce blue mana: 3 islands, a Mana Confluence, and a utility artifact that can produce blue mana. Your remaining lands (13) are split between plains and mountains.

For a limited format that doesn’t emphasize multicolor play, this configuration of lands/sources is usually about as far as you can stretch things. Now, here’s a question: is splashing for this powerful card a good idea? The answer ultimately depends on one’s goals. If the goal is to build the deck to cast the exciting spell, this post isn’t here to yuck anyone’s yum. However, if winning games is part of what makes limited fun for you, my hope is to convince you to avoid splashes like this when building your decks. Below, we’ll use a bit of math and simulation to see what happens when we build a deck to splash a double-pipped card (i.e., a card whose casting cost is something like 5UU; each “U” is called a “pip”).

Let’s start by naming some probabilities of interest:

• $Pr(\text{Keep}) =$ “the probability of drawing a viable opening hand”,
• $Pr(\text{Bomb}) =$ “the probability of seeing the bomb in the first 18 cards of your deck”,
• $Pr(\text{Splash}) = Pr(\text{2+ blue sources in first 18 cards}) =$ “the probability of seeing two or more blue sources in the first 18 cards of your deck”,
• $Pr(\text{Lands}) = Pr(\text{7+ lands in first 18 cards}) =$ “the probability of seeing at least 7 lands in the first 18 cards of your deck”,
• $Pr(\text{Cast}) = Pr(\text{Bomb} \cap \text{Splash} \cap \text{Lands}) =$ “the probability you can cast the bomb, after seeing 18 cards”, and
• $Pr(\text{Cast} \ | \ \text{Keep}) =$ “the probability you can cast the bomb after seeing 18 cards, given that the opening hand was viable”.

The natural way to think about these events is that we’re flipping over the first 18 cards of a shuffled deck.¹ Either we’re drawing them one at a time or surveiling/scrying.² This means we’re ignoring scenarios where your opponent puts cards from play back into your deck. We’re also assuming you’ll survive to see 18 cards: your opponent might be able to win in the first 7-ish turns, they might cause you to discard or mill one or more cards you need, etc. We’ll revisit this later, but let’s be as generous as possible to what this hypothetical deck is trying to do on paper, for now.

Deriving $Pr(\text{Cast})$ or $Pr(\text{Cast} \ | \ \text{Keep})$ is complicated, due to the fact that the events ($\text{Bomb}$, $\text{Splash}$, and $\text{Lands}$) are not independent. For example, drawing a island implies you’re also closer to having the lands needed on-time to cast your bomb. However, we can get around this by simulating the process, letting a computer shuffle the deck thousands of times, to get an estimate. Laying things out, this is what we’ll be giving to the computer to shuffle for each “game”:

B = bomb, A = artifact, C = creature, S = inst/sorc, I = island, P = plains, M = mountain, E = mana confluence

For our simulation, we’ll perform the following 10,000 times:

1. shuffle the deck
2. draw 7 cards from the top (the opening hand)
3. determine if the opening hand is viable:
   • mulligan if the hand contains 0-1 lands, or more than 4 lands; return to step 1
   • keep the hand if it contains 2 lands and 2+ playable spells (at least 1 being a creature)
   • keep the hand if it contains 3 lands and 2+ playable creatures
   • keep the hand if it contains 4 lands, and 3 playable spells (at least 2 being creatures)
   • mulligan all other scenarios; return to step 1
4. draw until you’ve seen 18 cards, and then count:
   • the total number of blue sources observed,
   • the total number of lands observed, and
   • whether the bomb was found

Note: in step 3, a “playable” card is one that can be cast using the lands available in the hand. Our utility artifact ("A2" in our list) can fix our colors, but we need to spend mana in order to put the artifact into play. This is has a cost, chiefly that we’re not able to play something that can block or attack in the early turns.

Now, here are the results. Under this list and heuristic for mulligans, the simulation suggests you’ll only be able to find a viable hand in 49/100 games. Notably, this means the chance that you have to mulligan twice in a row is $Pr(\text{Mull Twice}) = Pr(\text{Keep}^c) \cdot Pr(\text{Keep}^c) = (1 - 0.488)^2 = 0.262$! In instances where we see a viable opening hand, we’ll only be able to cast the bomb in 20/100 games.

This could sound appealing, but this means that in 80/100 games at least one of those conditions isn’t true. Maybe the rest of your deck functions, but a large share of the time your utility artifact isn’t helping you attack, and your islands are only adding generic mana (this issue is even more problematic when your deck includes double-pipped cards in your base colors, e.g., 2RR).

Something else that’s crucial to remember is that our simulation’s estimate for $Pr(\text{Keep})$ is lower than what we’d expect if we excluded the splash. Using the same simulation procedure with a purely 2-color deck (swapping the three islands for three basics of our base colors, and the utility artifact for a 2-mana creature), we’d estimate that $Pr(\text{Keep}) \approx$ 0.597, a much more comfortable proportion of viable hands. This matters quite a bit, given how costly a mulligan can be.

Mulligans hurt a deck’s win rate because they deprive players of vital early resources. However, cards that are stranded in your hand are a similar kind of deprivation. Stranded splash cards are lost opportunities to play spells that keep you alive. Outside the simulation, we should remember that one’s opponents are generally disinclined to wait for you to cast an expensive card. An early stumble to find your colors or taking a turn off to play a prophetic prism are moments that opponents capitalize on.

Lastly, I don’t want any readers come away from this post thinking they should mulligan less. Decisions to mulligan an opening hand are an unavoidable part of the game. Rather than dreading them, I think it’s best to view them as a tool that we can learn to apply appropriately. In this spirit, we can build decks that reduce the chances that a mulligan is necessary. In summary, try your best to play a two-color deck, and resist the urge to include an off-color card just due to its power. Your manabase (and I!) will thank you.