Dominaria’s focus on legendary cards accentuates a deck-building challenge: How many copies of a certain card should you put in your deck? Today, I’ll explain the math behind this, along with general guidelines for deciding how many copies to run.
The Probabilities of Drawing a Certain Number of Copies of a Card By a Certain Turn
Let’s start by running the numbers! If you disregard mulligans and just calculate the hypergeometric probabilities for 60-card decks—figuring out how many copies to run is not a big question for Commander players—you find the following:
There are several things of interest in this chart:
- Opening hand – Since you’re assumed to be on the play, the lefternmost numbers for “turn 1” represent the probability of drawing at least one copy of a certain card in your starting seven. You’re 39.9% to draw a certain 4-of in your opening hand.
- On the draw – If you’re on the draw, then you should move the probabilities one turn forward. For example, turn 2 on the draw is equivalent to turn 3 on the play.
- The rule of 8 – As a general rule of thumb for deck building: a deck list with four copies of a certain type of card has weak consistency (39.9% to draw at least one in your opening hand) while eight copies give reasonable consistency (65.4% for your opening hand). Take a deck like 8-Rack—it didn’t function reliably until Return to Ravnica introduced Shrieking Affliction as an alternative to The Rack. As a general deck building rule, if your strategy relies on a certain effect (like The Rack for 8-Rack or Slippery Bogle for Bogles) and you have no card selection, then you need at least eight virtual copies. That’s a solid minimum.
- About 50% to hold a 4-of by turn 4 – If your opponent runs four Settle the Wreckage, then they can Wrath your board (on turn 4 when they’re on the play) with a probability of 52.8%. It may be slightly less in practice due to mulligans and due to the fact that at least four of their cards need to be lands, but if you remember that it’s roughly 50-50, then you aren’t far off.
- Diminishing returns – Adding copy N to a deck adds more to the probability of drawing that card by a certain turn than adding copy N+1 to a deck. For example, the addition of a first copy to your deck increases your turn-3 probability to draw the card from 0% to 15%, whereas a second copy increases this probability only from 15% to 28%. This difference keeps decreasing. So if you only care about drawing at least one copy of a certain card, then the first one you add to your list is the most valuable. If your metagame is equal parts Affinity, Dredge, and Storm and all hate cards are equally good, then the best 3-card sideboard is one Stony Silence, one Rest in Peace, and one Rule of Law—a sideboard with three copies of the same hate card would be worse.
There are several things of interest in this chart:
- Legends as 3-ofs – If you play four copies of any given card, then over the course of a 5-round League you can expect to draw multiple copies of that card in 2-3 games (depending on how long you expect your games to last). With three copies, this number is nearly halved: As a specific comparison, the turn-4 risk of drawing multiples is 12.6% with four copies and 6.9% with three copies. Given that lists with three copies still yield reasonable odds to draw at least one, it’s often best to run legendary creatures or planeswalkers as 3-ofs.
- About 25% to hold another copy of a 4-of by turn 4 – If your opponent runs four Settle the Wreckage, then they’re holding multiples by turn 4 on the play with probability 12.6%. As this number is small, you might think: “If they Settle me on turn 4, then I’m not going to play around a second Settle—they only have two about 1/8th of the time.” But this a priori probability is irrelevant at that point—you need to update your beliefs after your opponent casts their first Settle by limiting yourself to the scenarios where they had the first one. Assuming that your opponent has the first Settle half of the time, this 12.6% averages over games where they don’t even have the first Settle (in which case the probability that they have multiples is 0%) and games where they drew at least one Settle (in which case they have another about 25.2% of the time). So a proper thought would be, “If they Settle me on turn 4, then I might still play around a second Settle—given that they drew the first, they have another about 1/4th of the time.”
- Pick the right interpretation – This Settle the Wreckage example emphasizes the importance of properly defining what you’re actually interested in, but if you’re not used to working with conditional probabilities, then it can be tough to figure out which number to consider when building decks. My suggestion: use the numbers in the chart for general deck evaluations and the elevated conditional probabilities for individual card evaluations. So if you built a deck with four Powerstone Shard and wonder if you can consistently fuel big Walking Ballistas or Pull from Tomorrows, then you should realize that you draw multiple Powerstone Shards only 12.6% of the time. But if you already built a big mana deck and are choosing between four Powerstone Shard or four Beneath the Sands as your 3-mana ramp spell, then you should realize that for the games where this choice actually matters (i.e., the games where you draw at least one of these cards by turn 4) you have at least two Powerstone Shards about 25.2% of the time. Likewise, a deck with four History of Benalia can’t craft its entire game plan around the 1/8 chance of playing back-to-back Sagas on turns 3 and 4, but if you’re trying to evaluate the third chapter, then you should realize that any History of Benalia you draw sees another about 1/4th of the time.
- Increasing returns – For small N, adding copy N+1 to a deck adds more to the probability of drawing multiple copies in the early game than adding copy N to a deck. This is the opposite of the evolution of the probability of drawing the first copy. As a result, if you really don’t want to draw multiples, then the first or second copy you add to a list is the most valuable. That’s another reason for running only one or two copies of sideboard cards like Stony Silence, Rest in Peace, or Rule of Law.
Guidelines for How Many Copies of a Card to Put in Your Deck
The decision on how many cards to run is not merely based on probabilities and percentages. It’s also based on how powerful a card is relative to its alternatives. And sometimes deck building is more art than science. With that in mind, I can give the following rules of thumb to guide your deck building decisions. In the spirit of Dominaria, I’ll use legendary cards as illustrative examples.
- 4 copies – You want to draw the card as often and as early as possible. The card is essential for the deck to work as it’s supposed to, and you don’t mind drawing multiples, even in your opening hand. You might even be happy to do so. It’s rare to play legends as 4-ofs, but you can make exceptions for cards that are extremely powerful and/or removal magnets. Example: Heart of Kiran in Mardu Vehicles or Karn Liberated in Modern Tron.
- 3 copies – You want to see the card at some point in the midgame, but you prefer not to draw multiples. A second copy is not necessarily dead, but it’s much less valuable than the first. Most commonly, this is because the card is expensive to cast, a situational answer, and/or a non-essential legendary. Example: Kari Zev, Skyship Raider in Mono-Red Aggro or Teferi, Hero of Dominaria in U/W Control.
- 2 copies – You’re not excited to see the card in your opening hand, and it’s not vital to your strategy. Often, the card is not much better than the alternatives, worthless in multiples, and/or crazy expensive to cast. Example: Josu Vess, Lich Knight in Mono-Black Control or Ugin, the Spirit Dragon in Modern Tron.
- 1 copy – You don’t want to draw the card naturally, but you want to have access to one in your deck. Most commonly, this is because it’s a kill condition for strategies that see their entire deck, because you hope to mislead your opponent into thinking you’re playing more copies than you actually are, or because it’s a tutor target in a toolbox-style deck. For example: Elesh Norn, Grand Cenobite in Gifts Ungiven decks or Shalai, Voice of Plenty in Chord of Calling decks.
I believe these are helpful guidelines, but of course they’re not the end-all. For example, the amount of card selection in your deck is also relevant: A deck with four Opt and four Serum Visions can find a 3-of just about as consistently as a nonblue deck can their 4-ofs.
Another big reason to run two or three copies is because you’re supplementing a 4-of that you want 6-7 virtual copies of. That line of thinking often stems from what I refer to as the axiomatic approach to deck building. (I started calling it that after working on axiomatic characterizations of allocation rules in cooperative game theory.)
The Axiomatic Approach to Deck Building
The axiomatic philosophy to deck building starts with a list of effects and how many cards of each effect (which could broadly refer to cards with a certain ability, type, mana cost, role, or other common factor) you want. Finding the right number totals for each effect is more important than the exact distribution over individual cards that give that effect.
If I want six or seven virtual copies of a certain effect, then I have to run two or three copies of a secondary card for redundancy. For example, Memnite is a virtual Ornithopter, and Abrade is a virtual Lightning Strike. This 6-7 virtual copy range is often adequate for diminishing-return effects that I’d like to see early on but that are not necessary to win.
As effects get more important to your game plan and better in multiples, you want to run more. For important effects that support your key cards, are not bad in multiples, might be destroyed by your opponent, and can be filled from a large card pool, 13-15 virtual copies is often a good target. Examples of such effects include artifacts for Toolcraft Exemplar, colored mana sources for early drops, spells with a mana cost of 2 or less, payoff cards in Affinity, or pilots for Heart of Kiran.
I’ve eagerly used axiomatic approaches in the past. For example, when building Modern Affinity, I stipulated a maximum number of colorless spells and 2-drops, a preferred number of interactive spells, and so on. When building tribal decks, I stipulated a minimum number of payoff cards and 1-drops, a preferred number of lands, and so on. The nice thing is that after putting down these axioms, the decks basically build themselves.
Since the sets of viable cards for different effects often intersect, there’s sometimes only one solution. If you were wondering why many B/R Vehicle decks run two Pia Nalaar: well, it could be that the deck designer wanted a limited number of 3-drops, a certain number of artifacts for Unlicensed Disintegration, and a certain number of ways to crew Heart of Kiran. Perhaps these and other constraints could only be met by running exactly two Pia Nalaar, who excels at ticking multiple boxes at the same time.
Ultimately, 2-ofs need not be the sign of an unfocused deck. There can be good reasons to play any number of copies of a card, whether it’s based on probabilities, individual card strength, axioms, or a combination of it all.