Navisd's picture
By: Navisd, Navi Dhaliwal
Nov 05 2014 12:00pm
0
Login or register to post comments
1618 views


Deck design is an oft explored topic in Magic writing. Whether the focus is going rogue, tweaking an existing deck, building on a budget, or finding the new format-breaking combo, Magic players love designing and discussing decks. But what if I told you that every article ever written on this subject, combined, has only tapped into 1% or less of total deck design space? This article will show you how the other 99% live through an object called the deck set, the set of all possible decks for a given format. 

This take on Magic is, admittedly, Melvinian. Maybe too Melvinian for some. It is abstract, formal, and structured. It likely will not lead to cutting edge deck tech or secrets to success at the game. It is mathematical, notation-heavy, and probably overly analytical. But it may also be, for those with similar aesthetics, remarkable and beautiful. It explores another way of enjoying and appreciating Magic outside of just playing it. To me at least, that’s pretty cool. By the end of this article, I hope some of you share that opinion.


 Introducing the Deck Set

Think about every possible deck you could make in Magic within a format, if you had access to a playset of every card. Think about every deck, not just every deck you might competitively play. Think about the deck of 60 Island and the deck of 342 Forest and the deck that won the last Pro Tour. Let’s call the set of all of these possible combinations the deck set. It looks something like this:

 

Here X is the set of all possible decks within a given format that has L unique cards, a minimum deck size of m, and a maximum deck size of M. The elements of X are vectors x which represent individual decks. Within each deck vector, each component corresponds to the number of copies of card i in that deck. To add some intuition to that definition, let’s consider some simple examples in the Momir Basic format.

Example 1: only non-snow basic lands allowed; exactly 60 cards (Momir Basic)

A deck in Momir Basic could look like this:

 

[12 Island, 12 Plains, 12 Forest, 12 Mountain, 12 Swamp]

Or it could look like this: 

[60 Island, 0 Plains, 0 Forest, 0 Mountain, 0 Swamp]

Or like this:

[1 Island, 1 Plains, 6 Forest, 3 Mountain, 49 Swamp]

Notice three things must be consistent across these deck vectors. First, each vector is of length five, which is the number of unique cards in the Momir Basic format. This corresponds to L. Second, order matters; in other words, the component i of each of these vectors must refer to the same card. So if the first component counts the number of Island in one vector then it should represent the number of Island in every other vector as well, not the number of Mountain or anything else. However, this is without loss of generality—it doesn’t really matter which slot represents a card as long as the same slot is used to represent it in every deck vector in the format. Finally, the sum of every component must equal 60 by the rules of Momir Basic. This corresponds to m and M. If we wanted to write the set of all such deck vectors possible in Momir Basic, that is, the deck set X of Momir Basic, it would be this:

 

How can we interpret this deck set? Well, in this case, we can actually calculate its size, which equals precisely the number of unique decks in the Momir Basic format. That number is exactly 455,126. Yes, over 400 thousand unique possible decks exist within the Momir Basic format. A meager three of those decks were presented earlier, which comprise around 0.0007% of the deck set of Momir Basic. This, by itself, is a pretty crazy fact.

The even more astonishing thing though, is what this implies about the metagame. There are 103,569,610,375 unique possible matchups of different decks within the Momir Basic metagame. That's 103 billion possible matchups of Momir before we even get into the randomness of the die roll, of card draw each turn, and of all the creature combinations!

If the format were changed to include snow-covered basics as well, the metagame would consist of over 78 quintillion (read: billion billion) possible deck matchups pairing over 12 billion unique decks. For an all-basics format, that's beautifully complex.* 

Island Plains Forest MountainSwampSnow-Covered Island Snow-Covered Plains Snow-Covered Forest Snow-Covered MountainSnow-Covered Swamp


 Exploring the Deck Set in Other Formats

You might feel like the deck set is infinite because Magic is a game of infinite possibilities, but we have shown that it is actually just very large. In every format in Magic, you are limited by the self-shuffle clause: you must be able to shuffle your own deck. (Magic Online likewise allocates finite memory to deck size). Furthermore, the all-time number of cards printed is finite. This is why M always exists. In many formats, you are also limited to four copies of most non-basic-land cards.

Since our original equation is the most general description of the deck set for a format, it does not yet include bounds on the number of copies of each card within that format. We can explore adding such bounds in the following examples.

Example 2: four card limit except basic lands; 60 card minimum (Standard)

 

Here, you can play up to M of any of the five basic lands. You can also play between zero and four of any of the L- 5 non-basic-land cards in Standard, as long as your total number of cards is between 60 and M. If L equals 5 and M equals 60, this reduces to the Momir Basic case. For L greater than 5 the set will be larger than the set for the Momir Basic case. To give an example, even if only one non-basic-land card were legal in Standard, you could still build 1,982,751 unique sixty-card decks.**

Example 3: four card limit except (snow) basics, Shadowborn Apostle, Relentless Rats; 60 card minimum (Modern, Legacy)

 

The interesting thing in this case is that the on-card rules of Magic can warp the deck set itself! Besides the basic lands, there are two other unique cards in Magic which you can play any number of copies of (as long as you can shuffle, so in fact any number less than or equal to M). Also, note that while the model is the same for Modern and Legacy, the value of L is different in each.

Example 4: four card limit except (snow) basics, Shadowborn Apostle, Relentless Rats; 60 card minimum; restricted list (Vintage)

 

For each of the r cards on the restricted list, there is just a binary: a deck has either zero or one copy. A banned list is not necessary in any of these models because L includes all the legal cards in a given format, by construction excluding banned cards. You can simulate banning an additional card i by changing the maximum number of copies of that card that can be played to zero. It's then possible to think about and compare the pre-banning and post-banning deck sets.

You may already notice some apparent limitations in these three models. First, they have no way to differentiate how cards are split between the sideboard and the main deck; they regard two decks with the exact same 75 cards as identical. Similarly, a parallel model would have no way to denote who the commander is in an EDH deck. Second, these types of models get much more complex when used to think about limited formats. It’s easy enough to formulate the set of all possible decks after you have drafted your 45 cards, but trying to formulate it before the draft is nontrivial. These issues can all be taken care of mathematically, but that's something I'll save for next time.

The biggest limitation though, is that by its nature the deck set itself tells us nothing about which decks are good or bad. It gives us no insight into the deck generating process itself--it just describes the set of our options. The all-lands and all-spells and other seemingly ridiculous decks all fall under its umbrella. The challenge in future articles then, will be understanding how the deck set is picked apart and partitioned by players looking to design decks. 

Shadowborn Apostle Relentless RatsShadowborn Apostle Relentless RatsShadowborn Apostle Relentless RatsShadowborn Apostle Relentless RatsShadowborn Apostle Relentless Rats


Toward a Card Economy

When you consider the sheer size of the deck set and then realize that only an extremely small subset of that whole set is actually played both casually and competitively, you have to wonder why. The fact that within different metagames certain archetypes emerge and certain cards appear more frequently than others is itself noteworthy. But when you add to that the fact that even across metagames cards with similar properties are played in similar numbers, then you’ve got something truly structurally amazing. Somehow, separate choices of numerous players over the set of all possible decks converge to just a handful of decks.

This result is well-suited to economic study, which frequently encounters the situation of multiple agents’ individual decisions yielding an equilibrium. In a series of articles, I’d like to work toward developing a microfounded game theoretic model of the metagame—one which considers how both the rules of the game of Magic and the expected choices of other players lead to decisions about which deck to play, and how such individual decisions then lead to aggregated outcomes for a format.

Next time, I’ll delve deeper into mathematically formalizing the notion of a format and address some of the limitations on the deck set as defined here and how the model can be upgraded to handle them.

Until then, may you come up with some never-before-seen Momir brews,


NSD


* The number of unique decks in Momir Basic is given by (59 choose 4). The percentage given is therefore just 3 / (59 choose 4) * 100. The number of unique matchups is calculated as ((59 choose 4) choose 2). The number of unique decks if snow-covered lands were legal is (59 choose 9). The number of unique matchups in the snow-covered lands Momir Basic format is ((59 choose 9) choose 2). 

** The number of unique sixty-card decks in the one non-basic-land card standard is  (59 choose 4) + (58 choose 4) + (57 choose 4) + (56 choose 4) + (55 choose 4); that is all the possible sixty-card decks running zero, one, two, three, and four copies of the non-basic-land card, respectively, summed.

2 Comments

A few errors to note by Navisd at Wed, 11/05/2014 - 12:46
Navisd's picture

Hello all,

I tried to contact the editor about this, but have not yet received a reply. I just wanted to clarify I actually underestimated some of the numbers above. I was accidentally only counting positive combinations rather than non-negative combinations. (That is, I listed the numbers given decks had to contain at least one copy of each of the land types). Here are the corrections:

Momir Basic: the number of possible decks is 635,376 (64 choose 60).
Momir metagame: the number of possible matchups is 201,851,013,000 (635,376 choose 2).
"Snow" Momir Basic: the number of possible decks is 56,672,074,888 (69 choose 60).
"Snow" Momir metagame: the number of possible matchups is 1,605,862,036,027,204,068,828 (1 sextillion / 1 thousand billion billion)(56,672,074,888 choose 2).

For the one non-basic-land card Standard, this changes the result to be: (64 choose 60) + (63 choose 59) + (62 choose 58) + (61 choose 57) + (60 choose 56) = 2,798,376 possible decks.

The best way to reach me is by JXClaytor at Thu, 11/06/2014 - 09:43
JXClaytor's picture

The best way to reach me is at puremtgoeditor@gmail.com.