Unlocking Legacy – Investigating Fetchlands

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Monday, May 5th – Christopher Coppola analyzes the deck thinning properties of fetchlands and dispels some misconceptions about their effects.

I. The Best Cards in Legacy

I have stated many times before that I believe fetchlands to be the “best” cards in Legacy, for a variety of reasons. They play a critical role in the design of most archetypes and offer irreproducible effects for a negligible cost. Onslaught was released two years before Legacy even existed, so they have always been a part of the design process in this format, but I remember how the design problem was completely changed in Vintage and Extended when fetchlands were first printed. Building a manabase in Vintage for a Mana Drain deck was a much different process, and the vast improvement in land configurations made these decks much more consistent and powerful (helped along the way by Thirst for Knowledge and Gifts Ungiven). Fetchlands are arguably even more important in Extended, where the almost-duals of Ravnica block are the starting point of every manabase. Wizards even recently changed their rotation schedule to keep Onslaught block in Extended for another rotation, and the fetchlands along with it.

If you have played Magic before you probably understand how good fetchlands are, for increasing a deck’s access to multiple colors and ensuring that it will be able to play the cards it draws. Fetchlands can double or triple the number of sources you have of a set of colors while keeping your land count the same, which has always been an amazing effect. In 1996 a set of fetchlands that came into play tapped were released in Mirage, and even with the added drawback they were still played in many decks. (watch Oath of Druids win a Pro Tour against Necropotence).

Fetchlands would be essential if they just rewrote the rules of manabase design, but they do even more than that. In Legacy, there are several other effects which are very useful. One is the increased number of cards that go to the graveyard, which is useful for pumping Nimble Mongoose, Werebear, and Mystic Enforcer, as well as feeding Rotting Giant, Jotun Grunt, or Tombstalker. Another is the protection they offer from Wasteland and Rishadan Port, allowing you to either to just get basic lands, or keep multiple colored sources in play until you can cast the spell you need to. Fetchlands also shuffle your deck, an effect which is commonly used in conjunction with Brainstorm and Sensei’s Divining Top.

II. Deck Thinning

Fetchlands also have another effect, deck thinning, the analysis of which is surprisingly inaccurate and full of misconceptions. One is likely to find pundits discussing this issue, rather than a successful tournament players who almost always play the maximum number of fetchlands. I attribute most of this confusion to a pair of articles written in 2003 with little reference to deck design . Monte Carlo methods may be unnecessary for the analysis of this topic, and may even be worse than a classical approach given the variety of questions asked about different game states.

In “A Modern Goblins Primer, Part I” I analyzed the power of fetchlands in conjunction with Goblin Ringleader, and showed that thinning your library with fetchlands results in drawing noticeably more threats. At the time I realized that this topic deserved much more attention, but I was focused on other issues. The use of Goblin Ringleader as an analytical device is just one part of a larger perspective for analyzing deck thinning, the correct one, which requires that the entire game be considered. This question is usually asked in the form of probabilities of drawing more spells at a single draw event, which is a completely useless metric. There are many draw events over the course of a game, and they combine to form your hand and board position. When you look at the game state on turn ten, it is due to a combination of all the draw steps from turns one to ten. You can’t point to a single draw step and say that that particular one resulted in drawing the extra spell instead of a useless land — they all contributed in a small amount, but it is difficult to picture this because cards exist as pure states of land or spell. However, the random card on the top of your deck is not pure state, it is a mixture of both, and it is the sum of many mixed events that form your hand at any given point in the game.

III. Probability

Probability can be a difficult topic for many people, so it will be useful to define the terms of simple situations to make sure the ideas are clear.

One question that many Magic players might find themselves asking when they first begin to build decks is “what is the probability that I will draw a four-of in my opening hand?” This is actually a pretty important number because it gives us a good idea of how random the game of Magic really is. It is also the probability that you will be able to Force of Will your opponent’s turn one play.

There are two ways to calculate the probability that you will have at least one Force of Will in your opening hand. The first would be to calculate the probability of drawing each number of Force of Wills, and multiply those probabilities by the degeneracy of each state, or the number of different card orderings that contain that number of cards. For simple questions like this, the second method is much faster, which is just to find the probability that you will draw no Force of Wills at all, which is the subtrahend of the number we are looking for. As I mentioned before, performing a calculation like this involves the combination of multiple draw events. This does make it a little more complicated since the properties of the deck change after each event. However, with a little combinatorics, we can calculate that the probability is 39.95%, if you have a sixty card deck with four Force of Wills.

This is a pretty easy number to interpret as well. It means that if we randomized our deck and drew an opening hand a large number of times, about sixty percent of them would have no Force of Wills in them, and the rest would have one or more.

Let’s do a slightly harder problem. Suppose we have a deck with twenty lands and forty spells. How many lands are we going to have in our opening hand?

We will apply the same methods to find the answer to this question. However, now we have to work a little harder to interpret the answer since we are going to get a small noninteger number. This is the average of the number of lands we would draw if we drew a large number of randomized opening hands from the deck. We can also ask what the specific probabilities are of drawing given numbers of lands, but this is only one piece of information about the deck and is more dangerous to work with. For example, Landstill and 43Lands have the same chance of drawing an opening hand with four lands in it, about 21.5%, despite having very different land counts in their opening hand — Landstill draws 2.92 lands on average, while 43Land draws 5.02 lands on average. This is why the average number of lands drawn is a more useful number, but it’s also statistical information, meaning we have to be more careful how we apply it.

Our hypothetical deck with twenty lands will draw an average of 2.33 lands per opening hand.

IV. Cumulative Draws

Now we can examine the issue of deck thinning in more detail. Fetchlands reduce the number of lands in your deck, increasing the probability that you will draw spells later in the game. There is only a small change in the probability that in individual draw event will yield a spell, but when many events are considered together, the change is substantial.

Draw spells are a very powerful way to amplify the increased probability of drawing spells. Combining deck manipulation with fetchlands generally has very strong effects, and this is one reason why.

Consider two Threshold decks that play seventeen lands — one with zero and one with eight fetchlands. Suppose all draw events are identical, and they both have thirty-five spells left in their decks. The Threshold deck without fetchlands has twelve lands left in its library, and the Threshold deck with fetchlands has nine left. On the play, on turn five, if both decks cast a Brainstorm hoping to fetch away the extra land they drew, what will happen?

The first deck will draw 2.23 spells, and the second will draw 2.38 spells. This is a subtle example since it is only turn five, but it means roughly one sixth of the time you would draw an extra spell with this particular Brainstorm. Every draw spell cast after this enjoys the same increased probabilities.

Consider two Landstill decks that play twenty five lands — one with zero and one with eight fetchlands. Again, suppose everything that happens in the game is the same, except the second deck has fetched four lands out of its library. The first deck has seventeen lands remaining in its library, and the second deck has thirteen. If both decks have twenty-five spells remaining, what will happen if they both cast Fact or Fiction on turn eight?

The first deck will reveal 2.9 spells, and the second will reveal 3.29 spells. Revealing only two spells off of Fact or Fiction can be quite a disappointing play, but revealing three or more puts the opponent into the difficult position of splitting up your cards. The probabilities of revealing only one or two spells are 7.9% and 25.4% without fetchlands, compared to 3.5% and 17.1% with fetchlands. Additionally, the probabilities of revealing four or five spells is 23.6% and 5.5%, compared to 32.7% and 10.6% with fetchlands. These are considerable differences, and in the case of Fact or Fiction I would say this is a critical point in making best use of the card, but again this is just one spell. Every other draw spell and draw step that Landstill gets cards off of will yield more spells than if the deck hadn’t used fetchlands.

What if we don’t even take into consideration draw spells? Drawing multiple cards at once does make a thinned deck much more noticeable, but the same effect will happen in a long game with many draw steps.

Consider two Aggro-like decks with twenty-two lands, one with zero fetchlands and one with ten, playing against each other. At some point in the late game, the decks have reached a stalemate since they cannot kill each others’ Tarmogoyfs, and each is hoping to draw out of it. Suppose we are on turn ten, and the second deck has fetched six times. How many turns will it take for the deck with fetchlands to draw one more spell than the opponent?

The answer is ten turns. After ten turns, the first deck will have drawn 6.36 spells, and the second deck will have drawn 7.37 spells. I have played many magic games at least this long in tournaments, and it is not unreasonable to conclude that with correct play, the deck that draws more threats or removal in this situation has a better chance of winning.

V. Probability Myths

This is a very different conclusion from the alternate theories offered by some others about the effects of deck thinning. Fetchlands actually have a substantial impact on the game by thinning one’s deck, and this is easily demonstrated in the examples I gave here as well as the more brief analysis I presented in my Goblins primer in 2007.

It is true that in a single draw event, fetchlands only slightly change the probability of drawing a spell instead of a land. Of course, this is an irrelevant statement since actual games of Magic depend on many draw events happening over the course of the game, and they must all be considered together in order to gain any competitive insight into deck design. Fetchlands substantially increase the probability of drawing extra spells, and this effect should be included in deck design even when the others are not present.

Christopher Coppola