I had a discussion recently with a friend about fetch lands. We all know that fetches are powerful – they provide untapped mana fixing, they provide access to more than two colors if you run shocks or battle lands, and they provide a free shuffle of your deck (for situations where you’ve put unwanted cards on top of it). We weren’t talking about that, though, we were talking about the other thing that fetches do: thin your deck.

If fetches have pulled basic lands out of your library, you have an increased chance to draw spells (instead of lands) later in the game. Spells are good! The question we were discussing wasn’t whether thinning was good or bad, but whether thinning was a big enough effect to care about. For me, at least, this is where the question becomes interesting. In order to decide if an effect is big enough to be interesting, we have to figure out what we’re going to compare it to, and this can be tricky.

When the question is “how many spells will we draw by a given turn?”, the thing that determines whether an effect is big or small is the intrinsic variance of Magic. If you take a deck and draw five hands from it, each hand will be different. In particular, they’ll have different mixes of lands and spells. Now imagine you’re drawing sets of cards from two different decks, one “thinned” and one with the usual mix of lands and spells. Would you be able to figure out, given the number of spells you drew, which one was thinned?

It’s easy to start looking at individual scenarios (I start with this hand, I make the following series of plays, at this point I’ve removed three basics from my deck, so my probability of drawing a spell has increased by X%), but a little trickier to move beyond that and ask, in an average game, how will decks with fetches perform differently from decks without them? To answer this, we construct a simple model:

The player starts with a deck with some number of fetch lands, fetchable lands, and other spells. We won’t worry much about the details of these cards, we’ll just assume that fetches can be used the turn they ETB, that they’re able to find any of the fetchable lands, and that all of the spells are the same.

The player starts with seven cards in their opening hand. I haven’t bothered with mulligans. On each turn the player draws a card. If they have a fetch in hand, they play it and crack it. If they don’t have a fetch, but have another land, they play that land. The player never casts spells, and has no maximum hand size. We run the game until the library is empty, and on each turn we track the number of lands they have in play and the number of spells they have in hand.

Because this model is simple, it’s easy to code it up in Python and run 10,000 games. I did this with a deck with 24 fetchable lands, 36 other spells, but no fetches, and another deck with 10 fetches, 14 fetchable lands, and 36 other spells. By running a large number of games we get a pretty clear idea of the distribution of lands in play and spells in hand on each turn in each scenario. The results are summarized in a handy plot:

The dark lines are the mean numbers of lands in play and spells seen. The shaded areas indicate the intrinsic variance of the game – most games won’t follow the dark lines exactly, but 2/3 of the time they’ll stay within the shaded regions. (For enthusiasts, these are roughly 1 sigma error bars, but the distribution of each quantity on a given turn is skewed, so they’re asymmetric.)

Now that we’ve generated the graph, let’s read it. We start by looking at Turns 1-10. Over those ten turns the deck with fetches and the deck with no fetches perform almost identically. The average numbers of spells and lands in play are very, very close together, and the red and blue shaded areas (where most games will fall) overlap almost completely. If you’re playing a format where games end by T10, don’t expect thinning to have any effect on your deck.

Looking at turns 10-20, some differences start to appear. This makes sense – as your remaining library gets smaller and the number of basics removed by the deck with fetches gets bigger, thinning starts to matter. How much? Not a ton. In one game with your deck that lasts for 20 turns, you won’t be able to tell the difference between a moderately good set of draws from the no fetches deck and a moderately poor set of draws from the deck with fetches. We see this from the graph because the shaded red and blue areas still substantially overlap. But, on average, by turn 20, the deck with fetches has seen one more spell than the deck that hasn’t.

Even by turn 30 there’s still a lot of overlap in the performance of the two decks, but by this point the deck with fetches does expect to have seen about two more spells than the deck without them. I’m skeptical that this would matter in such a slow game, but it’s a real effect.

The point here isn’t that you should stop using fetches, just that when you’re deciding whether to run them, you should focus on the big things they do exceptionally well: provide high quality fixing and a shuffle. If you’re setting up a recursion engine and using that to pull a crazy number of lands out of your deck, that’s great too. But don’t expect that just because you put a bunch of fetches in, your deck will suddenly become a paragon of consistency. The effect just isn’t that big!