Tuesday, December 1, 2015

The Optimal Number of Lands

Accepted wisdom is that 17 lands is the baseline for a limited 40-card deck. You increase or decrease based on curve, splash, and other considerations. But why is this accepted wisdom, and is it necessarily correct?

I have yet to see an actual rigorous mathematical argument in favor of 17 lands (if you know of one, please point me to it). My guess is that the best players arrived at it based on massive amounts of play. But large data sets have not readily been available for analysis until recently. Is it possible that the common wisdom isn't strictly correct? That we might be able to scrape a few more points of expected value (EV) out of the game by adjusting this baseline?

Let's look at some data first (thanks to Jack for churning these out):

So what we've got here is a table showing the chance of having a particular number of lands in your opening 7 cards depending on the number of lands in your 40-card deck. For example, the very first cell indicates 0.0258 for 0 lands in your opening hand with 15 lands in your deck. That means there's a 2.58% chance your opening hand will have no lands if you run 15 lands in your deck.

I've highlighted the rows for 17 and 18 lands, since those are the ones I'm most interested in comparing. If we make the reasonable assumption that opening hands with 0, 1, 6, or 7 lands are almost always instant mulligans, then we can sum the probabilities for those cases and compare them. Those totals are on the right of the table.

So with 17 lands, we insta-mulligan 12.14% of the time.
With 18 lands, we insta-mulligan 10.47% of the time.

That's 1.67% fewer mulligans, which adds up over a large number of games. How bad does it hurt us to mulligan? Here are some numbers from the current draft format, Battle for Zendikar:

Your win% with a 7-card hand is very close to 50%, either on the play or the draw. But as we can see from this data, mulligans hurt. Bad. You lose 12% equity on your first mulligan, and almost 30% on your second. 

And this is with the new scry rule in effect. This suggests that the penalty for mulligans is so severe, adjustments to our deck that minimize mulligans are particularly valuable. More on this in a second. First let's look at a couple of more interesting points from the first table.

With 17 lands, our chance of having 2 lands in our opening hand is 24.55%.
With 18 lands, that chance is 21.61%.

That's nearly a 3% lower chance of starting with 2 lands. 2-land hands are inherently risky, unless your curve is especially low. I keep most 2-landers, especially on the draw, but there is about a 15% chance you won't hit your third land drop on turn 3, which is often very punishing.

Most players would agree that the best starting hands, on average, contain either exactly 3 or 4 lands. 

In a 17-land deck, openers with 3 or 4 lands make up 54.91% of our hands.
In an 18-land deck, openers with 3 or 4 lands make up 57.30% of our hands.

Another incremental, but significant gain in value. What about 5-land openers, which, like 2-land openers, are questionable?

In a 17-land deck, openers with 5 lands make up 8.40%.
In an 18 land deck, openers with 5 lands make up 10.62%.

So we almost certainly lose some equity here. Most 5-land openers are relatively weak and risky. But this is the only category where we actually lose equity between 17 and 18 lands. 

With 18 lands in our deck, if we group 0-1-6-7 as a singular category, we gain equity in every single category except for 5-land openers.

0-1-6-7 lands: +1.67% 
2 lands: +2.94% 
3-4 lands: +2.39%
5 lands: -2.22%

So based on opening percentages and mulligans alone, running 18 lands seems strictly better. You have small but significant gains in every category of openers but one. But what are the other costs associated with running 18 lands over 17? Flood, of course. But how much equity do we lose to flood by having 1 extra land and 1 fewer spells?

That's very difficult to measure. Ideally we'd have statistics on game and match win % based on the number of lands in a given player's deck. But the way data is gathered on MTGO, we can only see cards played out, not ones stuck in players' hands or libraries.

Another statistic we can use is the number of lands or spells on average drawn by a particular turn given a particular number of lands. I've seen this analysis before, but can't find a good example currently. If you find one, or can calculate it, let me know. I'm inclined to think the loss of EV due to flooding between 17 and 18 lands is smaller than the loss due to suboptimal opening hands, but I can't actually back that up with hard numbers.

For me the main takeaway is this: Running 18 lands over 17 lands in your limited deck leads to significant gains in the average quality of your opening hands. However, it also probably reduces the quality of your draws in the mid to late game. Just how much is difficult to quantify.

Until then, I think it may actually be correct to run 18 lands in most draft decks, and I'm going to try it for a while to see how it works out.

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