Finding the Optimal Mana Curve via Computer Simulation
By Frank Karsten : 29/07/14
By Frank Karsten : 29/07/14
The concept of mana curve is a fundamental part of Magic theory. The essence is that your deck should contain a balanced mix of converted mana costs so that you can use all of your mana efficiently on every turn. The concept has become so ingrained in Magic theory that laying out decks according to piles per converted mana cost has become customary.
But what is the optimal mana curve? Is there even such a thing?
The answer to this question will depend on many factors. An important one is how many turns you expect a game to last, i.e., the overall speed of the format. Let’s take Standard for example. Based on my experience, most games will have effectively ended by turn five. Maybe it takes a few more turns for the game to truly end, but usually one player will have an insurmountable board presence at that point. So, the thing that matters most is curving out in the first five turns. The ideal is curving out perfectly with a one-drop on turn one, a two-drop on turn two, and so on (which amounts to spending 1+2+3+4+5=15 mana over the course of the first five turns). Such a perfect curve would typically beat an opponent who only manages to cast, say, 10 mana worth of cards in the first five turns.
Accordingly, I am going to determine the 60-card deck that maximizes the total expected amount of mana spent over the course of the first five of turns. I will also do that for faster formats (where only the mana spent till turn 3 or 4 matters), for slower formats (where you want to curve out till turn 6), and for various 40-card formats.
To do so, I wrote a computer program to enumerate all possible decks and to simulate tons of games with each of them to estimate each deck’s expected total mana spent. The full code can be found here. It’s an adaption of the program that I used to determine the optimal aggro deck about a year ago.
Our deck consists of arbitrary one-drops, two-drops, three-drops, and so on. I don’t care whether they are Thoughtseizes, Lightning Bolts, or Savannah Lions. Color requirements are not an issue. It’s about making the most of our mana, not the specifics of the cards.
We are on the play every game.
We are using an optimal mulligan strategy. It is determined via dynamic programming, as explained here.
On each turn, we play a land if possible. Then, we cast the highest cost spell that we have mana available for. If we have untapped lands remaining after that, we again play the highest cost spell that we can cast. And so on. This might cause us to play a three-drop on turn four rather than two two-drops, but that’s not unreasonable because the three-drop may have more impact on the game at that point.
On to the results!
The optimal curve for a turn-3 format:
17 One-Drops
13 Two-Drops
5 Three-Drops
25 Lands
This is meant for a turbo-fast Constructed format in which games typically last for only three turns. The expected mana spent for this deck over turns 1-3 was 5.57. 25 lands might seem like a lot, but it makes sense when you think about it—you really want to hit your first three land drops, and you only need a few spells to curve out well in a format that only lasts for three turns.
The optimal curve for a turn-4 format:
9 One-Drops
13 Two-Drops
9 Three-Drops
3 Four-Drops
26 Lands
An example of a turn-4 format might be Modern. Compared to the turn-3 format, the average mana cost in the deck has skewed upwards, and the land count has increased, too. The expected mana spent for this deck over turns 1-4 is 8.44.
The optimal curve for a turn-5 format
0 One-Drops
12 Two-Drops
9 Three-Drops
7 Four-Drops
3 Five-Drops
29 Lands
A good example of a turn-5 format is Standard. It is noteworthy that one-drops have completely disappeared. Apparently, when it comes to maximizing the total amount of mana spent, it is worth giving up one-drops entirely to focus on the “fatter” turns. The expected mana spent for this deck is 11.78.
It may be interesting to detail the optimal mulligan strategy for this deck. The complete strategy was rather complicated, but it was very close to the following rule of thumb:
Mulligan every 0-land, 1-land, 6-land hand, and 7-land hand.
For 2-land hands: Keep if it contains at least two cards costing 3 mana or less.
For 3-land hands: Keep if it contains at least one card costing 3 mana or less.
For 4-card hands: Keep if the sum of the converted mana cost of the spells is fewer than 13; mulligan otherwise.
For 5-card hands: Keep if the sum of the converted mana cost of the spells is 5, 6, 7, or 8; mulligan otherwise.
The optimal curve for a turn-6 format:
0 One-Drops
7 Two-Drops
11 Three-Drops
7 Four-Drops
5 Five-Drops
1 Six-Drop
29 Lands
An example of a turn-6 format might be Block Constructed. You can see the trend towards fewer early drops and more late drops for slower formats continues. The expected mana spent for this deck over turns 1-5 is 15.53.
The optimal curve for a turn-5 format:
0 One-Drops
8 Two-Drops
7 Three-Drops
4 Four-Drops
2 Five-Drops
19 Land
This would be the best mana curve for a blazingly fast Limited format. The expected mana spent for this deck over turns 1-5 is 11.92. This number is a little higher than the turn-5 Constructed format because there is less variance in the opening hand composition for a 40-card deck than for a 60-card deck.
The optimal curve for a turn-6 format:
0 One-Drops
5 Two-Drops
6 Three-Drops
5 Four-Drops
3 Five-Drops
1 Six-Drop
20 Lands
This would be the best mana curve for an average Limited format. The expected mana spent for this deck over turns 1-6 is 15.78. Although 20 lands still feels like a lot to me, the number makes sense because my computer algorithm only optimizes the first six turns. It doesn’t take into account the fact that Limited games can get into lengthy board stalls where mana flood is an issue.
Nevertheless, the mana curve looks close to what I generally strive for, so it’s comforting that this was the output.
At this point, I was curious what would happen if I would assume that I always draw first. The optimal deck in that situation is equal to the one above, except that it has one more two-drop and one fewer land. This outcome corresponds to a strategy that I frequently employ in Limited: boarding out a land when I’m on the draw.
The optimal curve for a turn-7 format:
0 One-Drop
1 Two-Drop
6 Three-Drops
5 Four-Drops
4 Five-Drops
3 Six-Drops
21 Lands
This would be the best mana curve for a somewhat slow Limited format. The expected mana spent for this deck over turns 1-7 is 19.94.
I love the singleton two-drop. It’s worth the inclusion for the small chance of using your mana on turn 2, but you never really want to draw two copies if the game goes long. This is similar to the rationale behind weird-looking one-ofs that you sometimes see in Constructed decklists---it’s frequently all about filling slots on the curve.
Based on some simple assumptions, my computer simulation was able to establish some baselines for how a good mana curve should look. The outcomes are reasonably similar to average curves from top-performing decks at recent Grand Prix tournaments, with two main exceptions: the tournament decks run more one-drops and fewer lands than my algorithm dictates.
I already explained the reasoning behind the large number of lands when the game is always assumed to end after a certain number of turns. Regarding the number of one-drops: An Elvish Mystic or Firedrinker Satyr on turn one can make a huge difference in establishing a quick board presence, especially when playing an aggro deck. For my algorithm, however, spending a mana on turn one is the same as spending a mana on turn five. This doesn’t properly account for the impact that a one-drop on turn one can have. As a result, the top-performing decks play more one-drops than the optimal decks under my assumptions.
For reference, let me give you the average mana curves from a small sample of top-finishing Modern, Standard, and Limited decks.
First, Modern. For the Top 4 decks from the recent Grand Prix tournaments in Minneapolis and Richmond, the average mana curve was:
13 One-Drops
12 Two-Drops
6 Three-Drops
5 Four-Drops
1 Five-Drop
23 Lands
Next, Standard. For the Top 4 decks from the recent Grand Prix tournaments in Chicago and Moscow, the average mana curve was:
4 One-Drops
11 Two-Drops
9 Three-Drops
6 Four-Drops
4 Five-Drops
1 Six-Drop
25 Lands
Finally, Limited. For the Top 4 decks from the M14 Limited Grand Prix tournaments in Oakland and Prague, the average mana curve was:
2 One-Drops
5 Two-Drops
7 Three-Drops
6 Four-Drops
2 Five-Drops
1 Six-Drop
17 Lands
There are various additional limitations to my analysis. For instance, combat tricks and removal spells (such as Titanic Growth and Lightning Strike) are rarely played on curve, so counting them as two-drops in your Limited decks can be misleading. Moreover, there are no card draw spells, activated abilities, mana producing creatures, or mana sinks in my simple format. Mechanics like bestow and strive are very strong because they allow you to make the most of your mana in both the early and the late game, but my assumptions are not rich enough to account for such flexible cards.
Nevertheless, because of these limitations, I was able to perform some actual analysis on the problem. I hope that the ideal mana curves I devised can give you a good baseline to start from when building decks in the future. Getting the perfect mana curve is not all there is to Magic, but it’s one of the most important things to keep in mind for deck construction.