I love math almost as much as I love Magic. Luckily, the two go hand in hand. You might not realize that when you play Magic, you end up crunching a lot of numbers. If you go in looking at a game of Magic through a mathematical lens, you will crunch those numbers better and level up your play.
In today’s article, we’re going to look at a subset of cards and record what we know about them. Then, we’re going to use that information to iteratively create a mathematical model that can help us understand that subset of cards better. In particular, subset in question is very important to Limited, and we can use the lessons we learn about this subset to help us evaluate all creatures in the entire game. (If we’re designing our own custom Magic set, we can also use these lessons as a development tool.) That subset is vanilla creatures.
Barring tribal synergies, vanilla creatures (i.e., creatures with no abilities) have three “knobs” that can be turned, as Magic R&D dubs them: power, toughness, and mana cost. When developing a set, R&D can adjust these knobs on a card to adjust the power level of the card in that format. Roughly, the par for a decent vanilla creature is one whose power and toughness add up to twice its converted mana cost. For example, a par vanilla 2-mana creature might be 0/4, 1/3, 2/2, or 3/1.
We can model this relationship mathematically by representing an appropriate mana cost (C) as a function of power (P) and toughness (T). Here’s what I gave Wolfram Alpha:
It helps to look at this relationship visually. The graph of the above function is pictured below. Note that a higher vertical value corresponds with a higher converted mana cost.
According to this graph, everything that sits on the curve is a solid, playable vanilla creature. Everything above the curve is over-costed, and everything below the curve is under-costed.
Now this model is pretty good, but it’s not perfect. Let’s try to iterate on it by looking at some cases where it falls short.
First of all, ask yourself this question: would you rather have a 1/1 for 1, a 4/4 for 4, or a 7/7 for 7? If you’ve drafted once or twice, you know that a 4/4 for 4 is the correct answer. A 1/1 for 1 is rarely going to impact the game enough to be worth a card, and while a 7/7 for 7 is playable sometimes, it’s hard to justify playing a card for that cost unless it will truly swing things in your favor. In other words, our first model breaks down at very high and very low mana costs; at the extremes, the ratio of cost to stats is too high. We can adjust our model by reducing the appropriate cost by an amount that increases the further you are from 4. Here’s our new function:
I chose the constant .05 by guessing and checking until the resultant graph was roughly consistent with my mental model. It’s not a flawless method, but it works. After all, we’re not trying to get our numbers perfect- we’re just trying to get them good enough to paint a realistic picture of a good vanilla creature. With that, here’s what our new graph looks like:
As the model dictates, if a vanilla 7/7 wants to be as good as a Rumbling Baloth, it needs to cost around 6 mana. Likewise, if a one mana vanilla creature wants to be good enough, 2/1 worth of stats gives it a pretty good shot.
We can continue to iterate forever, but we’re going to do it just one more time. You may have caught on by now that there are tons of different directions we can take this in, so we’ll have to pick one to hone in on.
I’m going to choose the differences between power and toughness. Up until this point, our models have treated the two stats exactly the same; a 4/2 will result in the same predicted cost as a 2/4. In reality, though, power is generally a bit more valuable than toughness. We can reflect this in our model by attaching a coefficient to the toughness variable (¾), then compensating for the shrunken cost by applying its reciprocal to the entire function:
And here’s the graph:
Again, this is all very hacky, and the numbers might be a bit off, but it works for our purposes. (In fact, I would invite anyone with more mathematical prowess than I to improve upon our models with more precise functions.) Now, a 2/1 is more valuable to our model than a 1/2, and all is right with the world.
As I said before, we could keep iterating. Of course, in the same way that it’s impossible to measure the exact length of a coastline, we’ll never arrive at a perfect mathematical model for decent vanilla Magic cards. Additionally, forces such as power creep and the nuances of formats constantly change the relationships at some level. But for our purposes, mere approximation provides a whole lot of insight.
What else can we do with models like this? Well, one idea is to extend the practice beyond the set of vanillas to French vanillas (i.e., creatures with no abilities other than an evergreen keyword ability). Of course, when you tack an evergreen keyword onto a vanilla creature, the value of having a high power, a high toughness, or a low mana cost is subject to change. For example, creatures with deathtouch care way less about power than most creatures, since they only need 1 power to kill in combat. Next week, we’ll go through all the current evergreen keywords and examine how each of them interacts with the three knobs in our vanilla model.
Follow us on Twitter: http://www.twitter.com/spellsnare_
Like us on Facebook: http://www.facebook.com/spellsnare